Bibliography - School of Physics - University of Melbourne

These notes were revised in 2006 by David Hoxley, incorporating 

suggestions from the 2005 demonstrating team. Thanks, guys. 

These notes were imported into L A TEXand substantially revised in 2005 by a 

team composed of: 

Simon Devitt 

John-Paul Goldby 

Chris Gurrie 

David Hoxley 

Steven Karataglidis 

Tracey Mackin 

Alaster Meehan 

Matthew Norman 

Tim Starling 

Melissa Wals 

The majority of the typesetting and revisions were done by Steven 

Karataglidis and Melissa Wals. Preparation of the final manuscript was done 

mostly by Melissa with red ink supplied by David Hoxley. 

The last major revision of the notes from which we draw heavily was in 

1999 with the assistance and hard work of the following people: 

Anton Barty, Chris Chantler, Jacinta den Besten, Phillip Fox, Amelia Liu, 

Leigh Morpeth, and Justine Tiller.

Contents 

1 Introduction 1 

1.1 Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 

1.2 Experimental techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

1.2.1 Computers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

1.3 The report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 

1.3.1 Preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 

1.3.2 Writing the report . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 

1.3.3 Assessment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

1.4 Sample report . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

1.5 Hints and tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

1.6 Deadlines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 

1.7 Illness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 

I Optics 13 

2 2-slit interference with single photons 15 

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 

2.2 Background theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 

2.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 

2.4 Experiment - Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.4.1 Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.4.2 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 

2.5 Experiment - Single Photon Counting . . . . . . . . . . . . . . . . . . . . 20 

i

ii 

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3 The Michelson interferometer 23 

3.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

3.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

3.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

3.2.2 Background theory . . . . . . . . . . . . . . . . . . . . . . . . . . 24 

3.2.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 

3.2.4 Experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . 26 

3.2.5 Conditions of interference - polarisation . . . . . . . . . . . . . . . 27 

3.2.6 Conditions of interference - coherence . . . . . . . . . . . . . . . . 27 

3.2.7 Micrometer calibration . . . . . . . . . . . . . . . . . . . . . . . . 28 

3.3 Fringe visibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

3.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 

3.3.2 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

3.3.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

3.4 Refractive index of air . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 

3.4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 

3.4.2 Background theory: optical path length . . . . . . . . . . . . . . . 30 

3.4.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 

3.4.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 

4 Fraunhofer diffraction 33 

4.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 

4.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 

4.3 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 

4.3.1 Fresnel or Fraunhofer? . . . . . . . . . . . . . . . . . . . . . . . . 34 

4.3.2 Diffraction by a single slit . . . . . . . . . . . . . . . . . . . . . . 34 

4.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 

4.4.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 

4.4.2 Single slit diffraction . . . . . . . . . . . . . . . . . . . . . . . . . 37 

4.4.3 Babinet’s principle . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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iii 

4.4.4 Multiple slits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 

4.5 Further work - circular and other diffracting objects . . . . . . . . . . . . . 42 

4.5.1 Overview and background theory . . . . . . . . . . . . . . . . . . 43 

4.5.2 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 

4.6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 

4.6.1 CCD camera operating instructions . . . . . . . . . . . . . . . . . 44 

4.6.2 How the CCD works . . . . . . . . . . . . . . . . . . . . . . . . . 45 


5 Holography 47 

5.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 

5.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 

5.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 

5.4 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 

5.4.1 Intensity of a wave . . . . . . . . . . . . . . . . . . . . . . . . . . 48 

5.4.2 Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 

5.4.3 Holograms and holography . . . . . . . . . . . . . . . . . . . . . . 49 

5.4.4 Movement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 

5.5 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 

5.5.1 Film handling and developing . . . . . . . . . . . . . . . . . . . . 51 

5.5.2 White light hologram . . . . . . . . . . . . . . . . . . . . . . . . . 53 

5.5.3 Transmission or off-axis hologram . . . . . . . . . . . . . . . . . . 54 

5.5.4 Time-averaged interferometry of a resonating tube . . . . . . . . . 55 

5.6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 

5.6.1 Spherical waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 

5.6.2 Interference of two waves . . . . . . . . . . . . . . . . . . . . . . 56 

5.6.3 Transmission function . . . . . . . . . . . . . . . . . . . . . . . . 56 


II General physics 59 

6 Magnets And magnetic fields 61

iv 

CONTENTS 

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 

6.2 The Hall effect and magnetic field probes . . . . . . . . . . . . . . . . . . 61 

6.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 


6.3 The Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 

6.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 

6.4.1 The semiconductor . . . . . . . . . . . . . . . . . . . . . . . . . . 64 

6.4.2 The electromagnet . . . . . . . . . . . . . . . . . . . . . . . . . . 65 

6.4.3 The Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 

6.4.4 The Hall probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 

6.5 Ferromagnetic materials and AC fields . . . . . . . . . . . . . . . . . . . . 67 

6.5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 


6.5.3 The apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 

6.5.4 Experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . 73 

6.6 Useful data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 


7 Fundamental constants 75 

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 

7.2 Charge to mass ratio of the electron . . . . . . . . . . . . . . . . . . . . . 75 

7.2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 

7.2.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 

7.2.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 

7.2.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 

7.2.5 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 

7.3 The photoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 

7.3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 

7.3.2 The failure of classical theory and Einstein’s postulate . . . . . . . 80 

7.3.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 

7.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

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7.4.1 Variation of stopping potential with frequency . . . . . . . . . . . . 84 

7.4.2 Calculation of the fundamental constants . . . . . . . . . . . . . . 84 

7.5 Useful data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 


8 Waves in waveguides 87 

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 

8.2 Acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 

8.2.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 

8.2.2 Stationary wave theory and the reflection of plane acoustic waves . 91 

8.2.3 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 

8.2.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 

8.3 Microwaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 

8.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 

8.3.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 

8.3.3 Modes in the guide . . . . . . . . . . . . . . . . . . . . . . . . . . 102 

8.3.4 Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 

8.3.5 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 

8.3.6 Polarisation of microwaves in the guide . . . . . . . . . . . . . . . 105 

8.3.7 Relation between guide and free space wavelengths . . . . . . . . . 106 

8.3.8 Impedance measurement . . . . . . . . . . . . . . . . . . . . . . . 106 

8.4 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 

8.4.1 The complex reflection coefficient K . . . . . . . . . . . . . . . . 107 

8.4.2 The Gunn diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 

8.4.3 The standing wavemeter . . . . . . . . . . . . . . . . . . . . . . . 110 

8.4.4 Reading the micrometer . . . . . . . . . . . . . . . . . . . . . . . 110 


9 Electron Spin Resonance 113 

9.1 A guide to background reading . . . . . . . . . . . . . . . . . . . . . . . . 113 

9.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 

9.3 Introduction: Angular Momentum in Quantum Mechanics . . . . . . . . . 114

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9.3.1 Orbital Angular Momentum . . . . . . . . . . . . . . . . . . . . . 114 

9.3.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 

9.4 Electrons in an external magnetic field . . . . . . . . . . . . . . . . . . . . 115 

9.5 Resonance absorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 

9.6 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 

9.6.1 Helmholtz coils . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 

9.6.2 RF oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 

9.6.3 Investigating the resonance . . . . . . . . . . . . . . . . . . . . . . 119 

9.6.4 Electron absorption . . . . . . . . . . . . . . . . . . . . . . . . . . 119 

9.7 Extension: electron diffraction . . . . . . . . . . . . . . . . . . . . . . . . 120 

9.7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 

9.7.2 Basic theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 

9.7.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 

9.7.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 

9.8 Useful data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125 

9.9 Wave function for the electron - a brief history . . . . . . . . . . . . . . . . 125 


III Nuclear physics 127 

10 Rutherford scattering 129 

10.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 

10.2 Introduction/background reading - importance of scattering . . . . . . . . . 130 

10.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 

10.3.1 Cross section . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 

10.3.2 Solid angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 

10.3.3 Differential cross section . . . . . . . . . . . . . . . . . . . . . . . 133 

10.3.4 Theoretical cross section . . . . . . . . . . . . . . . . . . . . . . . 134 

10.4 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 

10.4.1 Safety precautions . . . . . . . . . . . . . . . . . . . . . . . . . . 134 

10.4.2 Relationship between counts and source to detector distance . . . . 135

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10.4.3 Rutherford scattering . . . . . . . . . . . . . . . . . . . . . . . . . 139 

10.5 Useful data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 


11 β spectroscopy 143 

11.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 

11.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 

11.3 Background concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 

11.4 Experiment - the β decay of promethium . . . . . . . . . . . . . . . . . . . 146 

11.4.1 Calibrating the magnet . . . . . . . . . . . . . . . . . . . . . . . . 146 

11.4.2 Measuring the spectrum . . . . . . . . . . . . . . . . . . . . . . . 148 

11.4.3 The spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 

11.4.4 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 

11.5 Useful data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 


12 γ ray detection and spectroscopy 153 

12.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 

12.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 

12.3 γ radiation. How? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 

12.4 Calibration of the detector . . . . . . . . . . . . . . . . . . . . . . . . . . 154 

12.4.1 Detecting photons - the NaI(Tl) crystal . . . . . . . . . . . . . . . 154 

12.4.2 The Multi-Channel Analyser . . . . . . . . . . . . . . . . . . . . . 157 

12.4.3 The reference sources . . . . . . . . . . . . . . . . . . . . . . . . 157 

12.4.4 Gain of the detection system and MCA . . . . . . . . . . . . . . . 158 

12.4.5 Measuring spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 159 

12.4.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 

12.5 Energy resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 

12.6 Unknown sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 

12.7 MCA commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

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13 Energy loss of α particles through matter 163 

13.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 

13.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 

13.3 Measuring foil thickness . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 

13.3.1 Calibration of the system . . . . . . . . . . . . . . . . . . . . . . . 166 

13.3.2 Measurement of known foils . . . . . . . . . . . . . . . . . . . . . 167 

13.4 Making a thin foil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 

13.4.1 Background reading . . . . . . . . . . . . . . . . . . . . . . . . . 167 

13.4.2 Foil manufacture . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 

13.4.3 Thickness measurement . . . . . . . . . . . . . . . . . . . . . . . 170 

13.5 Rules for operating a high vacuum system . . . . . . . . . . . . . . . . . . 170 

13.6 MCA commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 


14 Errors 173 

14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 

14.2 Sources of error in experimental work . . . . . . . . . . . . . . . . . . . . 174 

14.2.1 Mistakes (!!) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 

14.2.2 Random errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 

14.2.3 Reading errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 

14.2.4 Systematic errors . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 

14.3 Graphs and errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 

14.3.1 Error bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 

14.3.2 Estimates of gradients and uncertainties . . . . . . . . . . . . . . . 179 

14.4 Excel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 

14.5 Combining errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181 

14.6 Significant figures and data/error presentation . . . . . . . . . . . . . . . . 183 

14.7 General expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 

14.7.1 General expression . . . . . . . . . . . . . . . . . . . . . . . . . . 184 

14.7.2 Sums and differences . . . . . . . . . . . . . . . . . . . . . . . . . 184 

14.7.3 Products and quotients . . . . . . . . . . . . . . . . . . . . . . . . 184

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14.7.4 Powers and functions . . . . . . . . . . . . . . . . . . . . . . . . . 184 

14.7.5 Simple and trigonometric functions . . . . . . . . . . . . . . . . . 184 

14.8 Concluding comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 


15 Excel tutorial 187 

15.1 Simple example, energy vs. ADC number . . . . . . . . . . . . . . . . . . 187 

15.1.1 Entering data into the worksheet . . . . . . . . . . . . . . . . . . . 187 

15.1.2 Making the plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 

15.1.3 Fitting the data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 

15.1.4 Absolute referencing . . . . . . . . . . . . . . . . . . . . . . . . . 188 

15.1.5 Using formulas to transform the data . . . . . . . . . . . . . . . . . 189 

15.2 Nonlinear fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 

15.2.1 The Solver Add-In . . . . . . . . . . . . . . . . . . . . . . . . . . 189 

15.2.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 


16 The Safe Use of Lasers 193 

16.1 General comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 

16.2 Precautions with lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

Chapter 1 

Introduction 

The experiments in the second year laboratory cover a diverse range of physical phenomena. 

There are three main experimental areas: optics, nuclear and general, the last of which encompass 

waves, quantum mechanics and magnetic fields. Through this diversity, we hope 

that you will develop the ability to quickly adapt to and show a critical approach towards the 

tasks confronting you. 

You will spend four weeks in each area undertaking a new experiment each week. Each 

experiment will involve you spending six hours in the lab spread over two consecutive days. 

You will be expected to work in a team with your fellow students, calling on your demonstrator 

to clarify or discuss various aspects of the experimental procedure, results or analysis. 

A formal report will also be written for each experiment within the time allotted. 

The next few pages will describe what is expected of you in the lab in more detail. Please read 

these carefully before you enter the class to ensure that you are prepared for the semester. 

These notes will be a useful reference for you throughout the semester, so we suggest that 

you continuously refer back to them. 

Notes and other information are available on our website and the LMS: 

http://www.ph.unimelb.edu.au/˜part2 

The demonstrators and myself hope you enjoy yourself in the second year lab this semester. 

Please feel free to email me at the below address with any queries or requests. I will use 

email as the principal means of communication with students (via the LMS), so make sure 

that you check it every few days. 

Stephen Marshall 

Part II Coordinator 

part2@physics.unimelb.edu.au 

1

2 CHAPTER 1. INTRODUCTION 

1.1 Safety 

The University of Melbourne has adopted the internationally recognised systems SafetyMAP 

(ISO 12001) and Environmental Management System (ISO 14001) to ensure a safe and environmentally 

friendly workplace for all staff, students, and visitors. As a student of the 

University you are responsible for adopting safe work and study practices, and you are required 

to comply with all relevant University and Departmental rules and procedures. 

Detailed information on University policy and procedures is provided in the Environment, 

Health and Safety Manual at http://www.unimelb.edu.au/ehsm 

The Laboratory Rules and Safe Work Procedures set out below in this practical manual 

must be adhered to at all times, and the direction of all staff and demonstrators must be 

followed. If you have any concerns about the safety or environmental impact of any aspect 

of these practical classes, please raise them with the staff member in charge. Report all 

injuries and incidents to the staff member. 

Safety rules for the Second Year Physics Laboratories: 

• No food or drink in the labs at any time. 

• No open-toed shoes in the lab. 

• Long hair should be tied back. 

In the first instance, your demonstrator will alert you to any potential dangers in the lab, and 

you should raise any concerns you have with them. You may also consult the Part II Lab 

Coordinator (Stephen Marshall) about any safety or environmental concerns you may have. 

Some experiments in the labs have special safety issues associated with them (e.g. they 

involve the use of high voltages or lasers). You will find directions for working with this 

equipment in the lab notes for that experiment. 

If you have any allergies or medical conditions that you think may be affected 

by chemicals, material or procedures to be used in these practical 

classes, fill in a “Medical Status - Voluntary Notification for Laboratory 

Classes Form” and give it to the staff member in charge so that any risk can 

be assessed and the work procedures modified. 

A risk assessment has been carried out for these practical classes and any identified risks 

have been minimised. The risk assessments are available for examination - contact the Part 

II Lab Coordinator should you wish to do so.

1.2. EXPERIMENTAL TECHNIQUES 3 

1.2 Experimental techniques 

Here are some helpful hints when working in the lab to make your experience in the lab more 

efficient, beneficial and enjoyable: 

• Share the workload with your partner. 

• Make sure your occupations within the experiment are varied, and that the same person 

is not always taking the measurements or using the computer. 

• Discuss your work with your partner or friends, but remember that copying is prohibited. 

• Before you ask your demonstrator a question, think about possible solutions yourself 

first. 

• Always check a circuit before turning the power on. 

• If something is not working, check to see if it is turned on. 

• Think about how many measurements to take and over what range. A quick check of 

the phenomena you will measure by quickly varying the conditions and watching the 

output will help you determine this. 

• Careless experimental techniques can damage or destroy vital and expensive equipment. 

Please take care with all equipment and report any damages to your demonstrator 

immediately. 

Remember that your demonstrator is there purely to teach, to help you come up with an 

answer, not to tell you the answer. Make full use of their knowledge and discussions. 

1.2.1 Computers 

Much of the analysis of your results will be done using Excel. You will often take your data 

and fit it to a known trend using the least squares method. More on how to do this using 

Excel is given in chapter 15. 

The computers themselves and their setup must not be tampered with. Severe penalties will 

be enforced if anyone is found doing so. 

1.3 The report 

Each experiment has been designed so that it can be finished with a completed report in the 

six hours allocated. However, this assumes that you have done adequate preparation 

before entering the laboratory.


1.3.1 Preparation 

Preparing for each experiment involves reading the experimental notes and familiarising 

yourself with the theory involved. Begin any report writing by attempting any Pre-lab questions 

asked in the notes and carrying out some of the calculations. If this is not done you 

will be wasting valuable time in the lab and therefore penalising yourself. 

Careful! You will find the Pre-lab questions scattered all the way through the notes, so read 

them carefully. 

1.3.2 Writing the report 

The report has two purposes. The first is to act as an experimental log. This includes an 

accurate description of what was done (not what was written in the manual), as well as an 

accurate description of all observations. This provides the bulk of the Procedure and Results. 

The second role is the experimental report itself. This explains the purpose of the experiment 

in the form of the Aim, the meaning of the results through the Analysis and Discussion and 

what was learnt as a result of the experiment in the Conclusion. Each prac consists of several 

experiments designed to investigate a small area of physics. Each experiment is worthy of 

its own report. Below is a little more detail about what is involved in writing a report. 

• Clearly state the experimental TITLE, DATE and NAME of your lab partner. 

• AIM AND THEORY 

The aim is a concise statement of the purpose of the experiment incorporating the 

following: 

– The theory to be tested 

– The quantities to be measured 

– Assumptions under which the experiment is to be done 

– Expected results including any theoretical values expected to be obtained 

– Definition of terms 

– Relevant formulae 

The theory can be easily incorporated into the aim. There is no need to copy sections 

of the manual as only a summary of the theory is required. 

We strongly advise you to write your background theory section first, and only then 

to write a brief statement of the specific aim (one or two sentences only). In this way, 

you will have defined all your terms before try to tell your reader what exactly you’re 

trying to find out. 

• PROCEDURE 

The procedure is an accurate record of what was done by you in the lab: 

– List all apparatus used 

– Draw circuit and block diagrams to show the experimental setup

1.3. THE REPORT 5 

– Show quantities measured 

– Briefly describe how quantities were measured 

– Record electronic settings if applicable 

This section should contain enough detail to reproduce your results without reference 

to the manual. Note: sometimes your procedure will be different to the manual. 

• RESULTS 

The results section is an accurate record of all observations, both visual and numeric: 

– Results are to be entered directly into your logbook in pen. Do not record your 

results on scraps of paper 

– Clearly present results - tables, colours, boxes 

– All measurements have errors. Explain where your errors come from. (Errors are 

explained in detail in chapter 14.) 

– Pertinent comments on experimental results or other observations 

– Mistakes should be explained, not erased 

– Remember to include units on all measurements 

• ANALYSIS AND DISCUSSION 

In this section the physical meanings of your results are discussed through the analysis 

of your data: 

– Graphs - clear and large 

– Calculations or sample calculations if many of the same analyses need to be made 

– Error analysis of your results 

– Compare your results with the theory. Is this the trend you expected? Is this 

value within the error bounds of your results? If not, why not? 

Through the analysis and discussion you must demonstrate your understanding of the 

physics involved and the limitations of the experiment. 

• CONCLUSIONS 

The conclusion is a brief summary of your findings and the importance they play in 

the role of physics. There should be no new information in this section; 

– Statement of final result 

– Success of theory - do your results concur with the theoretical ones? 

– Limitations of the experiment 

– Unexplained results 

– Suggestions for improving your results and technique for further investigation 

Ultimately, you must communicate what you learnt in the laboratory.


1.3.3 Assessment 

Marks are not given for the quantity of material written but for its quality. Comments, which 

show that you understand or have thought about what is going on, are valuable. Clarity of 

ideas, thoughts and understanding are essential for increasing your mark - a lack of these 

will detract from your mark. Your report should be legible but does not have to be a work of 

art. It is your ideas and experimental ability you will be graded on. 

Each report will be marked on a scale of 0 to 10. As a rough guideline the following marks 

are associated with the corresponding areas in the report: 

Pre-lab questions 2 

Aim/Theory and Conclusion 1.5 

Procedure 1.5 

Results and Experimental ability 2 

Analysis and Discussion 3 

1.4 Sample report 

See the next three pages, a sample experiment of “Measurement of resistance”, for an idea 

of how to set out your report. 

1.5 Hints and tips 

What follows in this section is a list of things that demonstrators tend to find are consistent 

problems with student reports. It may not make much sense until you’ve done one or two 

experiments, but it should make lots of sense from then on! Some are issues of presentation, 

others of content. Some of this will seem like simple repetition of material that appears 

elsewhere in the introduction - that’s because it is. It is a response to the sorts of mistakes 

students make repeatedly! The further on you go through the lab, the more the demonstrators 

will expect of you - and the less happy they will be when you don’t jump the basic hurdles! 

Scattered amongst the list (which was made up by a demonstrator as she was marking, so 

isn’t complete or necessarily in order of importance) are some ways of thinking about report 

writing that, we hope, will help you! 

1. All work should be IN PEN please - this includes diagrams. 

2. If you write separate aims for different sections of the experiment, you must write 

separate conclusions. Most of the pracs in the general lab (and several labs elsewhere) 

fall into at least two clear divisions, each one worthy of a separate (consecutive) “minireport”. 

3. When presenting a mathematical proof, your job is to justify each step. Basically, you 

will always have to include some words, not just a string of equations.

1.5. HINTS AND TIPS 7

8 CHAPTER 1. INTRODUCTION

1.5. HINTS AND TIPS 9


4. Make your aims specific. For example, not just “to investigate waves” but “to investigate 

acoustic standing waves in a tube, and thus to find. . . ” This means you have to 

read the prac carefully before you arrive. 

5. When answering questions, paraphrase the question in the answer so that your 

reader doesn’t have to refer to the manual to find out what you’re talking about - your 

report should be self-contained. 

6. Make your diagrams big and clear - some of the equipment is quite complex, and a 

tiny diagram makes it hard to see what’s going on. A good guide is to make every 

diagram not less than one-third of a page. 

7. When describing your observations, make liberal use of diagrams (e.g. “the signal 

looked like this: ... ”). 

8. Do not just present a list of numbers - you must always draw physical conclusions 

from what you calculate/measure. Sometimes that conclusion will be “this verifies the 

theory” (if so, you must have presented the theory so that your reader can make the 

comparison), but sometimes it will be “this value is bigger than that one - this does (or 

doesn’t) make sense for the following reasons...” 

9. You need to imagine a reader that is dumb, lazy and mean. Most readers don’t want 

to have to work very hard, but they want proof that you have. Also, they may one day 

be a competitor who is actively looking for mistakes - so get into the habit of justifying 

everything! 

10. Make sure your conclusion summarises your results. We are training you, in part, 

in how to write a scientific paper. Very few papers are read thoroughly - lots of readers 

do a quick scan over the method and turn straight to the conclusion to see what, 

specifically, you found out, and what if anything limited your results. If you spent 

three hours finding three numbers, you need to make sure those numbers are written 

down in the conclusion! 

11. ERROR ANALYSIS is always required and should be clear and complete, just like 

any other mathematical treatment you have to do in the course of the experiment. A 

number without an error attached is meaningless. There is never a time when you 

don’t have to do an error analysis. 

12. Don’t forget the importance of attaching units to all your numbers - again, without 

units your results are essentially meaningless. No reader can possibly judge the accuracy 

of your results without them. 

13. Method must be detailed - by that we mean it should include all the information you 

might need if you wanted to go back to this in three years when you’ve forgotten 

everything. So, things like the names and locations of any files you save are part of 

this, as is the name of the software you used and so on. 

14. It is useful to think of the lab report as a story - the story of the growth of your understanding 

throughout the prac. That means that you will write down everything you 

know at the beginning (i.e. context/theory) and all the stuff you find out by the end, as 

well as how you went about making your discoveries.

1.6. DEADLINES 11 

15. Don’t forget to define your terms - mathematical symbols are pointless if you don’t 

refer them back to the real world. 

1.6 Deadlines 

Your report is due at the end of the second session. This means you will have the six hours 

in the lab. The only extra time will be the night between the first and second sessions. It is 

therefore imperative that you write up as you go in order to finish. 

The demonstrator will assess your report and it will be available in advance of your next session. 

The books from the Monday - Tuesday class will be available after 2pm on Friday, 

those from the Wednesday - Thursday class will be available after 2pm on the following 

Monday and the books from the Thursday - Friday class will be available after 2pm on 

the following Wednesday. 

1.7 Illness 

If a class is missed due to an illness, please bring a medical certificate to your next class and 

a time will be arranged for the experiment to be made up. If classes are missed for other 

(previously known) reasons the makeup must be organised before the missed class. Please 

see your demonstrator or the coordinator to arrange this.

Part I 

Optics 

13

Chapter 2 

2-slit interference with single photons 

2.1 Introduction 

The double-slit experiment has as its origins an experiment performed in 1665 by Grimaldi, 

where he allowed light from the Sun to pass through two pinholes in an opaque screen. He 

had hoped to illustrate that when two circles of light overlap on a far screen, regions of 

darkness result. That he could not illustrate this was due more to the lack of coherence in the 

sources given that the primary source of light, the Sun, was too spatially extended to provide 

such coherence. 

Young returned to the experiment in 1805, relying on knowledge of diffraction from a single 

slit (refer to the Fraunhofer) to provide a spatially coherent primary source of light: the 

central maximum from the diffraction pattern that emerges from a single slit. (The alternative 

is to use a laser source, which is spatially coherent by construction, althought that is not 

strictly done here.) When that was passed to two identical slits in close proximity, Young 

was able to observe interference fringes. 

The experiment gives rise to two interesting questions: what are the conditions of interference? 

And, more importantly, can it be done with a single photon? 

The experiment itself consists of two parts: 

• Investigation using the laser source. This essentially reproduces the classic experiment 

as a calibration of the equipment; 

• Investigation using the low-intensity lamp. After calibration, the equipment is used to 

study interference with single photons. 

Note! A couple of important definitions.... 

Calibration (Oxford) n 1. The act or proccess of calibrating something; 2. each of a set of 

graduations on an instrument etc.; 

Calibrate (Oxford) v.tr. 2. Correlate the readings of (an instrument) with a standard. 

15

16 CHAPTER 2. 2-SLIT INTERFERENCE WITH SINGLE PHOTONS 

P 

S 

r 1 

S 1 

r 2 

a 

θ 

S 2 

y 

d 

D 

Figure 2.1: Geometry for Young’s double slit experiment. 

2.2 Background theory 

Consider the configuration shown in Fig. 2.1. A light source illuminates a single slit S, which 

then acts as a source of spherical waves. If the distance to the double slit is much greater 

than the separation of the slits, d >> a, then the path difference for the light arriving at the 

double slits S 1 and S 2 will be minimal and the two slits act as sources of coherent spherical 

waves. Light travelling to point P in the image plane will have a path difference, ∆r, which 

for small θ is given by 

∆r = aθ = ay 

D . (2.1) 

The phase difference, δ, induced by this path difference is then 

( ) 2π (ay ) 

δ = k∆r = , (2.2) 

λ D 

where λ is the wavelength of the light. The phase difference will modulate the intensity 

of the light measured at P due to the superposition of the light waves, with the resulting 

intensity distribution given by 

I = 2I 0 (1 + cosδ) . (2.3) 

Prelab Question (A): Show that the intensity distribution given in Eq. (2.3) leads to bright 

and dark fringes at y = mλD/a and y = (m + 1/2) λD/a respectively, where m is 

an integer. 

The spacing between successive bright or dark fringes is then 

y = λD a . (2.4)

2.3. APPARATUS 17 

Light sources 

Double slit and 

collimator 

Shutter 

Power 

Single slit and 

collimator 

Micrometers 

Detector box 

Figure 2.2: Schematic of the apparatus. (Not to scale.) 

However, this interpretation does not take into account that the fringe pattern is modulated 

due to diffraction effects from the slits themselves. That was the main criticism of the experiment 

at the time for which Fresnel and Lloyd suggested variations using either a biprism 

(Frensel) or a mirror (Lloyd) to create virtual, instead of real, sources. In this experiment, as 

we use real primary and secondary slits, we must take diffraction into account. 

The actual intensity distribution with angle is given by 

I(θ) = 4I 0 

( sin 2 β 

β 2 ) 

cos 2 α, (2.5) 

where α = (ka/2) sinθ and β = (kb/2) sin θ, with a and b being the distance between the 

two slits and the slit width, respectively. I 0 is the intensity at θ = 0. This is the reduction for 

N = 2 from the multiple slits diffraction irradiance pattern. (See the “multiple slits” section 

of the Fraunhofer experiment.) The cos 2 α factor is the interference which is modulated by 

the sinc 2 β factor for single slit diffraction. 

Prelab Question (B): Derive Eq. (2.5) from the many-slits distribution as given in the Frauhnhofer 

experimental notes. 

2.3 Apparatus 

The apparatus is from TeachSpin and is shown schematically in Fig. 2.2. It should be aligned 

with the detector box in front of you and to the right. At the left is the power source and light 

sources, which consist of a red diode laser (5 mW, λ = 670 ± 5 nm), and a low-intensity 

white lamp with green detachable filter. The use of the filter limits the wavelength to be in 

the region 540 to 560 nm. The power is from a DC transformer and has controls which allow 

for switching between the laser and the lamp and the variation of the intensity of the lamp. 

You should find the apparatus with the top cover CLOSED. If this is not the case, CONSULT 

YOUR DEMONSTRATOR IMMEDIATELY. (For the reasons why, read on.) 

The full length is the optic bench. Further along the bench, one finds the first micrometer. 

This is the position of the double slit. At the right end of the apparatus is the detector 

box and second micrometer. The second micrometer controls the position of an aperture 

situation in front of the detectors and is calibrated in (true) mm. The detectors themselves


are a photodiode, for use with the laser, and a photomultiplier tube (PMT) for single photon 

counting. The controls on the box govern the power for the PMT, as well as the outputs for 

both. 

IMPORTANT! OPERATION OF THE APPARATUS. Also at the detector box is the 

shutter. It is the black cylinder with the cable coming out of it at the top of the box at the end 

of the optic bench. This switches between the two detectors. When the shutter is FULLY 

down, the PMT is blocked and the photodiode detector is exposed to the bench. The reverse 

situation occurs when the shutter is up. Along the top of the bench is a cover and this should 

always be in place when measuring the intensity pattern. 

WARNING! NEVER remove the bench cover when the shutter is UP. 

The PMT is super-sensitive to photons and even a small amount of background light is 

enough to destory the very thin layer of phosphor at the entrance window to the dynode 

chain. (See the Radiation Lab notes for an explanation on how a PMT works.) Be ABSO- 

LUTELY SURE that the shutter is FULLY DOWN before removing the cover at any 

time. 

With the shutter down, and with the PMT bias turned off (toggle switch on power box), remove 

the bench cover by loosening the four latches and slowly sliding the cover away and 

up from the detector box end. You can now see the internal arrangement with the source slit 

and collimator sitting between the light sources and the first micrometer. The slits and collimators, 

both sets, are magnetically mounted on their respective posts and may be removed 

safely using the tweezers provided. Note that the first micrometer controls the position of 

the second collimator (that associated with the two slits), while the second micrometer sits 

just in front of the shutter and controls an aperture to allow for sampling of the intensity. 

2.4 Experiment - Laser 

You will note that the white light source/green filter may be raised and lowered in front of 

the laser. Be careful when moving the lamp: it is on a delicate mount and any motion should 

be slow and deliberate. 

Remove the single source slit and measure the slit width using the microscope provided. 

Measure also the widths of the double slits and their separation. Be sure to note the errors of 

the masurements. 

Question (a): What measurements may you perform in order to have better confidence in 

the widths of the slits? Discuss with your demonstrator. 

2.4.1 Alignment 

Once the slits have been replaced (be sure to place them in the appropriate orientation), we 

may align the apparatus. Begin by raising the lamp into line with the laser, and remove

2.4. EXPERIMENT - LASER 19 

the filter. Level the lamp and turn on the bulb to about 6 on the intensity scale. Adjust the 

position of the source slit such as the primary source reaches the double slit roughly centered. 

Turn the lamp off, replace the filter and move the lamp to the stored position. Turn on the 

laser and ensure that the beam reaches the source slit. Use the alignment screws to adjust 

the position of the beam until it is centered on the slit. What will you observe to guarantee 

alignment at the source slit? Make adjustments until the pattern reaches the double slit. 

The final bit of alignment concerns the collimator beyond the double slit. This should be 

positioned such that the vertical edges are parallel with the images of the two slits. Rotate 

the collimator until either slit instantly appears or disappears as you adjust the position of the 

collimator using the first micrometer. 

2.4.2 Measurement 

Using a white card, observe the image of the two slits just beyond the collimator. If the 

system is correctly aligned, you should see the double-slit interference pattern at the far end 

of the bench at the shutter’s position. Note the following collimator positions, they will be 

of great importance later: 

• Where both slits are blocked; 

• Where light emerges from the farther slit; 

• Where light emerges from both slits; 

• Where light emerges from the nearer slit; 

• Where both slits are blocked from the other side. 

Make observations on the images at the shutter for each of the cases. 

Question (b): What happens when you switch from one to two slits (and vice-versa) to a 

maximum or minimum at the shutter? 

You will note that at the shutter is another slit, of the same width as the source. This acts 

as an aperture for the detector, allowing for sampling the intensity pattern as one moves the 

micrometer. Do NOT touch this slit or adjust its position in its holder. 

Now replace the bench cover. You are now in a position to measure the intensity distribution 

of the pattern using the photodiode. The photodiode itself is a 1 cm 2 solid-state photodiode 

and generates a current upon illumination by a light source. The cable emerging from the top 

of the shutter carries the current from the photodiode and that should be place into the INPUT 

of the photodiode on the detector box. The OUTPUT carries the voltage as converted from 

the photodiode’s current. You may use the digital multimeter (on the 2V scale) to measure 

that voltage, which is proportional to the intensity of the light being sampled. 

With the cover on and the laser off, measure the zero offset of the diode. Turn the laser on, 

and position the first micrometer such that it is in the position where both slits are allowed


to emerge. Starting at the zero position of the second micrometer, measure the intensity at 

regular intervals between 0 and 10 mm. Be sure to choose a suitable interval such that any 

pattern is adequately resolved. You may enter the data directly into Excel, provided you 

printout a copy of the spreadsheet as well as the graph after analysis. 

At the central maximum, also measure the change in intensity by switching between having 

one slit allowed through and two. Does this correspond to your observations earlier? 

Using Excel, fit the assumed intensity distribution for double-slit diffraction [Eq. (2.5)]. 

Question (c): Why is this necessary? 

When fitting, choose appropriate parameters with which to vary the function to fit the data. 

Compare the fitted values to those expected. Plot the comparison of the fitted function to the 

data, as well as the single-slit pattern as derived from Eq. (2.5). 

2.5 Experiment - Single Photon Counting 

At this point, we now move our attention to counting single photons. With the PMT and laser 

power OFF and the shutter DOWN, remove the cover and raise the lamp to be level with the 

laser. When in operation that would ensure that the light from the lamp would traverse the 

same paths as the light from the laser did in the previous run. When the lamp is in position, 

switch it on and adjust the bulb intensity until you see it glow. It should not be raised above 

6 for any length of time, nor should it be required. Ensure that the light from the lamp runs 

the length of the bench; realigning should not be necessary. 

Once the lamp is on and in position, replace the cover. At this point, the room’s lights may be 

switched on. With the output of the PMT connected to the CRO, and the shutter still closed, 

raise the PMT voltage slowly. Each major division corresponds to 100 V; see how the pulses 

change as the voltage is raised. Note where the pulses, corresponding to the situation of zero 

light, emerge. There should be a (dark) count rate of around 1-10/sec, once you reach 4-5 on 

the dial. At this point, the voltage is set correctly and you may raise the shutter. The count 

rate should now be much higher; investigate what happens as you adjust the bulb intensity, 

bearing in mind that you should not go above 6. (This isn’t Spinal Tap...) 

Connect the output of the discriminator to the CRO also and look at those pulses in coincidence 

with those from the PMT output. What signal form does the output of the discriminator 

take? Adjust the discriminator level so that one signal is produced per count above 50 mV 

from the PMT. This allows for the correct setting of the trigger. 

Note that the PMT efficiency is 4%. Using the counter provided, measure the actual count 

rate at a maximum in the intensity. 

Prelab Question (C): How does the width of the (single-slit) maximum change with the 

wavelength of the incident light? 

Adjust the discriminator level such that the count rate is around 10 3 /second.

2.5. EXPERIMENT - SINGLE PHOTON COUNTING 21 

Prelab Question (D): What is the error in every measurement of counts using the PMT? 

(Hint: check the nuclear notes...) 

By making sure you have an adequate count rate, do the same measurement as above of the 

intensity distribution, only this time counting actual photons in 10-second intervals. 

Prelab Question (E): Given that the length of the bench is around 1 m, calculate the time 

it takes for a photon to traverse the length of the bench. Given this time-of-flight, 

calculate the average time between photons for a (true) count rate of 2 × 10 4 /second, 

as may correspond to the true count rate of the experiment at the central maximum. 

What does this say of the nature of the measurement? 

Analyse the data as before, making only necessary adjustments to the parameters given the 

change in the initial conditions. 

Question (d): Discuss the situation as observed. What do you conclude about the nature of 

the interference in this case?

Chapter 3 

The Michelson interferometer 

3.1 Abstract 

These experiments will examine some of the principles and applications of optical interference 

using the Michelson interferometer. The first sections look at the conditions of interference 

in the interferometer. Later sections will illustrate how the interferometer may be used 

to measure the separation of spectral lines in sodium and the refractive index of air. 


3.2.1 Overview 

Optical interference historically has provided some of the strongest evidence to support the 

wave theory of light. In addition, the interferometer constructed by Michelson and Morley 

in the 19 th century also provided the nail with which the coffin was closed on the concept 

of the ether 1 . Despite this seeming paradox, the phenomena arising from interference are 

most naturally explained if light is considered to be electromagnetic waves. The Michelson 

interferometer is an example of an amplitude splitting interferometer, where the wavefronts 

emitted from the light source remain undisturbed. A beamsplitter is used to produce two 

beams, each of which travel separate paths before being recombined. This not only allows 

individual adjustment of the length of each path, but also changes in the length of each arm 

can be measured accurately using the small wavelength of visible light. 

1 For a discussion of the ether, see Hecht [1]. The ether was assumed to be the medium in which electromagnetic 

waves propagated, by analogy with water waves propagating through water. 

23

24 CHAPTER 3. THE MICHELSON INTERFEROMETER 

3.2.2 Background theory 

3.2.2.1 Arrangement of the Michelson interferometer 

The Michelson interferometer uses quite a simple arrangement of mirrors and beamsplitters 

to produce interference fringes. Illuminating a diffusing ground-glass plate L with light 

from a discharge lamp (a mercury or sodium lamp, for example) produces an extended light 

source. Figure 3.1 illustrates the configuration. Light waves emitted from the source are 

incident on a glass plate, G 1 , the back of which is slightly silvered. The metal coating on G 1 

is not fully sufficient to reflect the light from L, so approximately half is transmitted through 

arm 2 to mirror M 2 , and the rest is reflected along arm 1 to M 1 . The beam traversing arm 2 

also passes through the compensator plate G 2 twice before returning to G 1 . G 2 is an exact 

copy of G 1 but without the silver coating. Part of the light from M 1 is transmitted through 

G 1 to the observer O, as is some of the light from reflected by G 1 from M 2 . In this way the 

beams from each arm interfere at O where the resulting pattern may be seen. 

M 

1 

L 

arm 1 

Source 

G 

G 

1 2 

arm 2 

M 

2 

O 

Figure 3.1: Schematic arrangement of the Michelson interferometer. 

Pre-lab Question 3.1 What is the purpose of the compensator plate, G 2 ? 

Pre-lab Question 3.2 If M 1 is moved a distance d, what is the change in the total path 

length for light travelling in arm 1 of the interferometer? 

The answer to Pre-lab question 3.2 is important with regards to the conditions of interference. 

3.2.2.2 Conditions of interference 

The conceptual rearrangement of the mirrors shown in figure 3.2 helps to explain how the 

fringes are formed. An observer at the detector will see both mirrors M 1 and M 2 simultaneously, 

each of which forms an image of the ground glass plate L, leading to the image planes 

L 1 and L 2 .

3.2. INTRODUCTION 25 

2d 

S 

d 

S 

1 

θ 

S 

2 

θ 

2dcos θ 

Detector 

M 

1 

M 

2 

L 

L 

L 

1 2 

Figure 3.2: Conceptual rearrangement of the Michelson interferometer. 

Consider a point S on the glass plate L. An image of S will be formed due to mirrors M 1 

and M 2 at S 1 and S 2 , respectively. If M 1 and M 2 have a relative separation d, then S 1 and S 2 

will have a relative separation of 2d and the path length D = 2d cos θ. Interference fringes 

will be observed at the detector, with destructive interference satisfying 

D = nλ . (3.1) 

Pre-lab Question 3.3 If the path length difference is an integral number of wavelengths, 

constructive interference would normally be expected. Why is destructive interference observed 

in this case? (You may need to refer to Hecht [1] to answer this one). 

Interference between two waves occurs also if two other conditions are met, namely: 

Coherence Two waves are said to be coherent if there is a constant phase difference, which 

may be zero, between them. 

Polarisation The polarisation of an electromagnetic wave is the vector indicating the plane 

of the wave’s electric field. 

Only waves which are coherent and have the same polarisation will interfere. 

Pre-lab Question 3.4 How does the arrangement of the interferometer preserve coherence 

and polarisation? 

3.2.3 Apparatus 

3.2.3.1 Interferometer 

The Michelson interferometer is shown schematically in figure 3.1. The mirror M 2 has fine 

controls to adjust the horizontal and vertical orientations, while M 1 can be displaced along 

arm 1 by adjusting the micrometer. The ground glass plate L has a line marked on it to act 

as a pointer for aligning the mirrors.


3.2.3.2 Light sources 

The light sources provided are a mercury lamp, a sodium lamp and a tungsten lamp. A filter 

(a piece of green cellophane) is placed over the mercury lamp to absorb all wavelengths 

except the one at 0.5461 µm, producing a light source that emits light of constant wavelength 

(green). Once switched on, the lamp will require a few minutes to achieve full brightness, 

and should not be switched off until the session has ended. 

• Wavelength of Hg lamp: λ = 0.5461 µm 

The tungsten lamp is a thermal light source, emitting light with wavelengths covering the 

entire visible region. 

3.2.3.3 Polaroids 

A polaroid is a device that is highly asymmetric in terms of the light that it transmits. The 

preferred transmission axis in the polaroid is known as the polarisation axis, and only the 

components of the electric field of the incident light that are parallel to the polarisation axis 

will be transmitted. The result of this is that the light transmitted by the polaroid will have a 

well defined polarisation. 

Question 3.1 How does the change in the polarisation of the transmitted light affect its 

intensity? 

There are two polaroids that can be mounted in the interferometer. Both are standard camera 

lens polaroids and may be rotated around 360 ◦ in their mounts. 

3.2.4 Experimental work 

WARNING: THE DISCHARGE LAMPS 

CAN BECOME QUITE HOT. HANDLE THEM 

ONLY BY THEIR STANDS. 

Switch on the mercury lamp and allow at least 5 minutes for it to warm up. Position the lamp 

so that the emitted light evenly illuminates the ground glass plate. Note that the ground glass 

plate has a line marked on it. This acts as a pointer to help in the alignment of the images 

from mirrors M 1 and M 2 . There should be three images of the pointer visible when looking 

into the interferometer. 

Question 3.2 Explain from where each of the three images comes.

3.2. INTRODUCTION 27 

3.2.4.1 Initial adjustment 

Set the position of the micrometer on M 1 to be about the centre of the range. (Ask the 

demonstrator for help if you have trouble reading the micrometer.) Adjust the mirror M 2 using 

its controls until two of the three images are aligned. Once properly aligned, interference 

fringes should become visible. Careful adjustment of M 2 should improve the visibility of 

the fringes and also alter their appearance. 

Question 3.3 What effect does adjusting the controls on M 2 have on the appearance of the 

interference fringes? 

Question 3.4 Explain the curvature of the fringes (again, you may find Hecht [1] useful). 

Discuss with your demonstrator. 

3.2.5 Conditions of interference - polarisation 

Adjust M 2 until at most about 10 slightly curved fringes are in the field of view. Place the 

polaroids in each arm of the interferometer with a slight rotational offset so that reflections 

off the surfaces of the polaroids will not distort the interference. Observe the effect on the 

fringes as you rotate one of the polaroids through 360 ◦ . What do you expect to observe? 

Does that agree with your observations? 

3.2.6 Conditions of interference - coherence 

Remove the polaroids from the interferometer and adjust M 2 so that there are about 20 

slightly curved fringes in the field of view and record the position of the micrometer on 

mirror M 1 . As the micrometer is adjusted the curvature of these fringes will change. Why? 

Rotate the micrometer so that the fringes become less curved and continue rotating until the 

fringes are curved in the opposite direction to that from which you started. Again record 

the position of the micrometer. Position the micrometer midway between the two positions 

and make minor adjustments to the micrometer position until parallel fringes are observed. 

Adjust M 2 until only one or two parallel fringes are observed. 

Question 3.5 What is the significance of adjusting the micrometer to achieve parallel fringes? 

What do parallel fringes suggest about the path difference between M 1 and M 2 ? 

Remove the mercury lamp (do NOT turn it off) and position the tungsten lamp so that the 

ground glass plate is evenly illuminated. Look into the interferometer and SLOWLY rotate 

the micrometer until fringes become invisible. They should appear within two full turns of 

the micrometer. If not, return to the original position and rotate in the other direction. Note 

the position of the micrometer where the fringes are clearest and record your observations. 

(Hint!) Note in particular the number of black and white fringes and the manner in which 

the colours disperse as you move away from the centre of the pattern.


Question 3.6 Explain why the colours in the pattern disperse as they do and why the fringes 

are observed such a small range of path difference. What property of the light source allows 

this pattern to become manifest? 

3.2.7 Micrometer calibration 

Switch the tungsten lamp off and replace it with the mercury lamp. Adjust M 2 until there 

are about 10 straight vertical fringes in the field of view and rotate the micrometer so that the 

pointer is on a fringe and record the position. Slowly rotate the micrometer until 20 fringes 

have passed the pointer and record the new position. Repeat this 10 times, always rotating 

in the same direction, so that a total of 200 fringes will have moved past the pointer. 

Question 3.7 Using equation 3.1, determine the change in path difference necessary to move 

the fringe pattern by 200 fringes. 

Question 3.8 What is the total change in the position of the micrometer that produced the 

observed change in the path difference? 

Question 3.9 What would be the change in the path difference if the micrometer were rotated 

by one unit? 

3.3 Fringe visibility 


Light emitted from the discharge lamps stems from atomic transitions in the gas of atoms 

contained in the lamp. The emitted light has a narrow wavelength band (or bands) corresponding 

to the energy (or energies) of the transition(s), such that 

E = hc 

λ 

(3.2) 

where E is the energy of the transition, λ is the wavelength of the light, h is Planck’s constant 

and c is the speed of light. For sodium, there are two transitions of similar energy (call them 

E 1 and E 2 ) which lead to the emission of light, with the separation of their wavelengths 

being ∆λ = 0.597 nm. 

The photons emitted due to two close atomic transitions each form an independent fringe pattern 

in the interferometer. The small difference in wavelengths means that the periodicity of 

the interference patterns will be slightly different, and hence at regular intervals the maxima 

of one pattern will be aligned with the minima of the other, causing a secondary interference. 

Regions where no fringes can be seen are referred to regions of zero fringe visibility. If the 

change in path difference between each region is denoted by ∆D, then the number of fringes

3.3. FRINGE VISIBILITY 29 

counted from λ 1 is m 1 and that counted from λ 2 is m 2 , where m 2 = m 1 + 1. The separation 

between the two emitted wavelengths can then be determined by measuring ∆D and using 

∆D = m 1 λ 1 = m 2 λ 2 = (m 1 + 1)λ 2 (3.3) 

∆λ = λ 1 − λ 2 = λ 1λ 2 

∆D ≈ λ2 

∆D 

(3.4) 

Pre-lab Question 3.5 Show that 

∆λ = λ 1λ 2 

∆D 

(3.5) 

Pre-lab Question 3.6 Equation 3.2 gives the energy of the atomic transition in terms of the 

wavelength of the emitted photon. For photons of energies E 1 and E 2 show 

∆E = E 2 − E 1 ≈ hc∆λ 

λ 2 (3.6) 


Sodium Lamp The sodium lamp is similar to the mercury lamp used earlier. Sodium only 

emits two very closely spaced wavelengths in the visible region, such that λ 1 ≈ λ 2 ≈ 

λ. 

• Wavelength of sodium lamp: λ = 0.5893 µm. 

3.3.3 Experiment 

Replace the mercury lamp with the sodium lamp and adjust M 2 until there are about 20 

fringes in the field of view. Starting with the micrometer at 0, rotate until the fringe pattern 

starts to fade. Slowly rotate the micrometer until the pattern is no longer visible and record 

the position of this zero visibility region, along with a reasonable error based on the range 

over which the region is observed. Continue rotating the micrometer finding and recording 

the position of each successive zero visibility region until the end of the micrometer scale is 

reached. 

Determine the change in the micrometer position needed to move from one region to the 

next. Any significant discrepancies here may indicate that a region of zero fringe visibility 

has been missed. Return to these positions on the micrometer to find any missed regions. 

Question 3.10 Using the position of the first and last region of zero visibility, and the number 

of such regions, calculate the average micrometer change necessary to move between 

successive regions. Use the calibration obtained earlier to determine ∆D, the average path 

difference change between regions of zero fringe visibility. 

Question 3.11 Calculate the wavelength separation of the sodium lines using equation 3.4 

and hence determine the energy difference between the two atomic transitions responsible 

for the emissions. Quote the result in eV. Is your answer reasonable? Why?


3.4 Refractive index of air 


It does have one. . . As the refractive index of a medium increases, the speed of light in the 

medium is reduced, which results in the effective optical path length for the light travelling 

through the medium increasing since the time taken to travel any given distance will increase. 

In an interferometer, the change in the optical path length generates a relative path difference 

if the refractive index in one arm of the interferometer is changes. The change in path 

difference can be used to measure the change in the refractive index of the original medium. 

3.4.2 Background theory: optical path length 

The optical path length is the distance that would be travelled by light in a vacuum in the 

time that it takes light to travel a distance s through a medium of refractive index n. If the 

refractive index of the medium varies along the path travelled, then the optical path length is 

given by 

∫ 

OPL = n(s)ds . (3.7) 

The integral may be done if the refractive index of the medium is homogeneous over a 

distance s, whence 

OPL = ns . (3.8) 

Pre-lab Question 3.7 If a cell of length s in arm 1 of the interferometer is evacuated, what 

is the change in the optical path length as the refractive index changes from that for air to 

that for vacuum? 


Gas cell The gas cell has a hand-operated pump attached to it that enables air to be removed 

from the cell. A release trigger located beneath the gauge allows the cell to be returned 

to atmospheric pressure. The length of the cell is (4.0 ± 0.1) cm. 

Telescope A telescope is available which may be mounted onto the interferometer to magnify 

the observed fringes. When mounting, the alignment of the telescope may not enable 

the fringes to be seen, whence some adjustments to the telescope may be needed. 

Mirror M 2 can also be adjusted to move the fringe pattern into the centre of the field 

of view. 


Using the sodium lamp as a light source, obtain clear circular fringes by refined adjustment 

of M 2 . Place the gas cell with pump attached into arm 1 of the interferometer and adjust the 

micrometer to ensure that the fringe pattern is not near a region of zero visibility. If the centre

BIBLIOGRAPHY 31 

of the rings is not clearly visible, mount the telescope onto the interferometer to magnify the 

pattern. 

Watch the pattern closely. Slowly pump down the gas cell, counting the number of dark 

fringes that appear from (or collapse into) the centre of the pattern. Continue pumping the 

cell until the fringe pattern no longer changes. Record the total number of fringes counted as 

the cell was evacuated. Let the cell back up to air by pressing and holding the release trigger 

for about 20 seconds. Repeat the procedure 10 times and determine the average number of 

fringes counted as the cell is evacuated, using the standard deviation in the numbers as the 

error. 

Question 3.12 Using equation 3.8, determine the change in the optical path difference resulting 

from the evacuation of the cell. 

Question 3.13 Using your answer to Pre-lab question 3.7, calculate the refractive index of 

air. Record the temperature in the room from the thermometer on the wall and determine the 

expected value of the refractive index at that temperature for the wavelength of sodium using 

the chart provided. Compare your measured value to that expected. (Hint - remember that 

the refractive index of the vacuum is n vacuum = 1). 

Bibliography 

[1] E Hecht. Optics. Addison-Wesley, 3 rd edition, 1998.

Chapter 4 

Fraunhofer diffraction 

4.1 Abstract 

The subject of this experiment is the phenomena of the diffraction of waves. While with 

the advent of quantum mechanics, and its basis of wave-particle duality, we know of the 

diffraction of particles (diffraction of electrons and neutrons are the most notable examples), 

it was the diffraction of light that was a major advance and led to the acceptance of the wave 

theory of light. The main focus of this experiment will be the far-field intensity distribution 

resulting from the diffraction of light from a single slit. Related to that will be the study of 

diffraction from a wire, and from a number of slits. Diffraction from other objects, and how 

all are related, will round off the experiment. 


The work of Fresnel and Fraunhofer regarding the diffraction of light around objects was 

instrumental in the acceptance of the wave theory of light. The principles of diffraction 

developed by them are applied to any wave/particle, be they photons, electrons, protons, 

neutrons, or even atoms and molecules. The language of diffraction is even used in subatomic 

physics for describing the dominant processes of the interaction of various systems. 

A large number of applications of diffraction involves waves incident on large periodic structures 

such as crystals or molecules. The observed patterns are dependent on the structure of 

the object causing the diffraction of the waves. With the knowledge of the patterns one may 

work backwards to reconstruct the structure of the diffracting object 1 . 

All of these applications share the same basic theory developed by Fresnel and Fraunhofer for 

the case of diffraction of light. It is this phenomenon, and Fraunhofer diffraction specifically, 

that we shall investigate here. 

1 Francis Crick and James Watson discovered the double-helix structure of the DNA molecule by studying 

its X-ray diffraction pattern. They shared the Nobel Prize for Medicine, along with Maurice Wilkins, in 1962 

as a result. This was the most celebrated example of the importance of diffraction as a tool for studying the 

structure of matter. 

33

34 CHAPTER 4. FRAUNHOFER DIFFRACTION 

4.3 Theory 

Francesco Grimaldi in the 1600s was the first to observe in detail the deviation of light 

from rectilinear propagation due to the obstruction by a solid, opaque, object. The resulting 

shadow is made up of bright and dark regions quite unlike anything one might expect from 

Newton’s theory of light particles. Grimaldi called this phenomenon “diffractio” . The 

effect is a general characteristic of waves occurring whenever a portion of the wave front is 

obstructed in some way. This is independent of the form of the propagating wave. It is, 

in fact, a result of the interference of the segments of the wavefronts propagating beyond the 

obstacle, when that obstacle alters part of the wave front in terms of amplitude, phase, or 

both. 

4.3.1 Fresnel or Fraunhofer? 

Consider an opaque screen Σ, containing a single, small aperture, which is illuminated by 

plane waves from a distant point source, S. A pattern is observed after Σ in a plane σ parallel 

and very close to Σ. At this point, a clear image of the aperture is visible. When one starts 

to move σ away from Σ, 

Fresnel diffraction results. Fringes begin to appear about the structure of the aperture in 

question. This is near-field diffraction. If one moves still further, to a distance far 

from Σ, 

Fraunhofer diffraction results. A continuous change in the fringes is effected to a point 

where the projected pattern is spread out considerably, and the pattern bears no discernible 

structure of the aperture. Beyond this, the change is only to the size and not 

the structure of the pattern as σ is moved further away. This is also known as far-field 

diffraction. 

The Fraunhofer condition is R > a 2 /λ, where R is the smaller of the two distances from S 

to Σ and from Σ to P (a point on σ), see figure 4.1. 

4.3.2 Diffraction by a single slit 

Consider plane waves incident on a slit of width d, as shown in figure 4.1. The contribution 

to the electric field dE at a point P on the screen plane σ, due to a segment of the slit dx, is 

given by 

dE = ε L 

sin (ωt − kr)dx , (4.1) 

R 

where ε L /R is the amplitude of the incident wave. The phase of the wave is dependent on 

r(x) which can, in the Fraunhofer approximation, be written as a linear function of x, such 

that 

r = R − x sin θ. (4.2) 

Integrating over x leads to the expression of the electric field at P , viz. 

E = ε L 

R 

∫ d/2 

−d/2 

sin [ωt − k (R − x sin θ)] dx , (4.3)

4.3. THEORY 35 

X 

Σ 

σ 

P 

d/2 

dx 

R 

−d/2 

Z 

L 

Figure 4.1: Plane waves incident on a slit of width d. The irradiance at point P is given by 

equation 4.6. 

from which, 

where 

E = ε Ld sin β 

sin (ωt − kR) , (4.4) 

R β 

β = kd 

2 

sin θ = 

πd 

λ 

sin θ . (4.5) 

The irradiance or time-averaged intensity, I(θ), is the observed quantity and it is given by 

I(θ) = ε 0 c 〈 E 2〉 

= ε ( ) 2 ( ) 2 

0c εL d sin β 

2 R β 

( ) 2 sin β 

= I(0) , (4.6) 

β 

where 〈 sin 2 (ωt − kR) 〉 = 1/2, and I(0) is the intensity at β = 0. This defines the intensity 

at any point on the detection plane in terms of measurable quantities. The distribution is 

shown in figure 4.2. 

Pre-lab Question 4.1 What is the relative intensity between the central maximum and the 

first subsidiary maxima? 

Pre-lab Question 4.2 Assuming that the separation, L, of the slit to the screen is known, 

derive a formula relating the irradiance to the distance along the diffraction pattern. Identify 

any quantities that are not known and which need to be determined from fitting the data. 

Pre-lab Question 4.3 Why is the slit in figure 4.1 machined to have bevelled (kinked) edges?


1 

0.8 

I(θ)/I(0) 

0.6 

0.4 

0.2 

0 

-10 -8 -6 -4 -2 0 2 4 6 8 10 

β 

Figure 4.2: Intensity distribution for the diffraction of light by a single slit. 

4.4 Experiment 

BEFORE BEGINNING, READ 

CAREFULLY CHAPTER 16 ON LASER 

SAFETY. 

You will be operating a laser during the course of this experiment and you must take great 

care in preventing stray reflections away from the apparatus. Stray reflections into people’s 

eyes may cause damage. Use the black card provided to block the beam when moving things 

on and off the optical bench. The beam can reflect off the posts and holders into the eyes of 

someone standing behind you. 

4.4.1 Calibration 

Turn on the laser and ensure that the beam runs down the length of the optic bench. Place the 

microscope objective onto the bench in front of the laser, making sure to prevent any stray 

reflections. Position the objective such that a smooth diverging (spherical) beam is produced 

and that it illuminates evenly the screen in front of the camera. Vertical and horizontal 

controls on the optic mount are available for small adjustments to the alignment. Place the 

aluminium plate with holes separated by 5 mm into the slot where the screen is mounted. 

Turn on the camera and computer, following the instructions given (see appendix, section 

4.6.1), and run the imaging software. Record the image of the dots from the plate.

4.4. EXPERIMENT 37 

You may need to adjust the plate to ensure that the line of dots is horizontal. Once a suitable 

line is produced, capture the image and take an intensity profile of the dots. The peaks in the 

spectrum should correspond to each of the holes in the plate, and there should be about 10 

well-defined peaks. Playing with exposure times may help in getting a good profile. 

Identify the pixels corresponding to each of the peaks in the profile, and determine the average 

pixel separation between them. From that determine the pixel calibration of the camera 

in mm/pixel. 

4.4.1.1 The CCD camera 

The CCD camera provides a simple and efficient way in which to capture intensity information 

and convert it into voltage information. The linear way in which light intensity is 

converted to voltage makes the camera ideal for quantitative applications 2 . The voltage information 

can be digitised, making storage and analysis of CCD images very easy. The 

applications of CCD are too numerous to list here; bear in mind that all digital cameras are 

CCDs. 

The appendix (section 4.6.2) describes the operation of the CCD. You should read the material 

and answer the following question 

Pre-lab Question 4.4 The potential wells in the substrate can only hold a limited number 

of electrons due to the Pauli exclusion principle. When the potential well is full, the pixel is 

said to be saturated. What will happen to electrons emitted from the SiO 2 substrate if the 

closest pixel is full? (This is known as “blooming”.) 

The camera you will be using is equipped with “anti-blooming” technology, which greatly 

improves the quality of the images. Your demonstrator will explain the principles involved. 

4.4.2 Single slit diffraction 

4.4.2.1 Initial adjustments and observations 

Remove the plate and return the microscope objective to the apparatus rack. We now need 

to make sure that the laser runs parallel to the optic bench. Using a mounted piece of tracing 

paper in one of the carriages, check that the position of the laser beam spot on the paper is 

unchanged as the carriage is moved along the length of the bench. Use the horizontal and 

vertical controls on the laser mounts to correct any misalignment. (There are two mounts for 

the laser, and so the adjustment may be a little tricky.) 

Once aligned, replace the tracing paper with the single slit mount. A diagram of the slit 

is shown in figure 4.3. It is mounted on an aluminium plate which is appropriate for an 

optic mount. The slit should be placed as close as possible to the laser so that the diffraction 

pattern can propagate over as long a distance as possible. Allow room for the variable neutral 

density filter to be inserted between the laser and the slit. 

2 It is noteworthy that Einstein got his first Nobel prize for the photoelectric effect, the basis for all CCD 

technology.


Diffracted beam 

Laser 

Slit width adjustment 

Figure 4.3: The single slit. 

4.4.2.2 Neutral density filter 

WARNING! THE NEUTRAL DENSITY FILTER MAY PRODUCE STRONG 

REFLECTIONS. TAKE CARE TO ENSURE THAT REFLECTIONS ARE 

DIRECTED AWAY FROM ANY PATH THAT MIGHT BRING THEM IN 

CONTACT WITH PEOPLE IN THE LAB. 

The neutral density filter is an optically dense non-polarising material used to reduce the 

intensity of incident light. It is mounted on a stand separate to the optic bench to allow for 

easy placement between the laser and the diffracting element. It is circular, and rotating it 

changes the density of material thus changing the attenuation 3 of the light. 

Align the slit with the beam by adjusting the horizontal and vertical controls on the mount to 

obtain a clear and symmetric diffraction pattern at the screen. 

Question 4.1 What effect does changing the width of the slit have on the pattern produced? 

Question 4.2 What effect does rotating the slit about the beam axis have on the pattern 

produced? 

4.4.2.3 Recording an image 

Adjust the slit so that the 0.3 mm wire barely slides through. Closing the slit and opening 

slowly until the wire passes through will set the appropriate width without crushing the wire 

in the jaws of the slit. Use the micrometer to measure the width of the wire, then close the 

micrometer with no object in place to record any offset. 

Scan the image of the diffraction pattern. 

3 “attenuation” means “reducing the intensity of”.


Question 4.3 Does the pattern produced on the monitor have as much detail as that seen 

on the screen? Consider your answer in the context of your answer to Pre-lab question 4.2. 

What is happening in the camera to reduce the resolution of the system? Discuss with your 

demonstrator. 

Rotate the slit to make sure that the entire diffraction pattern is horizontal. Adjust the ambient 

light in the lab to reduce background if necessary. 

The image may now be stored to produce a profile of the diffraction pattern. Check the 

following before saving the data: 

• Is there any saturation in the pattern? If so, can the saturation be reduced so that the 

whole pattern may be recorded? 

• Is the pattern symmetric, i.e. are the heights of the peaks to each side of the centre the 

same? 

The high intensity of the central maximum compared to the rest of the pattern precludes any 

real possibility of analysing the whole pattern. 

Question 4.4 Based on your answer to Pre-lab question 4.2, use Excel to fit the formula to 

the observed line profile. 

Run the program and extract the relevant parameters from your data. If you removed the 

central maximum from the fitting, use the parameters to recalculate the whole pattern and 

plot the whole pattern to the profile from the recorded image without the central maximum. 

Question 4.5 How well does the fitted profile match the data? 

4.4.3 Babinet’s principle 

The diffraction patterns produced by two complementary screens are the same, in the Fraunhofer 

limit, save for a small region in the centre of the pattern. This is Babinet’s principle. 

Its usefulness lies in the ability to study the structures of diffracting objects that are difficult 

to mount, by studying their complementary partners 4 . 

4.4.3.1 Background theory 

Two screens, Σ 1 and Σ 2 , are said to be complementary if the opaque regions in one match 

exactly the apertures in the other, such that if added together they form a completely opaque 

screen. If the electric field at the image plane due to screen Σ 1 is E 1 , and the field due to Σ 2 

is E 2 , then 

E 1 + E 2 = E 0 (4.7) 

where E 0 is the unobstructed field. 

4 see Hecht, [1], page 500.


Pre-lab Question 4.5 When E 0 = 0 it is clear that the diffraction patterns from the two 

screens are the same. Show what would happen for E 0 ≠ 0, and how the diffraction patterns 

are related. 

4.4.3.2 Experiment 

A piece of wire ≈ 0.3 mm thickness (the same as that used to set the slit width), attached to 

a circular mount is used as the complementary screen to the slit. Measure the thickness of 

the wire with the micrometer, being extra careful not to crush the wire. The mount may be 

rotated on its optic mount to allow for a horizontal pattern to be observed. 

Mount the wire and observe the pattern. Removing the neutral density filter will produce a 

clearer pattern. 

Question 4.6 How does the pattern produced by the wire differ from that produced by the 

slit? Is Babinet’s theorem satisfied? 

Put back the neutral density filter and check the alignment of the wire with respect to the 

laser to make sure that the diffraction pattern produced is horizontal. Scan the image, noting 

again the distance between the object and screen. Once completed, turn off the CCD camera 

(you won’t be needing it again). 

Question 4.7 With the central region removed, how does the diffraction pattern compare to 

the single slit pattern with the central maximum removed? What does this suggest? 

Question 4.8 Fit the expected intensity distribution and determine the width of the wire. 

How does it compare to the slit width you determined above? How do the values for I(0) 

compare? 

4.4.4 Multiple slits 

Diffraction from many slits combines both the ideas of diffraction and interference. While 

there is no actual physical distinction between the two, interference is generally reserved for 

instances where only a few wavefronts are involved whereas diffraction is the case corresponding 

to many wavefronts, as per the Huygens-Fresnel principle. 

4.4.4.1 Background theory 

For a system with N slits of width b and separated by a distance a, the integral in equation 4.3 

may be written as 

E = C 

∫ b/2 

−b/2 

F(x)dx + C 

∫ a+b/2 

a−b/2 

F(x)dx + · · · + C 

∫ (N−1)a+b/2 

(N−1)a−b/2 

F(x)dx, (4.8)


where F(x) = sin [ωt − k(r − x sin θ)] as before and C is a constant related to the amplitude 

of the wave. This leads to the following expression for the irradiance distribution at the image 

plane in the Fraunhofer limit, 

I(θ) = I 0 

( sin β 

β 

) 2 ( ) 2 sin Nα 

, (4.9) 

sin α 

where β = (πb/λ) sin θ, α = (πa/λ) sinθ, and I 0 = I(0)/N 2 . The distribution is shown in 

figure 4.4. 

1 

a = 3b; N = 8 

(sin(β)/β) 2 (sin(Nα)/sin(α)) 2 

(sin(β)/β) 2 

0.8 

I(θ)/I(0) 

0.6 

0.4 

0.2 

0 

-6 -4 -2 0 2 4 6 

β 

Figure 4.4: The irradiance distribution for 8 slits with the geometry given. 

The (sin (Nα)/ sin α) 2 factor can be considered to contain all the interference effects, with 

the overall diffraction pattern being produced with this interference being modulated by the 

irradiance distribution of a single slit. This leads to the following observations: 

1. Principal maxima will occur when (sin (Nα)/ sin α) = N, or when 

2. Minima will occur for sin(Nα) = 0; 

α = 0, ±π, ±2π,...; (4.10)


3. Local maxima in sin(Nα) will lead to subsidiary maxima. 

Pre-lab Question 4.6 Determine the values of α corresponding to the minima and subsidiary 

maxima in the diffraction pattern. For an aperture of N slits, how many subsidiary 

maxima will occur between principle maxima? How many minima will occur in the same 

region? 

A diffraction grating is effectively a large number of slits of equal spacing and equal width. 

The most useful aspects of gratings are derived from equation 4.10, with the substitution of 

α leading to 

a sin θ = mλ, (4.11) 

where m is an integer. This is known as the grating equation, and defines the following 

properties: 

1. The angular dependence of the principle maxima does not depend on the number of 

slits, provided N > 1. The positions of the principle maxima in the imaging plane 

should then be fixed. 

2. The angular position of the maxima depends on λ, which makes the diffraction grating 

a wavelength dependent device. This has numerous applications, particularly where 

a light source composed of many wavelengths needs to be split into its constituent 

colours. (Another is the subject of a prac in this laboratory. . . 5 ) 

4.4.4.2 Experiment 

There are two microscope slides. The first has groups of 1, 2, and 4 slits, each of equal width 

and spacing; the second has groups of 8 and 16 slits. 

Place the first of the microscope slides into the mount so that the pattern for the single slit is 

observed. Take note of the distribution and move the slide to observe the diffraction pattern 

of 2 slits. Continue with the other groups, taking note of the number of subsidiary maxima 

and minima. 

Question 4.9 Are the numbers of subsidiary maxima and minima consistent with your answers 

to Pre-lab question 4.6? 

Question 4.10 If a green laser were incident on the multiple slits concurrently with the red 

laser, how would the angular separation of the principle maxima change? 

4.5 Further work - circular and other diffracting objects 

Time and mojo permitting, you may wish to continue with the following sections. 

5 i.e. the Michelson experiment, chapter 3.

4.5. FURTHER WORK - CIRCULAR AND OTHER DIFFRACTING OBJECTS 43 

4.5.1 Overview and background theory 

Remember that the diffraction pattern produced by an object is directly related to the structure 

of that object. Hence, the importance of diffraction is obvious when one is dealing with 

a complex object whose structure is not readily apparent but produces a diffraction pattern 

that is far more easily studied. 

The theory of diffraction patterns produced by a circular object is far more complicated than 

the example of the single slit. Herein, we will only touch on the basics: you may wish to 

read section 10.2.5 of Hecht [1] if you want to know more. 

Plane waves incident on a circular aperture of radius a will form a circularly symmetric 

diffraction pattern on the screen σ at a distance L. That circular symmetry is a consequence 

of the electric field at σ being described by a class of functions known as Bessel functions, 

which themselves arise quite naturally when one is dealing with situations involving circular 

or spherical symmetry. The geometry is shown in figure 4.5. 

y 

000000000 

111111111 

000000000 

111111111 

000000000 

111111111 

000000000 

111111111 

000000000 

111111111 

000000000 

111111111 O 

000000000 

111111111 

000000000 

111111111 a 

000000000 

111111111 

000000000 

111111111 

000000000 

111111111 

000000000 

111111111 

000000000 

111111111 

x 

R 

L 

θ 

q 1 

y 

Figure 4.5: Geometry for the diffraction from a circular aperture. 

The resulting pattern has a large, high intensity, central region with secondary maxima forming 

rings around the centre. The area covered by the central maximum is called the Airy 

disc 6 . The secondary maxima are known collectively as Airy rings. The Airy disc extends to 

the centre of the first dark ring, with the radius given by 

q 1 = 1.22 Rλ 

2a , (4.12) 

where the variables are defined by the geometry (figure 4.5). Most of the intensity (84%) is 

concentrated in the centre, with the intensity falling away sharply. 

x 


Remove the mounted wire and replace it with the circular apertures. The three apertures, 

of varying radii, are mounted on a mounting similar to that for the wire. Align the smallest 

6 after Sir George Biddell Airy (1801–1892), who first derived the mathematical description of the intensity 

profile.


aperture to obtain a clear diffraction pattern at the camera screen. With the calipers provided, 

measure the diameter of the radius of the first dark ring. Measure also the distance between 

the aperture and the image. 

Question 4.11 Is the diffraction pattern as expected? Does it resemble that for the single 

slit? 

Question 4.12 What happens to the diffraction pattern as one varies the radius of the aperture? 

Question 4.13 What is the radius of the smallest aperture? 

Replace the circular apertures with the mount of the circular discs. Those are of the same 

radii as the apertures, so this too is a test of Babinet’s theorem. Align the laser with the 

smallest disc to obtain a clear diffraction pattern. 

Question 4.14 How does the observed pattern differ with that from the aperture? 

Question 4.15 Allowing for those differences, measure the radius of the first dark ring and 

determine the radius of the smallest disc. Is this consistent with the measured radius of the 

aperture? 

4.6 Appendices 

4.6.1 CCD camera operating instructions 

• Ensure that the camera cables are connected before switching on the CCD camera, 

which is powered through the CCD computer (black box). This is connected to the 

PC. 

• Run the Fraunhofer image collection software in the CCD software folder on the PC. 

This is called “MaximDL”. 

• There are three main windows in MaximDL that you’ll use, which may be found in 

the view menu: the Image Window, the CCD Control Window, and the Line Profile 

(LP) Window. 

• Under the CCD Control window, proceed to the Setup tab. Connect the camera (under 

software) and cool the CCD. It should reach 0 ◦ C. 

• Once the desired temperature is reached and is stable, take an image. This is done in 

the CCD/Expose tab. Feel free to play with the settings to achieve best results for the 

image. 

• You may also crop the image to the desired size. All future images are captured to this 

size.

4.6. APPENDICES 45 

• An intensity profile is obtained in the LP window. When that window is open, clicking 

on the captured image will place a horizontal line on the image. The profile along that 

line will be plotted. That profile may then be exported for analysis. 

• Save images as 12-bit FITS uncompressed (*.fit). Do not use compression as this is 

liable to lose information. 

• Powering down: from the CCD/Setup tab, turn the cooler off and disconnect the camera 

BEFORE closing the program and turning off the PC. 

4.6.2 How the CCD works 

The CCD camera consists of a two dimensional array of photodiodes coupled to the CCD 

array. A photodiode is a device that will absorb a photon and release an electron through 

the photoelectric effect. The photon energy required depends on the material used for the 

photodiode. For a material such as silicon dioxide (SiO 2 ) the energy required is 1.09 eV, 

corresponding to a photon with a wavelength of 1140 nm, which makes it ideal for detecting 

visible and near-infrared light. 

Figure 4.6: Schematic representation of CCD camera in a) initial condition with electrons 

trapped in potential well, b) intermediate state, and c) electrons displaced after one clock 

pulse.


To understand what the CCD array is and what it does consider the case of the one-dimensional 

CCD shown in figure 4.6. A silicon dioxide layer is deposited onto a p-type semiconductor 

silicon substrate. An array of metal electrodes is placed over the top of the SiO 2 layer. When 

high and low voltages are applied alternately to electrodes, a local potential well is formed in 

the silicon substrate in the vicinity of the high potential electrode. Electrons emitted from the 

silicon dioxide are confined to the potential wells. Reversing the potentials on the electrodes 

shifts the potential well, which takes the electrons with it. In this way the electrons can be 

shunted along the array to the last potential well where an output voltage is recorded, with 

the number of pulses counted since the initial shift indicating the original pixel for the data. 

A longer description is provided in the ST-237 CCD Camera manual. 



Chapter 5 

Holography 

5.1 Abstract 

Holography is an application of the interference property of light. From its limited beginnings, 

it is now enjoying widespread application from becoming a medium in the arts to the 

recording of minute deformations in materials from external forces. In the course of this experiment, 

you will make several holograms thereby learning the basics of holography. And 

this is the only experiment in the entire undergraduate physics program where you keep the 

results! 


Holography is an application of the interference property of light. By superimposing two 

sets of monochromatic and coherent wave fronts on a photographic plate, a microscopic interference 

pattern is recorded. The developed plate/film is known as a hologram and the photographed 

interference pattern stores both amplitude and phase information. When the hologram 

is placed in a beam of the same coherent, monochromatic light, the beam is diffracted 

through the fringes to produce wavefronts identical to those originally reflected from the 

object. Consequently, when viewed, the resultant wavefronts give a remarkably realistic, 

three-dimensional picture of the object. 

In this experiment, you will produce two or three holograms. Note that this experiment does 

not involve taking quantitative measurements or doing calculations, and as a result many 

students have difficulty writing the report. Remember that although you are not finding 

“a number”, you will be making observations on the holograms produced and so you are 

required to report an accurate qualitative description of the observations made and of 

the relevant physics ascribed to each. You must also take great care to keep a thorough 

informative record of what was done during the sessions, and explain clearly all experimental 

aims and procedures. As you have a lot of freedom in what you may wish to do, the record 

is of what you did. 

47

48 CHAPTER 5. HOLOGRAPHY 

5.3 Overview 

The principles of holography were first developed by Dennis Gabor in 1947 1 . Until 1960 

and the invention of the laser, application of this theory was limited (though not impossible). 

However, between then and the early 1970’s many useful and attractive applications for this 

process were discovered, making this one of the major new techniques, and physics success 

stories, of recent history. 

The production of holograms is quite simple, and the basic theory is relatively straightforward. 

Most of the background is given in the appendices, although a synopsis is given below. 

And you get to keep the holograms you produce to show off to your friends later! 

SAFETY NOTE! THE LASER USED IN THIS EXPERIMENT IS A 30 MW 

HE-NE LASER AND QUITE POWERFUL. TAKE GREAT CARE IN USING IT 

AS IT MAY CAUSE PERMANENT DAMAGE IF REFLECTED INTO THE EYE. 

5.4 Background 

The concept behind the hologram is relatively simple. Normal photography only records 

the intensity of an image, and all colours therein, and thus only a two-dimensional image is 

recorded. In order to produce the exact wavefront of the reflected light off any object, the 

relative phases of the light reflecting off the object, which vary according to its structure, 

must also be recorded. The observer then perceives the reconstructed image, when coherent 

light is passed through the diffraction grating (hologram), as three-dimensional. Gabor 

discovered the means as to how this relative phase may be recorded. 

5.4.1 Intensity of a wave 

The electromagnetic wave is constructed from transverse electric and magnetic fields; both 

are transverse to the direction of propagation and perpendicular to each other. The intensity 2 

of the wave is defined as the square of the amplitude of the electric field. For a plane wave 

of the form 

E = E 0 cos (k · r − ωt) (5.1) 

the intensity is given by 

I inst = ε 0 c ∣ ∣E 2∣ ∣ = ε 0 cE 2 0 cos 2 (k · r − ωt), (5.2) 

where E 0 is the amplitude of the electric field, k is the wave number and ω is the frequency. 

This intensity is known as the instantaneous intensity, being that given at time t. Normal 

1 for which he won the Nobel Prize in Physics in 1971. 

2 Often (as in Hecht, [1] section 3.3.2, page 50) the word irradiance is used instead of intensity. Both refer 

to the average energy per unit area per unit time.

5.4. BACKGROUND 49 

photography is an example where the time-averaged intensity, is recorded. That is given by 

I = ε 0 c 〈 |E| 2〉 

= ε 0 c 〈 E0 2 cos 2 (k · r − ωt) 〉 

= ε 0cE 2 0 

2 

(5.3) 

Note that in these experiments, spherical waves will be used to construct the holograms. The 

generalisation of these ideas to spherical waves can be found in appendix 5.6.1. 

5.4.2 Interference 

A problem arises in the recording of phase. Absolute phase is a little difficult to record: it 

doesn’t exist. Phase may only be discussed as between two waves, i.e., the relative phase is 

the relative angle between the two waves and interference is determined by the size of this 

relative phase. There are also relative phases due to reflections from different parts of the 

object and we need a way to be able to measure these. 

If one is to record a relative phase, then one needs two waves with which to record an 

interference pattern. The phase may then be recovered from this pattern. One of the waves 

is obvious: it is that reflected from the object. The other must be in phase (i.e. a relative 

phase of zero) with the wave used to reflect off the object if the changes in the phase after 

reflection are due only to the structure of the object itself. This reference beam is best 

constructed from the same light source to preserve the initial phase. How this is done is 

partly determined by which hologram we wish to produce. 

Appendix 5.6.2 develops the ideas of interference for plane waves. 

5.4.3 Holograms and holography 

Gabor coined the term holography from the (ancient) Greek word holos, meaning whole 3 . 

Holography, from Gabor’s earliest experiments through to modern applications, differs from 

standard photography in that it does not use lenses. Instead, the object is illuminated with a 

coherent monochromatic source of light, such as a laser, and the film is positioned to receive 

the light reflected from the object. As there is no lens, each point on the object reflects light 

to every point on the film. On its own, such an exposure would be rather boring: it would be 

a uniform (grey) and structureless picture. 

The same source also produces the reference beam and it too is reflected by a series of mirrors 

to illuminate the film with the same type of wave that illuminates the object. The interference 

of the two waves is formed in the plane of the film and it is that which is recorded. This interference 

pattern stores both amplitude and phase information. When light of the same form 

as the initial wave passes through it, the pattern acts as a diffraction grating, reconstructing 

the wavefronts as those reflected off the object. 

The transmission functions may be found in appendix 5.6.3. 

3 In modern Greek, the word is entirely different: olos!


Pre-lab Question 5.1 Why are we able to discern three dimensions with our sight? Why do 

we perceive the reconstructed image from the hologram to be three-dimensional? 

There are two main types of hologram: the transmission and the reflection. They are distinguished 

by the method used for recreating the wavefront. 

5.4.3.1 Transmission holograms 

The transmission hologram is one in which the initial wavefronts illuminate the grating (i.e. 

the film) and one perceives the (virtual) image as the transmission through the grating. This 

form of hologram requires that a coherent monochromatic source be used, although the wavelength 

of the light used to recreate the image need not be that of the light used to make the 

hologram. In that case, however, given that diffraction depends on wavelength, the image 

will change size if a different wavelength is used. 

5.4.3.2 Reflection holograms 

Reflection holograms, as the name suggests, require that the light used to recreate the image 

be on the same side as that of the observer, and the light is reflected back to the observer 

through the grating, forming the image. These holograms are necessarily thick, and are 

sometimes known as volume holograms. Such holograms are easily viewed with white light; 

those wavelengths which do not correspond to that which produced the hologram are absorbed 

by the plate. The reflection is then only of light with the same wavelength and the 

image is readily formed. 

5.4.3.3 In-line or off-axis? 

The first holograms produced were in-line. That refers to the reference beam being in the 

same line as that reflected off the object. The interference is seen as between the reference 

beam passing around the sample and the component of the beam which is distorted by the 

object. Coherence is not a problem as the same beam is used for both the object and reference. 

While these holograms may also be viewed using white light, it places constraints on the 

object. It must be small relative to the film, so that the reference beam may pass around it to 

reach the film, or the object must be transparent, so that the reference beam may pass straight 

through. Large, opaque objects will not work. Also, the beam used to reconstruct the image 

must be perpendicular to the hologram, making it difficult to see, as the observer tends to be 

in the way. 

Leith and Upatnieks each independently invented the off-axis hologram in 1962. Those 

require high coherence, since two beams are required. Those are normally from the same 

coherent source. Any type of object may be used, but the same coherent, monochromatic 

source is required to reconstruct the image.


5.4.4 Movement 

Note! The wavelength of the light used determines the scale of the interference. This is 

usually of the order of micrometres or less, hence any vibration in the apparatus during the 

course of the exposing of the hologram will destroy the interference pattern and no hologram 

will be recorded. The optic bench on which the apparatus sits is greatly damped to prevent 

any vibrations from external sources ruining the exposure. 

Pre-lab Question 5.2 What would be seen in the image if part of the object was changed 

minutely during the course of taking the hologram? Discuss with your demonstrator. 


During the course of the experiment, you will be making up to 3 holograms. The first is the 

easiest, and will teach you aspects of optical alignment necessary in setting up the subsequent 

holograms. It is also the most fun. 

5.5.1 Film handling and developing 

Before any hologram may be produced, some notes on the film used and the technique to 

develop it. 

5.5.1.1 Holographic film 

Both Geola PGF-01 (red sensitive) holographic film and emulsion-coated glass plates are 

used in this laboratory. The film is used for the off-axis holograms, while the plates are used 

for the white light. Note that care must be used in handling the glass plates: any undue force 

exerted will cause them to break. Also, take care to handle all film and plates along the edges 

to minimise any smudging of the emulsion which may ruin the hologram. 

The emulsion side of the film is facing away from you when you feel the clipped corner is on 

the top-right. Place the film with the emulsion side away from you in the brass film holder 

as shown in the figure 5.1. 

That film holder is then placed in the stand at the time of exposure. Take care when handling 

the brass plates to avoid moving the film away from its fixed position. 

The glass plates use a different method and holder. The holder is on its own stand and the 

glass plates slide straight into it. Great care must be taken to avoid breaking the glass as 

the experiment requires that this be a snug fit to prevent movement of the plate while the 

exposure is being taken. The emulsion side of the glass plate is slightly sticky to the touch 

when felt by a slightly damp finger-tip. (Do this at the edge; a finger-print at the centre of 

the plate does not do the hologram any good. . . )


Film 

Cut Corner 

Film Holder 

Figure 5.1: Orientation of the film in the film holder. 

5.5.1.2 Film exposure 

To produce high-quality holograms, the intensity of the reference beam should be at least 

three times and not more than ten times the intensity of the light reflected off the object 

beam, as measured at the position of the film 4 . Glaring reflections from the object on the film 

should therefore be avoided, but they may provide some nice contrast in certain situations. 

All beams should be in the same plane for optimum hologram production. Avoid touching 

ALL mirror surfaces, especially the spherical mirrors. 

The resultant image will have the same appearance as the object as seen through a window 

having the same size and position as the photographic film relative to the object in the original 

configuration. 

5.5.1.3 Developing 

ALL FILM HANDLING AND PROCESSING 

MUST BE DONE IN COMPLETE DARKNESS. 

In a darkroom, develop the film in CWC2 Developer for 4 minutes, gently agitating the 

film. Rinse the film in the running water bath for 2 minutes. Place the film in the bleach 

for 3 minutes, again providing gentle agitation. Rinse again in the running water bath for 2 

minutes. ONLY AFTER bleaching may the light be switched back on. Allow the film to dry 

in the drying cabinet, typically for about 5 minutes, before viewing. 

If you wish to make a phase hologram, you must bleach the film for about 50% more time 

4 The maximum fringe contrast would occur for equal intensity but this would result in distortion.


than it takes to clear. Rinse for 3 minutes and then dry. Drying dehydrates the film which 

shrinks the emulsion. You should allow a few minutes after drying for the film to absorb 

moisture and expand back to normal before viewing. 

The developer is mixed in equal parts of Part A (20 g catechol, 10 g ascorbic acid, 10 g 

sodium sulphite, 50 g urea, and water to make 1 litre) and Part B (60 g sodium bicarbonate 

in 1 litre of water). It should last the whole of the experiment, after which it should be 

discarded safely in the waste drum provided. 

The bleach consists of 35 g copper sulphate, 10 ml glacial acetic acid, 110 g potassium 

bromide, and made up to 1 litre of water. It looks green when good and cyan (deep blue) 

when bad, but it is highly unlikely that it will turn during the course of the experiment. It 

is provided ready mixed for you at the beginning and will also last the entire experiment. It 

should be discarded safely at the same time as the developer in the waste drum provided for 

it. 

5.5.2 White light hologram 

The first hologram to be produced is the white-light hologram. It utilises the laser, a single 

plane mirror, and a spherical mirror. The arrangement of the system is shown in figure 5.2. 

Glass plate 

Object 

LASER 

Reference beam 

Figure 5.2: Experimental set-up for the white light hologram. 

Object beam 

This is your opportunity to be a little creative: bring in your own object(s). Note that the 

glass plates are 63 × 63 mm 2 , and the plate must be quite close (almost touching) the object 

in situ. Care must be taken in handling both the object and film in darkness. Bear in mind 

that with the red laser, suitable objects must also be chosen. 

Pre-lab Question 5.3 What characteristics make an object suitable for recording a hologram 

using a laser of a given colour? Choose your object(s) accordingly and remember to 

bring them at the beginning of the first session. 

If you are not so inclined, a selection of objects which may be used are provided. You may 

wish to use one of the lenses provided in front of whichever object you photograph (highly 

recommended).


After setting up the experiment, take note of the intensity of the reference beam using the 

light meter provided. Take as many holograms of the object as number of partners involved, 

using an exposure time of 2 seconds. Develop using the method outlined and allow the 

holograms to dry completely before handling again. 

5.5.2.1 Viewing holograms 

As these are white-light holograms you may be able to see them in white light. The best 

source is the sun, so if the weather permits view your holograms outside. If that is not the 

case, a lamp is provided as a suitable point source. 

Question 5.1 Does the image appear three dimensional? What characteristics do you observe 

about the image? (The effects will be magnified if you used a lens as part of the 

object 5 .) 

Question 5.2 How many images may be seen? To what do these images correspond? Which 

is the clearer? 

5.5.3 Transmission or off-axis hologram 

The set up for the transmission hologram is shown in figure 5.3. 

Plane mirror 

Spherical mirror 

Plane mirror 

Object 

LASER 

Variable beamsplitter 

Spherical mirror 

Film 

Figure 5.3: The experimental setup for the transmission hologram. 

You may use a similar object or objects as with the white light. In this case, you may wish 

to use the dragon as supplied. 

5 And, yes, the pun IS intended.


Once you have set up figure 5.3, and ensured correct optical alignment, take measurements of 

the reference and object beam intensities at the film plane. Adjust the variable beam-splitter 

until a ratio of intensities between 3 and 10 is achieved. 

Question 5.3 What is the optimum intensity ratio for this setup? Discuss with your demonstrator. 

Using your exposure time from the first experiment as a guide, choose a new exposure time 

based on the intensity of the reference beam. Take 3 holograms, making minor adjustments 

in the chosen exposure time to allow for a variation in the exposure. Develop as before. 

5.5.3.1 Viewing holograms 

In this case, you must use the same monochromatic coherent source as that used for the 

reference beam. The reference beam itself in your setup is sufficient. NB! Care must be 

taken not to look directly into the reference beam as this is coming straight from the 

laser. Your demonstrator will show you how to observe the holograms. Adjust the angle 

of the film until you see the image of the object. The film should be roughly at the same 

angle relative to the reference beam when the hologram was taken, in which case the image 

should appear roughly the same distance from the film as the original object. 

You should be able to make a number of observations at this point. Can you see both images 

as with the white light? Does the image have the same 3-d properties as before? 

Using the best of the three holograms taken, cut one into halves. What do you observe? 

5.5.4 Time-averaged interferometry of a resonating tube 

If time permits, you may wish to try a second transmission hologram, that of a resonating 

tube. 

Replace the object from the previous experiment with the brass tube which has one end 

covered with thin paper and over which the other end has a small speaker, attached to a 

signal generator. Using the exposure time of the best hologram of the previous object, take 

three holograms at frequencies of 1500, 2100, and 3900 Hz. Note that while the stability of 

the signal may be an issue, you should still be able to see something in at least two of the 

holograms. 

Question 5.4 What do you expect to see in the holograms? Describe each. 


5.6.1 Spherical waves 

A plane wave is given by the electric field defined in equation 5.1. From that, the intensity, 

irradiance, interference properties, etc., may be derived. Similarly, a spherical wave may be


defined as 

E = E 0 

cos (kr − ωt). (5.4) 

r 

The instantaneous intensity of such a wave is then 

I inst = ε 0 c ∣ ∣ E 

2 ∣ ∣ = ε0 c E2 0 

r 2 cos2 (kr − ωt), (5.5) 

and the time averaged intensity becomes 

I = ε 0 c E2 0 

2r 2 . (5.6) 

A spherical wave thus decreases in intensity as the square of the radius to the source. 

5.6.2 Interference of two waves 

For any two waves, E 1 and E 2 , at any point in space or time they will add vectorially. The 

resulting intensity pattern will depend on how these vectors add. For E = E 1 + E 2 , the 

intensity is given by 

I inst = ε 0 c ∣ ∣E 2∣ ∣ = ε 0 c ∣ ∣(E 1 + E 2 ) 2∣ ∣ 

= ε 0 c ( E 2 1 + E 2 2 + 2E 1 · E 2 

) 

= ε 0 c ( E 2 1 cos 2 (k 1 x − ω 1 t) + E 2 2 cos 2 (k 2 x − ω 2 t) 

+ 2E 1 E 2 cos 2 (k 1 x − ω 1 t) cos 2 (k 2 x − ω 2 t) cos θ 12 

) 

, (5.7) 

for the case of two planes. The angle θ 12 is the relative phase between the two waves. 

5.6.3 Transmission function 

After developing, the transmission function of the hologram is something like 

t(x,y) = t 0 + βI = t 0 + βε 0 c ∣ ∣E 2∣ ∣ 

= t 0 + βε 0 c ∣ ∣E 2 1 + E 2 2 + 2E 1 · E 2 

∣ ∣ , (5.8) 

where t 0 and β depend on the transmission (or opaqueness) of the unexposed film, the sensitivity 

of the film to the light, and the developing process. 

If the reference wave is a plane wave pointing in the x direction, 

E 1 = E 01 cos (kx − ωt), (5.9) 

and the same wave is used to recreate the image, then the image wavefront is constructed as 

the transmission through the grating as defined by the transmission function. The hologram 

function is thus 

E holo = t(x,y)E 1 = (t 0 + βI)E 1 

= { t 0 + βε 0 c ( E 2 1 + E 2 2 + 2E 1 · E 2 

)} 

E1 

= t 0 E 1 + βε 0 c { E 2 1 cos 2 (k 1 x + ωt) + E 2 2 cos 2 (k 2 x + ωt) 

+ 2E 1 E 2 cos (k 1 x + ωt) cos (k 2 x + ωt) cos θ 12 } E 1 cos (k 1 x + ωt) . (5.10) 

This is a rather complicated beast, but it illustrates how the relative phase is recovered.



[1] E Hecht. Optics. Addison-Wesley, 4 th edition, 2002.

Part II 

General physics 

59

Chapter 6 

Magnets And magnetic fields 


The following two experiments examine the behavior of materials in two types of magnetic 

fields. The first day deals with the use of DC to observe the Hall Effect and constructing 

magnetic field probes. The second experiment deals with the study of AC fields by looking 

at the hysteresis of a material. This will allow us to calculate the permeability of a substance. 

6.2 The Hall effect and magnetic field probes 


The Hall effect was discovered in 1879 by Dr. E. H. Hall as part of an investigation into the 

nature of the force acting on a current carrying conductor in a magnetic field. It wasn’t until 

the introduction of the semiconductor in the 1950’s that the full usefulness of this effect was 

recognised. Not only does the Hall effect allow for characterisation of semiconductors useful 

for their implementation in computer chips and electronic equipment, but the Hall effect in 

semiconductors also allows us to build devices that can measure an unknown DC magnetic 

field. This latter property is what you will be attempting to use today. 


6.2.2.1 Lorentz force equation 

The magnetic force, F , on a charge q, moving in a region of magnetic field, B, with a 

velocity, ⃗v, is given by, 

⃗F = q⃗v × ⃗ B. (6.1) 

In general, due to the fact that we are dealing with electric charges, not only are magnetic 

fields present, but electric fields also. Recall that the electric force on a charge q, is given by, 

⃗F = q ⃗ E. (6.2) 

61

62 CHAPTER 6. MAGNETS AND MAGNETIC FIELDS 

Hence the net force on a charged particle in both electric and magnetic fields is given by, 

⃗F = q( E ⃗ + ⃗v × B). ⃗ (6.3) 

This is the Lorentz force equation. 

Question 6.1 Explain how the Lorentz force equation can be used to implement a velocity 

discriminator for a collection of charged particles with a random distribution of velocities 

(e.g. e − from a hot filament). 

6.3 The Hall effect 

We can now consider the implications of the Lorentz force when we examine a semiconducting 

material. Figure 6.1 represents a thin strip of semiconducting material placed in 

an external magnetic field B Z . This material has a current I Y flowing through it at right 

angles to B Z . In order to analyse this situation, we must consider two types of semiconductors. 

One type is called n-type where the mobile charge carriers have negative charge, the 

other is where the mobile charge carriers are positive (p-type conductors). In both types of 

z 

Bz 

x 

y 

z 

y 

x 

Iy 

Ex 

Figure 6.1: Current carrying specimen in a magnetic field. 

semiconductors, any moving charge can be characterised by a drift velocity v y in the same 

dimension as the current I Y . Since a charge has a drift velocity, it will experience a force 

in an electric field according to equation 6.1. In this case the right hand rule on the cross 

product shows the resultant force for the n-type semiconductor is down, while the force for 

the p-type semiconductor is up. Since the charges within the semiconductor are not free, but 

restricted in space by the boundary of the material, the charges will accumulate on the edges 

of the semiconductor. As the charges build up, an electric field E X will begin to appear. 

After a little thought it should be clear that for p and n-type semiconductors, the electric field 

produced will lead to a force on charge carriers opposite in direction to the force produced 

by the magnetic field. As the charge build up continues, E X will increase until the net force 

on the carriers is zero. Once this has been reached, we can re-write the force equation 6.3 as 

q(E X + v y × B Z ) = 0 (6.4) 

and hence 

E X = −v y × B Z . (6.5)

6.3. THE HALL EFFECT 63 

We now assign dimensions (X,Y,Z) to the specimen being considered. Assuming that the 

specimen has a conductivity σ, and n charge carriers with charge q per unit volume, we can 

define various properties of the conductor with respect to the electric and magnetic fields. 

We define the two dimensional current density j y as 

j y = nqu y = σE Y . (6.6) 

Where u y is the MEAN drift velocity u y = 〈v y 〉 of the carriers (in this case, for electrons, 

q = q e = 1.6 × 10 −19 ). The current is rightly defined as the sum of the current density over 

the cross sectional area, hence is an integral over the area of our semiconductor, 

∫ 

I Y = j y dA = j y XZ (6.7) 

AREA 

If the magnetic field is applied in the z-direction, then the force (in the x-direction) is F X = 

B Z u y q, and using equation 6.5, the electric field at equilibrium is given by E X = −B Z u y . 

Recalling that the electric potential of a system is given by the path integral over the electric 

field, we have, 

∫ 

V X = E X dX = XE X = −XB Z u y . (6.8) 

A similar expression holds for the relationship between V Y and E Y , with the negative sign 

implying that the potential across X is opposite to the direction of the applied field B Z . We 

now define the Hall coefficient as 

R H = E X 

j y B Z 

. (6.9) 

As will been seen later, the Hall coefficient provides an excellent indication of the charge 

concentration in the specimen. 

We also define the mobility of the charge carriers, given by, 

µ = u y 

E y 

. (6.10) 

The mobility of charge carriers gives an indication of how easy it is for the charge carriers to 

move through the specimen. 

Question 6.2 Derive the following expressions in terms of the measurable quantities; 

(a) σ from X, Y , Z, I Y and V Y . 

(b) R H from I Y , B Z , Z, and V X . 

(c) n from R H and q. 

(d) u y from V X , X and B Z . 

(e) µ from u y , V Y and Y . 

Question 6.3 Derive error expressions for all the terms in the previous question. Show a 

FULL derivation from Newton’s partial derivative expressions for errors for σ, in order to 

show that the standard product rule for errors can be applied to the other quantities.


This section of the experiment will act to characterise the specific semiconductor used in this 

lab. Once characterisation has been performed, and values of σ, R H , n, u y and µ have been 

found, we can use measurements of V X , I Y etc. to determine an unknown field B Z . 


The experiment we will be performing will be to measure V X , V Y , and I Y for various values 

of B Z . This will allow for determination of all characteristic quantities discussed earlier. 

6.4.1 The semiconductor 

Figure 6.2 shows a schematic diagram of the circuit that will be used. The variable resistor 

at the base of the circuit can be used to control the amount of current passing through the 

specimen. 

Figure 6.2: Electrical circuit for the semiconductor. 

The dimensions of the semiconductor (placed between the poles of the large soliton magnet) 

are: 

• X = 4.0 mm 

• Y = 16.0 mm 

• Z = 0.47 mm 

The resistance of the semiconductor is 385 Ω.


Question 6.4 What are the errors associated with the dimensions and the resistance as 

stated? Are they negligible compared with the other quantities measured? 

6.4.2 The electromagnet 

The specimen itself is enclosed in an epoxy package and is fixed between the magnet: Do 

not attempt to remove it. The magnet has been calibrated as B = 0.97 T @ 100 mA. 

SAFETY NOTE: SOME OF THE 

VOLTAGES USED IN THIS EXPERIMENT 

ARE HIGH!! 

The current I m , for the Hall effect magnet is provided by the soliton power supply, the maximum 

current for which is 120 mA. The direction of the induced field is down when the 

conventional current (red) enters the top terminal. 

The maximum safe control current I y for the specimen is 15 mA. 

Question 6.5 What is the corresponding maximum value for V Y ? Use the given resistance 

from the semiconductor. 

6.4.3 The Hall effect 

Measure the Hall potential difference, V X , when the magnet current I m is 100 mA for at 

least 10 values of the specimen current, I Y , by varying the resistor. Also take note of the 

applied specimen voltage, V Y . Then reverse the magnetic field and repeat the measurements 

at the same I Y values. Make sure the magnet current is returned to zero before changing 

terminals. 

Question 6.6 Determine whether the contacts on the specimen are placed symmetrically. 

Note: a yes/no answer is not appropriate. Explain physically how you determine this and 

physically what is going on (draw a diagram). Remember that the contacts used to measure 

V X and supply I Y are finite and are placed roughly in the centre of the Y Z and XZ planes 

respectively. 

Question 6.7 From the relationships derived in the Pre-lab questions, calculate the values 

for the following at each setting of I y and determine the averages and errors. 

• conductivity σ, 

• Hall coefficient R H , 

• charge density n, 

• mean drift velocity u y , 

• mobility µ.


Question 6.8 Using the background theory on the Hall effect and the sign of V X for a particular 

orientation of B Z , determine: 

• the sign of the charge carrier; 

• if the material is a p-type or n-type semiconductor. 

One interesting phenomena when looking at the magnetic fields is the way a current carrying 

wire in a magnetic field experiences a force. 

Question 6.9 Now that you know how the Hall effect works, predict the direction the wire in 

figure 6.3 will move. 

Figure 6.3: Current carrying wire in a magnetic field. 

Question 6.10 Using table 6.1 determine: 

• the dopant used (e.g. Ga,As). 

• the concentration of charge carriers. 

Impurity Concentration Type σ 

atom/m 3 

Ω −1 m −1 

nil zero intrinsic 2 

As 8 × 10 19 n 5 

As 1.5 × 10 21 n 90 

As 5 × 10 22 n 2000 

Ga 9 × 10 19 p 3 

Ga 8 × 10 20 p 30 

Ga 1 × 10 22 p 300 

Table 6.1: Conductivity of doped Germanium at 300 K 

6.4.4 The Hall probe 

Assuming that the Hall potential difference, V X , is a linear function of magnetic field, B Z , 

the specimen that we have characterised can be used as a Hall probe to measure an unknown

6.5. FERROMAGNETIC MATERIALS AND AC FIELDS 67 

magnetic field. The linearity can be determined from the magnet calibration. Plot a graph of 

B Z versus soliton current, I m . Proceed in the following way: 

• Start with 120 mA, decrease to zero; 

• reverse the supply polarity; 

• decrease to -120 mA then; 

• increase to zero. 

Discuss the features of the graph. Specifically, what equation(s) determined earlier show 

that the plot should be linear, and what possible effects could be producing non-linearity for 

large values of ±I m ? 

6.5 Ferromagnetic materials and AC fields 


The process of putting a ferromagnetic material through a cyclic change in magnetic field 

was first studied by Sir J.A. Ewing. By changing the magnetic field in this way, one is able 

to determine the work done in putting the material through this process, the permeability and 

magnetic polarisation of the magnetic material. 


6.5.2.1 Magnetic polarisation 

If we consider the electron as a current, I, flowing around the nucleus about an area, A, the 

atom and electron spin can be modelled by a magnetic moment. We then define the magnetic 

moment 

m = iA, (6.11) 

A 

e 

Nucleus 

Figure 6.4: Magnetic model of an atom. 

where A is in the direction of the normal as shown in figure 6.4. If there are N atoms per 

unit volume, then the magnetisation, M is defined as 

i


M = Nm = NiA. (6.12) 

The magnetisation is an indication of how many of the atoms are aligned in a particular 

orientation. 

6.5.2.2 Magnetic field intensity H and magnetic flux density B 

The magnetic flux density, B, is defined through Ampere’s circuital law 

∮ 

B.dl = µ 0 i (6.13) 

The magnetic field intensity, H, is defined as 

H = B µ 0 

− M (6.14) 

The assumption may be made that M ∝ H, and therefore 

M = χ m H (6.15) 

where χ m is the magnetic susceptibility. Thus, the relationship between the magnetic field 

intensity and the magnetic flux density is 

B = µ 0 (1 + χ m )H = µH = µ 0 µ r H, (6.16) 

where µ is the absolute permeability of the material. In free space, M (and thus χ m ) are 

zero, giving the absolute permeability of free space as µ 0 = 4π × 10 −7 Wb A −1 m −1 . µ r 

is the relative permeability. 

A general way of writing equation 6.14 is 

B = µ 0 (H + M) (6.17) 

6.5.2.3 Effects of materials in a magnetic field 

The motion of electrons in an atom is made up of two parts: orbital motion (around the 

nucleus) and spin. It is these components, both orientation and magnitude, which determine 

the type of magnetism the material will exhibit in a magnetic field (for further detail see 

Ohanian [4], chapter 33, page 806). 

• Diamagnetism - when in the absence of an external magnetic field, the electron spins 

are randomly orientated and produce a zero atomic dipole strength. Once inside a


magnetic field, magnetic moments are induced in the material, reducing the magnetic 

field in the material. This effect disappears with the removal of the field. 

• Paramagnetism - when in the absence of a magnetic field some of the atoms are 

aligned, but overall the magnetisation within the material is zero. Once the field has 

been introduced these atoms partially align producing a weak magnetic field. This 

effect disappears with the removal of the field AND thermal agitation. 

• Ferromagnetism - in this case the electrons are more strongly aligned and form domains 

(see figure 6.5) even without the introduction of a magnetic field. These domains 

are randomly aligned until the introduction of a magnetic field. As the external 

field is increased the domains tend to align with the external field until the material is 

essentially one large domain producing a very strong field. 

Figure 6.5: Magnetic domains in a ferromagnetic material (see Ohanian [4], chapter 33 page 

812.) 

So, when a ferromagnet is in an external field, which is strong enough to align all the domains, 

the material is said to be saturated and any further increase in the external field will 

not see a contribution from the material. 

Now let us investigate what happens to the material as we turn off the external field, reapply 

it in the opposite sense until saturation is obtained again and then turning off the field and 

reapplying it in the original sense. This is the process of recording the history of the material 

or obtaining the hysteresis loop (figure 6.6). 

• The first stage is from zero magnetisation in the material to the point of saturation, 

point P . 

• Next we decrease the external field, H, to zero, but there is a residual flux density, 

B, within the magnet, i.e. some domains are still in alignment. This is called the 

remanence. The remanence can be destroyed by thermal agitation. The temperature 

this occurs at is the Curie Temperature. 

• If the external field is reversed, then the aligned domains will begin to realign with the 

new field orientation. Firstly, the domains are randomly aligned and cancel, i.e. zero


Figure 6.6: Typical hysteresis loop for a ferromagnetic material. 

flux density. Then they will reach saturation at point P ′ . At the point of zero flux 

density, the strength of the applied field required to destroy the remnant magnetisation 

of the material is called the coercive force. 

• If the field is then decreased and then reversed once again, a closed loop can be obtained. 

To do this continuously, the magnetic field is supplied by an AC field. 

Pre-lab Question 6.1 In light of this information, and knowing the hysteresis of the material, 

what properties would a good permanent magnet require? Subsequently, what would be 

the shape of the hysteresis loop for this material? 

Examining equation 6.17 one can see that for a diamagnetic material M is small and in the 

opposite sense of H, while for a paramagnetic material M is in the same sense, but small. 

For a ferromagnetic material the relationship is not so simple: this was seen in figure 6.6, the 

hysteresis loop. In this case, the relationship between H and B is no longer linear and will 

depend on the previous history of the material or the direction the magnet has been cycled. 

One may still determine the relative permeability through the normal magnetisation curve. 

This curve can be obtained by cycling the magnet in the same direction but reducing the 

peak magnetic field intensity and plotting the positions of P and P ′ such as in figure 6.7. 

The gradient of the curve at zero will provide a good estimate for µ. 

The work done in each cycle in changing the magnetisation of a specimen of volume τ is 

given by 

∮ 

W = τ 

H.dB (6.18) 

where the integral is the area of the hysteresis loop. 

6.5.2.4 Magnetic circuits 

For the case of a transformer or a multi-turn device, Ampere’s circuital law can be written 

as


Figure 6.7: Normal magnetisation curve. 

∮ 

H.dl = Ni (6.19) 

where N is the number of turns on the coil. Hence for a transformer, if the cross-section 

A is large enough to make flux leakage negligible, then the magnetic field intensity will be 

constant around the loop, length L. Hence equation 6.19 reduces to HL = Ni. 

Figure 6.8: Dimensions of a transformer. 

The magnetic flux is defined by 

∫ 

Φ = B.dA (6.20) 

For a transformer 

A 

Φ = µANi 

L 

(6.21) 

6.5.2.5 Magnetic induction 

If the magnetic flux density, B, is varied with time, such as with an AC field, an electric field, 

strength ǫ, encircling the magnetic field is induced satisfying


∮ 

ǫ.ds = − dΦ 

dt 

(6.22) 

where s is the path enclosing the surface through which magnetic flux of density B is passing. 

This is Faraday’s law and is a statement of the conservation of energy. For a transformer, 

this expression becomes 

V = −N dΦ 

dt 

(6.23) 

provided the same flux is passing through each of the N coils. 

6.5.3 The apparatus 

Figure 6.9 shows the AC circuit, which will be used to produce a hysteresis loop and determine 

the magnetic properties of the core material in the transformer. 

The transformer used in the AC fields experiments has a demountable core to facilitate measurement 

of the cross section and path length. Care must be taken to ensure that the surfaces 

mate properly and that no air gap is introduced during reassembly. The transformer has 540 

turns in the primary coil and 50 turns in the secondary coil. 

Figure 6.9: Circuit diagram for an AC magnetic field in the magnetic core of a transformer. 

Consider the circuit for the secondary coil. The arrangement of the resistor and capacitor 

is a low pass filter, but is similar in principle to a voltage divider where the ‘resistance’ of 

the capacitor is now the impedance, Z c = 1/jωC, ω is the frequency and j 2 = −1 1 (see 

Ohanian [4], chapter 34, page 823 for a review of the theory). 

1 It is unfortunate that there are so few letters available to use as variable names - therefore throughout 

science, both i and j are used to mean the unit complex number. You will need to learn the appropriate 

meaning by context. In this experiment, i is the current.


Pre-lab Question 6.2 Derive an expression for the voltage across the capacitor in terms of 

the voltage across the secondary coil by considering the circuit as a voltage divider such as 

in figure 6.10. 

Figure 6.10: Voltage divider. 

An alternating current produces a time dependent voltage of the form V y = |V y |e −jωt . 

Pre-lab Question 6.3 Using the result from Pre-lab question 6.2 and differentiating equation 

6.20 and combining with 6.23 derive an expression for the magnetic flux density, B. 

R p is a variable resistor and can be used to qualitatively look at changing the shape of the 

loop. 

Pre-lab Question 6.4 Derive an expression for the magnetic field intensity with respect to 

the voltage across R p given by equation 6.19. 

Pre-lab Question 6.5 Explain why the display on the CRO is representative of a hysteresis 

loop. 

6.5.4 Experimental work 

SAFETY WARNING: LARGE VOLTAGES 

ARE USED AND MUST BE HANDLED 

CAREFULLY AND SLOWLY. 

• Wire up the circuit in figure 6.9 and obtain a hysteresis loop on the CRO. Comment on 

the shape of the loop and the effect of varying the resistor, R p . 

• Determine the values of the parameters used in the expressions for Pre-lab questions 

6.3 and 6.4 and use these to calibrate the horizontal and vertical axes on the CRO in 

suitable units.


• Draw a graph of the observed loop and indicate the values of the points of saturation, 

remanence and coercive force. 

• Think of a way to obtain a normal magnetisation curve. Plot this curve. 

• From the magnetisation curve calculate the relative permeability of the core material. 

• Calculate the work done in the transformer for one magnetic cycle. 

• Calculate the magnetic polarisation, M, and compare it with the magnetic field strength. 

What does this say about the type of material used as the core of the transformer? 

6.6 Useful data 

Quantity 

Value 

Electron charge 

−1.6 × 10 −19 C 

Absolute permeability of free space 4π × 10 −7 Wb A −1 m −1 

Units of magnetic flux density Wb m −2 (Tesla) 

Units of magnetic field intensity A m −1 

Units of magnetic polarisation A m −1 

Units of magnetic flux 

Wb 

Diamagnetism χ m negative, µ r < 1 

Paramagnetism χ m positive, µ r > 1 

Ferromagnetism χ m large, µ r ≫ 1 


[1] P Lorrain, D P Corson, and F Lorrain. Electromagnetic Fields and Waves. W.H. Freeman 

and Co., 3 rd edition, 1988. 

[2] S G Starling. Electricity and Magnetism. Longmans, Green and Co. Ltd., 8 th edition, 

1953. 

[3] D Halliday, R Resnick, and J Walker. Fundamentals of Physics. Wiley, 6 th edition, 2003. 

[4] H C Ohanian. Physics. 2 nd edition, 1989.

Chapter 7 

Fundamental constants 


Over the next two days you will investigate various properties of electrons. The first day will 

be investigating the charge to mass ratio of the electron and the second day will look at the 

photoelectric effect. As a result of these two experiments you will then be in a position to 

present your own experimental values for some important fundamental constants. 

7.2 Charge to mass ratio of the electron 


Faraday in 1833 demonstrated the quantisation of charge through a careful series of electrolysis 

experiments. Thomson ended a heated debate among physicists when he was able to 

identify the cathode rays seen in low pressure gas discharges as caused by negatively charged 

particles. These particles, electrons, were accepted as being fundamental constituents of matter, 

firmly banishing the notion that the atom was indivisible and fundamental. Thomson was 

able to measure the charge to mass ratio of this new particle with an experiment that made 

use of the same principles as the one you are about to perform. The value of e/m that Thomson 

determined was 1.0×10 11 C/kg. This is to be compared with the currently accepted value 

of (1.7588028 ± 0.0000054) × 10 11 C/kg. Since the properties of the electron depend almost 

solely on these two quantities we seek an accurate determination of their value. It would be 

of interest to compare the value of e/m that you calculate with Thomson’s first effort! 

7.2.2 Theory 

If a beam of electrons is accelerated through a known potential then their kinetic energy is 

given by the expression 

E k = 1 2 mv2 = eV (7.1) 

75

76 CHAPTER 7. FUNDAMENTAL CONSTANTS 

where m is the mass of the electron, e is the charge of the electron and V is the potential. 

If a magnetic field is applied perpendicular to the velocity of the electrons, they will then 

experience a Lorentz force mutually perpendicular to both the field, B, and the electron 

direction of motion (see the magnets experiment, section 6.3). The magnitude of this force 

is 

F = evB. (7.2) 

Since the electrons always experience a force perpendicular to their motion they move in a 

circle and thus experience a centripetal force with a magnitude given by 

F = mv2 

r 

(7.3) 

where r is the radius of the circle described by the electrons’ path. 

Pre-lab Question 7.1 What path would the electrons follow if their velocity were not totally 

perpendicular to the magnetic field (say at 45 ◦ )? 

Pre-lab Question 7.2 Using the equations above determine a formula for the ratio e/m 

in terms of the applied field strength, B, the radius of the circle, r, and the accelerating 

potential, V . 


The apparatus are displayed schematically in figure 7.1. 

7.2.3.1 Helmholtz coils 

The magnetic field for this experiment is generated by a pair of Helmholtz coils with the 

magnitude of the field generated at a volume in their centre given by 

( ) 3/2 4 Nµ 0 I 

B = 

5 a 

(7.4) 

where N is the number of turns in each coil, µ 0 = 4π × 10 −7 T m A −1 is the permeability of 

free space, I is the current through the coils and a is the radius of the coils. In this case the 

radius of the coils is 150 mm and each coil has 130 turns. 

7.2.3.2 The cathode ray tube 

The cathode ray tube is a 130 mm diameter helium filled glass bulb. This contains at its 

base the electron gun consisting of a heated cathode to produce the electrons that are then

7.2. CHARGE TO MASS RATIO OF THE ELECTRON 77 

Figure 7.1: The apparatus used to determine e/m 

accelerated towards the anode. The anode is made of a wire grill so that the electrons may 

pass through it and enter the body of the tube. The helium gas is retained at a pressure 

of 10 −2 Torr and is ionised by the electrons as they pass through it. As the helium ions 

recombine with their valence electrons to form atoms, they emit light, allowing the radius of 

the electron beam to be determined. 

Pre-lab Question 7.3 What happens to the kinetic energy of the electrons as they ionise the 

helium? How will this affect the quality of your results? 

7.2.3.3 Mirrored scale 

A mirrored scale is attached to the rear coil and will be illuminated when power is provided 

to the apparatus. This may be used to align the beam and its image and so eliminate errors 

due to parallax. 

7.2.3.4 Cathode ray tube power supply 

A power supply is provided for the cathode ray tube. This provides 6 V AC to heat the 

cathode and a variable DC voltage to accelerate the electrons.


7.2.3.5 Helmholtz coil supply 

Current to the Helmholtz coils is supplied by a current regulating power supply capable of 

providing 2 A at small voltages. 

Pre-lab Question 7.4 Why must both of these power supplies be kept as far from the cathode 

ray tube as possible? 


• Ensure that both the power supplies are connected as shown in figure 7.1. Before 

switching on the supplies check that the high voltage control and the current controls 

(supply and apparatus) are set to be minimum. 

• Switch on the supplies and allow the heater to warm up. Increase the accelerating 

potential until the electron beam can be seen. Turn the current control on the apparatus 

to maximum and adjust the current in the Helmholtz coils until the beam traverses a 

circle. If the beam travels in a spiral consult your demonstrator. Please note that the 

bulb is designed to rotate, so don’t be afraid to adjust the alignment, but carefully! 

• Adjust the accelerating potential to 300 V and adjust the Helmholtz coil current until 

the radius of the beam is 4.0 cm. Note the current, voltage and left and right radii. 

NOTE: Why are the two radii measurements different? How will this affect the way you 

measure radius? 

• Repeat this procedure at 5 V intervals until the beam no longer traverses a circle. Try 

as much as is practicable to adjust the current to keep the radius constant at 4.0 

cm. Detail the methods you employ to increase the accuracy of your results. 

7.2.5 Analysis 

Enter your data into the computer and use Excel to calculate the value of e/m for each 

voltage. Plot your results against potential. 

Pre-lab Question 7.5 How can you explain the shape of this graph? 

A better method for determining the value of e/m is to calculate the field variable. The field 

variable, χ, is a measure of the energy loss of the electrons due to their interactions with the 

gas. We may fit the function 

V = 

( e 

m) 

χ + δ (7.5) 

where e/m and δ are parameters and V and χ = [(Br) 2 /2] × 10 11 are variables. This allows 

a unique value of e/m to be determined while allowing for the effective potential loss due to

7.2. CHARGE TO MASS RATIO OF THE ELECTRON 79 

the presence of the gas with the parameter δ. Plot V against χ on another graph and overlay 

the fit. Hence determine a value for e/m and compare it to the accepted value. 

Pre-lab Question 7.6 Given the ionisation energy of He, how many electron-He collisions 

occur during the electron path through the tube on average?


7.3 The photoelectric effect 


At the middle of the nineteenth century the wave theory of light was yet unchallenged. However, 

in 1887, Hertz noticed that a spark induced in his circuit was stronger when he deliberately 

illuminated the detector with UV light. Although his study concentrated on the 

properties of electromagnetic radiation, his was the first observation of the liberation of electrons 

from the clean surfaces of metals under the action of radiant energy. This is known as 

the photoelectric effect. His colleague, Lenard, later confirmed that the emitted carriers were 

negatively charged. However, it wasn’t until Einstein’s revolutionary postulate in 1905, that 

all the observed features of this effect were reconciled with mainstream theory. 

7.3.2 The failure of classical theory and Einstein’s postulate 

Experiment showed that electrons ejected from the surface had small but finite speeds ranging 

from zero to some maximum value. By making the collecting plate negatively charged 

with respect to the illuminated plate, they could measure the force required to stop the most 

energetic of electrons. This is how the stopping potential, V 0 , is defined. 

Pre-lab Question 7.7 Cite instances where classical theory failed to account for the observed 

features of this phenomenon. You may wish to consult the suggested references described 

at the end of these notes. 

Einstein extended the quantum theory of Planck to the radiation field itself, hypothesising 

that the light existed in quanta of energy with magnitude hν. 

Thus the mechanism of the photoelectric effect became as follows: an electron absorbs a 

photon of energy hν and attains enough energy to escape the surface. Energy not used to 

overcome the binding of the electron to the atom becomes kinetic energy. Thus 

E k = eV 0 = hν − W 0 (7.6) 

where E k is the kinetic energy of the photoelectrons and W 0 is the work function particular 

to the material that is illuminated. The work function is the energy required for the electron 

to escape the surface. 


7.3.3.1 The mercury vapour lamp 

The spectral lamp you will use is a high intensity (100 W) mercury vapour lamp. Atoms are 

ionised by passing a current through the mercury vapour. The recombination of electrons

7.3. THE PHOTOELECTRIC EFFECT 81 

Table 7.1: Frequencies for the light spectrum issued from the mercury lamp. 

Colour Frequency (10 −1 PHz 1 ) 

yellow 5.19 

green 5.49 

blue 6.88 

violet 7.41 

deep violet 8.22 

and mercury ions produces a light spectrum composed of discrete wavelengths. The emitted 

frequencies are displayed in table 7.1. 

7.3.3.2 The diffraction grating 

The light from the lamp will be dispersed using a transmission diffraction grating 2 and focussed 

with a lens. The grating equation is a sin θ m = mλ, where a is the distance between 

the ruled grooves, m is the order of interference and θ m is the angle through which the order 

is deflected. A diagram illustrates this below. Note that each order is resolved in space, as is 

each spectral line. 

Figure 7.2: The transmission diffraction grating with the orders of interference shown. 

7.3.3.3 The photoelectric cell 

You are supplied with a photoelectric cell for measuring the stopping potential. The schematics 

inside the photoelectric cell look roughly like figure 7.3. Light from the mercury lamp 

strikes the surface of the metal, and those photons with sufficient energy will liberate electrons 

from the metal surface, which will then be collected by the detector. Each electron 

collected will increase the overall negative charge at the detector, and hence make it harder 

for electrons to jump the gap. Eventually, the number of electrons will reach a maximum. 

1 1 PHz = 10 15 Hz 

2 see Hecht [3], section 10.2.7, page 465.


detector 

light 

electrons 

metal 

Figure 7.3: Light is incident on the metal surface, which liberates electrons, which are then 

collected by the detector. 

This will take a finite amount of time (called the stopping time, t s ) and will have an associated 

stopping voltage V 0 (as in equation 7.6). In this experiment, both V 0 and t s will be 

measured. The red discharge button on the photoelectric cell grounds the built-up electrons, 

ready for a new measurement. 

7.3.3.4 Filters 

Three filters are required for this experiment: 

• A filter with 5 strips of various transmission strengths (from 20% to 100%, in steps of 

20%) 

• A yellow filter to prevent ambient light sources (UV from overhead fluorescent lights 

and also the violet line from the 3 rd order spectrum) from interfering with measurements 

of the stopping potential for the yellow line 

• A green filter for use with the green line for reasons cited above. 

All of these filters have magnetic strips for easy attachment to the photocell. 


A representation of the apparatus is given in figure 7.4 with the features you will need to be 

familiar with identified. 

• Allow the lamp to warm up for five minutes. This should give you the opportunity 

to identify the orders of interference produced by the transmission grating. Focus a 

spectral line of the 1 st order onto the white reflective mask of the photocell by moving 

the lens and grating back and forth. Fix the lens in place when you have obtained a 

sharp image.


Figure 7.4: The apparatus for the observation of the photoelectric effect. 

• Roll the light shield of the photocell (black tube) back and rotate the cell until the 

line is centred on the aperture of the photodiode. Tighten the locking screws on the 

photocell and grating. 

• Place the transmission filter on the front of the photocell to allow 100% of the light 

to enter the photodiode. If you have chosen the yellow or the green spectral lines 

remember to use the appropriate filters. Connect the voltmeter to the output of the 

photocell and switch the photocell on. Zero the photocell by pushing the red discharge 

button. 

Question 7.1 Record the stopping potential after a minute or so. Is it necessary to measure 

the stopping potential for all transmission filters? If not, why not? 

There are two effects that make the measurement of stopping time t s difficult. The first is the 

swiftness of the electron buildup - it is simply not viable to accurately measure time-spans 

of less than about a second by hand with a stopwatch. The second is the inaccuracy of the 

end-point. Note how the stopping potential is not constant, but instead fluctuates constantly. 

Certainly, in the act of measuring the stopping potential, it is difficult to tell when the final 

point is reached - and your choice of end-point will markedly affect the results. Instead, if 

we assume that the build-up is linear, we can instead choose some arbitrary factor κ 3 with 

which to multiply the stopping voltage, and then measure the time taken to reach κV 0 , which 

will be κt s . The factor κ needs to be small enough that the end-point is now well defined, yet 

3 0 < κ ≤ 1.


big enough that the length of time measured is at least a few seconds for 100% transmission. 

We suggest κ = 0.99. The quantity t s is restored by simply dividing κt s by κ. 

Zero the photocell again and record κt s . Repeat this procedure for all intensity filters. Readjust 

the apparatus for another spectral line and repeat the above procedure. 

Question 7.2 Explain what you observe in terms of operation of the photocell. 

Question 7.3 Make a plot of percent transmission versus stopping time t s for both the green 

and yellow lines. Is it linear? If not, why not? 

7.4.1 Variation of stopping potential with frequency 

Remove the transmission filter. Adjust the apparatus for all of the spectral lines you can 

observe (both 1 st and 2 nd order lines on both sides) and record the stopping potential for 

each line. Remember to use the appropriate filters. Consider also the effects of other light 

sources, for instance, overhead lighting. 

Question 7.4 What can you do to minimise these effects? 

Question 7.5 Average your results in Excel. Fit your data with Einstein’s equation. Determine 

h/e and W 0 from the parameters of this fit. Compare your result with the expected 

value of h/e = (4.135708 ± 0.000014) × 10 −15 J s C −1 . 4 

Question 7.6 Do your results agree better with quantum mechanics or classical mechanics? 

7.4.2 Calculation of the fundamental constants 

The third fundamental constant is introduced in the electron spin resonance experiment and 

is the Bohr magneton (µ b = e/2m e ). The value of this constant has been measured as 

(9.274096 ± 0.000065) × 10 −24 J T −1 . 

Now, use your results from the photoelectric effect determination of h/e, the previous practical 

exercise determination of e/m, and the result for the Bohr magneton cited, to calculate 

the values of h, m e and e. You should determine errors for each of these quantities so that 

you can effect a meaningful comparison with the accepted values of these constants. 

4 I know I know, we keep saying that errors should only be quoted to one significant figure. However, when 

a number has been measured using a huge number of readings, we sometimes give a second significant figure, 

especially if the first is a 1.

7.5. USEFUL DATA 85 


Quantity 

Value 

h 

6.6261 ×10 −34 J s 

e 

1.6022 ×10 −19 C 

m e 

9.1096 × 10 −34 kg 

e/m 

(1.7588028 ± 0.0000054) × 10 11 C kg −1 

h/e 

(4.135708 ± 0.000014) × 10 −15 J s C −1 

µ 0 4π × 10 −7 T m A −1 

µ b (9.274096 ± 0.000065) × 10 −24 J T −1 

ionisation energy of He I He 24.58 e V 


[1] R A Serway, C J Moses, and C A Moyer. Modern Physics. Saunders, 1989. 

[2] D Halliday, R Resnick, and J Walker. Fundamentals of Physics. Wiley, 6 th edition, 2003. 


Chapter 8 

Waves in waveguides 


In the following two experiments you will be studying the transmission of waves along 

waveguides. Even though sound waves and microwaves are physically very different, the 

fact that they are wave phenomena means that their transmission in waveguides may be 

treated with similar mathematical formalisms. 

The background theory for this experiment covers a lot of ground, and is quite mathematical. 

Don’t be intimidated by this - if you get confused, remind yourself of your basic knowledge 

of how standing waves behave in tubes 1 , and try to match the mathematics that is being 

presented with your prior understanding. 

8.2 Acoustics 

8.2.1 Theory 

Acoustics is defined as the study of vibration and sound. Sound is the propagation of elastic 

disturbances in a continuous medium; vibration refers to such disturbances in more simple 

systems, such as springs. You will be familiar with the situation of particles being displaced 

from their equilibrium positions, developing potential energy, and then being restored to 

their equilibrium positions, releasing energy. We may consider a particle that is displaced 

repeatedly such that it has the same displacement and velocity some number of times per 

second. If the displacements being considered are infinitesimal we may then consider differential 

elements of displacement. Such a treatment yields the wave equation for particle 

displacement. 

1 

c 2 ∂ 2 ξ 

∂t 2 = ∇2 ξ (8.1) 

Here ξ is the particle displacement and c is the wave velocity, defined as the speed at which 

1 see Ohanian [7], chapter 17 page 438. 

87

88 CHAPTER 8. WAVES IN WAVEGUIDES 

Quantity Symbol Quantity Symbol 

Particle displacement ξ Absolute pressure P 

Angular frequency ω = 2πf Equilibrium density ρ 0 

Sound pressure p Equilibrium pressure P 0 

Particle velocity v Bulk modulus B 

Wave velocity c Characteristic impedance Z 0 = ρ 0 c 

Wavelength λ 

Table 8.1: Quantities useful in characterising sound 

Real part 

ξ = A cos(ωt) + B sin(ωt) 

v = ∂ξ 

∂t 

= ω[−A cos(ωt) + B sin(ωt)] v 

a = ∂2 ξ 

∂t 2 = −ω 2 [A cos(ωt) + B sin(ωt)] 

Complex notation 

ξ = Ce iωt ,C = A − iB 

= ∂ξ 

∂t = iωCeiωt = iωξ 

a = ∂2 ξ 

∂t 2 = −ω 2 Ce iωt = −ω 2 ξ 

Table 8.2: Complex notation for the solution to the wave equation 

the wavefront moves through the medium. This equation has a solution of the form: 

ξ = A cos(ωt) + B sin(ωt) (8.2) 

describing simple harmonic motion, or pure tone. A and B are constants and ω is the 

angular frequency. Because any sound can be represented as a sum of simple harmonic 

terms the following discussion will be limited to single frequencies 2 . 

The quantities of displacement and wave velocity have already been encountered. Other 

useful terms used to characterise sound are given in table 8.1. 

If the motion is simple harmonic it is better to introduce complex notation 3 to represent the 

quantity ξ. In this case, as may be seen from table 8.2, ξ is a complex number, which is much 

easier to manipulate than the real or imaginary parts alone. 

8.2.1.1 Wave velocity, c 

The wave velocity c has already been encountered in the wave equation of motion. For media 

such as air or water, c is independent of frequency. For an ideal non-viscous fluid c may be 

related to the average or equilibrium density, ρ 0 via the relation 

2 We can use Fourier’s theorem to crack more complicated cases. 

3 It would be nice if we could choose standard meanings for i and j, e.g. set i 2 = −1, and use j for current. 

Sadly, the meanings of these individuate letters needs to be obtained from the context. j will sometimes be 

imaginary, a current density or something else. This is true of many letters in physics, so beware!

8.2. ACOUSTICS 89 

c = 

√ 

B 

ρ 0 

(8.3) 

where B is the adiabatic bulk modulus. The bulk modulus measures how the pressure of 

the gas changes, when the volume of the container which the gas occupies is changed. Bulk 

modulus can also be thought of as the inverse of the compressibility of a gas. For ideal gases, 

the bulk modulus is proportional to the average pressure P 0 . 

So the Laplace expression is obtained 

B = γP 0 (8.4) 

c = 

√ 

γP 0 

ρ 0 

(8.5) 

which is a standard relation for two gases, where γ is a ratio of heat capacities of the two 

gases. This particular invocation of Laplace’s equation is for air at standard temperature and 

pressure (1 atm and 0 ◦ C), c = 341 ms −1 . 

Given an ideal gas we may invoke the gas equation in the form 

P = NkT (where N = n V ) (8.6) 

where N is the number density of particles, k is Boltzmann’s constant and P and T are the 

equilibrium pressure and temperature respectively. If we write the density, ρ as Nµ where 

µ is the average mass of a single molecule it becomes apparent that c may be expressed as 

c = 

√ 

γkT 

µ 

(8.7) 

Pre-lab Question 8.1 Derive an expression for c 0 , the wave velocity at 0 ◦ C, given that you 

know the wave velocity at a temperature T . (Note: the expression should be in terms of T , 

T 0 and c T only). 

Pre-lab Question 8.2 Calculate the wave velocity of sound in hydrogen given that 

µ air = 29.0 a.m.u. 

µ hydrogen = 2.016 a.m.u. 

at STP. Remember that a.m.u. stands for atomic mass unit. What is one a.m.u. when expressed 

in kg? 

The expression for the root-mean-square (RMS) particle velocity of molecules in random 

thermal motion is


v RMS = 

√ 

3kT 

µ 

(8.8) 

Pre-lab Question 8.3 Calculate v RMS for hydrogen at standard temperature. Can you remark 

on any similarity between the wave velocity and the RMS velocity? 

The speed of sound may also be calculated if the frequency and wavelength have been 

determined via 

c = λf. (8.9) 

However, a correction must be made for standing waves in a tube as compared to plane 

waves in free space. This is both due to the mechanical friction and heat conduction. The 

expression 

c T = 

c 

1 − 0.37 

d √ ω 

(8.10) 

has been theoretically determined and empirically confirmed for air in a brass tube. Here c T 

is the speed of sound in free space at temperature T , d is the diameter of the tube in cm, 

and c is the speed of sound in the tube at that temperature. 

For plane acoustic waves the vector particle velocity is parallel to the direction of propagation 

of the wave. Such waves are called longitudinal (compressional or irrotational). 

8.2.1.2 Pressure 

Most acoustics experiments measure the sound or acoustic pressure, p, defined as 

p = P − P 0 (8.11) 

When there are no sound waves travelling in the tube, the pressure inside the tube is the 

average or equilibrium pressure, equal at every point inside the tube. However, when a 

standing wave exists inside the tube, there will be regions of over-density and under-density. 

P gives the instantaneous pressure at a point, and so p can be either positive or negative, 

depending on the position in the tube. 

For a plane wave travelling in the positive x-direction we solve a wave equation for pressure 

(analogous to the equation for displacement, equation 8.1) to obtain 

p + = a + e i(ωt−kx) (8.12) 

Here a + is a constant and k is the wavenumber, equal to 2π/λ. We may define the characteristic 

acoustic impedance, Z 0 of the material as the constant of proportionality linking 

pressure to particle velocity


Acoustic 

Electric 

Pressure P Voltage V 

Particle velocity V Current I 

Intensity I = PV Power P = IV 

Acoustic impedance Z = P/V Electrical impedance Z = V/I 

Table 8.3: This table makes an analogy between acoustic and electric parameters. 

p + = Z 0 v + (8.13) 

Z 0 is a real number that is a function only of the physical properties of the medium in which 

the wave travels. Similarly, for a plane wave travelling in the negative x-direction 

p − = a − e i(ωt+kx) = −Z 0 v − (8.14) 

Ideas of impedance are here introduced for reasons analogous to their use in electrical circuit 

theory. A list of such analogues is given in table 8.3. 

Let us conduct two thought experiments. 

The first is with electrical transmission lines. If we have two different wires joined together 

end-to-end, then if the two wires have similar impedances, a large amount of power will be 

transferred when electricity travels across the interface between the two wires. If however 

the impedances of the two wires are very different, only a little power will be transferred. 

Similarly, let us think about sound waves in a tube. The first medium is the air, and the 

second medium is the end piece of the tube (for example, lead or foam). The foam will have 

a very similar impedance to the air (much more so than the lead), and hence a large amount 

of power will be transferred when the sound wave hits the foam. Therefore, only a little of 

the sound wave will be reflected. With lead however, the impedance will be very different 

than air, and so very little power is transferred into the lead, and most of it is reflected. 

8.2.2 Stationary wave theory and the reflection of plane acoustic waves 

In practice a stationary wave system is due to the partial reflection of a travelling wave at a 

discontinuity in the medium. The study of such systems is most conveniently carried out in 

a tube. The advantage of studying wave phenomena in a tube is that the only type of wave 

propagated is a plane wave, despite the size of the source. This allows us to ignore any 

curvature. 

Consider the situation shown in figure 8.1. 

A plane wave travelling in the positive x-direction is partially reflected and partially transmitted 

at a plane of discontinuity in the medium. Since the frequency is preserved upon 

reflectance we may write the displacement as


Figure 8.1: Partial reflection at normal incidence 

ξ + = a + e i(ωt−kx) (8.15) 

ξ − = a − e i(ωt+kx) (8.16) 

The real part of the resultant displacement is 

ξ = Re(ξ + + ξ − ) 

= a + (1 + r) cos(ωt) cos(kx) + a + (1 − r) sin(ωt) sin(kx) (8.17) 

where 

r = a − 

a + 

= A max − A min 

A max + A min 

= Z 2 − Z 1 

Z 2 + Z 1 

(8.18) 

A max and A min are the displacement maxima and minima respectively, and Z 1 and Z 2 are the 

absolute values of the impedances - where Z 1 is the impedance of the original medium the 

wave is travelling in, and Z 2 is the medium which is being ‘reflected off’. The envelope, or 

standing wave profile for r = 1/3 is shown in figure 8.2. Note that the resultant envelope is 

the sum of two stationary waves with amplitudes a + (1 + r) and a + (1 − r) respectively, each 

with a period of 2π/ω, and have a relative displacement of maxima (and minima) of λ/4. 

You will be more familiar with the situations arising from an open or closed tube. In these 

cases, A min ≈ 0, giving an r of about 1. However, in the case of a closed tube, a stationary 

wave system is observed with displacement nodes located at λ/2, λ, 3λ/2 etc. from the 

end. In the situation with the open end, the displacement nodes are located at λ/4, 3λ/4, 

5λ/4 etc. from the end. These observations suggest that in order to completely characterise a 

stationary wave system not only a magnitude factor, but a phase factor needs to be supplied.


Figure 8.2: The resultant standing wave with r=1/3 

ξ − = a − e i(ωt+kx−φ) (8.19) 

Now the real part of the resultant displacement is 

ξ = Re(ξ + + ξ − ) 

= a + 

[ 

cos(ωt) cos(kx) + a+ sin(ωt) sin(kx) (8.20) 

+a − cos(ωt) cos(kx) cos(φ) − r cos(ωt) sin(kx) sin(φ) 

− r sin(ωt) sin(kx) cos(φ) − r sin(ωt) sin(kx) sin(φ) ] 

A stationary wave produced by partial reflectance/transmittance at a discontinuity in the 

medium is completely characterised by the complex reflection coefficient, K. K embodies 

both ideas about magnitude; it has magnitude r, and phase φ (see the appendix, section 8.4.1, 

for the derivation): 

K = re iφ = K ′ + iK ′′ , K ′ = r cos φ, K ′′ = r sin φ (8.21) 

So 

ξ = a + (1 + K ′ ) cos(ωt) cos(kx) + a + (1 − K ′ ) sin(ωt) sin(kx) (8.22) 

−a + K ′′[ sin(ωt) cos(kx) + cos(ωt) sin(kx) ] 

Pre-lab Question 8.4 There are two ways to express K, as re iφ , and K ′ + iK ′′ . Explain 

which way you think is most relevant for understanding the physics of the situation.


We may calculate the RMS of the real part of the resultant displacement profile upon reflection 

at a termination to calculate the theoretical standing wave profile: 

√ 

1 

τ 

∫ 0 

Using this average, this quantity is easily shown to be 

where 

ξ RMS = 

τ 

[Re(ξ)] 2 dt (8.23) 

√ 

(A RMS 

MAX )2 cos 2 [k(x − x 0 )] + (A RMS 

MIN )2 sin 2 [k(x − x 0 )] (8.24) 

A RMS = A √ 

2 

, φ = 2kx 0 (8.25) 

which we shall later use to plot against a measured wave profile. 

Figure 8.3: The resultant standing wave with phase for equation 8.24 

Equation 8.24 is completely defined by 4 parameters, where we can choose 2 of the 3 out 

of φ, k and x 0 (these are linked by equation 8.25) and 2 of the 3 out of r, A MAX and A MIN 

(these are linked by equation 8.18). Let us choose A RMS 

MAX , r, x 0 and k, three possible plots are 

displayed in figure 8.3, with choices of (0.5, 0.1, 0, 1) for the dashed line, (1,0.5,1,1) for the 

solid line and (0.5,0.5,2,1) for the dash-dotted line. By inspection of these it can be seen that 

x 0 is the distance between the discontinuity and the first minimum pressure (maximum 

displacement) node. 

K is real when x 0 = 0,λ/4,λ/2 etc. K is imaginary when x 0 = λ/8, 3λ/8, 5λ/8 etc. 

Pre-lab Question 8.5 Calculate K for the open and closed tubes. Remember that x 0 is 

defined as the distance from the discontinuity to the first pressure node. Recall also that 

pressure and displacement are π/2 out of phase, that is, a displacement node is a pressure 

antinode.


Figure 8.4: Experimental setup. 

The analogous expression to equation 8.24 for pressure is 

p RMS = 

√ [ 

(A RMS 

MAX )2 cos 2 k(x − x 0 ) − π ] 

2 

[ 

+ (A RMS 

MIN )2 sin 2 k(x − x 0 ) − π ] 

2 

(8.26) 

You will be measuring the pressure standing wave profile with a microphone, as in figure 

8.4. 


1 sound tube, with cart attached, on a ruler base 

1 signal generator 

1 CRO 

3 BNC cables 

1 BNC cable adaptor 

various ends for the sound tube 


8.2.4.1 Setup 

• Set up your equipment as in figure 8.5, using the BNC cables. Note: 

– On the signal generator, plug into “Output 50 Ω”, not “Output pulse” 

– For the first part of the experiment, keep the amplitude on the signal generator 

just below what is audible, to avoid a headache. 

– Check that the triggers on the CRO are both set to “AC”. 

• Check first that the signal coming from the signal generator is correctly read by the 

CRO. Ensure that you can read the scale in the vertical direction correctly (the “volts/div” 

knob controls this, it gives the volts per big division on the screen).


Figure 8.5: Connection diagram 

• Switch to channel B, and get the signal clearly on the screen. Roll the cart slowly up 

and down the rails, and notice how you get a varying function of voltage with respect 

to position of the cart. Notice also how the amplitude is not zero at the nodes, but only 

very close to (there is no real “zero” of a continuous quantity in experimental physics). 

8.2.4.2 Measurement of wave velocity 

Question 8.1 Set the signal generator to 500 Hz. Determine the positions of the minima in 

pressure along the sound tube, by reading off the ruler. The difference between two minima 

is half of one wavelength - how can you maximise the accuracy of this measurement? 

(a) Determine the wavelength of the standing wave in the tube. 

(b) Calculate the speed of sound in the tube, using the above value. 

(c) Calculate the theoretical speed of sound at 298 K, given that the CRC Handbook gives 

c 0 = 331.45 m s −1 . (Hint: use your answer to the first Pre-lab question). 

(d) Compare the speed of sound in the tube with the speed of sound in free space. 

Repeat parts 1 - 4 for two other frequencies, both in the range 1000-3000 Hz. 

8.2.4.3 Standing wave profiles 

In this section, we will measure the entire profile of the wave - not just the positions of the 

minima. The response of the microphone and speaker is dependent on the amplitude of the 

signal generator output. It is important that all your measurements are made in a region 

where the response is linear. As a result, you need to adjust the speaker level on the signal 

generator, so that the CRO reads 1.0 V peak-to-peak from channel A, and keep it at this level 

for the rest of the experiment. 

Obtain the standing wave profile at 1000 Hz for a CLOSED tube. Measurements should be 

taken from the END (cart pushed in all the way) of the tube in 2cm steps, with an extra point 

taken at the precise position of each minima


Question 8.2 Why do you think this is necessary for minima only and not maxima? This 

will become clearer once the profile has been plotted. 

Make a plot of the data you have taken. Now, using the measured values of A max , A min , 

λ and x 0 , overlay the theoretical plot (equation 8.26), and see how closely they match (DO 

NOT try to ‘fit’ for any parameters, simply compare the two!). 

You have been supplied with a range of end pieces for the sound tube, attach a few more and 

repeat steps 1 and 2. 

For each absorber, calculate the reflection coefficient K, from measurements of A max , A min , 

λ, and x 0 . 

Question 8.3 Make a summary table, and comment on the physics. 

Question 8.4 (Advanced) Find out the impedance of air, and using r = Z 2−Z 1 

Z 2 +Z 1 

and your 

knowledge of r, determine Z 2 for each end piece.


8.3 Microwaves 

8.3.1 Introduction 

In the laboratory we generally use only low frequency signals that are conveniently routed 

with wires or a coaxial cable. However the use of wires becomes intractable in the high 

frequency regime where field radiation in the space around the wires results in intolerable 

energy losses. In the high frequency realm guided microwaves are more appropriate for 

conveying electromagnetic signals. At 10 kHz the attenuation of a signal in a waveguide 

is 1/4 that in an equivalent coaxial cable. At 1 MHz and above, a coaxial cable has no 

application. 

8.3.2 Theory 

It should be no surprise that electromagnetic radiation may be propagated in a hollow metal 

tube. As Feynman facetiously remarks, if the tube is straight, we can see through it! 4 To 

understand pictorially how microwaves are propagated in a guide, refer to figure 8.6 below. 

Figure 8.6: Generation of electromagnetic disturbance in a guide 

Imagine that an oscillator exists and is connected to the end of the tube causing a deficiency 

4 Feynman lectures [4], Volume II section 24-2.

8.3. MICROWAVES 99 

of electrons on the bottom wall and a surplus on the top. An electric field exists between 

the top and the bottom walls of the conductor causing current to flow as shown in figure 

8.6(a). In this figure, magnetic fields trying to encircle these currents are prevented because 

of the presence of the solid metal wall, so they join up instead, as shown in figure 8.6(b). As 

the oscillator reverses its polarity, the magnetic fields generate other currents by induction 

further along the guide and so the disturbance is propagated. One cycle of this disturbance 

is shown in figure 8.6(c). A more mathematical treatment is required to extract all the interesting 

observable parameters from this phenomenon, specifically, what kind of waves can 

exist in a rectangular pipe? For electromagnetic radiation we can expect that both electric 

and magnetic fields will be perpendicular to the direction of propagation and that any mathematical 

form expressing these fields should satisfy Maxwell’s equations. We may construct 

the coordinate axis as shown in figure 8.7. 

Figure 8.7: The axes for the metal tube, including the dimensions of the rectangular tube, a 

and b. 

Since we expect no electric field in the z direction we may expect some variation in the 

electric field E across x or y. This function must satisfy the condition that the electric field 

goes to zero at the sides of the conductor since the currents and charges in a conductor 

conspire so that there is no tangential component of E at the surface. So we may have E y 

varying with x as shown in figure 8.8. 

Figure 8.8: The lowest mode for the y component of the electric field in the x direction. 

For a rectangular geometry, the waveform is generally simple harmonic so we may guess 

that the wave would have some form 

E y = E 0 sin(k x x)e i(ωt−kzz) (8.27)


The field is perpendicular to the top and bottom conductor surfaces and zero at the side walls 

if we choose 

k x a = nπ (8.28) 

where n is any integer. The z dependence indicates a wave travelling in the z direction with 

frequency ω and wave velocity v, where 

v = ω k z 

. (8.29) 

The divergence of E must be zero within the free space inside the conductor (by Maxwells 

equation). In this case E has only a y component, and it doesn’t change with y, so 

⃗∇. ⃗ E = 0 (8.30) 

is satisfied. The other Maxwell equations must also be satisfied, specifically the wave equation 

⃗∇ 2 E ⃗ 

∂ 2 E y = 

∂x + ∂2 E y 

2 ∂y + ∂2 E y 

2 ∂z 2 

Unless E y is zero everywhere this equation specifies that 

= 1 c 2 ∂ 2 E y 

∂t 2 (8.31) 

(−k 2 xE y ) + 0 + (−k 2 zE y ) = − ω2 

c 2 E y (8.32) 

k 2 x + k 2 z − ω2 

c 2 = 0 (8.33) 

k z = 

√ (ω ) ( ) 

2 n2 π 

− 

2 

c 2 a 2 

(8.34) 

k x is already fixed, so 

k z = 2π 

λ g 

(8.35) 

where λ g is the guide wavelength. 

We may define the free space wavelength as 

λ 0 = 2πc 

ω 

(8.36) 

and so


4π 2 

λ 2 g 

= 4π2 

λ 2 0 

− n2 π 2 

a 2 (8.37) 

1 

λ 2 g 

= 1 λ 2 0 

− n2 

4a2. (8.38) 

This is the fundamental relationship between free space and guided wavelengths. 

Associated with the electric fields there will be magnetic fields. Since 

c 2 ∇ × B = ∂E 

∂t , (8.39) 

the lines of B will circulate around the regions where ∂E/∂t is largest, that is, between the 

maxima and minima of E. 

Figure 8.9: Variation of E and B in the waveguide. 

We have assumed that k z has only one root but it should properly have two, a positive and 

negative one for the wave travelling in either direction. 

√ (ω ) 

2 

k z = ± − 

c 2 

( 

n2 π 2 

a 2 ) 

(8.40) 

We have also assumed that k z is real but if 

ω < nπc 

a 

(8.41) 

then the term in the square root is negative and we apparently have no solution anymore. 

However if we define


with 

k z = ±ik ′ (8.42) 

k ′ = 

√ (n2 ) ( ) 

π 2 ω 

2 

− 

a 2 c 2 

(8.43) 

Then for E y we have 

E y = E 0 sin(k x x)e i(ωt±ik′ z) 

(8.44) 

= E 0 sin(k x x)e ±k′z e iωt (8.45) 

Thus E y is seen to oscillate with time, but to decay smoothly with z as a real exponential. 

This tells us that for 

ω c = nπc 

a 

(8.46) 

or below, waves do not propagate down the guide, but decay from the source exponentially 

with k ′ . It is for this reason that ω c is called the cut-off frequency and 

λ c = 2a n 

(8.47) 

is called the cut-off wavelength. 

We may thus recast the relationship between free space and guide wavelength as 

1 

λ 2 g 

= 1 λ 2 0 

− 1 λ 2 c 

(8.48) 

If operation is below the cut-off frequency the wave is said to be evanescent. 

8.3.3 Modes in the guide 

We have been working with the transverse electric mode TE n0 . However in general, both 

n and m may vary, and we have TE nm , where n denotes the number of full variations of the 

electric field along the broad dimension, and m denote the number of full variations along 

the narrow dimension. These correspond to other solutions of the wave equations. 

Pre-lab Question 8.6 Draw the TE 10 , TE 01 , TE 11 ,and TE 20 modes (use figure 8.10 as a 

guide to what is required, but remember you are illustrating electrical rather than magnetic 

fields).


If in a mode TE mn the integers m and n are both non-zero, then the equation for the cut-off 

wavelength generalises to 

λ mn 

c = 

√ (n 

a 

2 

) 2 

+ 

( m 

b 

) 2 

(8.49) 

All modes with wavelength longer than or equal to λ c the cut-off wavelength exist simultaneously 

in the cavity. Thus the dimensions of the cavity can be tailored to exclude all modes 

but one, or one or two, etc. 

TM (transverse magnetic) modes are also possible. Some simple ones are shown in figure 

8.10. 

Figure 8.10: Simple TM modes. 

In TM modes a kind of propagation is produced whereby no component of the magnetic field 

is in the direction of propagation, while there is some component of the electric field in the 

z direction. 


8.3.4.1 The Gunn diode 

In this experiment, you can identify the Gunn diode as a metal box with a clear plastic top, 

inside of which you can see some electronics. It is a source of microwaves, for details on 

how this works, see the appendix (section 8.4.2). 

8.3.4.2 The cavity wavemeter 

The cavity wavemeter is used to measure the wavelength of the standing wave in the 

waveguide. The cavity wavemeter consists of a cavity terminated by a fixed short circuit and 

a moveable short circuit coupled to the main waveguide as shown in figure 8.11. 

If a short circuit is placed across a cavity, then at that point the voltage must be zero. Thus 

the only wave that can exist in the wavemeter is one that has a voltage minimum at both 

ends. Some of the allowable modes are shown in figure 8.12. 

In this case the wavemeter length is tuned so that only the first mode is possible. 

To obtain a reading of the wavelength of the standing wave in the main guide the micrometer 

is adjusted. When this is moved to the correct position the cavity resonates at the frequency


Figure 8.11: Schematic of a cavity wavemeter 

Figure 8.12: Allowable modes in the cavity wavemeter 

of the radiation in the guide, sucking energy from the propagation in the main guide. This is 

detected as a resonance dip in the standing wavemeter (the reading on the multimeter lowers 

suddenly). At this point, the wavelength (in centimetres) may be read off directly from the 

red scale on the micrometer, corresponding to a previous calibration. The wavemeter should 

be taken off resonance (detuned) after measurement to increase the available power in the 

guide. 

8.3.4.3 The standing wavemeter 

The standing wavemeter is a piece of equipment which measures the voltage of the standing 

wave at different points along the cavity. You can identify it as a rectangular tube with a 

BNC cable socket, which can slide up and down the tube, with a ruler attached. To learn 

more about the technical detail of how this works, see the appendix (section 8.4.3).



8.3.5.1 Familiarisation 

Figure 8.13: The microwave bench 

A block diagram of the microwave bench is shown in figure 8.13. Each square in the diagram 

corresponds with one piece of equipment on the bench. The separations in the diagram 

correspond to straight pieces of hollow waveguide, or in the case of the separation between 

standing wave meter and twist, or twist and variable end, simply two rings of metal screwed 

together. 

• Ensure the power to the Gunn diode is off, and carefully unscrew a few pieces of 

equipment, and examine inside. Inside the cavity wavemeter, can you see the cross as 

in figure 8.11? Carefully reassemble the equipment. 

• Connect the DC voltmeter to the standing wave detector. Adjust the cavity wavemeter 

until a resonance dip occurs (sudden rapid decrease from reading off the voltmeter). 

This corresponds to a reduction in output of ∼40% from the standing wavememeter. 

(See the appendix (section 8.4.4) for how to read the micrometer). 

Record λ g and calculate the frequency of the oscillation 5 . 

Question 8.5 Is this indeed in the microwave range of the electromagnetic spectrum (∼1- 

300 GHz)? 

8.3.6 Polarisation of microwaves in the guide 

• Attach a microwave horn to the free end of the cavity. Horns simply direct the microwaves 

to a narrow beam. Position the other horn with the crystal detector attachment 

some distance in front of the first horn. Connect the detector to a multimeter. 

• Place the wire grille between the two horns and observe the response in the crystal 

detector for grille orientations both parallel and perpendicular to the bench surface. 

5 See Ohanian [7], chapter 37, page 911.


Question 8.6 What conclusions may you draw from this exercise? (In what direction do the 

electric field vectors point in this system?) 

8.3.7 Relation between guide and free space wavelengths 

Retaining the set-up from the previous exercise replace the other horn with an aluminium 

plate ∼20 cm in front of the first horn. Attach a multimeter to the standing wave detector 

and observe the standing wave pattern by moving the carriage along the slot in the top of the 

waveguide. Refer to figure 8.14 to see how a standing wave is produced. 

Figure 8.14: Standing waves produced using reflection from an aluminium plate 

The radiation leaves the main guide via the horn and is reflected back from the plate producing 

a standing wave with wavelength λ a between the plate and the horn. Reflected radiation 

also enters the guide, producing a standing wave in the guide with wavelength λ g . Moving 

the plate simply translates the standing wave pattern laterally by the same amount. 

Question 8.7 Determine the guide wavelength λ g by measuring the distance between successive 

minima. Now measure λ a , the free space wavelength by keeping the standing wave 

detector stationary and moving the aluminium plate. Develop and describe your technique 

for minimising error. 

Question 8.8 Predict the value of λ c using the relationship between λ a and λ g . Compare this 

to the theoretical value of λ c calculated assuming that the TE 10 mode is being propagated. 

Describe the physics of what λ c represents. 

8.3.8 Impedance measurement 

Since microwaves are a wave phenomenon we may use an identical formalism as that described 

for acoustics in order to represent them. Thus we may assume identical expressions


for the phase change, φ = 2kx 0 , the reflection coefficient, r, and the impedance of a discontinuity, 

Z T /Z 0 , to characterise termination in this case. 

• Terminate the line with a short circuit, setting its micrometer to zero. 

Question 8.9 Plot the standing wave pattern for one wavelength, that is two maxima and 

minima, using the standing wave detector. 

Question 8.10 You should record the distance from the flange to the first minimum in order 

to calculate the true position of the first minimum. Calculate r, φ and K for the short circuit. 

Question 8.11 Repeat the measurement for the matched load and open guide. 

The short circuit may be described as the perfect mismatch, with almost all the microwave 

energy being reflected from the termination. The matched load is the perfect match, containing 

a dissipative wedge of iron that absorbs most of the microwave energy. 

Question 8.12 Do your results reflect these differences? 

Question 8.13 For electromagnetic waves, there is a π phase shift upon reflection from a 

short circuit. That is, φ EM = 2kx 0 − π. Given this, compare your results from this section 

with those from the acoustics section. 


8.4.1 The complex reflection coefficient K 

In the discussion above the complex reflection coefficient was assumed to apply to the displacement 

only. This quantity should be more properly labelled K DISP , since reflection is a 

process undergone also by the particle velocity and the pressure. From plane wave theory 

(and recalling figure 8.1) we may write 

∣ ∣ ∣ r ≡ 

ξ −∣∣∣ 

∣ = 

v −∣∣∣ 

ξ + 

∣ = 

p −∣∣∣ 

v + 

∣ 

(8.50) 

p + 

Where ξ ± , v ± and p ± are all complex. Let us define the full three complex reflection coefficients. 

K DISP = ξ − 

ξ + 

, 

K VEL = v − 

v + 

, 

K PRESS = p − 

p + 

. (8.51) 

We will now show that phase change in acoustic pressure upon reflection is π different to 

phase change in both displacement and velocity.


From table 8.2, we have 

therefore we have both 

v = iωξ (8.52) 

and 

v − = iωξ − (8.53) 

Dividing equation 8.53 by 8.54 we get 

v + = iωξ + (8.54) 

and since 

v − 

v + 

= ξ − 

ξ + 

⇒ K VEL = K DISP (8.55) 

then from equations 8.14 and 8.13 

v − = K VEL v + = K DISP v + ⇒ Z 0 v − = K DISP Z 0 v + (8.56) 

−p − = K DISP p + ⇒ K DISP = K VEL = −K PRESS (8.57) 

Q.E.D. 

Let us now define the form of φ = 2kx 0 and K = re iφ that is used in this experiment. 

We have an incident plane wave (equation 8.15) 

ξ + = a + e i(ωt−kx 0) 

(8.58) 

and another reflected plane wave (equation 8.16) 

ξ − = a − e i(ωt+kx 0) 

(8.59) 

since r = a − 

a + 

(equation 8.18) 

ξ − = ra + e i(ωt+kx 0) 

(8.60) 

Now, the reflected wave is given by K (actually K DISP ) times the original wave, so 

ξ − = Kξ + = Ka + e i(ωt−kx 0) 

(8.61)


so 

and so 

K = ra +e i(ωt+kx 0) 

a + e i(ωt−kx 0) 

(8.62) 

where 

K = re iφ (8.63) 

Q.E.D. 

φ = 2kx 0 (8.64) 

8.4.2 The Gunn diode 

Figure 8.15: Rectification by the crystal detector 

You will be using a solid state source to produce electromagnetic radiation of microwave 

frequencies. The central component of a Gunn diode is a piece of gallium arsenide crystal. 

Generally, in crystals, the repetition of a regular array in space results in allowable energy 

states for the electrons, depending on their wave number, k. In the absence of an applied 

electric field, the electrons will choose energy levels as low as possible given a certain temperature. 

The net electron flow will be zero since there is no preferred direction. If an 

external voltage is applied, the electron velocity will increase linearly with the voltage until 

electron collisions with the lattice are energetic enough to damage the crystal. 

In GaAs, the linear increase of electron velocity with voltage obtains until an internal field 

of 3 kV/cm is achieved. At this point the electrons gain enough energy to occupy a second 

energy level. In this state, the electrons have a higher effective mass and a lower velocity. 

Since the velocity is lower, the current is appreciably smaller. Thus even though the voltage is 

rising the current is decreasing, meaning that the crystal is behaving as a negative resistance. 

Oscillations result.


The oscillation mechanism is complex and involves many modes. The usual mode involves 

a dipole distribution of charge on either side of a non-uniformity in the crystal. This dipole 

domain drifts toward the anode and dissipates. The field across the sample rises, causing 

a new dipole domain to grow. The frequency of this oscillation depends critically on the 

domain drift velocity (or applied voltage) and the lateral dimension of the sample. These 

parameters may be tuned to achieve electromagnetic disturbances of a microwave frequency. 

8.4.3 The standing wavemeter 

The standing wavemeter consists of a probe mounted in a slot through the waveguide wall. 

This probe is connected to a silicon crystal slab connected directly to a thin tungsten wire. 

Passing microwave energy induces a current in the probe. However since the microwave 

disturbance is symmetrical about zero, any meter detection of this current would read zero 

since the fluctuations are so frequent. The crystal-wire configuration rectifies this signal. 

Electron flow through the silicon crystal is hampered by the scarcity of electrons. Tungsten 

as a metal, has surplus electrons, so current can flow readily in this direction only, which 

rectifies the signal as shown. The average signal is a measure of the microwave energy in the 

cavity. 

We read off the RMS signal due to the frequency of the fluctuations. 

8.4.4 Reading the micrometer 

This is a foolproof plan to read the micrometer on the cavity wavemeter, the final answer of 

which we shall call M. You will have to adjust the method accordingly for when you can 

only just see a line under the thimble - but reading through this guide (including the example 

at the end) should give you a good working understanding of reading (a) this micrometer in 

particular, and (b) a good idea at how to read any other micrometer you may come across. 

First of all, rotate the micrometer until it is at such a point that you would like to record the 

result. 

big black line 

sleeve 

thimble 

Figure 8.16: Parts of a micrometer


Step 1 What is the biggest red number that you can read, (it is written on the sleeve)? Call 

it N. This is the number of centimetres. Note that N < M. 

Step 2 How many small division markers on the right hand side of the big black line can 

you count that come after the big division marker corresponding to your N? 

If there are 0 then N < M < N + 0.2 

If there is 1 then N + 0.2 < M < N + 0.4 

If there are 2 then N + 0.4 < M < N + 0.6 



Record L, such that (N + L) < M < (N + L + 0.2). 

Step 3 Is there a small division marker on the left hand side of the big black line after the 

final marker on the right hand side? 

If the answer is yes, (N + L + 0.1) < M < (N + L + 0.2) 

If the answer is no, (N + L) < M < (N + L + 0.1) 

Record l, such that l < M. 

Step 4 Which division on the thimble is coincident with the big black line? Call the red 

number associated with this line (which is written on the thimble) R. 

Step 5 The final answer, M, is given by 

M = l + R 

1000 . (8.65) 

Hence, we have measured the number of centimetres to 3 decimal places. 

Example 

Step 1 N = 2, and M > 2. 

Step 2 There are 2 ⇒ L = 0.4, and M is between 2.4 and 2.6. 

Step 3 Yes ⇒ l = 25, and M is between 2.5 and 2.6. 

Step 4 R = 54 

Step 5 M = 2.5 + 54 

1000 

= 2.5 + 0.054 = 2.554 cm 


[1] A W Cross. Experimental Microwaves. Marconi Instruments Ltd., Sanders Division, 

1977.


02 

25 

50 

30 

60 

Figure 8.17: Set the micrometer up so it looks like this. 

[2] R Bowers. A solid-state source of microwaves. Scientific American, 215, 1966. 

[3] P Lorrain and D R Corson. Electromagnetic Fields and Waves. W.H. Freeman and Co., 

2 nd edition, 1970. 

[4] R P Feynman, R B Leighton, and M Sands. The Feynman Lectures, volume II. Addison- 

Wesley, 1964. 

[5] E Hecht. Optics. Addison-Wesley, 2 nd edition, 1987. 

[6] D K Cheng. Field and Wave Electromagnetics. Addison-Wesley, 2 nd edition, 1992. 

[7] H C Ohanian. Physics. 2 nd edition, 1989.

Chapter 9 

Electron Spin Resonance 

9.1 A guide to background reading 

While these notes are designed to be fairly self-contained, students are directed to the cited 

references for a more detailed treatment of the theory as some results are presented without 

detailed proofs. In particular a cursory reading of the pertinent sections of “Introduction to 

Solid State Physics” by C. Kittel [6], which is available from the Part 2 office, may provide 

some valuable insights into the phenomena of electron diffraction. 

Electron spin resonance and Zeeman splitting can be found in any number of quantum mechanics 

books. Specifically “Modern Physics” by Serway, Moses and Moyer [7] or “Physics 

of Atoms and Molecules” by B. Bransden and C. Joachain [2] provides detailed treatment 

for 2 nd or 3 rd year level. 

For an introduction to the quantum mechanics of angular momentum and spin, see Quantum 

Physics of Atoms, Molecules, Solids, Nuclei and Particles by R. Eisberg and R. Resnick 

[1] or any numer of second year textbooks. A discussion on Helmholtz coils can be found 

in Experimental Methods in Magnetism by H. Zijlstra [3]. For a detailed discussion of the 

experimental considerations of ESR see Electron Paramagnetic Resonance - Techniques and 

Applications by R.S. Alger [4]. For additional information on LC resonator circuits see 

Introductory Electronics for Scientists and Engineers by R.E. Simpson, section 2-12 [5]. 


In 1896, Zeeman observed that atomic spectral lines were split when the atom giving rise 

to the spectrum was placed into an external magnetic field. And in 1922, Stern and Gerlach 

performed their canonical experiment of passing silver atoms through a magnetic field, observing 

the original beam split into two in the presence of the field. The answer to these 

two disparate observations came in 1925, when Uehlenbeck and Goudsmit postulated that 

the splitting of atomic spectra was due to an intrinsic angular momentum they denoted spin. 

This property couples to the orbital angular momentum of the electrons giving rise to the 

observed splitting in the spectral lines in the presence of an external magnetic field. That 

spin-orbit coupling is a fundamental force in atomic and subatomic physics. While such a 

113

114 CHAPTER 9. ELECTRON SPIN RESONANCE 

feature has been incorporated extensively in the Schrödinger equation to describe phenomena 

(nuclear physics couldn’t work without it), a true understanding of spin-orbit coupling 

came in 1929 with Dirac and his equation. Spin was the first quantum observable introduced 

which has no classical analogue. It is developed in this section. 

Note, however, that Dirac actually sought, as did Schrödinger before him, a correct wave 

theory for the electron which incorporated the spin of the electron. Spin itself is not a consequence 

of the Dirac equation; the spin-orbit coupling is. 

In this experiment we will study the Zeeman splitting of spectra from a molecule, diphenylpicra-hydrazyl 

(DPPH), which has an unpaired electron on one of the nitrogen atoms. It has 

features which allow for the spin of the electron to be studied in isolation. 

9.3 Introduction: Angular Momentum in Quantum Mechanics 

9.3.1 Orbital Angular Momentum 

The quantum mechanical analogue of classical angular momentum is orbital angular momentum. 

For a particle moving in a circular path about a fixed point in space, its angular 

momentum is defined as in the classical case, viz. 

L = R × P . (9.1) 

The quantum mechanical properties of orbital angular momentum are, however, different. 

The properties of momentum, and how they relate to the position of a particle, require the 

orbital angular momentum take quantized values, which are zero and positive integers, ie 

L = 0, 1, 2,.... 

As angular momentum is a vector, we may define its projection against an arbitrary axis, 

with a given direction. In doing so, we find that the values its projection may take are also 

integer, but this time including negative values. Picture the positive values as corresponding 

to the angular momentum vector pointing in the same general direction as the axis, and the 

negative values with the angular momentum pointing in the general opposite one. 

9.3.2 Spin 

The above picture, however, is incomplete. We may observe the above projections only in 

the presence of a magnetic field, which splits an energy level corresponding to a given L into 

its 2L + 1 levels, from −L to +L in integer steps. For a particle moving in a magnetic field, 

such splitting is due to an induced magnetic moment, viz. 

µ = e ↕. (9.2) 

2mc 

For an atom, comprising of (an even number of) Z electrons, the total magnetic moment is 

the sum of each magnetic moment induced by each orbiting electron. This amounts to a split

9.4. ELECTRONS IN AN EXTERNAL MAGNETIC FIELD 115 

into 2L + 1 levels dictated by 

E = MBµ B , (9.3) 

where B is the strength of the external field and µ B is a constant of proportionality known as 

the Bohr magneton: 

µ B = e 

2mc . (9.4) 

The M values are known as the magnetic, or projection, quantum numbers and may take 

values from −L to +L in integer steps. Thus, for even-Z atoms, there are an odd number of 

split states. 

A major problem arises in the case of atoms with an odd number of electrons: the number 

of split levels is observed to be even. That leads to the contradictory conclusion that L must 

be half -integer. This was observed most strikingly in 1922 by Stern and Gerlach. They 

passed a beam of silver atoms through a magnetic field and observed that the beam split 

into 2. (Remember that silver has Z = 47 which means that there is one odd electron in its 

configuration.) If there are 2L+1 = 2 levels into which the angular momentum may project, 

then that would mean an L value that is half -integer, specifically L = 1 2 . 

The solution came in 1925 when Uehlenbeck and Goudsmit postulated that the electron must 

contain an intrinsic angular momentum, which they denoted by spin, taking the value 1 2 for 

the electron. That then induces an additional magnetic moment in the presence of a magnetic 

field given by 

µ s = g s 

e 

2mc s (9.5) 

where s is the spin of the electron. Agreement with experiment is achieved when g s = 2. 

The total angular momentum j is then taken as the vector sum of the orbital and spin angular 

momenta, and this resolves any discrepancy with experiment. 

As with orbital angular momentum, one may then project the spin vector onto an arbitrary 

axis. What one finds is that the spin magnetic values may take values anywhere from −s to 

+s in integer steps, as with L. For the electron, s = 1 which means it has projection values 

2 

of m s = ± 1 ; the positive we denote as “spin-up” and the negative “spin-down”, refering to 

2 

the direction the vector takes against the axis. That we have two such values explains the 

splitting into two of the silver atoms in the Stern-Gerlach experiment. 

Two important facets of spin should be noted: 

1. It is a fundamental property of particles; 

2. There is no classical analogy for it. 

In fact spin was the first observable in the atomic regime for which there was no classical 

counterpart. 

9.4 Electrons in an external magnetic field 

What happens when an electron is placed in a uniform magnetic field? If we passed a beam 

of electrons through a uniform field we would see the beam split into two consistent with


E 

E 0 

0 

B=0 B= 

E−∆Ε 

Ε+ ∆Ε 

0 

−∆Ε 

+∆Ε 

Figure 9.1: Energy splitting for an electron in a uniform magnetic field B, with direction as 

indicated. Note that the value of ∆E is, from Eq. (9.6) negative. 

the original Stern-Gerlach experiment. A stationary electron in a uniform field would feel 

the magnetic field through the spin magnetic moment and precess about the direction of the 

field. This will induce a change in the energy of the electron, depending on the orientation 

of the electron with the field, viz. 

∆E = −µ s · B. (9.6) 

In this situation, therefore, the external field itself defines the axis against which the spin is 

measured. To minimise energy, the electron would wish to align its spin with the field. If it 

is unable to do so, precession occurs. If no precession is observed, then the electron’s spin 

will either be parallel (m s = + 1) or anti-parallel (m 2 s = − 1 ) to the field. 

2 

Prelab Question (A): Using Eq. (9.6) show that when an electron is placed in an external 

field, its energy changes by 

∆E = ± 1 2 g sµ B B . (9.7) 

The change in energy depends on whether the electron is in the spin-up or spin-down state. 

The minus sign in Eq. (9.6) means that the spin-up electron will be in a lower energy state 

than that in the spin-down. (Why?). But that change in energy is only effected when there 

is an external magnetic field, ie., for a stationary electron the only mechanism by which it 

interacts with the external magnetic field is through its spin. 

Once the field is applied the electron will either be aligned with or against the field (spin-up 

or down). To make the electron switch between these states requires the additional energy 

of ∆E ′ = g s µ B B. That may be done by the absorption or emission of a photon with this 

energy. By measuring that energy we may then determine the value of g s from experiment. 

The whole scenario is illustrated in Fig. 9.1. 

9.5 Resonance absorption 

We consider the situation in which the only source of photons in the system are those that are 

controlled in the experiment. The experiment should account for background lighting and


N 

N 

Figure 9.2: The DPPH molecule showing the isolated unpaired electron in the molecular 

configuration. 

control the wavelength of the photons required in the absorption in the above situation to be 

outside that of background light. 

Prelab Question (B): Calculate the value of the external magentic field necessary such that 

a photon, wavelength λ = 450 nm, has the required energy to flip the spin of the electron. 

Why then may we ignore the background light when performing the experiment? 

The source of electrons in this experiment is the organic molecule diphenyl-picra-hydrazyl, 

or DPPH (Fig. 9.2). This molecule is convenient in that it has one valence, unbonded electron 

on the second N atom. The interaction of that electron with the mean Coulomb field 

generated by the other electrons in the molecule ascribe an energy E 0 to it. 

Prelab Question (C): Why can’t truly free electrons be used in this experiment? Why is a 

beam of electrons or a metal inappropriate? 

Choosing a sample in which an electron may be unpaired is not the only consideration. Another 

may be the situation in which the electron in its natural state in the molecular configuration 

is predominantly in the spin-down level. One requires such in order to see a measureable 

effect. One would also like the lifetime of the spin-down state to be relatively short in order 

to eventually decay to the lower energy spin-up state. (This is an important point.) 

Under these conditions, we may then provide a source of photons, frequency f, in an experiment. 

When the photon is at the resonance frequency for a given magnetic field strength, 

then the spin-up electron will flip its spin; the resonance energy is given by hf = g s µ B B. 

By scanning through a range of frequencies, one may then observe that resonance. 

Note that in this experiment the electron does not exhibit any orbital motion, hence the 

response to the magnetic field of the electron will be solely due to the electron’s spin. 


Fig. 9.3 shows the experimental configuration required to measure g s . 

9.6.1 Helmholtz coils 

NB! There are LARGE coils which carry current on the desk.


inductor with 

sample 

Oscillator 

power 

supply 

Coil power 

supply 

AC 

I 

B 

Helmholtz coils 

(in series) 

000000000 

111111111 

000000000 

111111111 

RF 

−12 0 +12 

−12 0 +12 

Y 

000000000 

111111111 

000000000 

111111111 

1 Ω 

TO CRO 

freq. 

meter 

Oscillator adaptor 

Figure 9.3: The apparatus used for the measurement of ESR. The AC supply is 50 Hz, and 

the voltages indicated on the Oscillator supply and adaptor are in V. The field generated by 

the coils is in the direction as indicated. 

The Helmholtz coils are connected to an AC power supply (50 Hz), so the current will vary 

sinusoidally with time. 

Prelab Question (D): What do Helmholtz coils produce? Discuss with your demonstrator, 

if unsure. 

Prelab Question (E): Let R be the radius of a pair of Helmholtz coils separated by a distance 

R. If x denotes the distance from the centre of the left hand coil to any point 

along that axis, calculate the magnetic field produced at points x = 0.2R and x = 

0.5R. Remember that the field produced by one ccoil with n turns carrying a current 

I is given by 

µ 0 nIR 2 

B = 

(9.8) 

2 (x 2 + R 2 ) 3/2 

Question (a): Draw a voltage vs time graph for the voltage across the resistor for two full 

periods of the AC signal. Assuming a peak-to-peak voltage of 5 V across the coils, 

draw the B vs time graph for the coils. Why do we use a 1 Ω resistor? 

9.6.2 RF oscillator 

The RF oscillator provides the photons needed to examine resonance absorption. It coverts a 

signal into a magnetic field and back again. The field produced preiodically bathes anything 

within the coil in a sea of photons of the frequency selected using the knob on the unit.


The oscillator can produce photons with frequencies between 30 and 130 MHz depending 

on the coil attached. (The smaller the coil, the higher the frequency.) On the rear of the 

oscillator you can connect a micro-ammeter to the socket marked “I/µA”. The ammeter 

then monitors the current flowiing the through the unit and the frequency. 

9.6.3 Investigating the resonance 

Included is what is termed a “tank” circuit. It consists of a variable capacitor connected to a 

coil of wire similar to the one on the RF unit. The circuit is actually an LC circuit and will 

resonate at the frequency determined by the values of capacitance and inductance. 

The tank circuit is simply used to demonstrate resonance, and the settings at which such 

resonance are NOT used for the remaining sections of the experiment. 

Turn on the RF unit and use the ammeter to determine the oscillation frequency. Then bring 

the tank circuit up to the RF unit such that the coils are ALMOST touching (you will have 

to take the RF unit out of the Helmholtz coils to do this). Connect the tank circuit up to the 

CRO to monitor the voltage across the circuit and then adjust the knob on top fo the trank 

circuit until it resonates wth the RF unit. This will occur when the voltage across the CRO 

reaches a maximum. 

Note: the LC tank circuit has a limited range corresponding to the range of the variable 

capacitor. If you don’t observe a maximum you may have to change the frequency of the RF 

unit. 

When you have determined the resonance point, use the ammeter to examine the current 

through the RF unit. What happens as you move in and out of the resonance? Explain. 

9.6.4 Electron absorption 

The sample of DPPH is contained in a vial. Place the sample within the coil of the RF unit, 

itself in the centre of the Helmholtz coils. 

Draw the B vs time graph through the coils that indicates the strength of the uniform field 

seen by the electrons in the DPPH sample. Under this plot draw the current you would expect 

to measure through the RF oscillator. Assume that the maximum value for B is 0.50 T and at 

a give frequency the resonance condition is satisfied whenever B = ±.025 T. Discuss with 

your demonstrator. 

Question (b): The relaxation time of the electrons back to the ground state should be short, 

compared to the frequency of the sweeping B field. Why is it so? 

Position the Helmholtz coils correctly using the dial caliper ensuring that they are connected 

corretly and in series with the resistor and AC supply. “A” identifies the beginning of the 

coild and Z the end. The mean diameter of the coils is 13.6 cm and the number of turns in 

each is 320.


Connect the RF unit to the power supply, place the sample in the center of the coil. Position 

the RF coil as centrally as possible into the Helmholtz coils. 

After everything is connected examine the voltage across the resistor and the current through 

the RF unit simultaneously. Adjust the current through the coils. What happens to the current 

through the RF unit? 

Question (c): Why do the features on the plot change? Discuss with your demonstrator. 

You should now be able to determine how best to measure B when resonance occurs. 

Question (d): How should the maximum current through the coils be chosen such that you 

reduce the error when measuring B at resonance? 

Measure the voltage at which resonance occurs, as a function of RF frequency. Use the 

small bar magnet to observe any changes when you change the magentic field by placing 

the magnet near the sample such that it is parallel and then perpendicular to the coil axis. 

Explain. 

Question (e): You should have enough information to determine g s . How does it compare 

to the nominal g s = 2? 

Question (f): Why to we observe a width on the resonance peak? 

9.7 Extension: electron diffraction 

This section contains some extra work for efficient or ambitious students. It should only be 

attempted when all the work in the previous section has been completed. You can still get 

full marks having done only the previous section. 


The postulate that matter may also possess wave-like properties was one of the most stunning 

and unifying hypotheses of this century. In 1924 de Broglie, working from arguments based 

on the properties of wave packets, hazarded that the wavelength of these matter waves could 

be obtained from the same relationship that held for light, namely 

λ = h p 

(9.9) 

where λ is the wavelength, p is the momentum and h is Planck’s constant 1 . 

Davisson and Germer obtained the first experimental evidence for matter waves by reflecting 

slow electrons from the face of a single nickel crystal. The wavelength that they were able to 

1 Some useful data, including a value for h, are provided in section 9.8.

9.7. EXTENSION: ELECTRON DIFFRACTION 121 

determine from using the periodic array of nickel atoms as a diffraction grating was strikingly 

similar to that calculated from the de Broglie relation. Further experimentation by Thomson 

yielded diffraction patterns from the transmission of electrons through thin polycrystalline 

foils. The fact that non-relativistic electron wavelengths are comparable to interatomic distances 

in solids means that electron diffraction experiments (such as Transmission Electron 

Microscopy, or TEM) are powerful probes of the structure of solids. 

9.7.2 Basic theory 

9.7.2.1 Wavelength 

The wavelength of a beam of electrons is given by 

λ = h p = h mv . (9.10) 

Thus in the non-relativistic limit the wavelength is inversely proportional to the electron 

velocity v. Conservation of energy requires that for a charge accelerated through a potential 

the change in kinetic energy plus the change in electrical potential energy while traversing 

from one point to another must equal zero. Thus we may write 

( ) mv 

2 

2 

2 + mv2 1 

+ (eV 2 − eV 1 ) = 0 (9.11) 

2 

where V 1 and V 2 are the potentials at points 1 and 2 respectively. Applied to the electron 

transversing an electron gun in an electron diffraction tube V 2 is V a , the anode voltage and 

V 1 equals zero. So for negative charges 

mv 2 

2 

= eV a . (9.12) 

Hence the wavelength is given by 

λ = 

h 

√ 2meVa 

, (9.13) 

where m and e are the mass and charge of the electron respectively. When the values of the 

constants are substituted into equation 9.13 and the potential is in volts then the wavelength 

in nanometres is given by 

λ = 1.2264 √ 

Va 

. (9.14)


9.7.2.2 Electron diffraction 

Pre-lab Question 9.1 Given the situation of electrons being reflected from a set of planes 

and the geometry shown below (figure 9.4) derive the condition for constructive interference, 

known as the Bragg condition, 

2d sin θ b = nλ (9.15) 

Figure 9.4: The reflection of light from a set of planes. The Bragg condition for constructive 

interference may be derived purely from the geometry shown. 

The experiment you will undertake replicates Thomson’s method of transmitting electrons 

through a thin film of randomly oriented crystallites to investigate a ring diffraction pattern. 

The periodic array of atoms in the 3-dimensional solid forms a space lattice. Diffraction from 

such a space lattice is somewhat more involved than the situation just depicted, however, a 

close analogue of the Bragg equation still holds. 

2d n sin θ b = λ (9.16) 

Here d n is the distinct interplanar distance for a particular set of planes described by the 

periodic atomic arrangement. A diffraction pattern for a single crystal with a cubic structure 

is shown below (figure 9.5). R 1 and R 2 arise from two different sets of planes, and when 

analysed, will yield two separate d values, d 1 and d 2 . 

The angle between the direct and diffracted beam is shown in figure 9.6. You will be measuring 

ring radius as a function of anode voltage. 

Pre-lab Question 9.2 Using the geometry shown in figure 9.6, derive an expression relating 

the ring radius to the scattering angle, 2θ b . You will be measuring D = 2R with a pair of 

dial calipers, so take care when you define this quantity. 

Pre-lab Question 9.3 Using the Bragg relation and the expression determined above, derive 

an expression for d n , the interatomic spacing in terms of R.

9.7. EXTENSION: ELECTRON DIFFRACTION 123 

Figure 9.5: The electron diffraction pattern from a single crystal of a substance with a cubic 

structure. The radii shown correspond in real space to two separate sets of crystal planes and 

yield two values for the interplanar spacing, d. 

Pre-lab Question 9.4 The specimen you will use in this experiment is graphitised carbon. 

Assume that the form of carbon is a simple cubic structure. The atomic mass of carbon is 12 

and its density is 2.3 g cm −3 . Calculate the spacing of adjacent carbon atoms. 

Figure 9.6: The geometry of the electron diffraction experiment. 


HIGH VOLTAGES ARE USED IN THIS EXPERIMENT. BEFORE SWITCHING 

THE APPARATUS ON CHECK THAT ALL THE VOLTAGES ARE TURNED 

DOWN. IF THIS IS UNCLEAR CONSULT YOUR DEMONSTRATOR.


• Ensure that the apparatus is wired up as shown in figure 9.7. 

• Switch on the high voltage supply and wait for one minute for the cathode heater to 

stabilise. 

• Select “CT”, and slowly increase the high voltage, V a , to the approximate value of 

2.7 kV. This is the knob which you will be adjusting for this experiment. Remember, 

this is the anode voltage, and corresponds to equation 9.14. 

• You must ensure that the current on the multimeter does not exceed 0.2 mA, and you 

can modify this by adjusting V b , the bias voltage supply. Current overload causes 

the target to glow a dull red, so inspect the target at intervals during the experiment 

to ensure that this is not happening. Adjust the bias voltage supply V b to produce a 

clear ring image and uniform central spot. If V b is initially connected the wrong way 

reverse it, but TURN DOWN THE HIGH VOLTAGE BEFORE DOING THIS. Note 

that adjusting the knob for V b also changes the value of V a , so ensure that you are 

measuring the ring radius for the correct value of V a ! 

• Sketch the observed ring pattern. Roam around the neck of the tube with a bar magnet 

and try to increase the clarity of the ring pattern. If there is no improvement, remove 

the magnet. Sketch the ring pattern with the magnet in different positions. Describe 

and explain what you observe. 

Question 9.1 Deduce which end of the magnet is North. 

• Using the dial calipers measure both ring diameters for 12 different anode voltages. 

You may note that the rings are distorted. To minimise errors, four measurements 

should be taken in the four quadrants and averaged. Watch for drift in the voltage and 

adjust when required. Record all your measurements including the voltages. 

• Explain why the ring diameters increase with decreasing voltage. 

• Graph Va 

−1/2 against sin θ b using Excel, and calculate the interatomic spacings from 

the gradient. Compare your lattice spacings to the actual values of d 11 = 0.123 nm 

and d 10 = 0.213 nm. From your derived values of interplanar distance suggest how 

the carbon atoms are more likely to be arranged. Use a diagram to illustrate your 

conclusions. 

9.7.4 Discussion 

Question 9.2 Why does diffraction from graphitised carbon produce a ring pattern? What 

does a single crystal diffraction pattern look like? Explain the difference. 

Question 9.3 Could you perform a similar experiment with protons? What parameter would 

you have to change? 

Question 9.4 Would equation 9.13 still be valid for a 20 MeV beam of electrons? What 

expression would be more appropriate?


Figure 9.7: The circuit used in the electron diffraction experiment. 


Quantity 

Value 

h 

6.626 × 10 −34 J s 

µ b 9.273 × 10 −24 A m 2 

µ 0 1.2566 × 10 −6 H m −1 

Avogadro’s number 6.02217 × 10 23 

Nuclear magneton µ N = 5.05 × 10 −27 J T −1 

9.9 Wave function for the electron - a brief history 

Once the spin of the electron had been discovered, Schrödinger actually sought to write down 

a wave equation for the electron incorporating spin and wrote down (first) what we now 

know as the Klein-Gordon (KG) equation, which is a relativistic version of the Schrödinger 

equation. (It is important to remember that Schrödinger first wanted to find a relativistic wave 

equation for the electron.) Schrödinger noted that the KG equation could not accommodate 

in any reasonable way the spin of the electron and remain relativistic. (Thus the Klein- 

Gordon equation remains the relativistic wave equation for particles with integer spin, such 

as the α particle, which has a spin of 0.) Dirac resolved the problem of a relativistic wave 

equation for the electron in 1929 and, with it, found the source of the spin-orbit force, which 

was known to affect the electron orbits in the atom.



[1] R Eisberg and R Resnick. Quantum Physics of Atoms, Molecules, Solids, Nuclei and 

Particles. Wiley, 2 nd edition, 1985. 

[2] B H Bransden and C J Joachain. Physics of Atoms and Molecules. Longman, 1983. 

[3] H Zijlstra. Experimental Methods in Magnetism. North-Holland Pub. Co., 1967. 

[4] R S Alger. Electron Paramagnetic Resonance - Techniques and Applications. Interscience 

Publishers, 1968. 

[5] R E Simpson. Introductory Electronics for Scientists and Engineers. Allyn and Bacon, 

1987. 

[6] C Kittel. Introduction to Solid State Physics. Wiley, 7 th edition, 1996. 

[7] R A Serway, C J Moses, and C A Moyer. Modern Physics. Saunders, 1989.

Part III 

Nuclear physics 

127

Chapter 10 

Rutherford scattering 

10.1 Abstract 

One of the breakthroughs by Einstein in 1905 dealt with Brownian motion, that seemingly 

random motion by small particles on the surface of water. He demonstrated that this proved 

the existence of atoms and molecules; the “random” motion was caused by collisions with 

the molecules in the water. Together with the discovery of the electron 9 years earlier by 

J.J. Thompson, the questions were asked as to the structure of the atom itself. 

The one that was naturally assumed was the so-called “plum pudding” model, where the 

electrons were embedded in positively charged matter. (It had been established by chemists 

that atoms had to be electrically neutral.) While this assumption had not been put to the test, 

it was already known to have at least one flaw: would not the electrons and positive charges 

cancel each other out through interaction? A test came in 1911 by Rutherford and his team, 

who conducted an experiment by elastically scattering α particles 1 from a thin gold (Au) foil. 

As the α particles are positively charged, it was assumed that the scattering would proceed 

via the Coulomb interaction from the positively charged centres in the atom. 

There were two surprises: first, some of the α particles were elastically scattered at an angle 

of 180 ◦ . Second, and far more surprisingly, most of them did not scatter at all. The only 

interpretation, and the correct one, was that the positive charge was concentrated in the very 

centre of the atom, with the electrons in orbit at a great distance from it, and that most of the 

atom was, in fact, free space. 

This experiment reproduces that first experiment, with the view to measuring the distribution 

of elastically scattered α particles from which one may infer the existence of the nucleus. We 

may also, as a result, find the thickness of the gold foil we use. 

1 Remember that an α particle consists of two protons and two neutrons, i.e. an He 2+ ion. 

129

130 CHAPTER 10. RUTHERFORD SCATTERING 

10.2 Introduction/background reading - importance of scattering 

“. . . is devoted to the theory of scattering and, more generally, collision processes. It is 

impossible to overemphasise the importance of this subject.” [1], chapter 7. 

Imagine yourself looking at the entrance to a tunnel at the side of a mountain. It is dark, it 

is wet, and you are wondering if there is anything in the tunnel. You can’t see directly into 

it because of the lack of light and you don’t want to go near it as you’ll be soaked. So you 

send in a car (no driver, this one’s robotic...). If shortly thereafter you hear a loud crash and 

observe a wheel flying out of the tunnel back towards you, then you may safely infer that 

there was something in the tunnel. So you send in another car. And another. And another 

after that. After picking up the pieces of debris lying in front of you from each collision, a 

picture soon emerges of what that object may have been. The velocity at which the pieces 

came flying back tells you of the impulse involved in the collision and hence you may infer 

something of the object’s mass. The time you record when you hear the crash tells you 

something of where that object may have been in the tunnel. 

This type of experiment is typical of scattering experiments at the quantum level. We can’t 

see directly the object about which we wish to find information, so we send something in to 

scatter from it and pick up the pieces. Doing this repeatedly and controlling the kinematic 

conditions then allows us to work backwards and infer the structure of the object. This has 

direct correlation to diffraction studies in optics, where studying diffraction patterns allows 

us to work backwards to the structure of the diffracting object. 

Rutherford, and his students Geiger and Marsden, scattered low-energy α particles from a 

gold foil to find out about the structure of atoms; in particular, the structure of the positively 

charged half of them. Most of nuclear and particle physics has been done as variations on 

this theme. The quarks were discovered in 1964 by deep inelastic scattering of high energy 

electrons off the proton, which translated to elastic scattering of electrons off the quarks 

inside. 

10.3 Geometry 

In order to perform this experiment it is first necessary to understand the geometry of the 

experiment. This then provides the framework by which we may understand the scattering. 

Figure 10.1 defines the variables for the scattering, with the relevant lengths for the 

arrangement given in table 10.1. 

10.3.1 Cross section 

The quantity we will measure is the number of scattered α particles reaching the detector. 

However, that number is affected by variables whose values are unique to this experiment: 

Number of nuclei The number of Au nuclei from which scattering may take place is defined

10.3. GEOMETRY 131 

only for this experiment. It depends on the thickness of the foil, the area of the annulus, 

and the density of Au. 

Flux of α particles The number of α particles produced by the source must be known. This 

depends on the activity of the source. 

Counting time The number is affected by the time alloted for counting. This has bearing 

also on the statistics needed. 

θ 

Scattering angle 

D S−F 

αsource 

θ R 

D S−0 

θ D 

000 111 

000 111 

000 111 

000 111 

detector 

Au foil 

l = 6.90 cm 

d = 1 − 20 cm 

Figure 10.1: The geometry of the experiment. The various distances are given in the table 

10.1. 

Apparatus Quantity Value 

Au foil Inner radius r 1 2.05 cm 

Outer radius r 2 2.50 cm 

α source diameter 0.63 cm 

detector diameter 0.8 cm 

Table 10.1: The values of the parameters defined in figure 10.1. 

In order to produce a measurable quantity (or semi-measurable) that may be compared to 

results of other experiments, and also to theory, we need to define a quantity that is more 

general, and does not depend on the quantities defined above. Defining such a measurable 

quantity also allows us to make the experiment reproducible, even for ourselves. This comes 

from the problem that when we perform the experiment the conditions under which the 

experiment is performed is unique to that time. Even if the experiment is performed with the 

same equipment at a later date, the results may be different as the thickness of the foil may 

have changed (deformities in the metal), the activity of the source may have changed (decay 

laws playing their part), or the time taken may be different.


The first thing we can do is instead to think of the probability per unit time of scattering. The 

probability is found by dividing the number of particles detected by the number given from 

the source. Taking it per unit time removes the problem involving the counting time. If we 

now normalise it to the probability per unit time per nucleus, then we obtain an observable 

which we can compare to results of other experiments. That observable is the cross section. 

The cross section may be thought of as an “area” within the nucleus as the particle sees it. If 

the particle hits that area then, on average, it will scatter through the angle θ and reach the 

detector. As the name suggests, it has the units of area. When a particle passes through a 

thin target, the probability for a scattering event is, in terms of the size of the beam, 

blocked area in beam 

Probability for scattering = 

beam area 

− dN (nA dx)σ 

= 

N A 

= nσ dx. (10.1) 

The blocked area in beam indicates the number of particles that have actually been scattered 

away from the beam direction. dN is the number of scattered particles, N is the number of 

incident particles, n is the number of nuclei per unit volume in the target, dx is the thickness 

of the target, and A is the area of the target struck by the beam. σ is the scattering cross 

section, and, from equation 10.1, it is clear that it must have the dimension of area. In fact, it 

is customary to quote the cross section in units of barn (1 barn = 10 −28 m 2 ), which comes 

straight from the saying “couldn’t hit the side of a ...” 

10.3.2 Solid angle 

The solid angle comes from considering how position is defined using spherical coordinates. 

The effective area of the detector viewed from the target is dependent on the distance to the 

detector from the target. This is illustrated in figure 10.2. The solid angle, dΩ, subtended at 

the surface from the point P may be calculated as 

dΩ = 

dS · ˆr 

r 2 , (10.2) 

where the vector dS is the (infinitesimal) area vector normal to the surface. In terms of the 

angular coordinates (θ,ϕ) of a spherical system, the solid angle is 

dΩ = sin θ dθdϕ. (10.3) 

The units of solid angle are steradians (abbreviated “sr”). There are 4π steradians in a sphere. 

P 

dΩ 

r 

dS 

Figure 10.2: Solid angle.

10.3. GEOMETRY 133 

Pre-lab Question 10.1 Write down the expression for ∆Ω, the solid angle subtended by the 

detector in the foil, in terms of the following quantities 

(a) d, the distance from the centre of the annulus to the detector; 

(b) ∆S, the area of the detector; and 

(c) r = (r 1 + r 2 )/2, the mean radius of the annulus. 

Remember that the solid angle may be defined as the dot product of two vectors. You may 

wish to draw a diagram showing the directions of those vectors. 

Solid angle comes into its own when one considers that a form of the cross section depends 

on angle. . . 

10.3.3 Differential cross section 

The cross section, as discussed above, is known as the total (or sometimes integral) cross 

section. The scattering depends on energy and, especially, angle, and when we wish to 

discuss the angular dependence the concept of differential cross section comes into being. 

This is the cross section associated with the scattering of particles through an angle θ into 

an infinitesimal solid angle, dΩ, subtended at the detector. Following the same argument as 

before, we find that the total number of particles scattered into a solid angle ∆Ω from a thin 

target ∆N sc (θ,ϕ), is given by 

∆N sc (θ,ϕ) 

N 0 

= nx dσ (θ,ϕ)∆Ω (10.4) 

dΩ 

where dσ/dΩ is the differential cross section. The relationship between the differential and 

total cross sections is 

∫ 2π ∫ π 

dσ 

σ = dϕ (θ,ϕ) sin θ dθ . (10.5) 

0 dΩ 

0 

Normally, the particle beam reaching the target is unpolarised, which means the differential 

cross section is not dependent on ϕ, hence 

∫ π 

dσ 

σ = 2π (θ) sin θ dθ . (10.6) 

0 dΩ 

That is the case in this experiment. The unit of the differential cross section is barn/steradian 

(b/sr). 

In terms of quantities that can be measured in this experiment, the differential cross section 

is 

dσ 

dΩ (θ) = ∆n s 

Nn i ∆Ω , (10.7) 

where n i and n s are the incident and scattering flux of particles per unit area, N is the number 

of Au nuclei, and ∆Ω is the solid angle subtended at the detector.


Pre-lab Question 10.2 The number of atoms, N, in the foil must be known. This can be 

found by calculating the weight of Au in the scattering annulus and using Avogadro’s number. 

As the (exact) thickness is unknown, we approximate the number by 

N = ct atoms, (10.8) 

where c is a constant and t is the foil thickness in µm. Calculate the constant and check the 

value with your demonstrator. 

10.3.4 Theoretical cross section 

One of the earliest major successes of quantum mechanics was the confirmation that the 

theoretical differential cross section derived from quantum mechanics using the Coulomb 

interaction was the same as that derived by Rutherford classically. That cross section is 

given by 

( ) 

dσ 

dΩ (θ) = Z1 Z 2 e 2 2 

1 

, (10.9) 

16πǫ 0 E K sin 4 θ 2 

where Z 1 is the charge (atomic number) of the target nucleus, Z 2 is the charge of the projectile, 

and E K is the kinetic energy of the projectile. e and ε 0 are the usual electron charge and 

permittivity of free space. 

Pre-lab Question 10.3 For your experiment 

dσ 

dΩ (θ) = 

k 

sin 4 θ 2 

. (10.10) 

Determine the constant k for your experiment to give the final value in m 2 . Check that value 

with your demonstrator. 

Pre-lab Question 10.4 Write down an expression for the scattering angle θ in terms of the 

foil-detector distance d. 


First, some words about safety... 

10.4.1 Safety precautions 

At all times, be wary of the following: 

10.4.1.1 Gold foil 

The Au foil is extremely fragile and must on no account be touched. Be sure to handle the 

foil frame only by the edges, and be very careful not to scratch or tear it.


10.4.1.2 Vacuum system 

Working with vacuum systems is a highly specialised art (and the subject of another experiment). 

While the system you will use is very simple, general rules apply: 

1. Do not touch the inside of the evacuated detector chamber. Any contamination will 

result in a loss of vacuum. 

2. Ensure that the exhaust from the pump is adequately ventilated. (The exhaust hose 

should go outside.) 

3. Finally, make sure that the whole vacuum chamber is sealed when the pump is running. 

It is inefficient and stresses the pump if a valve is left open or there is an air leak in the 

system. 

10.4.1.3 Detector 

The surface of the detector is extremely sensitive and must never be touched. Damage to 

the surface would render the detector useless and its replacement cost is several thousand 

dollars. 

The bias should be applied and taken off in slow, even steps to prevent dielectric breakdown. 

Take at least 30 seconds to raise or lower the bias. 

There should be no bias on the detector during large changes in pressure. Make sure the bias 

is OFF whenever the chamber is pumped down or let up to air. Also turn the bias off when 

moving the detector arm in case of leaks around the sliding O-ring. 

ALWAYS check the detector arm is clamped in place BEFORE the chamber is pumped 

down. 

10.4.1.4 Radiation 

There is a low risk of radiation exposure in this experiment. The decay length of α particles 

in air is around 3 cm - less than the distance from the source to the open face of its holder 

- and the source itself is almost always sealed inside the vacuum system, out of which no 

α particles can leak. Common sense in handling the source (especially during the spacer 

plate to foil changeover) should still be the norm, and don’t forget to wash your hands after 

handling radioactive sources. 

10.4.2 Relationship between counts and source to detector distance 

In the first part of the experiment, you will study the relationship between the distance from 

the source to a detector and N, the number of counts detected in a given time. Actually, this 

should look familiar to you.... 

Pre-lab Question 10.5 What relationship do you expect this to follow? Why?


10.4.2.1 Detector and electronics 

The electronics of the detection system are given in the block diagram shown in figure 10.3. 

Detector Pre−amp Amp SCA 

Bias 

Counter 

Figure 10.3: Block diagram of the electronics. SCA = Single Channel Analyser. The scaler 

and clock together form a counter, which is a single unit. 

The detector is a solid-state Si detector and is, in essence, a type of miniature ionisation 

chamber. The active volume is formed by the depletion region of a large area p-n junction. 

It is shown schematically in figure 10.4. The detector is a p-n junction where the charged 

particle enters the detector through the p-type layer. This layer is as thin as possible in order 

to minimise the energy loss through the material from ionisation. A reverse bias voltage 

applied as shown in figure 10.4 increases the volume of the depletion region. 

A charged particle entering the depletion region creates electron-hole pairs (free charges) 

which then move in the electric field induced by the positive and negative charges created 

by the application of the bias voltage in the p- and n-type layers. The mean energy loss of a 

charged particle required to produce one electron-hole pair in Si is 3.5 eV, and so 2.87 × 10 5 

such pairs are produced on average per MeV energy deposited. 

The effect of the total energy lost in the depletion layer is to produce a potential V dependent 

on the total amount of charge collected in the capacitor: V = Q/C, where Q is the charge 

collected. The capacitor recharges through a resistor after the discharge. A charge-sensitive 

preamplifier produces a signal whose amplitude is proportional to the charge collected and 

hence also to the energy of the detected charged particle. 

p−type 

n−type 

incident 

particle 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

+ 

− 

− 

− 

− 

− 

− 

− 

− 

− 

− 

EMF 

depletion 

layer 

Figure 10.4: Detector and response as a function of time.


That signal is then passed onto the (high-gain) amplifier which produces a pulse with amplitude 

0 − 10 V, still proportional ultimately to the energy of the detected particle, and 

whose shape characteristics are such that it may be analysed in subsequent hardware. In 

our case, that is the Single Channel Analyser. The Single Channel Analyser (SCA) acts as 

a filter transmitting only those pulses whose amplitudes fall within a predefined window to 

the counter. The two modes of operation are “window” and “normal”. The Lower-Level 

Discriminator (LLD) sets the lower voltage level of the pulses to accept. In normal mode, 

the Upper-Level Discriminator (ULD) sets the upper voltage level, while in window mode, 

the ULD scale is reduced by a factor of 10, and that new level then acts as the window above 

the LLD for the acceptable pulses. 

The Counter simply counts the number of signals it receives, incrementing by one whenever 

a signal reaches it. In conjunction with the SCA it forms an Analogue to Digital Converter. 

A timer connected to the counter allows for counting to be done within preset times. 

The distribution of counts in this detection system is Gaussian, and for N counts, the uncertainty 

is ± √ N. 

10.4.2.2 Foil and source 

The source of α particles is 241 Am which produces α particles of general discrete energy. 

The ones of interest to us are those with 5.48 MeV of energy. The source is secured at the 

end of the scattering chamber by a plate which exposes a limited area of the source. As 

protection from contamination, the active material is enclosed between a backing layer of 

silver and a 3 mm thick gold window layer. The window reduces the energy of the emitted 

particles to 4.66 MeV with a spread of 0.46 MeV. 

The Au foil that you will use is evaporated onto a mylar backing (mylar is essentially all C 

and H). For the purposes of scattering, we can ignore the mylar due to its much lower mass 

and number compared to the Au. 

The Au foil is extremely fragile and must ON NO ACCOUNT be touched. Great care must 

be taken in handling it. 

There are two inserts that fit between the source and the detector. One has the actual foil 

attached, with an annulus cut into an Al backing plate. The second is simply an open annulus 

which acts as a spacer, to ensure that the source is placed at the same relative distances as 

when the foil is in place. 

10.4.2.3 Vacuum system 

The scattering chamber is shown schematically in figure 10.5. The pressure in the scattering 

chamber must be reduced since the range of α particles in air at atmospheric pressure is only 

about 3 cm. The pump provided will reduce the pressure in the chamber to about 10 −2 Torr 

in 5 minutes, greatly increasing their range. 

As shown in figure 10.5, O-ring seals are used at various locations. The shaft slides through 

O-rings and care should be taken when moving the shaft under vacuum to avoid deformation, 

which would cause leaks. At all times turn the detector bias DOWN before moving the


To vacuum 

pump 

Air intake 

valve 

Protective clamp 

Detector 

To electronics 

241 

Am source 

Au foil 

annulus 

O−ring seals 

Figure 10.5: The scattering chamber. 

shaft. 

10.4.2.4 Data collection 

Place the spacer plate in the chamber and evacuate the apparatus. 

Question 10.1 Why is the experiment done under vacuum? 

Once the chamber is evacuated, SLOWLY raise the bias voltage of the detector to 30 V and 

view the signal out of the detector on the CRO, with the detector at a distance of 1 cm. Set 

the LLD and ULD on the SCA and check that the electronics are working. 

Question 10.2 To what did you set the LLD and ULD? Why? 

Measure the counts reaching the detector over a wide range of detector-plate distances, from 

1 cm to 20 cm, using a suitable counting time. 

Question 10.3 What determines a suitable count time? Check with your demonstrator before 

proceeding. 

With the count time reset to 5 minutes, record the number reaching the detector at a distance 

of 1 cm. 

10.4.2.5 Analysis 

Plot the inverse root of the count versus distance. Make sure to include the errors you have 

already calculated.


Question 10.4 What do you expect this relationship to be? 

Using this calibration graph, determine the expected number of counts when the detector is 

sitting in the plane of the foil for 5 minutes. 

We must determine the incident flux on the foil. This can be calculated after the number of 

α particles passing to the detector have been measured. The flux incident on the foil is 

n i = C 0 

A DET 

[ 

DS−O 

D S−F 

] 2 

cos θ R , (10.11) 

where C 0 is the number of counts per second at zero distance, A DET is the area of the detector 

in units of m 2 , D S−O is the distance from the source to zero distance, D SF is the source-foil 

distance, and θ R is defined in figure 10.1. 

Pre-lab Question 10.6 With the aid of diagrams, derive equation 10.11. 

From your 5 minute measurement, determine the average incident flux of α particles on the 

foil, as described in Pre-lab question 10.6. 

10.4.3 Rutherford scattering 

We now turn our attention to recreating the Rutherford experiment. The results will be able 

to distinguish between the plum pudding and nuclear models of the atom (we hope!), and, 

by assuming the correct model, we may be able to determine the thickness of the Au foil. 

Lower the bias on the detector to zero, SLOWLY, and let the chamber up to air. Replace the 

spacer plate with the plate holding the Au foil, and evacuate the chamber. Restore the bias 

to the detector for measurements. Note that due to the thin foil and low source strength, long 

count times will be required. In fact, this point is so important, we take the rare opportunity 

to repeat a question for your consideration: 

Question 10.5 What determines a suitable count time? Check with your demonstrator before 

proceeding. 

At least 4 data points must be taken (the more the better), and so it is strongly recommended 

that at least one point be taken in the first session, one point early in the morning, one around 

lunch time, and two during the second session 2 . 

Consult with your demonstrator and select a range of distances for your measurement. Use 

figure 10.6 to determine suitable distances and count times. As you take the data, consider 

the analysis and prepare the calculations using Excel. 

2 This assumes that the first session is in the afternoon, which is nearly always the case.


1.0 

Counts (arb units) 

0.8 

0.6 

0.4 

total 

detector orientation 

inverse square law 

0.2 

0.0 

0 5 10 15 20 

d (cm) 

Figure 10.6: Expected number of counts as a function of detector-foil distance. Use this 

diagram to decide on counting times. 

10.4.3.1 Analysis 

By equating the data with the theoretical (Rutherford) cross section, one may determine the 

thickness of the foil. Assuming that the measured and calculated cross sections are equal 

dσ 

dΩ∣ = dσ 

∣ 

THEORY 

dΩ 

∣ 

EXP 

k 

sin 4 (θ/2) = 1 ∆n s 

t cn i ∆Ω . (10.12) 

By plotting the ratio of the theoretical cross section to t times the experiment cross section, 

as a function of d, one may infer an average value for the thickness of the foil. 

Be sure to include a sample calculation showing how you converted from the raw data to the 

cross section. 

Question 10.6 What value have you obtained for the thickness of the foil? 

Question 10.7 Using this value, re-calculate the experimental cross section and plot as a 

function of the scattering angle. How does it compare with Rutherford’s formula?


10.4.3.2 Further considerations 

You may wish to discuss, qualitatively, the effects of the following on the results: 

• the finite width of the scattering foil, and the corresponding spread in the scattering 

angle; 

• the effect of alpha particles striking the walls of the chamber; 

• the change in the alpha particle energy as it passes through the foil; and 

• the variation in mean path length of the particles through the foil as a function of 

scattering angle. 


Quantity 

Value 

Permittivity of free space, ε 0 

8.8542 × 10 −12 F/m 

Energy of α particles emitted by 241 Am 5.48 MeV 

Mean energy and spread of 5.48 MeV α particles 

after passing through 3 µm of Au 

4.66 ± 0.23 MeV 

Atomic weight of Au 

196.97 g/mole 

Density of Au 

19.3 g.cm −3 

Atomic number of Au 79 

Atomic number of α 2 

Avogadro’s number 

6.02205 × 10 25 mole −1 

1 eV 1.60219 × 10 −19 J 

e 

1.60219 × 10 −19 C 

d 

) 

x 

a 

dx a 

a 2 +x 2 


[1] J J Sakurai. Modern Quantum Mechanics. Addison-Wesley, 1994.

Chapter 11 

β spectroscopy 

11.1 Abstract 

Of the four fundamental forces in nature, the most beguiling is the weak nuclear force. Its 

sole purpose is to mediate β decay and electron capture processes in the nucleus, and is the 

only mechanism by which the neutrino may (or may not, as is much more likely) interact. 

The purpose of this experiment is to observe β decay, which has become one of the most 

fundamental processes of nature by which we can, amongst other things, understand the 

standard model of particle physics. 


The three (main) nuclear decays were established by the 1930s. The first was designated α, 

the second β, the third γ. Rutherford, in a very elegant experiment, proved that α particles 

were 4 He nuclei. The ubiquitous high energy photon proved to be the γ radiation. The β 

particles are the electron (β − ) and positron (β + ) and only one or the other may be produced 

as a result of β decay. 

The early studies of β decay were most controversial. β decay is the process by which 

a nucleus decays leaving the daughter nucleus with the same mass but changing the proton 

and neutron numbers by 1 (Z ±1,N ∓1) after the emission of either the electron or positron. 

Figure 11.1: Early beta spectrum from the decay of Ra. The measurement is from 1927 [1]. 

143

144 CHAPTER 11. β SPECTROSCOPY 

Figure 11.1 shows an early beta spectrum (1927) as measured from radium 1 . The astounding 

observation was that this spectrum was continuous. Only two particles were known in nature 

at the time and, with the neutron around the corner, the only known processes for the decay 

were 

p → n + e + 

n → p + e − . 

The two-body kinematics dictated that the spectrum of β particles should be discrete at a 

fixed value. The observation of a continuous spectrum led to a crisis in the physics community: 

how could this be? It appeared that conservation of energy was violated. A famous 

exchange of letters between Heisenberg and Pauli discussed whether conservation of energy 

was finally observed to be violated at the nuclear level; after all, quantum physics gave us 

uncertainty which was revolutionary. Heisenberg was uncertain 2 and thought that energy 

conservation might have been violated. Pauli was, however, adamant: energy conservation, 

he said, was inviolate at all levels - there had to be a third, as yet unobserved, particle taking 

some energy away, which would then lead to a continuous β spectrum. He was first to postulate 

the neutrino. He postulated the neutrino in 1931; it wasn’t discovered until 1953 by F. 

Reines and C. L. Cowan. Reines was awarded the Nobel Prize in 1995 for their discovery. 

As a result, the corrected underlying processes of nuclear β decay are 3 

For nuclear β ± decay itself the processes are 

p → n + e + + ν 

n → p + e − + ¯ν . 

A 

ZM N → A Z∓1M ′ N±1 + e ± + 

{ ν¯ν . 

(In this equation, M is the decaying nucleus of mass A with Z protons and N neutrons.) 

The decay can only occur if the conditions are energetically favourable, i.e., that the binding 

energy of M is higher than that of the daughter nucleus M ′ . While the mass number is 

conserved, there is a very slight difference in the rest masses of the parent and daughter 

nuclei from the change of a proton to a neutron or vice-versa. 

11.3 Background concepts 

This brief section is based on Fermi’s theory of β decay. However, all descriptions (Fermi, 

Gamow-Teller, field-theoretic, electroweak, etc.) of β decay are beyond the scope of second 

year level and we only present the concepts involved and which variables are defined to 

facilitate measurement. 

The Fermi model assumes that the decay occurs according to the model presented in figure 

11.2. The interaction effecting the decay is in this case considered a contact interaction, 

1 The neutron was discovered by Chadwick in 1932. 

2 Sorry. 

3 where ν is a neutrino, which is chargeless. It is important to note that until very recently the neutrino was

11.3. BACKGROUND CONCEPTS 145 

n (p) 

p (n) 

000 111 

000 111 

000 111 

e + (e − ) 

ν (ν) 

Figure 11.2: Diagrammatic representation of the Fermi model of β ± decay. The brackets 

denote the process of neutron decay. 

i.e. where the interaction between the particles participating in the reaction all interact at a 

point 4 . 

By considering this picture, one can define a new variable of measurement, which is derived 

from a knowledge of the probability of the process, and the kinematics involved. That 

variable is called the Kurie variable and is defined by 

K(Z,p) = 

√ 

w(p) 

p 2 F(Z,p) = 

8π 

h 7/2 c 3/2 |M if| (E 0 − E), (11.1) 

where 

2πη Ze2 

F(Z,p) = 

1 − e−2πη,η = 

4πǫ 0 v , (11.2) 

v is the velocity of the electron, and w(p) and |M if | are measures of the probability of the 

decay for a particular β momentum p and F(Z,p) is known as the Fermi function. E is the β 

energy. One then expects that a plot of K against E to be a straight line with negative slope. 

As w(p) is related to the number of counts at each β energy and momentum, one can replace 

the probability by the number of counts for a given time and define an experimental Kurie 

variable by 

K(Z,p) = 

√ 

N(p) 

p 2 F(Z,p) . (11.3) 

Pre-lab Question 11.1 What is the value of K when the β energy is equal to E 0 ? 

Pre-lab Question 11.2 What is then the probability of a β particle being emitted with an 

energy E 0 ? 

assumed to have zero mass. The neutrino mass was finally discovered (∼ O(0.1eV)) in measurements taken 

with the very large neutrino detectors at Kamiokande (Japan) and Sudbury (Canada). For all practical purposes, 

a zero mass neutrino is still a good approximation. 

4 the more modern description involving the full weak theory of β decay involves the exchange of the W ± 

and Z particles.


Evacuated chamber 

r 

e 

− 


147 

Pm source 

Geiger tube 

Figure 11.3: Magnetic spectrometer showing the configuration with the 147 Pm source and 

detector. The radius of the path the β traverses is r = 3.80 ± 0.01 cm. 

Pre-lab Question 11.3 How does the intercept with the energy axis change with neutrino 

mass? 

11.4 Experiment - the β decay of promethium 

You will be studying the β decay of 147 

61 Pm which decays into 147 

62 Sm. The process is 

147 Pm → 147 Sm + e − + ¯ν , 

so, for this decay the β particles are electrons. The decay from promethium may lead to 

several states in samarium, with the total number of counts from the decay being distributed 

among the various states that are accessible. This will be reflected in the energy spectra 

collected. 

Pre-lab Question 11.4 What do you think might influence the count rate going into each 

state? Discuss with your demonstrator. 

11.4.1 Calibrating the magnet 

To measure the energy of the electrons in the experiment we use a magnetic spectrometer 

connected to a Geiger-Müller tube. The spectrometer is shown in figure 11.3. A particle of 

mass m, charge q, and velocity v, moving in a uniform magnetic field B, experiences a force 

F where 

F = qv × B. (11.4)

11.4. EXPERIMENT - THE β DECAY OF PROMETHIUM 147 

If B is perpendicular to v then the particle will move in a circle of radius r where 

mv 2 

r 

= qvB (11.5) 

with momentum p = qrB. One can therefore select the momentum of the charged particles 

by adjusting the magnetic field given that the path taken from source to detector is fixed. 

In order to measure the magnetic field, we use a duplicate spectrometer which has identical 

dimensions to the actual spectrometer but without the source or detector present. Mount the 

duplicate spectrometer between the poles of the electromagnet. Initialise the Hall probe: 

• Place the tip of the probe in the zero field case and switch on the meter. 

• When the number 1973 appears on the display, press the ENTER/RESET button. 

• At this point three dashes should appear on the display. 

• Press the PB button. 

• Press the orange UP/DOWN keys until KGAUSS is selected. 

• The Hall probe is now ready for use. 

• NB! Do NOT switch off the meter from this point until ALL of the data have been 

taken. If the meter is switched off it will have to be re-initialised. 

Once the Hall probe is ready for use, insert it into the duplicate chamber without switching 

off the meter. Remove any remnant field by degaussing the chamber. If the magnitude of the 

field is still above 10 G, degauss again. If the magnetic field persists, see your demonstrator. 

Question 11.1 Explain how the degausser works. 

Connect the magnet to the stabilised DC supply, making sure the current is flowing into the 

red plug of the magnet and out of the black. Use the multimeter to measure the current. 

Before switching on the supply check that: 

• The current is turned to a maximum. 

• The voltage is at a minimum. 

• The multimeter is on the 0 − 2 A DC scale. 

Measure the magnetic field as a function of current from 0 to 500 mA in 20 mA steps. Note 

that you should: 

• Never reduce the current: backtracking at any stage will render the calibration useless. 

• Never change scales on the hall probe or multimeter. 

• Allow a few seconds for the field to stabilise before taking a measurement.


~ 1 kV 

cathode 

β 

anode 

electron cascade 

Figure 11.4: Operation of a Geiger-Müller tube. 

Question 11.2 Why would backtracking render the calibration useless? 

You may notice a small black button on the side of the Hall probe. When the magnetic field 

is entering the Hall probe from this side, the gaussmeter will give a positive reading. 

Question 11.3 What is the direction of the magnetic field for this direction of current? 

After you have completed the measurement, slowly reduce the current to zero and remove 

any remnant field. Find the calibration of the Hall probe and check the slope and intercept 

of the calibration with your demonstrator. 

Question 11.4 What is the maximum kinetic energy you will measure? 

Question 11.5 To what magnetic field, and thus current, does this energy correspond? 

11.4.2 Measuring the spectrum 

Now that the Hall probe has been calibrated, you are now able to measure the magnetic field 

inside the spectrometer by measuring the electromagnet’s current. By measuring the number 

of counts as a function of current one finds the β spectrum. 

11.4.2.1 The Geiger-Müller Tube 

A schematic of the detector used in this experiment is given in figure 11.4. The tube consists 

of an easily ionisable gas contained within a thin-walled chamber. A high voltage is applied

11.4. EXPERIMENT - THE β DECAY OF PROMETHIUM 149 

between the central anode and the walls of the chamber creating a very high electric field 

within the chamber. After entering the detector, a β − ionises the gas. Each collision liberates 

a secondary electron which is accelerated towards the anode, together with the original β − . 

The high electric field provides extra energy to the electrons as they travel towards the anode. 

The mean free path of electron in the gas is very small, so the electrons undergo many 

collisions before reaching the anode. So the original β − initiates an electron cascade which 

results in a large charge at the anode. This charge is collected by the detector electronics. 

As the electrons move towards the anode, the positively charged ions move towards the 

cathode but much more slowly as they are much, much heavier. Hence if another β − enters 

the detector before a significant number of positive ions have been discharged at the cathode 

no cascade is possible as the ions will recombine with the secondary electrons induced by 

the second β − . So the tube is rendered useless until the event has been cleared from the 

chamber. This is known as dead time, denoted by τ. It comes about as a combination of the 

detector and electronics, which require a finite time to collect the charge from the detector 

coming from each detected particle. 

Pre-lab Question 11.5 Show that if the tube has a dead time τ, and N counts are observed 

in a time t, the true number of events, N ′ , is given by 

N ′ = 

N 

1 − τ t N . (11.6) 

The dead time for the detector you will be using has been measured, at a count rate of 

∼ 1000 counts/sec, to be 20.0 ± 0.5 µs. To a good approximation, the dead time may be 

considered constant for the spectra you will measure. 

11.4.3 The spectrometer 

Ensure that you have disconnected the power supply and degaussed the magnet. Carefully 

remove the Hall probe, switch it off, and replace it in its case. Replace the duplicate chamber 

with the actual spectrometer containing the source and detector. 

NB!! All the data for this part of the experiment must be taken at once. It cannot be split 

over two days. It is recommended that you take the data on the first day and do the analysis 

on the second day, leaving time on the second day to take more data if the need so arises. 

Question 11.6 Why must all the data be taken at once? 

Connect the DC supply to the magnet. 

Question 11.7 In which direction are the β particles moving as a result of the magnetic 

field? 

Ensure that the vacuum pump is connected to the chamber and switch on the pump and 

gauge. After a few minutes the pressure in the chamber should be below 10 −2 Torr. Connect 

up the circuit shown in figure 11.5. Set the electronics to the following settings and then 

switch on the NIM bin:


DET. 

AMP 

SCA 

COUNT 

H.V. 

CRO 

TIMER 

Figure 11.5: Electronics for the detection of β particles. 

Amplifier Course gain = 16; fine gain = 6; input = positive; output = unipolar. 

SCA ULD = 10 V; LLD = 0.20 − 0.30 V. 

Counter Discriminator = 3.0 V. 

Timer 100 s. 

Start measuring the number of counts per 100 s in the current range 0 to 400 mA, taking 

suitable steps across the range. Remember NEVER to backtrack, always increasing the 

current, and allowing for a few seconds before counting each time you change the current to 

allow the field to stabilise. 

As the data are taken, observe the output pulses from the amplifier on the CRO. 

Question 11.8 Draw, on scaled axes, the shape of the output. At which current do they 

appear smallest? 

Check, for a current ∼ 100 mA, that the output does not exceed the upper level on the SCA. 

Discuss the shape of the pulses in terms of the operation of the detector. 

Question 11.9 Are the pulses consistent with the quoted dead time? 

After the first set of measurements, increase the current to 500 mA and take one more datum. 

This will serve as the background measurement, which must be subtracted from each of your 

previous measurements. 

Question 11.10 Why is this a suitable current at which to take such a background reading? 

11.4.4 Analysis 

You should now have a table of the measurements along with the appropriate uncertainties. 

The analysis of these data requires a series of calculations and great care must be taken to 

ensure the transformations are done correctly. The steps you make should be made clear; be 

sure to comment on all aspects of the analysis. 

Use Excel to correct the number of counts for background and dead time, and calculate the 

electron kinetic energies from the deduced momenta. Plot the corrected counts against the


kinetic energy (in keV). Explain the shape of the spectrum, discussing its features in terms 

of both the physics and the apparatus used. 

From this spectrum, and again using Excel, transform the spectrum into a Kurie plot. Fit the 

data to a straight line in the region of interest and obtain a value for E 0 . 

Question 11.11 How does your deduced value for E 0 compare with the quoted value for the 

decay of 147 Pm? 

Question 11.12 Is the experiment sensitive enough to allow for a determination of the neutrino 

mass? (??) Discuss, if you are game enough. . . 


Quantity 

Accepted value for E 0 , for the decay of 147 Pm 

h 

c 

ε 0 

Value 

224.5 ± 0.4 keV 

6.6256 × 10 −34 Js 

2.997925 × 10 8 ms −1 

8.85416 × 10 −12 Fm −1 

1.6022 × 10 −19 C 

e 

1 eV 1.6022 × 10 −19 C 

electron rest mass E e 

511 keV c −2 


[1] K Heyde. Basic Ideas and Concepts in Nuclear Physics. Institute of Physics Publishing, 

2 nd edition, 1999.

Chapter 12 

γ ray detection and spectroscopy 

12.1 Abstract 

How does one detect radiation? This series of experiments is designed to give you some 

familiarity with detectors, in this case, a gamma-ray detector. Along the way, you may learn 

a little bit about how a nucleus decays. . . 


A piece of “something” is placed in front of you. Placing something else next to it elicits a 

response in that something else. At once you may infer that something had been transferred 

between the first something and the other something, giving that measurable response in the 

other. Depending on our understanding of the other we may gain an understanding of what 

was transferred. 

This is, in essence, what happens in a detector when a radioactive source is placed next to it. 

Our ability to detector whichever radiation being produced depends on the response of the 

detector to that radiation, and our understanding of the processes involved in that response. 

No experiment may be done without this knowledge. 

For this experiment, we will examine the response of a typical γ ray detector to incident γ 

rays. In achieving this goal, we need to understand how photons interact with matter, and 

what happens when it does. The techniques and principals involved are adaptable to all cases 

of particle detection. 

12.3 γ radiation. How? 

We need to understand how a γ source works, in order to understand the spectrum of photons 

produced by it. First, and most importantly, γ radiation is the product of another nuclear 

decay and/or reaction - ALWAYS! A γ is produced as a result of a transition between an 

excited state of a nucleus to a lower energy state in that nucleus, proceeding down to the 

153

154 CHAPTER 12. γ RAY DETECTION AND SPECTROSCOPY 

706.41 

57 

Co 

0.18% 

EC 

366.74 

57 

Fe 

136.47 

14.41 

99.82% 

Figure 12.1: Decay from 57 Co by electron capture to 57 Fe showing the three possible γ rays 

from 57 Fe produced: 136.47, 122.06, and 14.41 keV. All energies in the figure are in keV. 

ground state. It is impossible to produce a lump of matter in which all, or indeed any, of the 

nuclei are in an excited state. 

To produce a γ ray within a source, one starts off with a radioactive source which would 

decay in one of the other modes (α decay, β decay or electron capture) leaving the daughter 

nucleus in an excited state which then decays to its ground state by γ emission. A radioactive 

source is defined as a source of nuclei which may decay by those modes with a suitably long 

half-life. That half-life must be long enough (at least a few hours) in order to produce a 

mass of it which can be stored and used. Figure 12.1 shows the production of a 14.41 keV γ 

from 57 Fe as a result of the decay of 57 Co. The half-life of 57 Co is 271 days so this is a very 

suitable candidate of very low energy γ rays. The other sources in this experiment operate 

by similar decays and transitions. 

12.4 Calibration of the detector 

You will be using NaI(Tl) scintillators as γ detectors in this experiment. In order to interpret 

the results accurately, we must understand how photons interact with matter. This is not so 

easy: the photon is not charged and so it does not interact with matter by ionising the atoms 

in the material (as is the case with charged particles). The thallium (Tl, Z = 81) dopant is 

critical in the detection process. 

The circuit for this part of the experiment is shown in figure 12.2. 

12.4.1 Detecting photons - the NaI(Tl) crystal 

The photon interacts with matter in three principle ways, depending on its energy.

12.4. CALIBRATION OF THE DETECTOR 155 

NaI 

Pre− 

amp Amp MCA 

CRO 

Figure 12.2: Circuit for detecting γ rays and calibrating the detection system. 

hν’ 

hν 

θ 

Figure 12.3: Kinematics of Compton scattering. 

φ 

T 

12.4.1.1 Photoelectric effect 

In this process 1 , a photon is absorbed by an atomic electron and imparts all of its energy 

to it. The electron is ejected from the atom with most of its kinetic energy coming from 

the photon. (Some of it is absorbed in the recoil of the nucleus, but for the most part this 

is negligible given the mass of the nucleus.) The kinetic energy of the ejected electron is 

thus E = E γ − E B , where E B is the binding energy of that electron to the atom. A little 

energy also comes from the X-ray that is produced from the transition when the hole left by 

the (inner shell) electron is filled by an outer shell electron. The probability of this process 

∼ Z 4 , so a high-Z material is most useful in the detection. Hence the Tl dopant. 

The primary photoelectron then ionises the surrounding material producing light in the visible 

spectrum from the subsequent transitions. 

12.4.1.2 Compton scattering 

In this process, the photon undergoes scattering on free electrons. (In the detector material, 

the process is on the outermost electrons which are very loosely bound and as such may be 

considered “free” to a good approximation.) The process is illustrated in figure 12.3. 

The energy of the scattered electron depends on the angle of scattering and is given by 

E e = E γ 

α(1 − cos θ) 

1 + α(1 − cos θ) , (12.1) 

1 for which Einstein won the Nobel Prize coming from his three masterworks of 1905 (the other two were 

Brownian motion and Special Relativity - for which he did NOT win a prize!!)


Phosphor screen 

Dynodes 

1100 

01 

01 

00 11 01 

00 11 01 

1100 

01 

01 

00 11 01 

00 11 01 

To base 

and electronics 

NaI(Tl) 

crystal 

Cathode 

Light guide 

Anode 

Figure 12.4: Configuration of NaI(Tl) detector with photomultiplier tube. The dynodes of 

the photomultiplier are shown. 

where E γ is the photon energy, E e is the scattered electron energy, and α = E γ /(m e c 2 ); 

m e = 511 keV/c 2 is the rest mass energy of the electron. 

The scattered electron then ionises the surrounding material and also liberates visible-light 

photons. The scattered γ undergoes more and more scattering events, scattering more electrons 

which produce more light, until the original photon expends all its energy. 

12.4.1.3 Pair production 

If the incident photon has an energy of more than twice the rest mass energy of an electron, 

the third process is possible. Pair production takes place in a region of a high Coulomb 

field, where the photon is converted into an electron-positron pair. The Tl nucleus provides 

that strong Coulomb field. The electron then ionises the surrounding material as in the other 

cases while the positron collides with another electron causing annihilation into two 511 keV 

photons, which then also may be detected. 

The upshot of all three processes is that an electron is produced which ionises the surrounding 

material producing visible light [E ∼ O(1 eV)] photons. The NaI crystal has a refractive 

index of ∼ 2 so the probability of total internal reflection from the mirrored sides of the 

detector is quite high. The far side of the detector has a glass window which is optically 

coupled to a photomultiplier tube. The configuration is shown in figure 12.4. 

The light produced in the crystal is guided to the phosphor screen at the entrance to the 

photomultiplier tube. Each photon liberates an electron via the photoelectric effect. That 

electron then hits the cathode, which is the first dynode in the chain. Each dynode is under 

high voltage (V ∼ 1 kV, typically) and so each collision has the effect of creating an electron 

shower. That shower of electrons proceeds to the next dynode, producing another shower per 

electron. There are typically, 10 to 14 dynode stages in a photomultiplier tube and, overall, 

this gives a cascade of electrons which provides a gain of ∼ 10 7 electrons at the anode per 

electron produced at the phosphor. The charge at the anode is then collected and passed 

to the electronics through the base. As the number of visible-light photons entering the

12.4. CALIBRATION OF THE DETECTOR 157 

photomultiplier tube is proportional to the energy of the original γ being detected, the charge 

collected at the end of the tube is also proportional to the energy of that γ. 

When detecting the photons as a function of energy, one obtains a spectrum that is quite 

structured. As all the energy is deposited by a photoelectric event in the detector, one obtains 

a peak (called the “photopeak”) at the energy of the incident photon. Likewise, one obtains 

a peak at the energy of the incident photon after pair production. In that case, however, 

two more peaks are observed: a) when one of the annihilation 511 keV photons escapes 

the detector, at 511 keV below the full energy peak; and b) when both annihilation photons 

escape the detector, at 1022 keV below the full energy peak. The energy of the electron 

after a Compton scattering is continuous from zero energy up to a maximum, and thus it is 

reflected in the photon spectrum as a continuous distribution up to an edge (the Compton 

edge) which lies just below the photopeak. 

Pre-lab Question 12.1 Calculate the maximum energy given to the scattered electron as 

a result of Compton scattering of a photon energy E γ . To what scattering angle does this 

energy correspond? Calculate the maximum energy of the electron for the case of an incident 

662 keV photon. 

Pre-lab Question 12.2 Calculate the energy of a Compton scattered photon for an incident 

photon of energy E γ when the scattering angle is 180 ◦ . Is it possible only for the scattered 

photon to be detected? 

Pre-lab Question 12.3 Show that as E γ → ∞, the energy of photons scattered through 

180 ◦ approaches 255 keV, and is independent of E γ . 

12.4.2 The Multi-Channel Analyser 

The centre-piece of the electronics is the Multi-Channel Analyser (MCA). It may sometimes 

be a stand-alone piece of equipment but is more likely to be a card in a PC, as it is here. 

When the signal is produced as a result of the detection of a photon, it is amplified by the 

pre-amplifier/amplifier system to produce a positive (unipolar) or positive/negative (bipolar) 

pulse whose amplitude is proportional to the energy of the photon. The MCA accepts this 

pulse and performs a comparison of the voltage of the signal with an internal voltage stepping 

up from zero volts until agreement. The number of steps defines the channel and the counter 

in that channel is incremented by 1. The results of such counting are displayed on the screen 

for analysis. It is possible then to take an entire spectrum at once with such a system. 

All MCAs accept pulses to 10 V full-scale, and divide the scale into the channels defined 

by the Analogue-to-Digital-Converter (ADC). These are usually in 2 k steps; for our system 

there are 1024 channels, defining 10 mV channels. 

12.4.3 The reference sources 

To calibrate the detector, we need sources for which we know the energies of the γ rays 

produced. Those sources are listed in table 12.1.


Table 12.1: Radioactive sources used for the calibration, and the energies of the photons 

produced. 

Source E γ (MeV) 

137 Cs 0.032 

0.662 

22 Na 0.511 

1.275 

60 Co 1.173 

1.332 

12.4.4 Gain of the detection system and MCA 

Starting with the detector, identify each piece of equipment and follow the wiring as per 

figure 12.2. The electronics, including the high voltage (HV) power supply, are modules 

contained in the NIM bin. Ensure that the wiring from the detector to the MCA is complete, 

and that the HV is connected to the detector. With the HV set to zero switch on the bin, the 

CRO and the computer (MCA). Slowly increase the voltage on the HV to 800 V. 

Place the 137 Cs source in front of the detector. Observe the signal produced by the preamplifier 

on the CRO to ensure that a signal is produced. Reconnect the amplifier to the preamplifier 

and observe the signals output from the amplifier. 

Question 12.1 Describe the pulses emerging from the amplifier and note any differences, 

giving possible reasons. 

Recall that the MCA accepts voltages between 0 and 10 V. The gain on the amplifier should 

be set such that the voltages of the pulses produced fall within this range. Note that the 

photon energies you may encounter in this experiment are no more than 2 MeV. 

Question 12.2 The highest energy photon emitted by the 137 Cs source is 0.662 MeV. Assuming 

a linear calibration of the MCA (1024 channels), at approximately which channel will 

the photopeak appear if we wish to measure energies of ∼ 2 MeV? 

Adjust the amplifier gain such that you affect this situation. Note the amplifier settings. 

The overall gain of the detection system is also affected by the HV applied to the photomultiplier 

tube. 

Pre-lab Question 12.4 Why does the high voltage affect the gain of the system? 

Investigate the effect of the HV supply on the gain by making very gradual changes to the 

applied voltage and observe the effect on the output voltages on the CRO. 

Question 12.3 What is the effect of small drifts in the applied HV on the overall spectrum? 

Question 12.4 What action may you take to minimise the effect of drift?

12.5. ENERGY RESOLUTION 159 

There is one more quantity to consider. When one particle is detected it triggers off the 

electronics in sequence to the MCA to record the event. As this happens the electronics are 

working to process the event. Another particle entering the detector during this time will not 

be detected - the detector is dead for this time. Dead time is recorded at the bottom of the 

MCA screen. Generally, this will not be a major problem but you should keep a record of it. 

Dead time is discussed in detail in the β spectroscopy experiment. 

12.4.5 Measuring spectra 

The commands for operating the MCA software are given in appendix 12.7. Reset the HV 

to 800 V and allow a few minutes for the power supply to stabilise. Collect a spectrum for 

around 10 min using the 137 Cs source. The collection time is nothing special, but it leads to 

a very important concept and a visiting question from the α scattering experiment: 

Question 12.5 What determines a suitable count time? 

Save the spectrum and print one copy for each of the partners in the group. Record the 

channel numbers of all of the features in the spectrum. Collect spectra also for the 22 Na and 

60 Co sources. 

12.4.6 Calibration 

Once you have identified the features in each spectrum, and especially the positions of the 

peaks and the errors therein, you may calibrate the detector. Using the known energies of 

the photons in the three sources given in table 12.1 plot a graph of energy against channel 

number. 

Question 12.6 What is the relationship between energy and channel number? Use this to fit 

an appropriate function to the data and hence find the calibration of the detector. 

Use the calibration to determine the energies of the other features in the spectra: Compton 

edges and pair production peaks. Compare the measured energies with your expectations. 

12.5 Energy resolution 

The accuracy of the measurement of the spectrum is dictated by the energy resolution of 

the system. That accuracy determines the amount of detail that may be identifiable. The 

resolution is defined as the width of the peaks in the spectrum, and is embodied in the full 

width half maximum (FWHM) - the width as measured at half the number of counts in the 

peak. The resolution is then 

Resolution = FWHM 

E γ 

. (12.2)


Measure the FWHM of the photopeaks in your spectra and calculate the resolution for each 

of the six peaks, including errors as you go. Plot a function of resolution against E γ and 

comment. What function of the detection system causes the resolution observed? Think 

about it, and remember how the detector works to observe a photon. 

12.6 Unknown sources 

Now that the system is calibrated, and potential sources of error identified, you are in a 

position to use it to identify unknown sources. Collect the spectra of the unknown sources 

given to you and measure the energies of any features you identify in the spectra. Compare 

those energies to the ones listed in table 12.2. What are the sources? 

Note that you should use those features of the spectra which are readily identifiable. 

12.7 MCA commands 

Alt-F1 Start data acquisition. 

Alt-F2 Stop data acquisition. 

Alt-F5 Move data from the MCA card into the computer’s memory buffer. 

Alt-F, Alt-S Save the data stored in the memory buffer into a file. Only a spectrum stored in 

the memory buffer can be saved. 

Arrow keys Left and right keys move the cursor. Channel number of the cursor’s position 

is indicated at the bottom of the screen. Page-up and Page-dn are fast movements. 

Up arrow changes the vertical scale from logarithmic to linear; the down arrow key is 

the reverse. 

F3 Changes to expanded view. The expansion region is changed by the +/- key on the 

keypad. 


[1] G F Knoll. Radiation Detection and Measurement. Wiley, 3 rd edition, 2000. 

[2] W R Leo. Techniques for Nuclear and Particle Physics Experiments. Springer-Verlag, 

1987.


Table 12.2: Table of various γ sources with the energies in keV. 

Energy Source Energy Source Energy Source 

32 137 Ba 303 133 Ba 779 152 Eu 

35 125 Sb 344 152 Eu 812 56 Ni 

53 133 Ba 353 226 Ra 894 142 Ba 

53 226 Ra 356 133 Ba 898 88 Y 

77 142 Ba 364 142 Ba 949 142 Ba 

80 131 I 364 131 I 964 152 Eu 

81 133 Ba 381 381 Sb 1001 142 Ba 

97 75 Se 401 75 Se 1064 207 Bi 

121 75 Se 425 142 Ba 1078 142 Ba 

122 152 Eu 428 125 Sb 1086 152 Eu 

136 75 Se 463 125 Sb 1120 226 Ra 

176 125 Sb 511 e + − e − 1173 60 Co 

186 226 Ra 570 207 Bi 1204 142 Ba 

232 142 Ba 600 142 Ba 1238 226 Ra 

242 226 Ra 601 125 Sb 1277 22 Na 

245 152 Eu 607 125 Sb 1333 60 Co 

225 142 Ba 609 226 Ra 1378 226 Ra 

265 75 Se 636 125 Sb 1408 1408 Eu 

270 56 Ni 637 131 I 1562 56 Ni 

273 133 Ba 662 137 Cs 1764 226 Ra 

280 75 Se 723 131 I 1770 207 Bi 

284 131 I 750 56 Ni 1836 88 Y 

295 226 Ra 769 226 Ra

Chapter 13 

Energy loss of α particles through matter 

13.1 Abstract 

Nature has been both kind and unkind to us for giving us charged particles with which to 

perform experiments in our endeavour to understand it. Charged particles interact with matter 

through the Coulomb force with the greatest ease, making detection easy, but which also 

means that such particles interact in air. In order to perform experiments with charged particles 

in a controlled environment one needs to create vacuums. 

The experiment is thus twofold: a) to understand how charged particles, in our case α particles, 

interact with matter; and b) how to create high vacuums. 


Much of what we know about the atomic and subatomic universes comes from studying 

the interactions of charged particles with the subject (call it “target”) under investigation. 

The problem is, that needs to be done in a controlled way such that the charged particles 

are interacting ONLY with that target. The very thing that allows us to easily study the 

results of the interactions and therefore the structure of the targets, the Coulomb force, is 

also responsible for making it incessantly difficult to perform experiments in air. It is not 

possible to control the flux under those conditions as: 

• the charged particles lose energy through inelastic collisions with atomic electrons; 

and, 

• the direction of the flux changes through elastic scattering with the surrounding nuclei. 

Other reactions can occur, of course, (Cherenkov radiations, nuclear reactions, bremsstrahlung 1 ). 

For our purposes, the most probable interaction effecting loss of energy is the interaction between 

the charged particles and the atomic electrons, causing either ionisation or atomic 

excitation. 

1 braking radiation 

163

164 CHAPTER 13. ENERGY LOSS OF α PARTICLES THROUGH MATTER 

Energy loss itself is embodied in the quantity dE/dx, the energy loss per unit distance (It 

is, by definition, negative.) Two calculations of this quantity exist: that of Bohr, derived 

classically, and that of Bethe and Bloch, which is derived from quantum mechanics and is 

the correct formulation. The classical formulation is sufficient for low energy α particles. It 

is: 

{ } 

dE 

dx = e 4 γ 2 

−4πz2 m e v N mv 3 

2 e ln 

ze 2¯ν , (13.1) 

where 

ze is the charge of the α particle; 

m e is the mass of the electron; 

m is the mass of the α; 

γ is the relativistic boost; 

v is the velocity of the α; and, 

ν is the average orbital frequency of the bound electron in the atom. 

Taking E α = mv 2 /2, we can subsume most of the constants (including the introduced mass 

of the α) into a single beast and obtain 

dE 

dx = −c 1 

E ln [c 2E]. (13.2) 

If the material is sufficiently thin that the energy loss is small compared to the total energy, 

the log factor is almost constant over the foil thickness and we can rewrite the equation as 

∆E 

T ≈ 

k 

E m , (13.3) 

where ∆E is the energy lost over thickness of foil T ; E is the energy of the incident α 

particle; k and m are constants of the foil. 

Taking the log of both sides, one obtains 

log ∆E = log k + log T − m log E . (13.4) 

Thus, if we plot a curve of ∆E as a function of E on a log-log scale, we should get a straight 

line. 

13.3 Measuring foil thickness 

With a knowledge of energy loss of charged particles through a known thickness of foil, one 

may use equation 13.4 to determine the actual function, and from that determine thicknesses 

of other foils of the same material. 

To do this, we need to measure the energy loss as a function of energy through foils of known 

thickness. The apparatus for such a measurement is shown in figure 13.1. The geometry of

13.3. MEASURING FOIL THICKNESS 165 

O−ring seal 

Air intake 

valve 

Foil holder 

226 

Ra 

α 

Solid state 

detector 


Foil 

To vacuum pump 

Figure 13.1: Detection system for energy loss of α particles in foil.


SSD 

Pre− 

amp Amp MCA 

Bias 

CRO 

Figure 13.2: Circuit for analysing the detector signals. 

the system is such that all α particles detected must have passed through a thickness of foil. 

The detector itself is the same as that used in the Rutherford Scattering experiment. As 

with the Rutherford scattering, the experiment must be performed under high vacuum. The 

source is a sample of 226 Ra which produces α and β particles on its way to decay to 206 Pb, 

the spectrum of which is shown in Table 13.1. 

Table 13.1: Spectrum of α and β particles produced by the 226 Ra source. Energies are in 

keV. 

Particle Energy 

α 7687.1 

6002.6 

5489.7 

5305 

4748.5 

β 3280 

1161 

690 

170 

The signal from the detector is passed to the circuit shown in figure 13.2. 

13.3.1 Calibration of the system 

Identify all the components in figures 13.1 and 13.2 and set it up. Make sure that the bias 

voltage is off when handling the detector in air, and ensure that for this part of the experiment 

there is no foil in the foil holder. Assemble the detecting chamber and, with the air inlet 

valve closed, pump down the chamber until the pressure is below at least 0.1 Torr. Once it 

has dropped below this pressure, slowly turn up the bias voltage to 70 V. NB! Do NOT adjust 

the bias voltage under pressures GREATER than 0.1 Torr. 

The MCA system is as with the other experiments, measuring voltages in the 0−10 V range, 

and is based on a 1024-channel ADC. The MCA commands are given in appendix 13.6. 

Given that we are measuring α particles with energies up to 7.8 MeV, adjust the gain of 

the amplifier such that the highest energy α produces a pulse under the maximum voltage. 

Record your setting. 

Collect a spectrum with no foil in place. Ensure a long enough time for collection.

13.4. MAKING A THIN FOIL 167 

Question 13.1 What determines a suitable count time? 

Determine which peaks are due to detection of α particles and which are due to the β’s. 

Record the channel numbers of only the α peaks and use those to calibrate the system. Be 

sure to take note of the dead time (see β and γ experimental notes) and ensure that it is 

minimised. 

13.3.2 Measurement of known foils 

Take various energy spectra of α particles as they pass through known thicknesses of foils, 

using the foils supplied. Track any changes to the spectrum with foil thickness using a couple 

of representative peaks. There are two things for you to consider: 

1. Any changes in the positions of peaks; 

2. Any changes to the full-width-half-maximum (FWHM) of said peaks. 

Plot a curve of ∆E against E for a few points on a log-log scale and determine the energyloss 

function (equation 13.4) for your foils. 

Question 13.2 Discuss possible reasons why there may be changes to the FWHM with foil 

thickness. Can a point be reached where energy resolution becomes too poor for the recognition 

of α peaks? 

13.4 Making a thin foil 

As all experiments involving charged particles involve the use of vacuum systems, this part 

of the experiment requires the use of ultra-high vacuum pumps. The application is to make 

your own thin foil, of the same material used in the previous section, and use the results of 

α energy loss to measure the thickness of the foil you make. This takes all of the second day 

- make sure you hit the ground running! 

13.4.1 Background reading 

The book by Roth, Vacuum Technology [2], is available from the part 2 office and all answers 

to the questions may be found there. Chapters 5, 6, and section 7.1, 7.2, address the 

techniques studied in this part of the experiment and you should take the time to read/skim, 

and familiarise yourself with, those sections. NB! Do NOT transcribe large chunks of the 

book into the prac: the knowledge gained should form part of the report as is relevant to the 

experiment being done!


Bell jar 

Baffle 

valve 

Roughing 

valve 

Air inlet 

Backing 

valve 

Diffusion 

pump 

To outside 

Rotary 

pump 

Figure 13.3: Diagram of the evaporation system. 

13.4.2 Foil manufacture 

The aluminium foil you will make will utilise the techniques of evaporation. The source 

material (Al) is placed on a metallic boat made of heat resistant material, such as titanium. 

That boat is heated by passing an electric current through it. The material sits within a 

depression in the boat. The depression is thinner than the surrounding material, creating a 

region of higher resistance relative to the rest of the circuit and so heats up the fastest. The 

source material is heated to boiling point at which time the resultant vapour expands in a 

hemisphere above the boat condensing on whatever surface it first strikes. The procedure is 

done under high vacuum to ensure that no oxidation of the material takes place. 

The evaporation system is shown in figure 13.3. The process takes place inside a glass 

bell jar which is evacuated by a system of a rotary pump and diffusion pump. Next to the 

bell jar is a high-current power supply consisting of variable (240 V, 10 A) to (3 V, 80 A) 

step down transformer connected to the boat clamps via 150 A cables. NB! The bell jar is 

very expensive and delicate. Only the demonstrator should handle it, and the perspex shield 

should always be placed around the jar when it is evacuated to guard against shrapnel in the 

unlikely event of an implosion. 

Take note of the rules for operating a vacuum system as given in appendix 13.5. Identify all

13.4. MAKING A THIN FOIL 169 

the components of the vacuum system and have the demonstrator remove the bell jar if not 

already done. The titanium boat will have been handled prior to use so it is necessary for 

you to clean it before placing it in situ. Clean the boat using isopropyl alcohol and lint-free 

tissues. (Remember those latex gloves!) After cleaning, carefully place the boat in its holder 

in the evaporation chamber, being careful not to touch it with bare hands. 

Pre-lab Question 13.1 Why is it necessary to keep the inside of a vacuum chamber clean? 

How can contamination be minimised? 

Pre-lab Question 13.2 What is outgassing? 

Place the 0.5 g Al slug into the boat, making sure it is clean. Clean the mylar film holder, 

again with isopropyl alcohol, and mount it in the clamp ∼ 7 cm above the boat in the chamber. 

Inspect the inside of the vacuum chamber for any contamination. If you find any, remove 

them with the alcohol before replacing the bell jar and perspex shield. 

Pre-lab Question 13.3 Note the rubber seals on the jar. Discuss the function of the vacuum 

seal and the main factors involved. 

Pre-lab Question 13.4 Briefly discuss the function of rotary and diffusion pumps and their 

operating vacuums. 

Pre-lab Question 13.5 How does a thermistor gauge work? Over what range of pressures 

can such thermal gauges operate? 

Pre-lab Question 13.6 What is the definition of a Torr? 

To reduce the pressure to below 10 −4 Torr, we use the diffusion pump. Before we can use the 

diffusion pump, we must evacuate the bell jar to a pressure of 10 −2 Torr. Close all of the air 

inlet valves, checking that both the baffle and backing valves are closed, and then open the 

roughing valve. Switch on the rotary pump and wait until the pressure is below 10 −2 Torr. 

Once reached, open the backing valve and switch on the diffusion pump heater. 

Once heated (about 15 minutes), close the roughing valve, open the baffle valve (and worry 

about why the roughing valve should be closed FIRST), and observe the change in pressure. 

Comment on any rapid changes during this step. 

When the pressure is below 10 −5 Torr you can start the evaporation. Turn on the power 

supply and raise the current until the boat glows bright red (∼ 50 − 70 A). Once the Al has 

melted, slowly increase the current until the boat glows white hot and you can the bell jar 

being coated with Al. Continue evaporation until you can see that all of the Al source has 

evaporated (roughly 2 minutes). 

You have made the foil and can now begin the process of bringing the bell jar back up to air. 

First, close the baffle valve and then switch off the diffusion pump heater. The bell jar is now 

isolated from the pumps so you can use the air inlet closest to the jar to bring the chamber 

back up to air. When the diffusion pump has cooled (∼ 20 minutes), close the backing valve, 

turn off the rotary pump, and bring the vacuum side of the rotary pump to air using the right 

most air inlet valve.


13.4.3 Thickness measurement 

You have a new foil. Now measure its thickness using the energy loss information you 

measured for aluminium foils. Ensure that the mylar backing is facing the 226 Ra α source. 

Compare your measured value with what you may expect if all of the slug had been evaporated 

(Al density is 2.7 g/cm 3 ) during which it uniformly coats a hemisphere of radius equal 

to the boat-mylar distance. 

Note that the α particles have also passed through mylar (a compound of C and H). If time 

permits, take measurements to correct for that absorption using the second mylar-only foil 

available for use. Would you expect there to be a large correction? 

13.5 Rules for operating a high vacuum system 2 

• Always vent a mechanical pump to air when the power is off. The presence of a 

residual vacuum in the pump causes oil to be drawn into the pump contaminating its 

chambers. 

• Do not permit a mechanical pump to exhaust a high vacuum system below a pressure 

of about 10 −2 Torr, unless the pump is separated from the high-vacuum chamber with 

a trap stopping the oil vapour from entering the chamber. 

• Always keep the inside of a high vacuum system clean. In particular, grease from 

fingers can result in considerable contamination and gloves should ALWAYS be used 

when handling anything destined for the inside of high vacuum equipment. 

• Do not run a mechanical pump at high vacuum for long periods of time. The motors 

are not generally designed for long term use with the additional load, and the pump 

will eject much oil, in addition to gas, into the system. 

• Diffusion pumps should be cooled to a safe intermediate temperature before being 

vented to air. Venting at too high a temperature results in oxidation of the pump fluid 

and excessive carryover of the fluid into the mechanical pump. 

• Always check that the cooling water supply to diffusion and turbomolecular pumps is 

turned on prior to operation. It is also a good idea to have thermal protection to guard 

against interruption to the cooling water supply. 

• In liquid-N 2 trapped systems the trap should be cool enough to condense the diffusion 

pump oil prior to turning on the diffusion pump. Maximum backstreaming of pump 

oil occurs during startup and shutdown of the diffusion pump. 

• Do not vent a liquid N 2 trap to air whilst cold. 

• When the chamber is vented to atmosphere it is advisable to use a dry, inert, gas such as 

nitrogen to minimise moisture adsorption on the surface of the vacuum system. Never 

vent from the fore-line of the diffusion pump. 

2 Not all of the rules apply here; for example, we do not have a liquid N 2 trap on our diffusion pump. 

However, we mention it for the sake of reference.

13.6. MCA COMMANDS 171 

• Ionisation gauges, in particular hot cathode gauges, should not be turned on until the 

pressure is below 10 −3 Torr. 

13.6 MCA commands 

Alt-F1 Start data acquisition. 

Alt-F2 Stop data acquisition. 

Alt-F5 Move data from the MCA card into the computer’s memory buffer. 

Alt-F, Alt-S Save the data stored in the memory buffer into a file. Only a spectrum stored in 

the memory buffer can be saved. 

Arrow key Left and right keys move the cursor. Channel number of the cursor’s position is 

indicated at the bottom of the screen. Page-up and Page-dn are fast movements. Up 

arrow changes the vertical scale from logarithmic to linear; the down arrow key is the 

reverse. 

F3 Changes to expanded view. The expansion region is changed by the +/- key on the 

keypad. 


[1] W R Leo. Techniques for Nuclear and Particle Physics Experiments. Springer-Verlag, 

1987. 

[2] A Roth. Vacuum Technology. Elsevier Science, 1990.

Chapter 14 

Errors 


This chapter has been constructed to show you that error analysis really is quite simple and 

is of great importance to your report and your development as an experimental physicist. 

Topics covered include sources and types of error, how to calculate and combine errors and 

how to display your final result. Errors are often thought of as troublesome, but given the 

appropriate background knowledge they are really quite easy. So please take the time to read 

through the following guide and master those errors once and for all. 

When scientists present the results of experimental measurements, they should almost always 

specify a possible error associated with the quoted results. It is very poor experimental 

technique to say that, “this is my best value, but . . . I have no idea, nor do I care about how 

accurate it is.” In first year laboratory you were introduced to the concept of errors. In 

second year laboratory we not only expect you to think about and mention the errors, but 

you are required to understand the errors, estimate their magnitude using some relatively 

simple formulae, reduce them by developing good experimental technique, and discuss them 

at length throughout your report. The failure to consider, estimate, reduce or discuss the 

error associated with any quantity you obtain is very poor experimental technique. Hence, 

you should always state your final result plus a confidence limit. 

x + △x units 

Your final result is your best estimate for the value being investigated, whereas the confidence 

limits are the upper and lower limits between which you declare that you confidently expect 

the true value to lie. 

The confidence limit or ‘error’ conveys very significant information about the experiment and 

the result obtained, thus its consideration is an indication of a sound experimental technique. 

We shall now consider the different types of errors, how to estimate and combine them and 

how to present them appropriately. 

173

174 CHAPTER 14. ERRORS 

14.2 Sources of error in experimental work 

14.2.1 Mistakes (!!) 

First, let us dispose with human errors that are just silly blunders; misreading a scale, recording 

a number incorrectly or a making a mistake in calculation. These are not really errors in 

the scientific sense at all. The best advice we can give is - don’t make them! 

14.2.2 Random errors 

Random errors become evident when one takes repeated readings of a physical quantity and 

a small spread of readings is obtained, with readings equally likely to be in either direction 

from the average. The cause could be fluctuations in experimental conditions, unavoidable 

instrument variations, or your own unconscious subjective bias in the setting or reading of 

instruments. 

Random errors can be investigated by taking repeated measurements. 

14.2.2.1 Repeated readings 

If you need to know the value of a quantity as accurately as possible, it is obvious that you 

need to check the result a few times. Repeating the measurement allows you to calculate the 

best value for the quantity being investigated and investigate the magnitude of the random 

errors by considering the statistical variation in the data. 

The best estimate of the true value of the result is the arithmetic mean of the measured result, 

〈X〉 = 1 N 

∑ 

X i . (14.1) 

i 

An estimate in the uncertainty of the result comes from the standard deviation σ of the data, 

σ 2 = 

∑i (X i − 〈X〉) 2 

. (14.2) 

(N − 1) 

where N is the number of points in the data set. For errors with a normal distribution we 

expect 68% (= 1 − 1/e) of the recorded measurements to be within one standard deviation 

of the mean [2]. 

14.2.3 Reading errors 

Both the accuracy and precision of any real measuring device is limited 1 . In first year laboratories 

you learnt that the precision is limited to perhaps to a quarter of the smallest scale 

1 The distinction between accuracy and precision can be illustrated by making an analogy with target archery. 

Precision is how close the arrows are together; accuracy is how close they are to the bullseye.

14.2. SOURCES OF ERROR IN EXPERIMENTAL WORK 175 

on the instrument (ruler). The instruments used to measure quantities include rulers with 

Vernier scales, calipers and digital voltmeters. Each of these improves the precision with 

which values can be obtained; however, a reading error is still always present. 

14.2.3.1 Parallax 

A quick little reminder of a thing called parallax. It is important to align yourself with the 

instrument appropriately if you wish your results to be accurate. 

Figure 14.1: Parallax errors arising from the measurement of the length of an object. The 

apparent length, r 2 , in this case is shorter than the real length, r 1 . The apparent length would 

be the real length if the measurement was taken with the ruler in contact with the object. 

14.2.3.2 Limit of accuracy 

Figure 14.2: The first scale can be read to the nearest 0.5 mm, while the second scale can be 

read to the nearest 0.025 mm. 

Limit of accuracy = a quarter of the smallest division 

The accuracy of the result will depend upon the accuracy of the measuring device you use. In 

figure 14.2 we consider the length of a block of material. It is quite obvious that the accuracy


of our result is greater in the second case. In the Second Year laboratories the accuracy of 

our result should be down to a quarter of the smallest division. This applies to both rulers 

and needle metres. If the results depend greatly upon the accuracy of this result you will find 

a Vernier scale or digital meter (both discussed below) more appropriate. 

14.2.3.3 The Vernier scale 

A Vernier scale 2 is used whenever one needs to make a measurement of a distance or angle 

to a greater accuracy than that obtainable through direct visual reading of a linear scale. 

The Vernier uses the linear reading scale of the measuring apparatus (i.e. a ruler) along with 

another scale which is scaled by a factor of 9:10 compared to the linear scale. 

Reading a Vernier scale is best seen with an example (figure 14.3). 

In order to read this scale: 

Figure 14.3: Vernier scale 

• Read the linear part of the measurement, in this case, the ruler marking imediately to 

the left of the ‘zero’ of the Vernier. This is the first approximation to the measurement 

and yields a result of 2.5 mm. 

• Next, determine which marking on the Vernier scale most nearly lines up with the 

markings on the linear scale. Do this by scanning your eyes across the scale and 

judging whether the Vernier scale is to the left or the right of the linear scale. When 

we cannot say left or right, we know that we have the alignment. In this case the 2.5 

on the Vernier. We add this to the first approximation to find the reading. So the result 

is 2.75 mm. 

• Finally, the smallest scale on the Vernier scale is 0.005 mm. This is the uncertainty in 

the measurement. Thus our final result would be 

(2.750 ± 0.005) mm 

For further details on reading a Vernier, see section 8.4.4, “Reading the micrometer”. 

2 Invented by the French mathematician and inventor Pierre Vernier (1850-1637).

14.2. SOURCES OF ERROR IN EXPERIMENTAL WORK 177 

14.2.3.4 Digital voltmeter 

If you are using a digital voltmeter to record a voltage, it might be reading 1.004 volts, and 

the limit of the reading is 0.001 volts (which is the smallest difference between readings 

which the device can resolve). We would then be tempted to claim the result to be (1.004 ± 

.001)V. Which would seem reasonable. However the accuracy of the digital metres is less 

than the number of digits displayed would suggest, typically the accuracy is 1 digit plus some 

fraction of the reading (typically 0.3%). In this example the error is calculated as follows: 

△V = 3 × V + limit of reading 

1000 

e.g. △V = 3/1000 × 1.004 + 0.001 = 0.004012 ∴ error = 0.004. 

Thus the final result plus confidence limit is (1.004 ± 0.004) V. 

14.2.4 Systematic errors 

It is fairly easy to assess the size of the random errors that we have been discussing. Averaging 

a number of readings tends to cancel out the effects of such errors, and the spread of 

the readings allows us to estimate the size of the random errors involved. There is however 

another type of error - known as systematic errors - the effects of which are much more 

difficult to assess. Systematic errors are errors that systematically shift the measurements in 

one direction away from the true value. Thus repeated readings do not show up the presence 

of systematic errors, and no amount of averaging will reduce their effects. Both systematic 

and random errors are often present in the same measurement, and the different effects they 

have are contrasted in figure 14.4. 

There are no rules for eliminating systematic errors, or even detecting them, but the following 

points may be of some assistance. 

• Apparatus 

– If at all feasible, meters should be checked against a standard, or at least, against 

another and hopefully better meter. Zero settings should be checked and adjusted 

if necessary. 

– Given values should be treated with suspicion, and checked if possible; especially 

resistors that can vary 10% or more from their nominal value. 

– Instructions for use of the apparatus should be read and followed. Make sure 

you know how to read the scales on the instruments. Identifying features of the 

apparatus should be recorded. 

– If possible, vary the conditions at measurement slightly to avoid systematic errors. 

Use different rulers or section of a ruler to measure lengths or swap rulers if 

possible. The variation of conditions should be used in a discussion of systematic 

errors in your report. 

• Observations


Figure 14.4: Schematic diagram showing the different types of errors and how they are 

manifest in the results. 

– Measurements should be repeated by different observers to detect experimenter 

bias. 

– Corrections for instrument bias should be made if necessary (e.g. incorrect zero 

setting on a metre). 

– Any necessary precautions (warm up times, step to avoid backlash in screw transports, 

temperature controls etc.) should be observed. 

• Analysis 

– The experiment should be carefully analysed for likely sources of error and steps 

should be taken to minimise these. 

– Theoretical arguments used in the derivation of relations for indirectly measured 

quantities should be checked for assumptions, and the assumptions checked for 

validity in the particular experiment. 

– Results should be checked for feasibility, and silly results 3 recalculated. If the 

results are still unreasonable then measurements must be taken again. 

3 Note we say recalculated, not discarded - sometimes the “silly” results contain a lot of physics. Be careful 

- if you torture the data enough, it will confess to anything!

14.3. GRAPHS AND ERRORS 179 

14.3 Graphs and errors 

14.3.1 Error bars 

If estimates of uncertainties in measured values are available then these should be indicated 

on the plot in the form of error bars. These are bars passing through the estimated point 

running from the lower limit to the upper limit of the range covered by the uncertainties. 

Thus if (〈x〉, 〈y〉) is the estimated point and △x and △y are the uncertainties then the error 

bar in the x direction runs from 〈x〉 − △x to 〈x〉 + △x, and the bar in the y directions runs 

from 〈y〉 − △y to 〈y〉 + △y as shown in figure 14.5. 

One may choose error bars with a magnitude of one or two standard deviations, remember 

to indicate the chosen convention in the description of the plot. In many practical cases the 

uncertainty of one of the quantities (usually the independent variable often placed on the 

x-axis) is much less than that of the other. In such cases the smaller error bar is often omitted 

and a note made in the report. If more than one data set is plotted on the same graph, a 

different coloured pen or different symbols should be used. 

(,+ ∆y) 

(− ∆x,) 

(,) 

(+ ∆x, ) 

(, − ∆y) 

Figure 14.5: Error bars 

14.3.2 Estimates of gradients and uncertainties 

In many experiments the result is obtained from the gradient of a straight line. The straight 

line is usually a line of best fit to a set of experimental points. However, random errors will 

most probably cause the plotted points not to lie precisely on any one straight line, although 

the trend of the data may be clear. 

There are thus two questions to be answered: 

• What is the best estimate for the gradient of the line? 

• What is the uncertainty in the estimated gradient? 

There are a number of ways to estimate the gradient of a straight line from a plot of experimental 

points. We shall consider two such methods; best fit by eye, and least squares 

fitting.


14.3.2.1 Best fit by eye 

Figure 14.6: Line of best fit by eye. 

In many cases it is sufficient to draw the single straight line which seems to lie closet to the 

largest number of experimental points. 

If error bars of length one standard deviation have been drawn, then such a line should pass 

through at least 2/3 of these error bars. If the error bars are two standard deviations long, 

the line should pass through 90% of the bars. 

The gradient can then be calculated using “rise over run” and the equation of a straight line: 

y = mx + c 

m = rise 

run = y 2 − y 1 

x 2 − x 1 

The uncertainty in the gradient can be calculated using the ‘maximum and minimum gradient 

approach’. Two lines are drawn, a steepest and most shallow line that fit the data, (with fit 

meaning that they pass through 67% or 90% of the error bars for one and two standard 

deviation error bars respectively). The gradient is calculated for each line and the average of 

these is used as the estimate of the gradient. 

gradient = 1 2 

(steepest gradient + shallow gradient) 

The uncertainty can be taken as 

gradient = 1 2 

(steepest gradient - shallow gradient) 

14.3.2.2 Least squares fit 

This technique is one of the most accurate techniques for obtaining the gradient of a line of 

best fit. Excel uses this technique to find the line of best fit in its curve fit algorithm. This

14.4. EXCEL 181 

technique was developed by Gauss and assumes that all of the error is concentrated in the y 

co-ordinate, and asks 

“For a line of the form y = mx + c, what values of m and c yield the smallest 

value for the sum of errors (differences between the point it predicts and the 

experimental point) squared?” 

This technique minimises the total distance between the points on a line of best fit and the 

experimental point to obtain the best fit. 

14.4 Excel 

Excel is a powerful and elegant software program used with the 2 nd year physics laboratories 

within the school of physics. Excel, as you can see in the tutorial (chapter 15), enables you 

to take a data set and fit an equation to it. 

Figure 14.7 is obtained in the tutorial. In this experiment, particles with particular energy 

are binned in a particular ADC channel number. Excel is used to find out the relationship. 

Clearly this is roughly linear so we guess that the equation to be used in the transform is of 

the form y = mx + c. 

Figure 14.7: Line of best fit obtained using Excel in the tutorial, chapter 15. 

14.5 Combining errors 

In the majority of experiments there is more than one source of error. For instance, several 

random and systematic errors may be present in measurements of a single quantity. Further-


more, it is likely that a number of measurements of different quantities (each of which has 

an error associated with it) will have to be combined to calculate the required results. Thus 

it is important to know how errors are combined to get the overall error in an experiment. 

The basis of this combination is the following formula, which really makes a lot of sense. 

Suppose that your final result Z is a function of some experimentally determined quantities 

(x,y,...). It seems reasonable that if x and y have small errors in them, then the final result, 

being a function of those variables will also have some error in it. In the mid 1600’s Sir 

Isaac Newton developed calculus (and partial derivatives) to consider such variations. The 

general formulae for the error in Z due to experimental uncertainties (△x, △y,...) in the 

parameters is 

(△Z) 2 ≈ 

( ) 2 δZ 

(△x) 2 + 

δx 

( ) 2 δZ 

(△y) 2 + ... (14.3) 

δy 

The squared terms in this function arise as a result of statistical theory, which is beyond the 

scope of these notes see [2] for more detail. 

Every confidence limit we quote can be calculated using the parameters in equation 

14.3. The errors △x, △y,... can be random errors such as the standard 

deviation from repeated measurements, reading errors from a Vernier scale, tolerance 

limits from a digital voltmeter or uncertainties in the gradient of a graph. 

All these ‘errors’ combine together to give an error in the final result. 

The example below illustrates the use of this equation and calculation of the final error. 

14.5.0.3 Worked example 

Consider the use of a diffraction grating to determine the wavelength of an unknown emission 

line. The formula is: 

Thus 

mλ = d sin θ. (14.4) 

λ = d sin θ 

m . (14.5) 

We can now work out the errors. We are trying to find out, if there are small errors (standard 

deviation, reading errors . . . ) in the variables m, d and θ, what the resultant error in λ is. We 

calculate the partial derivatives of λ with respect to d, m and θ and then use equation 14.3) 

δλ 

δd = sin θ 

m , 

δλ 

δm 

= 

−d sin θ 

m 2 , 

δλ 

δθ = d cos θ 

m 

(14.6) 

Thus

14.6. SIGNIFICANT FIGURES AND DATA/ERROR PRESENTATION 183 

(△λ) 2 = sin2 θ 

m 2 (△d)2 + d2 sin 2 θ 

(△m) 2 + d2 cos 2 θ 

( π 

) 2 

(△q) 2 (14.7) 

m 4 m 2 180 

The last term was converted into radians. This allows you to enter the angle in degrees. 

Alternatively you could not include the (π/180) 2 term and enter the angle in radians. If we 

are given the quantities 

d = 1710 ± 2 nm, 

m = 1, 

q = 17.30 ± 0.01 degrees, 

note that m is a DISCRETE variable and so has no associated error with it. When all the 

terms are evaluated we get λ = 508.511 ± 0.6595 nm = (508.5 ± 0.7) nm. Note also that we 

only quote the error to one significant figure. This leads us to a discussion about. . . 

14.6 Significant figures and data/error presentation 

The correct presentation of the final result from the previous problem is 508.5 ± 0.7 nm. 

Hence we are confident that our result is between 507.8 nm and 509.2 nm. Please take note 

of the following: 

• Results can be rounded up or down depending upon the magnitude of the error. Ask 

your demonstrator if you are unsure. 

• The ‘best guess’ can only be as accurate as the error. There is no point is quoting 

508.5110 ± 0.6. The 0.0110 is not important. 

• Usually only the first significant figure in the error is of any use. Stating the wavelength 

is 508.5 ± 0.7 will suffice. Clearly giving details of the error to the n th degree is not 

required. i.e. ± 0.6595. 

• Displaying the result as 5.085×10 2 ±6.595×10 −1 nm indicates a lack of appreciation 

for what the result is actually telling us. This error presentation method is far from 

elegant and indicates that we can calculate the error but do not really know what the 

important points to take from the calculation are. The form of (5.085 ± 0.007) ×10 2 

nm would be more appropriate. 

Calculating an error is only half the exercise. Once the error has been calculated 

we must then discuss not only the magnitude of the error and its significance but 

sources of error and what was done to minimise them. As we said earlier, a best 

result without an error is next to useless, but an error calculation without an 

explanation of what it means is almost as pointless.


14.7 General expressions 

Our final result Z is a function of experimental parameters (x,y,...). These formulae reduce 

to a simple form in many common cases. We encourage everyone to derive the following so 

we can appreciate that the following formulae are based upon the idea that if the variables x 

and y differ then our final result Z(x,y) will differ. 

14.7.1 General expression 

If the result Z depends upon (x,y,...), the corresponding error is calculated as follows: 

(△Z) 2 ≈ 

( ) 2 dZ 

(△x) 2 + 

dx 

( ) 2 dZ 

(△y) 2 + ... (14.8) 

dy 

14.7.2 Sums and differences 

If Z = x + y + ... or Z = x − y − ... 

then 

(△Z) 2 ≈ (△x) 2 + (△y) 2 + ... (14.9) 

14.7.3 Products and quotients 

If Z = x × y × ... or Z = x ÷ y ÷ ... 

then 

( ) 2 △Z 

= 

〈Z〉 

( ) 2 △x 

+ 

〈x〉 

( ) 2 △y 

+ ... (14.10) 

〈y〉 

14.7.4 Powers and functions 

If Z = x N 

then 

( ) ( ) 

△Z △x 

= |N| 

〈Z〉〈x〉 

(14.11) 

14.7.5 Simple and trigonometric functions 

If Z = f(x) 

then

14.8. CONCLUDING COMMENTS 185 

( ) △Z 

= 

〈Z〉 

df 

∣dx∣ 

( ) △x 

〈x〉 

(14.12) 

Note: this last category includes all the trigonometric functions. Substituted angles must be 

in radians unless we have included the (π/180) 2 term in the error. 

A few of the errors to be calculated in the second year laboratories will require a combination 

of the above formulae. If you wish to use these formulae, then you must go through your 

specific case step by step. Alternatively you could use equation 14.3. 

14.8 Concluding comments 

We have looked at the source and types of errors, and how to record, reduce and calculate 

them. Error analysis is neither mysterious nor tricky 4 . As we have mentioned before, we 

must always state our final result plus confidence limit with correct accuracy and significant 

figures. 

It is a bad look to obtain a value without considering a confidence limit. The confidence limit 

or ‘error’ conveys very significant information about the experiment and the result obtained. 

Once the error has been calculated it should be displayed appropriately and finally discussed 

in depth. 


[1] L Kirkup. Experimental methods. Wiley, 1994. 

[2] P Bevington and D K Robinson. Data Reduction and Error Analysis for the Physical 

Sciences. McGraw-Hill Science/Engineering/Math, 3 rd edition, 2002. 

4 Although it can be very tedious.

Chapter 15 

Excel tutorial 

Welcome to the 2 nd year labs. This semester you will encounter a number of new concepts 

and experiments that will enable you to improve your experimental technique. In addition to 

becoming confident with your ability as a experimental physicist, you will hopefully develop 

an appreciation of the physical significance of the results you are taking, the associated errors 

and how to analyse these results. 

To this ends, we shall be using Excel. Though you should have already used Excel in Part I 

labs, in this tute you shall be introduced to the more powerful aspects of this program. 

15.1 Simple example, energy vs. ADC number 

Let us enter the data from table 15.1 and create a graph of it. We can then create a linear fit 

for this data, and plot this over the top of the experimental data to observe the accuracy of 

the fit. Finally we will transform this data to obtain another quantity. 

15.1.1 Entering data into the worksheet 

After entering Windows, open Excel. A data worksheet will then open - enter the data 

contained in table 15.1 into two columns in the worksheet. Save your worksheet (File → 

Save As → yourFilename.xls). 

15.1.2 Making the plot 

We shall now fit the data. Select the two columns (including titles) and choose Insert → 

Chart, or click the graph icon on the toolbar. This will begin a chart wizard - your first 

choice is almost always XY (Scatter) as graph type (the 5 th choice in the list), and the next 

screen ensures you have the correct data in the correct arrangement (does it look like you 

expect?). Now, enter the chart title, and axes labels. The final screen asks you if you would 

prefer to place your new chart as a “A new sheet” or “As object in”. The latter choice is 

187

188 CHAPTER 15. EXCEL TUTORIAL 

ADC Number Energy (MeV) 

313 18.207 

399 19.186 

452 20.653 

538 21.632 

574 23.100 

611 24.155 

613 24.079 

713 26.602 

814 29.048 

907 31.495 

1069 33.941 

1130 36.387 

1227 38.833 

1347 41.279 

1430 43.725 

1509 46.171 

1600 48.617 

Table 15.1: Data. 

pre-selected, however the former, I find, always makes your program neater (try both, and 

see how the latter choice may cover your data, and generally gets in the way). 

15.1.3 Fitting the data 

To fit the plot, choose Chart → Fit the trendline. Within this window, there are two tabs, 

Type (which in this case is linear) and Options (please select both Display equation on chart, 

and Display R-squared value 1 on chart. I found that E = 0.0239ADC + 9.5876, with 

r 2 = 0.9969. Note how the chart is linked in a ‘live’ fashion to the data - observe this by 

changing one point, and checking to see how the point on the graph changes. 

15.1.4 Absolute referencing 

When formulas are used in Excel, it applies special rules to copy these formulas around the 

worksheet. For example, try typing =B1*A4 into cell C1. Both B1 and A4 can be entered 

using the keyboard, or by using the arrow keys or clicking to highlight the necessary cells, 

after the equals sign has been entered. Now copy cell C1 (Ctrl+c or Edit → Copy) and paste 

it into cell D1 (Ctrl+v or Edit → Paste). Note how the formula is no longer what it was 

originally, but instead has changed to =C1*B4. This is because Excel reads the formula not 

as “B1*A4”, but more as “multiply the cell one to the left with the cell two to the left and 

three down”. The application of dollar signs to formulae is summarised in table 15.2. 

The addition of dollar signs ($) into the formula will change the way Excel views the formula. 

1 The R 2 value gives the quality of the fit, it is called the (linear) correlation coefficient. If R 2 is 1, the fit is 

perfect, if R 2 is 0, there is absolutely no correlation. For more detail, see [1], section 11.2, page 198 - 201.

15.2. NONLINEAR FITTING 189 

For example, change the contents of cell C1 to =$B1*A4. Now the column B is fixed in B1 

- try this by again copying cell C1, and pasting into cell D1. The new formula in D1 should 

be =$B1*B4. The formula in C1 now reads “multiply the cell in column B, the same row 

with the cell two to the left and three down”. 

Formula Excel Reads 

=B1*A4 “multiply the cell one to the left with. . . ” 

=$B1*A4 “multiply the cell in column B, the same row with. . . ” 

=B$1*A4 “multiply the cell in row 1, one column to the left, with. . . ” 

=$B$1*A4 “multiply the cell B1, with. . . ” 

Table 15.2: Absolute referencing. 

15.1.5 Using formulas to transform the data 

We shall now calculate the momentum associated with each channel as follows. This is 

achieved using a transform. Select cell E1, and type 938.3 (this is E 0 ). Select the cell in the 

first row of column C, and type =SQRT(B1*B1+2$E$1*B1). ‘Fill down’ by selecting cells 

C1 through C17, and either pressing Ctrl + d, selecting Edit → Fill down, or clicking on the 

small crosshairs at the bottom right corner of cell C1, and dragging down to the end of C17. 

Excel evaluates the formulae so now column C displays the momentum. 

15.2 Nonlinear fitting 

In simple experiments, a measurable quantity is recorded for various values of an experimental 

parameter. This information is then plotted, and usually a straight line results. The slope 

and vertical intercept are then determined, either by hand (drawing a graph and then using a 

ruler to draw a straight line through it) or by some computer program. 

But what if the data were more complicated than your average straight line? What if the 

theory predicted a slightly curved graph, which couldn’t be simply factored out by squaring 

one of the variables, or even predicted a wildly squiggly plot? Then, a non-linear curve fit is 

needed, which is extremely difficult to do by hand. Excel is good at this, and in this section, 

we will use the example of a damped sinusoid to demonstrate Excel’s ability to perform 

nonlinear fitting. This ability is not a standard feature of Excel, however it is easy to install. 

15.2.1 The Solver Add-In 

If you have already installed the solver add-in, then a quick check in the Tools menu will 

show you that the option Solver exists. If it is not there, select Add-Ins (also under the Tools 

menu). A window should appear with a number of add-ins available; select Solver Add-In, 

and click OK. Your computer should take less than a minute to install this, and no disk should 

be required. On University computers, you may need to repeat this process each time you 

log in - let your demonstrator know if this is the case.

190 CHAPTER 15. EXCEL TUTORIAL 

Figure 15.1: Solver parameters 

15.2.2 Example 

Figure 15.2: Solver results 

Open damped sinusoid.xls, (you can download it from the part 2 labs website 2 ) and you 

should see a number of columns of data on the left with titles t, y obs, y fit, and error; 

parameters on the top right, and two graphs with data points and a curve through them. This 

worksheet gives a good example of how non-linear fitting works. The function 

y = Ae −γt cos(ωt/100 + φ) − B (15.1) 

will be used, where t is the independent variable, y is the dependent variable, and the parameters 

γ, ω, φ and B are to be obtained by fitting. 

There are a number of features of this spread sheet with which you should be familiar 

• Columns I and J show the value of the parameters for which the data was generated 

from 

• Columns L and M show the value of the parameters subsequently determined by Excel’s 

non-linear fit 

• Column A contains the time data 

• Column B uses equation 15.1 and the parameters in column J to determine an ‘observed 

y’ 

• Column C gives Column B, rounded to 2 significant figures 

2 www.ph.unimelb.edu.au/˜part2


• Column D show the values of y as a result of the fit, i.e. equation 15.1 using column 

M. (Column B could be used, but then we would end up with zero error). 

• Cell M7: yields the χ 2 , which is given by the sum of the squares of the errors (column 

E) (This must be squared, to cancel out the possibility of some negative and some 

positive errors yielding exactly zero error.) 

Open the Solver (Tools menu). In the target cell, you should see the cell ‘fit chi sq’ (which is 

M7). This cell contains the value we would like to minimise for; ensure that ‘Equal To: Min’ 

is selected. The cells to be modified to ensure that this is minimised are the fit parameters, 

namely fit w, fit phi, fit A, fit gamma, and fit B. Click Solve, and Excel quickly brings up a 

new window titled “Solver Results”, which should announce that “Solver found a solution. 

All constraints and optimality conditions are satisfied”. Ensure that “Keep Solver Solution” 

is selected, and check OK. This first run isn’t so exciting, as the fit was probably almost 

perfect in the first place. 

Now try and make Excel do some actual work. Start by modifying the values in column M 

by small amounts; note how the fit on the graphs no longer matches the data points. It should 

become clear at this stage why there are two graphs, not one - the smaller range has enough 

space to see individual data points, and the bigger range shows the whole exponentially 

damped cosine at once. 

If you modify the fit parameters by larger amounts, Excel may not find a solution. This can 

easily be checked by eyeballing the graphs to check whether the fit matches the points. 

Other ways to play with this could be to change the function entirely. What about sine instead 

of cosine? What about log rather than exponential? 

Obviously, in a real life situation, there would be little reason to generate data using a function, 

and then try and fit to it, as we have done here. It is more likely that you could take 

some data, (something more complicated than which would yield your average straight line 

plot), get a function from theory, calculate the rough parameters, then use Excel to fit to that. 

A final suggestion - use your partners! Many of you will have much stronger computer skills 

than your partners. Take the opportunity to teach them - they can then teach you some physics 

in return. Avoid simply doing it for them - it is bad karma, and infuriates demonstrators. 


[1] P Bevington and D K Robinson. Data Reduction and Error Analysis for the Physical 

Sciences. McGraw-Hill, 2 nd edition, 1992.

Chapter 16 

The Safe Use of Lasers 

16.1 General comments 

A characteristic of all laser beams is the low divergence and high power density over long 

distances. A small (5 mW) helium/neon laser can have an apparent brightness 1,000 times 

that of the sun! Laser wavelengths range from ultraviolet through the visible to the far infrared 

regions of the spectrum. 

Many lasers are capable of inflicting biological damage, principally through the heat generated 

by the interaction of light with matter. Very high power lasers may also produce a 

thermally induced sonic shock wave which may damage tissue some distance from the site 

of beam exposure. 

The most vulnerable organ is the eye, although skin is also prone to damage. Lasers which 

produce light in the visible (400 to 700 nm) and near IR (700 to 1400 nm) regions of the 

spectrum are particularly hazardous since the eye will focus the beam onto the retina where 

a burn could cause impairment of vision or even blindness. For wavelengths in the UV (100 

to 400 nm) and far IR (∼1400 nm) regions which cannot penetrate the lens, the lens and 

cornea are most at risk. 

Although the skin can tolerate a great deal more exposure than the eye, the greatest hazards 

arise from the IR and near UV regions. Skin burns may result from continued exposure, 

leading to the formation of cancers. 

In addition to the hazards of direct exposure, exposure to the specular reflections pose similar 

risks. 

Finally, lasers usually require high voltage power supplies with the attendant risk of electric 

shock 

193

194 CHAPTER 16. THE SAFE USE OF LASERS 

16.2 Precautions with lasers 

NEVER LOOK DIRECTLY AT THE BEAM OR 

REFLECTED BEAMS. 

Avoid exposing the skin. 

When viewing the pattern on a diffuse screen, view from the same side as the laser to avoid 

direct illumination through pin-hole leaks.

Multiplier Prefix Symbol 

10 −24 yocto y 

10 −21 zepto z 

10 −18 atto a 

10 −15 femto f 

10 −12 pico p 

10 −9 nano n 

10 −6 micro µ 

10 −3 milli m 

10 −2 centi c 

10 −1 deci d 

10 deca da 

10 2 hecto h 

10 3 kilo k 

10 6 mega M 

10 9 giga G 

10 12 tera T 

10 15 peta P 

10 18 exa E 

10 21 zetta Z 

10 24 yotta Y 

10 36 undecillion - 

10 100 googol - 

10 googol googolplex - 

A list of multipliers. 

Those without symbols are not SI approved.

Bibliography - School of Physics - University of Melbourne

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