An Introduction to Neutron Scattering - Spallation Neutron Source

y 

Roger Pynn 

Los Alamos 

National Laboratory 

LECTURE 1: Introduction & Neutron Scattering “Theory”

Overview 

1. Introduction and theory of neutron scattering 

1. Advantages/disadvantages of neutrons 

2. Comparison with other structural probes 

3. Elastic scattering and definition of the structure factor, S(Q) 

4. Coherent & incoherent scattering 

5. Inelastic scattering 

6. Magnetic scattering 

7. Overview of science studied by neutron scattering 

8. References 

2. Neutron scattering facilities and instrumentation 

3. Diffraction 

4. Reflectometry 

5. Small angle neutron scattering 

6. Inelastic scattering

Why do Neutron Scattering? 

• To determine the positions and motions of atoms in condensed matter 

– 1994 Nobel Prize to Shull and Brockhouse cited these areas 

(see http://www.nobel.se/physics/educational/poster/1994/neutrons.html) 

• Neutron advantages: 

– Wavelength comparable with interatomic spacings 

– Kinetic energy comparable with that of atoms in a solid 

– Penetrating => bulk properties are measured & sample can be contained 

– Weak interaction with matter aids interpretation of scattering data 

– Isotopic sensitivity allows contrast variation 

– Neutron magnetic moment couples to B => neutron “sees” unpaired electron spins 

• Neutron Disadvantages 

– Neutron sources are weak => low signals, need for large samples etc 

– Some elements (e.g. Cd, B, Gd) absorb strongly 

– Kinematic restrictions (can’t access all energy & momentum transfers)

The 1994 Nobel Prize in Physics – Shull & Brockhouse 

Neutrons show where the atoms are…. 

…and what the atoms do.

The Neutron has Both Particle-Like and Wave-Like Properties 

• Mass: m n = 1.675 x 10 -27 kg 

• Charge = 0; Spin = ½ 

• Magnetic dipole moment: μ n = - 1.913 μ N 

• Nuclear magneton: μ N = eh/4πm p = 5.051 x 10 -27 J T -1 

• Velocity (v), kinetic energy (E), wavevector (k), wavelength (λ), 

temperature (T). 

• E = m n v 2 /2 = k B T = (hk/2π) 2 /2m n ; k = 2 π/λ = m n v/(h/2π) 

Energy (meV) Temp (K) Wavelength (nm) 

Cold 0.1 – 10 1 – 120 0.4 – 3 

Thermal 5 – 100 60 – 1000 0.1 – 0.4 

Hot 100 – 500 1000 – 6000 0.04 – 0.1 

l (nm) = 395.6 / v (m/s) 

E (meV) = 0.02072 k 2 (k in nm -1 )

Comparison of Structural Probes 

Note that scattering methods 

provide statistically averaged 

information on structure 

rather than real-space pictures 

of particular instances 

Macromolecules, 34, 4669 (2001)

Thermal Neutrons, 8 keV X-Rays & Low Energy 

Electrons:- Absorption by Matter 

Note for neutrons: 

• H/D difference 

• Cd, B, Sm 

• no systematic A 

dependence

Interaction Mechanisms 

• Neutrons interact with atomic nuclei via very short range (~fm) forces. 

• Neutrons also interact with unpaired electrons via a magnetic dipole 

interaction.

Brightness & Fluxes for Neutron & 

X-Ray Sources 

Brightness 

(s -1 m -2 ster -1 ) 

Neutrons 10 15 

Rotating 

Anode 

Bending 

Magnet 

10 16 

10 24 

Wiggler 10 26 

Undulator 

(APS) 

10 33 

dE/E 

(%) 

Divergence 

(mrad 2 ) 

Flux 

(s -1 m -2 ) 

2 10 x 10 10 11 

3 0.5 x 10 5 x 10 10 

0.01 0.1 x 5 5 x 10 17 

0.01 0.1 x 1 10 19 

0.01 0.01 x 0.1 10 24

Φ = number of 

σ 

= 

totalnumber 

dσ 

number of 

= 

dΩ 

2 

d σ 

= 

dΩdE 

of 

number of 

incident neutrons per 

Cross Sections 

cm 

neutrons scatteredper 

per second 

second / Φ 

neutronsscatteredper 

second into 

dΩ 

Φ dΩ 

neutronsscatteredper 

second into 

Φ dΩ 

dE 

2 

dΩ 

& dE 

σ measured in barns: 

1 barn = 10 -24 cm 2 

Attenuation = exp(-Nσt) 

N = # of atoms/unit volume 

t = thickness

Scattering by a Single (fixed) Nucleus 

If v is the velocity of 

passing through an area dS per second after scattering is 

v dS ψ 

scat 

Since the number of 

2 

= v dS b 

2 

dσ 

v b dΩ 

= = b 

dΩ 

ΦdΩ 

2 

2 

the neutron (same before and after scattering), 

the 

/r 

2 

= v b 

2 

dΩ 

incident neutrons passing through unit areasis 

soσ 

total 

= 4πb 

2 

: 

• range of nuclear force (~ 1fm) 

is scattering is 

elastic 

• we consider only scattering far 

from nuclear resonances where 

neutron absorption is negligible 

: 

number of 

Φ = vψ 

incident 

neutrons 

2 

= v

Adding up Neutrons Scattered by Many Nuclei 

0 

) 

( 

, 

) 

( 

) 

' 

( 

, 

2 

i 

2 

) 

' 

( 

i 

'. 

2 

) 

'.( 

i 

i 

scat 

R 

. 

k 

i 

i 

' 

by 

defined 

is 

Q 

r transfer 

wavevecto 

the 

where 

get 

to 

dS/r 

d 

use 

can 

we 

R 

r 

that 

so 

away 

enough 

far 

measure 

we 

If 

R 

- 

r 

1 

R 

- 

r 

b 

- 

is 

wave 

scattered 

the 

so 

e 

is 

ave 

incident w 

the 

R 

at 

located 

nucleus 

a 

At 

0 

0 

0 

i 

0 

k 

k 

Q 

e 

b 

b 

e 

b 

b 

d 

d 

e 

e 

b 

d 

dS 

vd 

vdS 

d 

d 

e 

e 

j 

i 

j 

i 

i 

i 

i 

R 

R 

. 

Q 

i 

j 

j 

i 

i 

R 

R 

. 

k 

k 

i 

j 

j 

i 

i 

R 

. 

k 

k 

i 

r 

k 

i 

i 

scat 

R 

r 

k 

i 

R 

. 

k 

i 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

− 

= 

= 

= 

Ω 

= 

Ω 

>> 

Ω 

= 

Ω 

= 

Ω 

∴ 

⎥ 

⎥ 

⎦ 

⎤ 

⎢ 

⎢ 

⎣ 

⎡ 

= 

− 

− 

− 

− 

− 

− 

∑ 

∑ 

∑ 

∑ 

σ 

ψ 

σ 

ψ

Coherent and Incoherent Scattering 

The scattering length, b i , depends on the nuclear isotope, spin relative to the 

neutron & nuclear eigenstate. For a single nucleus: 

b 

i 

b b 

i 

= 

but 

j 

δb 

2 

i 

= 

dσ 

∴ 

dΩ 

b 

δb 

= 

= 

+ δb 

b 

2 

= 

b 

i 

b 

0 

i 

+ 

− 

2 

b 

where δb 

averages to zero 

and 

b 

∑ 

i, 

j 

( δb 

e 

2 

i 

+ δb 

δb 

δb 

= 

i 

b 

2 

r r r 

−iQ.( 

R −R 

Coherent Scattering 

(scattering depends on the 

direction of Q) 

i 

j 

j 

j 

) 

) + δbδb 

vanishes 

− 

+ ( 

b 

b 

2 

i 

2 

j 

− 

unless 

b 

2 

) N 

Note: N = number of atoms in scattering system 

i 

= 

Incoherent Scattering 

(scattering is uniform in all directions) 

j

Nuclide 

1 H 

2 H 

C 

O 

Al 

Values of σ coh and σ inc 

s coh 

1.8 

5.6 

5.6 

4.2 

1.5 

s inc 

80.2 

2.0 

0.0 

0.0 

0.0 

Nuclide 

V 

Fe 

Co 

Cu 

36 Ar 

s coh 

0.02 

11.5 

1.0 

7.5 

24.9 

• Difference between H and D used in experiments with soft matter (contrast variation) 

• Al used for windows 

• V used for sample containers in diffraction experiments and as calibration for energy 

resolution 

• Fe and Co have nuclear cross sections similar to the values of their magnetic cross sections 

• Find scattering cross sections at the NIST web site at: 

http://webster.ncnr.nist.gov/resources/n-lengths/ 

s inc 

5.0 

0.4 

5.2 

0.5 

0.0

so 

or 

ie 

Coherent Elastic Scattering measures the Structure 

Factor S(Q) I.e. correlations of atomic positions 

dσ 

= 

dΩ 

Now 

where 

b 

∑ 

i 

2 

e 

r 

N. 

S( 

Q) 

r r 

−iQ. 

R 

r 1 

S( Q) 

= 

N 

∫ 

r 

g( R) 

= 

i 

= 

∫ 

r 

dr 

' 

∑ 

i≠0 

r 

dr. 

e 

∫ 

for an assembly of similar atomswhere 

r r 

−iQ. 

r 

∑ 

r r r 

δ( 

R− 

R + R 

i 

i 

r r 

δ( 

r −R 

) = 

0 

∫ 

) 

i 

∫ 

r 

dr. 

e 

r r 

−iQ. 

r 

r 

r 1 r r 2 

−iQ. 

r r 

S( 

Q) 

= ∫ dr. 

e ρN 

( r) 

N 

r r r r 

−iQ.( 

r −r 

') 

r r 1 

dr. 

e ρN 

( r) 

ρN 

( r') 

= 

N 

r r r r r 

−iQ. 

R 

S( 

Q) 

= 1+ 

dR. 

g( 

R) 

. e 

ρ 

r 

( r) 

r 

dR 

∫ ∫ 

whereρ 

r 

dre 

r 

is a function of R only. 

N 

r 1 

S( 

Q) 

= 

N 

r r 

−iQ. 

R 

N 

ρ 

is 

N 

thenuclear 

r 

( r) 

ρ 

∑ 

N 

i, 

j 

e 

r r r 

−iQ.( 

R −R 

r r 

( r − R) 

i 

j 

) 

ensemble 

number density 

g(R) is known as the static pair correlation function. It gives the probability that there is an 

atom, i, at distance R from the origin of a coordinate system at time t, given that there is also a 

(different) atom at the origin of the coordinate system

S(Q) and g(r) for Simple Liquids 

• Note that S(Q) and g(r)/ρ both tend to unity at large values of their arguments 

• The peaks in g(r) represent atoms in “coordination shells” 

• g(r) is expected to be zero for r < particle diameter – ripples are truncation 

errors from Fourier transform of S(Q)

Neutrons can also gain or lose energy in the scattering process: this is 

called inelastic scattering

Inelastic neutron scattering measures atomic motions 

The concept of a pair correlation function can be generalized: 

G(r,t) = probability of finding a nucleus at (r,t) given that there is one at r=0 at t=0 

G s (r,t) = probability of finding a nucleus at (r,t) if the same nucleus was at r=0 at t=0 

Then one finds: 

2 ⎛ d σ ⎞ 

⎜ 

d dE ⎟ 

⎝ Ω. 

⎠ 

2 ⎛ d σ ⎞ 

⎜ 

d dE 

⎟ 

⎝ Ω. 

⎠ 

coh 

inc 

= b 

= b 

where 

r 1 

S( 

Q, 

ω) 

= 

2πh 

2 

coh 

2 

inc 

∫∫ 

k' 

r 

NS( 

Q, 

ω) 

k 

k' 

r 

NSi 

( Q, 

ω) 

k 

r 

G( 

r, 

t) 

e 

r r 

i( 

Q. 

r −ωt) 

r 

drdt 

and 

(h/2π)Q & (h/2π)ω are the momentum & 

energy transferred to the neutron during the 

scattering process 

r 1 

Si 

( Q, 

ω) 

= 

2πh 

∫∫ 

G 

s 

r 

( r, 

t) 

e 

r r 

i( 

Q. 

r −ωt 

) 

r 

drdt 

Inelastic coherent scattering measures correlated motions of atoms 

Inelastic incoherent scattering measures self-correlations e.g. diffusion

Magnetic Scattering 

• The magnetic moment of the neutron interacts with B fields 

caused, for example, by unpaired electron spins in a material 

– Both spin and orbital angular momentum of electrons contribute to B 

– Expressions for cross sections are more complex than for nuclear scattering 

• Magnetic interactions are long range and non-central 

– Nuclear and magnetic scattering have similar magnitudes 

– Magnetic scattering involves a form factor – FT of electron spatial distribution 

• Electrons are distributed in space over distances comparable to neutron wavelength 

• Elastic magnetic scattering of neutrons can be used to probe electron distributions 

– Magnetic scattering depends only on component of B perpendicular to Q 

– For neutrons spin polarized along a direction z (defined by applied H field): 

• Correlations involving B z do not cause neutron spin flip 

• Correlations involving B x or B y cause neutron spin flip 

– Coherent & incoherent nuclear scattering affects spin polarized neutrons 

• Coherent nuclear scattering is non-spin-flip 

• Nuclear spin-incoherent nuclear scattering is 2/3 spin-flip 

• Isotopic incoherent scattering is non-spin-flip

Magnetic Neutron Scattering is a Powerful Tool 

• In early work Shull and his collaborators: 

– Provided the first direct evidence of antiferromagnetic ordering 

– Confirmed the Neel model of ferrimagnetism in magnetite (Fe 3 O 4 ) 

– Obtained the first magnetic form factor (spatial distribution of magnetic 

electrons) by measuring paramagnetic scattering in Mn compounds 

– Produced polarized neutrons by Bragg reflection (where nuclear and magnetic 

scattering scattering cancelled for one neutronspin state) 

– Determined the distribution of magnetic moments in 3d alloys by measuring 

diffuse magnetic scattering 

– Measured the magnetic critical scattering at the Curie point in Fe 

• More recent work using polarized neutrons has: 

– Discriminated between longitudinal & transverse magnetic fluctuations 

– Provided evidence of magnetic solitons in 1-d magnets 

– Quantified electron spin fluctuations in correlated-electron materials 

– Provided the basis for measuring slow dynamics using the neutron spin-echo 

technique…..etc

Ene rgy Trans fe r (me V) 

10000 

100 

1 

0.01 

10-4 

10-6 

Ne utro ns in Co n de ns e d Matte r Re s e arc h 

SSPSS pa lla tio - Chopper n - Ch opp Spectrometer 

er 

ILL - w itho u t s pin -ec ho 

ILL - w ith s p in -e ch o 

Elastic Scattering 

[Larger Objects Resolved] 

Micelles Polymers 

Proteins in Solution 

Viruses 

Metallu rgical 

Systems 

Colloids 

Aggregate 

Motion 

Polymers 

and 

Biological 

Systems 

Coherent 

M odes in 

Glasses 

and Liquids 

Molecular 

M otions 

Mem branes 

Proteins 

Critic al 

Sca ttering 

Diffusive 

Modes 

Spin 

Wa ves 

Elec tron- 

phonon 

Intera ctions 

C rystal and 

Magnetic 

Structures 

Crystal 

Fields 

Slower 

M otions 

Res olved 

Molecular 

Reorie nta tion 

Tunneling 

Spectroscopy 

Surface 

Effects ? 

Itine rant 

Magnets 

Hyd rog en 

Modes 

Molecular 

Vibrations 

La ttice 

Vibra tions 

and 

A nharmonicity 

Amorphous 

Systems 

N eutron 

Induced 

Excitations 

Momentum 

Distributions 

Accurate Liquid Structu res 

Precision Crystallography 

Anharmonicity 

0 .0 01 0.01 0 .1 1 10 100 

Q (Å -1 ) 

Neutron scattering experiments measure the number of neutrons scattered at different values of 

the wavevector and energy transfered to the neutron, denoted Q and E. The phenomena probed 

depend on the values of Q and E accessed.

Next Lecture 

2. Neutron Scattering Instrumentation and Facilities – how is 

neutron scattering measured? 

1. Sources of neutrons for scattering – reactors & spallation sources 

1. Neutron spectra 

2. Monochromatic-beam and time-of-flight methods 

2. Instrument components 

1. Crystal monochromators and analysers 

2. Neutron guides 

3. Neutron detectors 

4. Neutron spin manipulation 

5. Choppers 

6. etc 

3. A zoo of specialized neutron spectrometers

References 

• Introduction to the Theory of Thermal Neutron Scattering 

by G. L. Squires 

Reprint edition (February 1997) 

Dover Publications 

ISBN 048669447 

• Neutron Scattering: A Primer 

by Roger Pynn 

Los Alamos Science (1990) 

(see www.mrl.ucsb.edu/~pynn)

y 

Roger Pynn 

Los Alamos 

National Laboratory 

LECTURE 2: Neutron Scattering Instrumentation & Facilities

Overview 

1. Essential messages from Lecture #1 

2. Neutron Scattering Instrumentation and Facilities – how is 

neutron scattering measured? 

1. Sources of neutrons for scattering – reactors & spallation sources 

1. Neutron spectra 

2. Monochromatic-beam and time-of-flight methods 

2. Instrument components 

3. A “zoo” of specialized neutron spectrometers

Recapitulation of Key Messages From Lecture #1 

• Neutron scattering experiments measure the number of neutrons scattered 

by a sample as a function of the wavevector change (Q) and the energy 

change (E) of the neutron 

• Expressions for the scattered neutron intensity involve the positions and 

motions of atomic nuclei or unpaired electron spins in the scattering sample 

• The scattered neutron intensity as a function of Q and E is proportional to 

the space and time Fourier Transform of the probability of finding two atoms 

separated by a particular distance at a particular time 

• Sometimes the change in the spin state of the neutron during scattering is 

also measured to give information about the locations and orientations of 

unpaired electron spins in the sample

What Do We Need to Do a Basic Neutron Scattering Experiment? 

• A source of neutrons 

• A method to prescribe the wavevector of the neutrons incident on the sample 

• (An interesting sample) 

• A method to determine the wavevector of the scattered neutrons 

– Not needed for elastic scattering 

• A neutron detector 

Incident neutrons of wavevector k 0 Scattered neutrons of 

wavevector, k 1 

Q 

k 0 

k 1 

Sample 

Detector 

Remember: wavevector, k, & 

wavelength, λ, are related by: 

k = m n v/(h/2π) = 2π/λ

Neutron Scattering Requires Intense Sources of Neutrons 

• Neutrons for scattering experiments can be produced either by nuclear 

fission in a reactor or by spallation when high-energy protons strike a heavy 

metal target (W, Ta, or U). 

– In general, reactors produce continuous neutron beams and spallation sources produce 

beams that are pulsed between 20 Hz and 60 Hz 

– The energy spectra of neutrons produced by reactors and spallation sources are different, 

with spallation sources producing more high-energy neutrons 

– Neutron spectra for scattering experiments are tailored by moderators – solids or liquids 

maintained at a particular temperature – although neutrons are not in thermal equilibrium 

with moderators at a short-pulse spallation sources 

• Both reactors and spallation sources are expensive to build and require 

sophisticated operation. 

– SNS at ORNL will cost about $1.5B to construct & ~$140M per year to operate 

• Either type of source can provide neutrons for 30-50 neutron spectrometers 

– Small science at large facilities

About 1.5 Useful Neutrons Are Produced by Each Fission Event in 

a Nuclear Reactor Whereas About 25 Neutrons Are Produced by 

spallation for Each 1-GeV Proton Incident on a Tungsten Target 

Artist’s view of spallation 

Nuclear Fission 

Spallation

Neutrons From Reactors and Spallation Sources Must Be 

Moderated Before Being Used for Scattering Experiments 

• Reactor spectra are Maxwellian 

• Intensity and peak-width ~ 1/(E) 1/2 at 

high neutron energies at spallation 

sources 

• Cold sources are usually liquid 

hydrogen (though deuterium is also 

used at reactors & methane is sometimes 

used at spallation sources) 

• Hot source at ILL (only one in the 

world) is graphite, radiation heated.

The ESRF* & ILL* With Grenoble & the Beldonne Mountains 

*ESRF = European Synchrotron Radiation Facility; ILL = Institut Laue-Langevin

The National Institute of Standards and Technology (NIST) Reactor Is a 

20 MW Research Reactor With a Peak Thermal Flux of 4x10 14 N/sec. It Is 

Equipped With a Unique Liquid-hydrogen Moderator That Provides Neutrons 

for Seven Neutron Guides

Neutron Sources Provide Neutrons for Many 

Spectrometers: Schematic Plan of the ILL Facility

A ~ 30 X 20 m 2 Hall at the ILL Houses About 30 Spectrometers. 

Neutrons Are Provided Through Guide Tubes

Proton 

Radiography 

800-MeV Linear 

Accelerator 

LANSCE 

Visitor’s Center 

Los Alamos Neutron Science Center 

Manuel Lujan Jr. 

Neutron Scattering 

Center 

Proton Storage 

Ring 

Weapons 

Neutron Research

Neutron Production at LANSCE 

• Linac produces 20 H - (a proton + 2 electrons) pulses per second 

– 800 MeV, ~800 μsec long pulses, average current ~100 μA 

• Each pulse consists of repetitions of 270 nsec on, 90 nsec off 

• Pulses are injected into a Proton Storage Ring with a period of 360 nsec 

– Thin carbon foil strips electrons to convert H - to H + (I.e. a proton) 

– ~3 x 10 13 protons/pulse ejected onto neutron production target

Components of a Spallation Source 

A half-mile long proton linac… 

…and a neutron production target 

….a proton accumulation ring….

A Comparison of Neutron Flux Calculations for the ESS SPSS (50 Hz, 5 MW) 

& LPSS (16 Hz, 5W) With Measured Neutron Fluxes at the ILL 

Instantaneous flux [n/cm 2 /s/str/Å] 

Flux [n/cm 2 /s/str/Å] 

10 17 

10 16 

10 15 

10 14 

10 13 

10 12 

10 11 

10 10 

2.0x10 14 

1.5x10 14 

1.0x10 14 

5.0x10 13 

ILL hot source 

ILL thermal source 

ILL cold source 

0 1 2 3 4 

Wavelength [Å] 

0.0 

0 500 1000 1500 2000 2500 3000 

Time [μs] 

l = 4 Å 

poisoned moderator 

decoupled moderator 

coupled moderator 

long pulse 

peak flux 

poisoned m. 

decoupled m. 

coupled m. 

long pulse 

Flux [n/cm 2 /s/str/Å] 

10 17 

10 16 

10 15 

10 14 

10 13 

10 12 

10 11 

10 10 

ILL hot source 

ILL thermal source 

ILL cold source 

average flux 

poisoned m. 

decoupled m. 

coupled m. 

and long pulse 

0 1 2 3 4 

Wavelength [Å] 

Pulsed sources only make sense if 

one can make effective use of the 

flux in each pulse, rather than the 

average neutron flux

Simultaneously Using Neutrons With Many Different Wavelengths 

Enhances the Efficiency of Neutron Scattering Experiments 

k i 

k i 

k f 

k f 

I 

I 

Dl res = Dl bw 

Dl bw a T 

Dl res a DT 

Potential Performance Gain relative to use of a Single Wavelength 

is the Number of Different Wavelength Slices used 

l 

l 

I 

Neutron 

Source Spectrum 

l

The Time-of-flight Method Uses Multiple Wavelength Slices 

at a Reactor or a Pulsed Spallation Source 


Distance 

Moderator 

Dl res = 3956 dT p / L 

Dl bw = 3956 DT / L 

DT 

When the neutron wavelength is determined by time-of-flight, ΔT/δT p different 

wavelength slices can be used simultaneously. 

dT p 

DT 

Time 

L

The Actual ToF Gain From Source Pulsing 

Often Does Not Scale Linearly With Peak Neutron Flux 

low rep. rate => large dynamic range 

short pulse => good resolution — 

neither may be necessary or useful 

large dynamic range may result in intensity 

changes across the spectrum 

at a traditional short-pulse source the wavelength 

resolution changes with wavelength 

k i 

k i 

Q 

k i k f 

k i 

Q 

k f 

k f 

I 

I 

k f 

l 

l 

l

A Comparison of Reactors & Spallation Sources 

Short Pulse Spallation Source 

Energy deposited per useful neutron is 

~20 MeV 

Neutron spectrum is “slowing down” 

spectrum – preserves short pulses 

Constant, small δλ/λ at large neutron 

energy => excellent resolution 

especially at large Q and E 

Copious “hot” neutrons=> very good for 

measurements at large Q and E 

Low background between pulses => 

good signal to noise 

Single pulse experiments possible 

Reactor 

Energy deposited per useful neutron is 

~ 180 MeV 

Neutron spectrum is Maxwellian 

Resolution can be more easily tailored 

to experimental requirements, except 

for hot neutrons where monochromator 

crystals and choppers are less effective 

Large flux of cold neutrons => very 

good for measuring large objects and 

slow dynamics 

Pulse rate for TOF can be optimized 

independently for different 

spectrometers 

Neutron polarization easier

Why Isn’t There a Universal Neutron Scattering Spectrometer? 

Mononchromatic 

incident beam 


• Conservation of momentum => Q = k f – k i 

• Conservation of energy => E = ( h 2 m/ 8 p 2 ) (k f 2 - ki 2 ) 

Elastic Scattering — detect all 

scattered neutrons 

Inelastic Scattering — detect 

neutrons as a function of 

scattered energy (color) 

• Scattering properties of sample depend only on Q and E, not on neutron l 

Many types of neutron scattering spectrometer are required because 

the accessible Q and E depend on neutron energy and because resolution and 

detector coverage have to be tailored to the science for such a signal-limited technique.

Brightness & Fluxes for Neutron & 

X-ray Sources 

Brightness 

(s -1 m -2 ster -1 ) 

Neutrons 10 15 

Rotating 

Anode 

Bending 

Magnet 

10 16 

10 24 

Wiggler 10 26 

Undulator 

(APS) 

10 33 

dE/E 

(%) 

Divergence 

(mrad 2 ) 

Flux 

(s -1 m -2 ) 

2 10 x 10 10 11 

3 0.5 x 10 5 x 10 10 

0.01 0.1 x 5 5 x 10 17 

0.01 0.1 x 1 10 19 

0.01 0.01 x 0.1 10 24

• Uncertainties in the neutron 

wavelength & direction of travel 

imply that Q and E can only be 

defined with a certain precision 

Instrumental Resolution 

• When the box-like resolution 

volumes in the figure are convolved, 

the overall resolution is Gaussian 

(central limit theorem) and has an 

elliptical shape in (Q,E) space 

• The total signal in a scattering 

experiment is proportional to the 

phase space volume within the 

elliptical resolution volume – the better the resolution, the lower the count 

rate

Examples of Specialization of Spectrometers: 

Optimizing the Signal for the Science 

• Small angle scattering [Q = 4π sinθ/λ; (δQ/Q) 2 = (δλ/λ) 2 + (cotθ δθ) 2 ] 

– Small diffraction angles to observe large objects => long (20 m) instrument 

– poor monochromatization (δλ/λ ~ 10%) sufficient to match obtainable angular 

resolution (1 cm 2 pixels on 1 m 2 detector at 10 m => δθ ~ 10 -3 at θ ~ 10 -2 )) 

• Back scattering [θ = π/2; λ = 2 d sin θ; δλ/λ = cot θ +…] 

– very good energy resolution (~neV) => perfect crystal analyzer at θ ~ π/2 

– poor Q resolution => analyzer crystal is very large (several m 2 )

Typical Neutron Scattering Instruments 

Note: relatively massive shielding; long 

flight paths for time-of-flight spectrometers; 

many or multi-detectors 

but… small science at a large facility

Components of Neutron Scattering Instruments 

• Monochromators 

– Monochromate or analyze the energy of a neutron beam using Bragg’s law 

• Collimators 

– Define the direction of travel of the neutron 

• Guides 

– Allow neutrons to travel large distances without suffering intensity loss 

• Detectors 

– Neutron is absorbed by 3 He and gas ionization caused by recoiling particles 

is detected 

• Choppers 

– Define a short pulse or pick out a small band of neutron energies 

• Spin turn coils 

– Manipulate the neutron spin using Lamor precession 

• Shielding 

– Minimize background and radiation exposure to users

Most Neutron Detectors Use 3 He 

• 3 He + n -> 3 H + p + 0.764 MeV 

• Ionization caused by triton and proton 

is collected on an electrode 

• 70% of neutrons are absorbed 

when the product of gas 

pressure x thickness x neutron 

wavelength is 16 atm. cm. Å 

• Modern detectors are often “position 

sensitive” – charge division is used 

to determine where the ionization 

cloud reached the cathode. 

A selection of neutron detectors – 

thin-walled stainless steel tubes 

filled with high-pressure 3 He.

Essential Components of Modern Neutron 

Scattering Spectrometers 

Horizontally & vertically focusing 

monochromator (about 15 x 15 cm 2 ) 

Pixelated detector covering a wide range 

of scattering angles (vertical & horizontal) 

Neutron guide 

(glass coated 

either with Ni 

or “supermirror”)

Rotating Choppers Are Used to Tailor Neutron 

Pulses at Reactors and Spallation Sources 

• T-zero choppers made of Fe-Co are used at spallation sources 

to absorb the prompt high-energy pulse of neutrons 

• Cd is used in frame overlap choppers to absorb slower 

neutrons 

Fast neutrons from one pulse can catch-up with 

slower neutrons from a succeeding pulse and spoil 

the measurement if they are not removed.This is 

called “frame-overlap”

Larmor Precession & Manipulation of the Neutron Spin 

• In a magnetic field, H, the neutron 

spin precesses at a rate 

υ 

L 

= −γH 

/ 2π 

= −2916. 

4H 

Hz 

where γ is the neutron’s gyromagnetic 

ratio & H is in Oesteds 

• This effect can be used to manipulate 

the neutron spin 

• A “spin flipper” – which turns the spin 

through 180 degrees is illustrated 

• The spin is usually referred to as “up” 

when the spin (not the magnetic 

moment) is parallel to a (weak ~ 10 – 

50 Oe) magnetic guide field


Next Lecture 

1. Diffraction by a lattice 

2. Single-crystal diffraction and powder diffraction 

3. Use of monochromatic beams and time-of-flight to measure powder 

diffraction 

4. Rietveld refinement of powder patterns 

5. Examples of science with powder diffraction 

• Refinement of structures of new materials 

• Materials texture 

• Strain measurements 

• Structures at high pressure 

• Microstrain peak broadening 

• Pair distribution functions (PDF)

LECTURE 3: Diffraction 

by 

Roger Pynn 

Los Alamos 

National Laboratory


This Lecture 

1. Diffraction by a lattice 

2. Single-crystal compared with powder diffraction 

3. Use of monochromatic beams and time-of-flight to measure powder 

diffraction 

4. Rietveld refinement of powder patterns 

5. Examples of science with powder diffraction 





dσ 

dΩ 

⎛ 

⎜ 

⎝ 

Q 

= 

dσ 

⎞ 

⎟ 

dΩ 

⎠ 

where 

the 

i, 

j 

From Lecture 1: 

number of neutrons scattered through angle 2θ 

per second into dΩ 

number of incident neutrons per square cm per second 

coh 

k 0 

= 

k’ 

∑ 

b 

coh 

i 

b 

coh 

j 

e 

r r r r 

i( 

k −k 

') 

. ( R −R 

wavevector 

transfer 

2θ 

Incident neutrons of wavevector k 0 

0 

i 

j 

) 

= 

∑ 

i, 

j 

b 

coh 

i 

b 

coh 

j 

r 

Q is defined by 

e 

r r r 

−iQ. 

( R −R 

i 

r r r 

Q = k '−k 

For elastic scattering k 0 =k’=k: 

Q = 2k sin θ 

Q = 4π sin θ/λ 

Sample 

2θ 

j 

) 

0 


Scattered neutrons of 

wavevector, k’

Neutron Diffraction 

• Neutron diffraction is used to measure the differential cross section, dσ/dΩ 

and hence the static structure of materials 

– Crystalline solids 

– Liquids and amorphous materials 

– Large scale structures 

• Depending on the scattering angle, 

structure on different length scales, d, 

is measured: 

2π / Q 

= d = λ / 2sin( 

θ ) 

• For crystalline solids & liquids, use 

wide angle diffraction. For large structures, 

e.g. polymers, colloids, micelles, etc. 

use small-angle neutron scattering

Diffraction by a Lattice of Atoms 

. 

G 

vector, 

lattice 

reciprocal 

a 


equal 

is 

Q, 

r, 


scattering 

when the 

occurs 

only 

lattice 

(frozen) 

a 

from 

scattering 

So 

. 

2 

) 

.( 

then 

If 

. 

2 

. 

Then 

ns. 

permutatio 

cyclic 

and 

2 

Define 

cell. 

unit 

the 

of 

on vectors 

translati 

primitive 

the 

are 

, 

, 

where 

can write 

we 

lattice, 

Bravais 

a 

In 

. 

2 

) 

.( 

such that 

s 

Q' 

for 

zero 

- 

non 

only 

is 

S(Q) 

s, 

vibration 

thermal 

Ignoring 

position. 

m 

equilibriu 

the 

from 

thermal) 

(e.g. 

nt 

displaceme 

any 

is 

and 

i 

atom 

of 

position 

m 

equilibriu 

the 

is 

where 

with 

1 

) 

( 

hkl 

* 

3 

* 

2 

* 

1 

* 

3 

2 

0 

* 

1 

3 

2 

1 

3 

3 

2 

2 

1 

1 

, 

) 

.( 

π 

πδ 

π 

π 

M 

j 

i 

Q 

a 

l 

a 

k 

a 

h 

G 

Q 

a 

a 

a 

a 

V 

a 

a 

a 

a 

a 

m 

a 

m 

a 

m 

i 

M 

j 

i 

Q 

u 

i 

u 

i 

R 

e 

N 

Q 

S 

hkl 

ij 

j 

i 

i 

i 

i 

i 

i 

i 

j 

i 

R 

R 

Q 

i j 

i 

= 

− 

+ 

+ 

= 

= 

= 

∧ 

= 

+ 

+ 

= 

= 

− 

+ 

= 

= ∑ 

− 

− 

r 

r 

r 

r 

r 

r 

v 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

v 

r 

r 

r 

r 

r 

r 

r 

r 

r 

v 

r 

r 

a 1 

a 2 

a 3

For Periodic Arrays of Nuclei, Coherent Scattering Is Reinforced Only in 

Specific Directions Corresponding to the Bragg Condition: 

λ = 2 d hkl sin(θ) or 2 k sin(θ) = G hkl

Working 

⎛ dσ 

⎞ 

⎜ ⎟ 

⎝ dΩ 

⎠ 

and W 

is 

Bragg Scattering from Crystals 

through the 

( 2π 

) 

V 

where the unit - cell structure factor 

r 

r r 

iQ. 

d −Wd 

F ( Q) 

= b e e 

hkl 

Bragg 

d 

= 

∑ 

d 

N 

d 

0 

the Debye - Waller 

3 

math (see, for example, Squires' book), we find : 

r r 

δ ( Q − G ) F 

r 2 

( Q) 

∑ 

hkl 

hkl 

is 

hkl 

factor that 

given 

accounts 

for thermal 

motions 

atoms 

• Using either single crystals or powders, neutron diffraction can be used to 

measure F 2 (which is proportional to the intensity of a Bragg peak) for 

various values of (hkl). 

• Direct Fourier inversion of diffraction data to yield crystal structures is not 

possible because we only measure the magnitude of F, and not its phase => 

models must be fit to the data 

• Neutron powder diffraction has been particularly successful at determining 

structures of new materials, e.g. high T c materials 

by 

of

We would be better off 

if diffraction measured 

phase of scattering 

rather than amplitude! 

Unfortunately, nature 

did not oblige us. 

Picture by courtesy of D. Sivia

Reciprocal Space – An Array of Points (hkl) 

that is Precisely Related to the Crystal Lattice 

a* 

• • • • • • • • • • • 

• • • • 

k 

• • • • • • • 

k 0 

• • • • • • • • • • • 

G hkl 

• • • • • • • • • • • 

O 

• • • • • • • • • • • 

b* Ghkl = 2p/dhkl (hkl)=(260) 

a* = 2p(b x c)/V 0 , etc. 

A single crystal has to be aligned precisely to record Bragg scattering

Powder – A Polycrystalline Mass 

All orientations of 

crystallites possible 

Single crystal reciprocal lattice 

- smeared into spherical shells 

Typical Sample: 1cc powder of 

10mm crystallites - 10 9 particles 

if 1mm crystallites - 10 12 particles

Powder Diffraction gives Scattering on 

Debye-Scherrer Cones 

Incident beam 

x-rays or neutrons 

Bragg’s Law l = 2dsinQ 

Powder pattern – scan 2Q or l 

Sample 

(220) 

(200) 

(111)

Measuring Neutron Diffraction Patterns with a 

Monochromatic Neutron Beam 

Since we know the neutron wavevector, k, 

the scattering angle gives G hkl directly: 

G hkl = 2 k sin θ 

Use a continuous beam 

of mono-energetic 

neutrons.

Neutron Powder Diffraction using Time-of-Flight 

Pulsed 

source 

L o = 9-100m 

F 


bank 

Sample 

L 1 ~ 1-2m 

l = 2dsinQ 

2Q - fixed 

Measure scattering as 

a function of time-offlight 

t = const*l

Time-of-Flight Powder Diffraction 

Use a pulsed beam with a 

broad spectrum of neutron 

energies and separate 

different energies (velocities) 

by time of flight.

Counts 

10.0 0.05 155.9 CPD RRRR PbSO4 1.909A neutron data 8.8 

Scan no. = 1 Lambda = 1.9090 Observed Profile 

X10E 3 

.5 1.0 1.5 2.0 2.5 

Compare X-ray & Neutron Powder Patterns 

X-ray Diffraction - CuKa 

Phillips PW1710 

• Higher resolution 

• Intensity fall-off at small d 

spacings 

• Better at resolving small 

lattice distortions 

1.0 

D-spacing, A 

2.0 3.0 4.0 

10.000 0.025 159.00 CPD RRRR PbSO4 Cu Ka X-ray data 22.9. 

Scan no. = 1 Lambda1,lambda2 = 1.540 Observed Profile 

Counts 

X10E 4 

.0 .5 1.0 1.5 

1.0 

D-spacing, A 

2.0 3.0 4.0 

Neutron Diffraction - D1a, ILL 

l=1.909 Å 

• Lower resolution 

• Much higher intensity at small 

d-spacings 

• Better atomic positions/thermal 

parameters

There’s more than meets the eye in a powder pattern* 

Intensity 

Rietveld Model 

I c = I b + SY p 

I o 

I c 

Y p 

TOF 

*Discussion of Rietveld method adapted from viewgraphs by R. Vondreele (LANSCE) 

I b

The Rietveld Model for Refining Powder Patterns 

I c = I o {Sk h F 2 h m h L h P(D h ) + I b } 

I o - incident intensity - variable for fixed 2Q 

k h - scale factor for particular phase 

F 2 h 

- structure factor for particular reflection 

m h - reflection multiplicity 

L h - correction factors on intensity - texture, etc. 

P(D h ) - peak shape function – includes instrumental resolution, 

crystallite size, microstrain, etc.

How good is this function? 

Protein Rietveld refinement - Very low angle fit 

1.0-4.0° peaks - strong asymmetry 

“perfect” fit to shape

Examples of Science using Powder Diffraction 





High-resolution Neutron Powder Diffraction in CMR manganates* 

• In Sr 2LaMn 2O 7 – synchrotron data indicated 

two phases at low temperature. Simultaneous 

refinement of neutron powder data at 2 λ’s 

allowed two almost-isostructural phases (one 

FM the other AFM) to be refined. Only 

neutrons see the magnetic reflections 

• High resolution powder diffraction 

with Nd 0.7Ca 0.3MnO 3 showed 

splitting of (202) peak due to a 

transition to a previously unknown 

monoclinic phase. The data showed 

the existence of 2 Mn sites with 

different Mn-O distances. The 

different sites are likely occupied 

by Mn3+ and Mn4+ respectively 

* E. Suard & P. G. Radaelli (ILL)

Texture: “Interesting Preferred Orientation” 

Loose powder 

Metal wire 

Random powder - all crystallite orientations 

equally probable - flat pole figure 

Pole figure - stereographic projection of a 

crystal axis down some sample direction 

(100) random texture 

(100) wire texture 

Crystallites oriented along wire axis - pole figure 

peaked in center and at the rim (100’s are 90º 

apart) 

Orientation Distribution Function - probability 

function for texture

Texture Measurement by Diffraction 

Non-random crystallite 

orientations in sample 

Incident beam 

x-rays or neutrons 

Sample 

(220) 

Debye-Scherrer cones 

• uneven intensity due to texture 

• different pattern of unevenness for different hkl’s 

• intensity pattern changes as sample is turned 

(200) 

(111)

Texture Determination of Titanium Wire Plate 

(Wright-Patterson AFB/LANSCE Collaboration) 

Possible “pseudo-single crystal” turbine blade material 

Ti Wire - 

(100) texture 

• Bulk measurement 

• Neutron time-of-flight data 

• Rietveld refinement of texture 

TD 

• Spherical harmonics to L max = 16 

• Very strong wire texture in plate 

ND 

Ti Wire Plate - does it 

still have wire texture? 

RD 

ND 

TD 

reconstructed (100) pole figure

Definitions of Stress and Strain 

• Macroscopic strain – total strain measured by an extensometer 

• Elastic lattice strain – response of lattice planes to applied 

stress, measured by diffraction 

d - d 

e hkl = 

d 

hkl 0 

• Intergranular strain – deviation of elastic lattice strain from 

linear behavior 

• Residual strains – internal strains present with no applied force 

• Thermal residual strains – strains that develop on cooling from 

processing temperature due to anisotropic coefficients of 

thermal expansion 

0 

s 

e I =ehkl - 

E 

applied 

hkl

Neutron Diffraction Measurements of Lattice Strain* 

Text box 

The Neutron Powder 

Diffractometer at LANSCE 

Neutron measurements :- 

• Non destructive, bulk, phase sensitive 

• Time consuming, Limited spatial resolution 

s 

-90° Detector 

measures e parallel 

to loading direction 

Incident neutrons 

* Discussion of residual strain adapted from viewgraphs by M. Bourke (LANSCE) 

+90° Detector 

measures e ^ 

to loading direction 

s

Why use Metal Matrix Composites ? 

F117 (JSF) 

Landing Gear 

Higher Pay Loads 

High temperatures 

High pressures 

Reduced Engine Weights 

Reduced Fuel Consumption 

Better Engine Performance 

MMC Applications 

Rotors 

Fan Blades 

Structural Rods 

Impellers 

Landing Gears 

National AeroSpace Plane 

Engine

W-Fe :- Fabrication, Microstructure & Composition 

• 200 mm diameter continuous Tungsten fibers, 

• Hot pressed at 1338 K for 1 hour into Kanthal 

( 73 Fe, 21 Cr, 6 Al wt%) – leads to residual strains after cooling 

• Specimens :- 200 * 25 * 2.5 mm 3 

10 vol.% 

20 vol.% 

30 vol.% 

70 vol.% 

Normalized Intensity 

Powder diffraction data includes Bragg peaks 

from both Kanthal and Tungsten. 

K 

W 

d spacing (A) 

K 

W

Neutron Diffraction Measures Mean Residual Phase 

Strains when Results for a MMC are compared to an 

Undeformed Standard 

Residual Strain (me) 

Residual strain (me) 

4000 

3000 

2000 

1000 

0 

-1000 

-2000 

-3000 

-4000 

0 20 40 60 80 100 

Symbols show data points 

Angle to the fiber (deg) 

Kanthal 

10 vol.% W 

20 vol.% W 

30 vol.% W 

70 vol.% W 

Tungsten 

10 vol.% W 

20 vol.% W 

30 vol.% W 

70 vol.% W 

Curves are fits of the form =< e 11 >cos 2 a + < e 22 >sin 2 a 

Pressure Vessels and Piping, Vol. 373, “Fatigue, Fracture & Residual Stresses”, Book No H01154

Load Sharing in MMCs can also be 

Measured by Neutron Diffraction 

• Initial co-deformation results 

from confinement of W fibers 

by the Kanthal matrix 

• Change of slope (125 MPa) is 

the typical load sharing 

behavior of MMCs: 

– Kanthal yields and ceases to bear 

further load 

Applied stress [MPa] 

– Tungsten fibers reinforce and 

strengthen the composite 

50 

0 

-1000 -500 0 500 1000 1500 2000 2500 3000 

400 

350 

300 

250 

200 

150 

100 

Kanthal 

Tungsten 

Elastic lattice strain [με]

Deformation Mechanism in U/Nb Alloys 

Atypical “Double Plateau Stress Strain Curve. 

Applied Stress (MPa) 

500 

400 

300 

200 

100 

Tensile Axis 

0 

0 0.02 0.04 0.06 

Q || 

-90° Detector 

U-6 wt% Nb 

Typical Metal 

Macroscopic Strain 

NPD Diffraction Geometry 

Q ⊥ 

+ 90 ° Detector 

Diffraction Results 

• Important Programmatic Answers. 

1. Material is Single Phase Monoclinic (a’’). 

2. Lack of New Peaks Rules out Stress Induced Phase Transition. 

3. Changes in Peak Intensity Indicate That Twinning is the Deformation Mechanism.

Pair Distribution Functions 

• Modern materials are often disordered. 

• Standard crystallographic methods lose the aperiodic 

(disorder) information. 

• We would like to be able to sit on an atom and look at 

our neighborhood. 

• The PDF method allows us to do that (see next slide): 

– First we do a neutron or x-ray diffraction experiment 

– Then we correct the data for experimental effects 

– Then we Fourier transform the data to real-space

Obtaining the Pair Distribution Function* 

Raw data 

* See http://www.pa.msu.edu/cmp/billinge-group/ 

Structure function 

2 

G( r) 

∫ Q[ 

S( 

Q) 

1] 

sinQrdQ 

0 

∞ 

= − 

π 

PDF

Structure and PDF of a High Temperature 

Superconductor 

The structure of La 2-x Sr x CuO 4 

looks like this: (copper [orange] 

sits in the middle of octahedra 

of oxygen ions [shown shaded 

with pale blue].) 

The resulting PDFs look like this. 

The peak at 1.9A is the Cu-O 

bond. 

So what can we learn about 

charge-stripes from the PDF?

Effect of Doping on the Octahedra 

• Doping holes (positive 

charges) by adding Sr 

shortens Cu-O bonds 

• Localized holes in stripes 

implies a coexistence of 

short and long Cu-O inplane 

bonds => increase in 

Cu-O bond distribution 

width with doping. 

• We see this in the PDF: σ 2 

is the width of the CuO 

bond distribution which 

increases with doping then 

decreases beyond optimal 

doping 

Polaronic 

Fermiliquid 

like 

Bozin et al. Phys. Rev. Lett. Submitted; 

cond-mat/9907017

LECTURE 3: Surface Reflection 

by 

Roger Pynn 

Los Alamos 


Surface Reflection Is Very Different From 

Most Neutron Scattering 

• We worked out the neutron cross section by adding scattering 

from different nuclei 

– We ignored double scattering processes because these are usually very weak 

• This approximation is called the Born Approximation 

• Below an angle of incidence called the critical angle, neutrons 

are perfectly reflected from a smooth surface 

– This is NOT weak scattering and the Born Approximation is not applicable to 

this case 

• Specular reflection is used: 

– In neutron guides 

– In multilayer monochromators and polarizers 

– To probe surface and interface structure in layered systems

This Lecture 

• Reflectivity measurements 

– Neutron wavevector inside a medium 

– Reflection by a smooth surface 

– Reflection by a film 

– The kinematic approximation 

– Graded interface 

– Science examples 

• Polymers & vesicles on a surface 

• Lipids at the liquid air interface 

• Boron self-diffusion 

• Iron on MgO 

– Rough surfaces 

• Shear aligned worm-like micelles

What Is the Neutron Wavevector Inside a Medium? 

[ ] 

materials. 

most 

from 

reflected 

externally 

are 

neutrons 

1, 

n 

generally 

Since 

2 

/ 

1 

: 

get 

we 

) 

A 

10 

(~ 

small 

very 

is 

and 

), 

definition 

(by 

index 

refractive 

Since 

material 

a 

in 

r 


the 

is 

and 

vevector 

neutron wa 

is 

where 

4 

/ 

) 

( 

2 

Simlarly 

. 

/ 

2 

so 

) 

( 

0 

) 

( 

/ 

) 

( 

2 

: 

equation 

s 

r' 

Schrodinge 

obeys 

neutron 

The 

(SLD) 

Density 

Length 

Scattering 

nuclear 

the 

called 

is 

1 

where 

2 

: 

is 

medium 

the 

inside 

potential 

average 

the 

So 

nucleus. 

single 

a 

for 

) 

( 

2 

) 

( 

: 

by 

given 

is 

potential 

neutron 

- 

nucleus 

the 

Rule, 

Golden 

s 

Fermi' 

by 

given 

with that 

S(Q) 

for 

expression 

our 

Comparing 

2 

2 

- 

6 

- 

0 

0 

2 

0 

2 

2 

2 

2 

0 

. 

2 

2 

2 

2 

< 

= 

= 

= 

− 

= 

− 

= 

= 

= 

= 

− 

+ 

∇ 

= 

= 

= 

∑ 

π 

ρ 

λ 

ρ 

πρ 

ψ 

ψ 

ρ 

ρ 

ρ 

π 

δ 

π 

- 

n 

n 

k/k 

k 

in vacuo 

k 

k 

V 

E 

m 

k 

mE 

k 

e 

r 

in vacuo 

r 

V 

E 

m 

b 

volume 

m 

V 

r 

b 

m 

r 

V 

r 

k 

i 

i 

i 

o h 

h 

h 

h 

r 

h 

r 

r 

r

Typical Values 

• Let us calculate the scattering length density for quartz – SiO 2 

• Density is 2.66 gm.cm -3 ; Molecular weight is 60.08 gm. mole -1 

• Number of molecules per Å 3 = N = 10 -24 (2.66/60.08)*N avagadro 

= 0.0267 molecules per Å 3 

• ρ=Σb/volume = N(b Si + 2b O ) = 0.0267(4.15 + 11.6) 10 -5 Å -2 = 

4.21 x10 -6 Å -2 

• This means that the refractive index n = 1 – λ 2 2.13 x 10 -7 for 

quartz 

• To make a neutron “bottle” out of quartz we require k= 0 I.e. 

k 0 2 = 4πρ or λ=(π/ρ) 1/2 . 

• Plugging in the numbers -- λ = 864 Å or a neutron velocity of 

4.6 m/s (you could out-run it!)

Only Those Thermal or Cold Neutrons With Very Low 

Velocities Perpendicular to a Surface Are Reflected 

α 

α’ 

nickel 

for 

) 

( 

1 

. 

0 

) 

( 

: 

Note 

quartz 

for 

) 

( 

02 

. 

0 

) 

( 

sin 

) 

/ 

2 

( 

A 

10 

05 

. 

2 

quartz 

For 

4 

is 

reflection 

external 

for total 

of 

value 

critical 

The 

4 

or 

4 

sin 

' 

sin 

i.e. 

4 

) 

sin 

(cos 

) 

' 

sin 

' 

(cos 

4 

Since 

Law 

s 

Snell' 

obey 

Neutrons 

' 

/cos 

cos 

n 

i.e 

' 

cos 

' 

cos 

cos 

: 

so 

surface 

the 


parallel 

locity 

neutron ve 

the 

change 

cannot 

surface 

The 

/ 

critical 

critical 

critical 

1 

- 

3 

0 

0 

2 

0 

2 

2 

2 

0 

2 

2 

2 

2 

2 

0 

2 

2 

2 

2 

0 

2 

0 

0 

0 

0 

0 

A 

A 

k 

x 

k 

k 

k 

k 

k 

k 

k 

k 

k 

k 

k 

n 

k 

k 

k 

n 

k 

k 

o 

o 

critical 

critical 

z 

z 

z 

z 

zl 

zl 

λ 

α 

λ 

α 

α 

λ 

π 

πρ 

πρ 

πρ 

α 

α 

πρ 

α 

α 

α 

α 

πρ 

α 

α 

α 

α 

α 

≈ 

≈ 

⇒ 

= 

= 

= 

− 

= 

− 

= 

− 

+ 

= 

+ 

− 

= 

= 

= 

= 

= 

−

Reflection of Neutrons by a Smooth Surface: Fresnel’s Law 

n = 1-λ 2 ρ/2π 

T 

T 

R 

R 

I 

I 

T 

R 

I 

k 

a 

k 

a 

k 

a 

a 

a 

a 

r 

r 

r 

& 

= 

+ 

= 

+ 

⇒ 

= 

(1) 

0 

z 

at 

& 

of 

continuity 

ψ 

ψ 

) 

/( 

) 

( 

/ 

by 

given 

is 

e 

reflectanc 

so 

sin 

sin 

sin 

sin 

) 

( 

) 

( 

(3) 

& 

(1) 

cos 

cos 

: 

Law 

s 

Snell' 

(2) 

& 

(1) 

(3) 

sin 

sin 

) 

( 

(2) 

cos 

cos 

cos 

: 

surface 

the 


parallel 

and 

lar 

perpendicu 

components 

Tz 

Iz 

Tz 

Iz 

I 

R 

Iz 

Tz 

R 

I 

R 

I 

T 

R 

I 

T 

R 

I 

k 

k 

k 

k 

a 

a 

r 

k 

k 

n 

a 

a 

a 

a 

n 

nk 

a 

k 

a 

a 

nk 

a 

k 

a 

k 

a 

+ 

− 

= 

= 

= 

′ 

≈ 

′ 

= 

+ 

− 

=> 

′ 

= 

=> 

′ 

− 

= 

− 

− 

′ 

= 

+ 

α 

α 

α 

α 

α 

α 

α 

α 

α 

α 

α

What do the Amplitudes a R and a T Look Like? 

• For reflection from a flat substrate, both a R and a T are complex when k 0 < 4πρ 

I.e. below the critical edge. For a I = 1, we find: 

1 

0.5 

-0.5 

-1 

0.005 0.01 0.015 0.02 0.025 

Real (red) & imaginary (green) parts of a R 

plotted against k 0 . The modulus of a R is 

plotted in blue. The critical edge is at 

k 0 ~ 0.009 A -1 . Note that the reflected wave is 

completely out of phase with the incident wave 

at k 0 = 0 

2 

1.5 

1 

0.5 

-0.5 

-1 

0.005 0.01 0.015 0.02 0.025 

Real (red) and imaginary (green) parts 

of a T . The modulus of a T is plotted in 

blue. Note that a T tends to unity at 

large values of k 0 as one would expect

V(z) 

One can also think about Neutron Reflection from a Surface as a 

substrate 

z 

1-d Problem 

Film Vacuum 

V(z)= 2 π ρ(z) h 2 /m n 

k 2 =k 0 2 - 4π ρ(z) 

Where V(z) is the potential seen by 

the neutron & ρ(z) is the scattering 

length density

Fresnel’s Law for a Thin Film 

• r=(k 1z-k 0z)/(k 1z+k 0z) is Fresnel’s law 

• Evaluate with ρ=4.10 -6 A -2 gives the 

red curve with critical wavevector 

given by k 0z = (4πρ) 1/2 

• If we add a thin layer on top of the 

substrate we get interference fringes & 

the reflectance is given by: 

r 

= 

r 

01 

1+ 

+ r 

r 

01 

12 

r 

12 

i2k 

1z 

i2k 

and we measure the reflectivity R = r.r* 

e 

e 

t 

1z 

t 

0.01 0.02 0.03 0.04 0.05 

• If the film has a higher scattering length density than the substrate we get the 

green curve (if the film scattering is weaker than the substance, the green curve is 

below the red one) 

• The fringe spacing at large k 0z is ~ π/t (a 250 A film was used for the figure) 

-1 

-2 

-3 

-4 

-5 

Log(r.r*) 

0 

1 

2 

substrate 

k 0z 

Film thickness = t

Kinematic (Born) Approximation 

• We defined the scattering cross section in terms of an incident plane wave & a 

weakly scattered spherical wave (called the Born Approximation) 

• This picture is not correct for surface reflection, except at large values of Q z 

• For large Q z , one may use the definition of the scattering cross section to 

calculate R for a flat surface (in the Born Approximation) as follows: 

R 

= 

= 

L 

because 

number of 

number of 

sin α 

From the definition 

dσ 

dΩ 

x 

L 

= 

σ 

y 

ρ 

2 

k 

x 

= 

= 

k 

0 

neutrons reflected by a sample of 

neutrons incident 

L 

x 

L 

y 

1 

cosα 

r 

r 

∫ dr∫ 

dr 

'e 

sin α 

of 

so 

iQ.( 

r −r ') 

r v r 

∫ 

dk 

2 

x 

4π 

Q 

on sample ( = ΦL 

L 

2 

2 

z 

It is easy to show that this is the same as the Fresnel form at large Q 

0 

1 

sin α 

sin α 

dα. 

a cross section we get for a smooth substrate : 

= 

dσ 

dΩ 

= 

dΩ 

ρ 

= −k 

L 

L 

x 

x 

L 

y 

y 

L δ ( Q 

x 

∫ 

) δ ( Q 

size 

x 

dσ 

dΩ 

y 

) 

y 

L 

2 

0 

so 

x 

L 

y 

sinα 

) 

dk 

k 

x 

dk 

y 

sin α 

R = 16π 

2 

ρ 

z 

2 

/ Q 

4 

z

Reflection by a Graded Interface 

ion. 

justificat 

physical 

good 

have 

must 

data 

ty 

refelctivi 

fit 


refined 

models 

that 

means 

This 

n. 

informatio 

phase 

important 

lack 

generally 

we 

because 

(z), 

obtain 

o 

uniquely t 

inverted 

be 

cannot 

data 

ty 

reflectivi 

: 

point 

important 

an 

s 

illustrate 

equation 

e 

approximat 

This 

(z). 

1/cosh 

as 

such 

/dz 

d 

of 

forms 

convenient 

several 

for 

ly 

analytical 

solved 

be 

can 

This 

) 

( 

Q 

large 

at 

form 

correct 

the 

as 

well 

as 

interface, 

smooth 

a 

for 

answer 

right 

get the 

we 

, 

R 

ty 

reflectivi 

Fresnel 

by the 

prefactor 

the 

replace 

we 

If 

parts. 

by 

ng 

intergrati 

after 

follows 

equality 

second 

the 

where 

) 

( 

16 

) 

( 

16 

: 

gives 

of 

dependence 

- 

z 

the 

keeping 

but 

viewgraph 

previous 

the 

of 

line 

bottom 

the 

Repeating 

2 

2 

z 

F 

2 

4 

2 

2 

2 

2 

ρ 

ρ 

ρ 

ρ 

π 

ρ 

π 

ρ 

∫ 

∫ 

∫ 

= 

= 

= 

dz 

e 

dz 

z 

d 

R 

R 

dz 

e 

dz 

z 

d 

Q 

dz 

e 

z 

Q 

R 

z 

iQ 

F 

z 

iQ 

z 

z 

iQ 

z 

z 

z 

z

The Goal of Reflectivity Measurements Is to Infer a 

Density Profile Perpendicular to a Flat Interface 

• In general the results are not unique, but independent 

knowledge of the system often makes them very reliable 

• Frequently, layer models are used to fit the data 

• Advantages of neutrons include: 

– Contrast variation (using H and D, for example) 

– Low absorption – probe buried interfaces, solid/liquid interfaces etc 

– Non-destructive 

– Sensitive to magnetism 

– Thickness length scale 10 – 5000 Å

Direct Inversion of Reflectivity Data is Possible* 

• Use different “fronting” or “backing” materials for two 

measurement of the same unknown film 

– E.g. D 2 O and H 2 O “backings” for an unknown film deposited on a quartz 

substrate or Si & Al 2 O 3 as substrates for the same unknown sample 

– Allows Re(R) to be obtained from two simultaneous equations for 

R 

1 

2 

and 

R 

2 

2 

– Re(R) can be Fourier inverted to yield a unique SLD profile 

• Another possibility is to use a magnetic “backing” and 

polarized neutrons 

SiO 2 

Unknown film 

H2O or D2O 

Si or Al 2 O 3 substrate 

* Majkrzak et al Biophys Journal, 79,3330 (2000)

Vesicles composed of DMPC molecules fuse creating almost a perfect 

lipid bilayer when deposited on the pure, uncoated quartz block* 

(blue curves) 

When PEI polymer was added only after quartz was covered by the lipid 

bilayer, the PEI appeared to diffuse under the bilayer (red curves) 

4 

Reflectivity, R*Q z 

10 -8 

10 -9 

Neutron Reflectivities 

Lipid Bilayer on Quartz 

Lipid Bilayer on Polymer 

on Quartz 

0.00 0.02 0.04 0.06 0.08 0.10 0.12 

Q z [Å -1 ] 

* Data courtesy of G. Smith (LANSCE) 

Scattering Length Density [Å -2 ] 

2 10 -6 

0 

-2 10 -6 

-4 10 -6 

-6 10 -6 

-8 10 -6 

2 O 

D 2 O 

Scattering Length Density 

Profiles 

Head 

σ = 5.9 Å 

Hydrogenated 

Tails 

Head 

Length, z [Å] 

Polymer 

σ = 4.3 Å 

Quartz 

Quartz 

0.0 20.0 40.0 60.0 80.0 100.0

4 

R*Q z 

10 -7 

10 -8 

10 -9 

1.3% PEG lipid 

in lipids 

Neutron Reflectivities 

Pure lipid 

Lipid + 9% PEG 


in lipids 

0 0.05 0.1 0.15 0.2 0.25 

Q z [Å -1 ] 

Polymer-Decorated Lipids at a Liquid-Air Interface* 

mushroom-tobrush 

transition 

*Data courtesy of G. Smith (LANSCE) 


in lipids 

neutrons see contrast between 

heads (2.6), tails (-0.4), 

D 2 O (6.4) & PEG (0.24) 

x-rays see heads (0.65), but all 

else has same electron density 

within 10% (0.33) 

SLD [Å -2 ] 

Interface broadens as PEG 

concentration increases - this is 

main effect seen with x-rays 

R/R F 

8 10 -6 

6 10 -6 

4 10 -6 

2 10 -6 

0 

10 1 

10 0 

10 -1 

10 -2 

10 -3 

SLD Profiles of PEG-Lipids 

Black - pure lipid 

Blue - 1.3% PEG 

Green - 4.5% PEG 

Red - 9.0% PEG 

40 60 80 100 120 140 160 

Length [Å] 

X-Ray Reflectivities 

Lipid + 9% PEG 

Pure lipid 

0 0.1 0.2 0.3 

Q 

z 

0.4 0.5 0.6

~650Å 

~1350Å 

Reflectivity 

Non-Fickian Boron Self-Diffusion at an Interface* 

10 

0.1 

0.001 

10 -5 

10 -7 

10 -9 

10 -11 

0 0.02 0.04 0.06 0.08 0.1 0.12 

11 B 

10 B 

Si 

Boron Self Diffusion 

Q (Å -1 

) 

Unnanealed sample 

2.4 Hours 

5.4 Hours 

9.7 Hours 

22.3 Hours 

35.4 Hours 

*Data courtesy of G. Smith (LANSCE) 

-2 ) 

Scattering Length Density (Å 

8 10 -6 

7 10 -6 

6 10 -6 

5 10 -6 

4 10 -6 

3 10 -6 

2 10 -6 

1 10 -6 

Non-Standard Fickian Model 

10 -4 

10 -5 

10 -6 

10 -7 

10 -8 

10 -9 

10 -10 

10 -11 

Data requires density 

step at interface 

0 

-800 -600 -400 -200 0 200 400 600 800 

Distance from 

11 B- 10 B Interface (Å) 

0.045 0.05 0.055 0.06 0.065 0.07 0.075 0.08 

-2 ) 

Scattering Length Density (Å 

8 10 -6 

7 10 -6 

6 10 -6 

5 10 -6 

4 10 -6 

3 10 -6 

2 10 -6 

1 10 -6 

Fickian diffusion 

doesn’t fit the data 

Standard Fickian Model 

0 

-600 -400 -200 0 200 400 600 800 

Distance from 

11 B- 10 B Interface (Å)

Polarized Neutron Reflectometry (PNR) 

Non-Spin-Flip Spin-Flip 

++ measures b + Mz - - measures b - Mz +- measures Mx + i My -+ measures Mx – i My

X-Ray Reflectivity 

Polarized Neutron Reflectivity 

10 0 

10 -1 

10 -2 

10 -3 

10 -4 

10 -5 

10 -6 

10 0 

10 -1 

10 -2 

10 -3 

10 -4 

Structure, Chemistry & Magnetism of Fe(001) on MgO(001)* 

0 0.1 0.2 0.3 0.4 0.5 

Q [Å -1 ] 

R ++ (obs) 

R ++ (fit) 

R -- (obs) 

R -- (fit) 

R SF (obs) 

R SF (fit) 

10 -5 

0.00 0.02 0.04 0.06 

Q [Å 

0.08 0.10 0.12 

-1 ] 

X-Ray 


*Data courtesy of M. Fitzsimmons (LANSCE) 

Co-Refinement 

α-FeOOH (1.8µ B ) 

Fe(001) (2.2µ 

B 

) 

fcc Fe (4µ B ) 

H 

X-Ray 

β = 3.9(3) 

β = 5.2(3) 

β = 4.4(2) 

β = 3.1(3) 

1.9(2)nm 

10.7(2)nm Fe 

2.1(2)nm 

0.20(1)nm 

a-FeOOH 

0.44(2)nm 

0.07(1)nm 

interface 

0.03(1)nm 

MgO 

TEM 


β = 4.4(8) 

β = 8.6(4) 

β = 6.9(4) 

β = 6.2(5) 

H

Reflection from Rough Surfaces 

z 

z = 0 

k 1 

θ 1 

• diffuse scattering is caused by surface roughness or inhomogeneities in 

the reflecting medium 

• a smooth surface reflects radiation in a single (specular) direction 

• a rough surface scatters in various directions 

• specular scattering is damped by surface roughness – treat as graded 

interface. For a single surface with r.m.s roughness σ: 

R 

= 

Fe R 

1 

t Izk z 

−2k 

σ 

2 

x 

θ 2 

k 2 

k t 1

When Does a “Rough” Surface Scatter Diffusely? 

• Rayleigh criterion 

path difference: Dr = 2 h sing 

phase difference: Df = (4ph/l) sing 

boundary between rough and smooth: Df = p/2 

that is h < l/(8sing) for a smooth surface 

γ 

γ 

γ 

γ 

where g = 4 p h sin g / l = Q z h 

h

fi fi fi 

Q =kf -ki 

Qx-Qz 

Transformation 

qi 

Time-of-Flight, Energy-Dispersive Neutron Reflectometry 

ki fi 

z 

where 

fi 

Q 

-ki fi 

kf fi 

qf 

2 p 

Q x = l *(cos qf -cos qi ) 

2 p 

Q z = 

l *(sin qf +sin qi ) 

For specular reflectivity qf =qi. Then, 

Qz= 

4 psin( q) 

=2k z 

l 

Vandium-Carbon Multilayer — specular & diffuse scattering 

in θ f -TOF space and transformed to Q x -Q z 

x 

Raw data in θf -TOF space for a single layer. 

Note that large divergence does not imply poor Qz resolution 

TOF l . L / 4 

Raw Data from SPEAR

The Study of Diffuse Scattering From Rough Surfaces Has 

Not Made Much Headway Because Interpretation Is Difficult 

The theory (Distorted Wave 

2 2 

from a rough surface, works in some cases but breaks down when ξ / k >> 1, 

where ξ is the range of 

In some 

cases (e.g. faceted surfaces) one would also expect the approximation 

using an " average 

Born Approximation) 

correlations 

in thesurface 

surface" wavefunction 

used to describe scattering 

for perturbation 

theory 

R 

R 

micro−rough 

facet 

= 

R 

k z 

of 

to break down. 

= 

R 

smooth 

smooth 

e 

−2 

k 

2 

0 

s 

e 

2 

−2k 

t 

0k1 

s 

a 

2

Observation of Hexagonal Packing of Thread-like Micelles 

Under Shear: Scattering From Lateral Inhomogeneities 

NEUTRON 

BEAM 

INLET 

HOLES 

46Å 

QUARTZ or SILICON 

TEST SECTION 

TEFLON LIP OUTLET 

TRENCH 

RESERVOIRS 

TEFLON 

Up to 

Microns 

Thread-like micelle 

H2O 

Specularly reflected beam 

Flow 

direction 

Quartz Single 

Crystal 

z 

x 

Scattering pattern 

implies hexagonal 

symmetry

LECTURE 5: Small Angle Scattering 

by 

Roger Pynn 

Los Alamos 


This Lecture 

5. Small Angle Neutron Scattering (SANS) 

1. What is SANS and what does it measure? 

2. Form factors and particle correlations 

3. Guinier approximation and Porod’s law 

4. Contrast and contrast variation 

5. Deuterium labelling 

6. Examples of science with SANS 

1. Particle correlations in colloidal suspensions 

2. Helium bubble size distribution in steel 

3. Verification of Gaussian statistics for a polymer chain in a melt 

4. Structure of 30S ribosome 

5. The fractal structure of sedimentary rocks 

Note: The NIST web site at www.ncnr.nist.gov has several good resources 

for SANS – calculations of scattering length densities & form factors as well 

as tutorials

Small Angle Neutron Scattering (SANS) Is Used to 

Measure Large Objects (~1 nm to ~1 μm) 

• Complex fluids, alloys, precipitates, 

biological assemblies, glasses, 

ceramics, flux lattices, long-wave- 

length CDWs and SDWs, critical 

scattering, porous media, fractal 

structures, etc 

Scattering at small angles probes 

large length scales

Two Views of the Components of a Typical 

Reactor-based SANS Diffractometer

The NIST 30m SANS Instrument Under Construction

Where Does SANS Fit As a Structural Probe? 

• SANS resolves structures 

on length scales of 1 – 1000 

nm 

• Neutrons can be used with 

bulk samples (1-2 mm thick) 

• SANS is sensitive to light 

elements such as H, C & N 

• SANS is sensitive to 

isotopes such as H and D

What Is the Largest Object That Can Be Measured by SANS? 

• Angular divergence of the neutron beam and its lack of monochromaticity contribute to finite 

transverse & longitudinal coherence lengths that limit the size of an object that can be seen 

by SANS 

• For waves emerging from slit center, 

path difference is hd/L if d

Instrumental Resolution for SANS 

Traditionally, neutron scatterers tend to think in terms of Q and E resolution 

4π 

Q = 

sin θ 

λ 

For SANS, ( δλ/ 

λ) 

For equal source - sample & sample - detector distances 

apertures at source and sample of h, δθ 

The smallest v alue of θ 

At this 

δQ 

rms 

~ ( δθ 

The largest 

value of θ , angular 

rms 

⇒ 

/ θ 

min 

δQ 

rms 

Q 

)Q 

observable 

is 

~ δθ 

This is equal to the transvers e coherence length for the neutron and achieves 

a maximum of about 5 μm 

at the ILL 40 m SANS instrument using 15 Å neutrons. 

Note that at the largest va lues of θ , set by the detector size and distance from the 

sample, wavelengt h resolution 

2 

2 

~ 5% andθ 

is small, 

min 

= 

δλ 

λ 

2 

determined 

resolution 

rms 

2 

+ 

cos 

rms 

4π 

/ λ ~ ( 2π 

/ λ) 

h / L 

object is ~ 2π/ 

δQ 

sin 

dominates. 

2 

θ. 

δθ 

so 

by the direct 

dominates 

rms 

= 

2 

θ 

2 

δQ 

2 

Q 

2 

5/ 

12h/L. 

and 

~ λh 

/ L . 

= 

0. 

0025 

of 

L 

+ 

and 

beam size : θ 

δθ 

θ 

equal 

min 

2 

2 

~ 1. 

5h 

/ L

ecall that 

where 

The Scattering Cross Section for SANS 

b 

nuclear density 

2 

ds 

dO 

= 

b 

2 

NS( 

Q) 

r r 1 

r 

−iQ. 

r 

where 

r 

S( 

Q) 

= 

r 

dr. 

e 

is the coherent nuclear scattering length and n 

r 

( r) 

is 

• Since the length scale probed at small Q is >> inter-atomic spacing we may use the 

scattering length density (SLD), ρ, introduced for surface reflection and note that 

n nuc (r)b is the local SLD at position r. 

• A uniform scattering length density only gives forward scattering (Q=0), thus SANS 

measures deviations from average scattering length density. 

• If ρ is the SLD of particles dispersed in a medium of SLD ρ 0 , and n p (r) is the 

particle number density, we can separate the integral in the definition of S(Q) into 

an integral over the positions of the particles and an integral over a single particle. 

We also measure the particle SLD relative to that of the surrounding medium I.e: 

N 

∫ 

nuc 

n 

nuc 

r 

( r ) 

2 

the

SANS Measures Particle Shapes and Inter-particle Correlations 

dσ 

dΩ 

= 

= 

b 

3 

∫ d R ∫ 

space space 

2 3 

∫ d r ∫ 

space space 

d 

3 

R' 

n 

P 

d 

3 

r' 

n 

r 

( R) 

n 

N 

P 

r 

( r ) n 

r 

( R') 

dσ 

= ( ρ − ρ 

dΩ 

0 

) 

2 

N 

e 

v 

F( 

Q) 

r 

( r '). 

e 

r r r 

iQ.( 

R− 

R') 

r 

F( 

Q) 

2 

r r r 

iQ.( 

r −r 

') 

2 

where G P is the particle - particle correlation 

function (the probability 

that the re 

r 

v 2 

is a particle at R if there's 

one at the origin) and F( Q) 

is the particle form factor 

= 

∫ 

( ρ − ρ 

N 

d 

particle 

3 

∫ 

P 

space 

x. 

e 

d 

0 

3 

) 

r v 

iQ. 

x 

R. 

G 

2 

∫ 

d 

particle 

P 

3 

x. 

e 

r v 

iQ. 

x 

r 

( R). 

e 

These expressions are the same as those for nuclear scattering except for the addition 

of a form factor that arises because the scattering is no longer from point-like particles 

2 

r r 

iQ. 

R 

:

Scattering for Spherical Particles 

particles. 

of 

number 

the 

times 

volume 

particle 

the 

of 

square 

the 


al 

proportion 

is 

)] 

r 

( 

) 

r 

G( 

when 

[i.e. 

particles 

spherical 

ed 

uncorrelat 

of 

assembly 

an 

from 

scattering 

total 

the 

0, 

Q 

as 

Thus, 

0 

Q 

at 

) 

( 

3 

) 

( 

cos 

sin 

3 

) 

( 

: 

Q 

of 

magnitude 

on the 

depends 

only 

F(Q) 

R, 

radius 

of 

sphere 

a 

For 

shape. 

particle 

by the 

determined 

is 

) 

( 

factor 

form 

particle 

The 

0 

1 

0 

3 

0 

2 

. 

2 

v 

r 

r 

r r 

r 

δ 

→ 

→ 

= 

→ 

≡ 

⎥ 

⎦ 

⎤ 

⎢ 

⎣ 

⎡ − 

= 

= ∫ 

V 

QR 

j 

QR 

V 

QR 

QR 

QR 

QR 

V 

Q 

F 

e 

r 

d 

Q 

F 

sphere 

V 

r 

Q 

i 

2 4 6 8 10 

0.2 

0.4 

0.6 

0.8 

1 

3j 1 (x)/x 

x 

Q 

and 

(a) 

axis 

major 

the 

between 

angle 

the 

is 

where 

) 

cos 

sin 

( 

: 

by 

R 

replace 

particles 

elliptical 

For 

2 

/ 

1 

2 

2 

2 

2 

r 

ϑ 

ϑ 

ϑ b 

a 

R + 

→

Examples of Spherically-Averaged Form Factors 

R 

H 

Fo rm Fact or fo r a Ve s icle 

2 

2 

R j1( 

QR) 

− r j1( 

Qr) 

F( 

Q, 

R, 

r) 

= 3V0 

2 3 3 

Q ( R − r ) 

Form Factor for Cylinder with Q at angle q to cylinder axis 

4V 

F ( Q, 

θ, 

H , R) 

= 

r 

0 

t = R - r 

sin([ QH / 2] 

cosθ 

) J1( 

QR sinθ 

) 

2 

QH cosθ 

( QR sinθ 

) 

R

Determining Particle Size From Dilute Suspensions 

• Particle size is usually deduced from dilute suspensions in which inter-particle 

correlations are absent 

• In practice, instrumental resolution (finite beam coherence) will smear out 

minima in the form factor 

• This effect can be accounted for if the spheres are mono-disperse 

• For poly-disperse particles, maximum entropy techniques have been used 

successfully to obtain the distribution of particles sizes

Correlations Can Be Measured in Concentrated Systems 

• A series of experiments in the late 1980’s by Hayter et al and Chen et al 

produced accurate measurements of S(Q) for colloidal and micellar systems 

• To a large extent these data could be fit by S(Q) calculated from the mean 

spherical model using a Yukawa potential to yield surface charge and screening 

length

Size Distributions Have Been Measured 

for Helium Bubbles in Steel 

• The growth of He bubbles under neutron irradiation is a key factor limiting 

the lifetime of steel for fusion reactor walls 

– Simulate by bombarding steel with alpha particles 

• TEM is difficult to use because bubble are small 

• SANS shows that larger bubbles grow as the steel is 

annealed, as a result of coalescence of small bubbles 

and incorporation of individual He atoms 

SANS gives bubble volume (arbitrary units on the plots) as a function of bubble size 

at different temperatures. Red shading is 80% confidence interval.

Radius of Gyration Is the Particle “Size” Usually 

Deduced From SANS Measurements 

result. 

general 

this 

of 

case 

special 

a 

is 

sphere 

a 

of 


form 

for the 

expression 

that the 

ified 

easily ver 

is 

It 

/3. 

r 

is 

Q 

of 

ues 

lowest val 

at the 

data 

the 

of 

slope 

The 

. 

Q 

v 

ty) 

ln(Intensi 

plotting 

by 

or 

region) 

Guinier 

called 

- 

so 

(in the 

Q 

low 

at 

data 

SANS 


fit 

a 

from 

obtained 

usually 

is 

It 

. 

/ 

is 

gyration 

of 

radius 

the 

is 

r 

where 

... 

6 

1 

... 

. 

sin 

. 

sin 

cos 

2 

1 

.... 

) 

. 

( 

2 

1 

. 

) 

( 

: 

Q 

small 

at 


form 

the 

of 

definition 

in the 

l 

exponentia 

the 

expand 

and 

particle 

the 

of 

centroid 

the 

from 

measure 

we 

If 

2 

g 

2 

3 

3 

2 

g 

6 

0 

2 

2 

0 

3 

3 

2 

0 

0 

2 

2 

0 

3 

2 

3 

0 

. 

2 

2 

∫ 

∫ 

∫ 

∫ 

∫ 

∫ 

∫ 

∫ 

∫ 

= 

≈ 

⎥ 

⎥ 

⎦ 

⎤ 

⎢ 

⎢ 

⎣ 

⎡ 

+ 

− 

= 

⎥ 

⎥ 

⎥ 

⎥ 

⎦ 

⎤ 

⎢ 

⎢ 

⎢ 

⎢ 

⎣ 

⎡ 

+ 

− 

= 

+ 

− 

+ 

≈ 

= 

− 

V 

V 

g 

r 

Q 

g 

V 

V 

V 

V 

r 

Q 

i 

V 

r 

d 

r 

d 

R 

r 

e 

V 

r 

Q 

V 

r 

d 

r 

d 

r 

d 

d 

Q 

V 

r 

d 

r 

Q 

r 

d 

r 

Q 

i 

V 

e 

r 

d 

Q 

F 

r 

g 

π 

π 

θ 

θ 

θ 

θ 

θ 

r 

r 

r 

r 

r 

r 

r 

r

Guinier Approximations: Analysis Road Map 

* Viewgraph courtesy of Rex Hjelm 

2 

P(Q) = 

d ln P(Q) 

d ln(Q) 

Generalized Guinier approximation 

1; α = 0 

απ Q –α ; α = 1,2 ΔM 

2 2 

α0 exp RαQ – 

3 – α 

Derivative-log analysis 

= Q 

P(Q) 

d P(Q) 

dQ 

= –α – 

2 

3 – α R α 2 Q 2 

• Guinier approximations provide a 

roadmap for analysis. 

• Information on particle 

composition, shape and size. 

• Generalization allows for 

analysis of complex mixtures, 

allowing identification of domains 

where each approximation 

applies.

Contrast & Contrast Matching 

Water 

O D2O HO 2O * Chart courtesy of Rex Hjelm 

Both tubes contain borosilicate beads + 

pyrex fibers + solvent. (A) solvent 

refractive index matched to pyrex;. (B) 

solvent index different from both beads 

and fibers – scattering from fibers 

dominates

Hydrogen 

Isotope 

Isotopic Contrast for Neutrons 

Scattering Length 

b (fm) 

1 H -3.7409 (11) 

2 D 6.674 (6) 

3 T 4.792 (27) 

Nickel 

Isotope 

Scattering Lengths 

b (fm) 

58 Ni 15.0 (5) 

60 Ni 2.8 (1) 

61 Ni 7.60 (6) 

62 Ni -8.7 (2) 

64 Ni -0.38 (7)

Verification of of the Gaussian Coil Model for a Polymer Melt 

• One of the earliest important 

results obtained by SANS was 

the verification of that r g ~N -1/2 

for polymer chains in a melt 

• A better experiment was done 

3 years later using a small 

amount of H-PMMA in D-PMMA 

(to avoid the large incoherent 

background) covering a MW 

range of 4 decades

SANS Has Been Used to Study Bio-machines 

• Capel and Moore (1988) used the fact that 

prokaryotes can grow when H is replaced 

by D to produce reconstituted ribosomes 

with various pairs of proteins (but not 

rRNA) deuterated 

• They made 105 measurements of interprotein 

distances involving 93 30S protein 

pairs over a 12 year period. They also 

measured radii of gyration 

• Measurement of inter-protein distances 

is done by Fourier transforming the form 

factor to obtain G(R) 

• They used these data to solve the 

ribosomal structure, resolving ambiguities 

by comparison with electron microscopy

Porod Scattering 

function. 

n 

correlatio 

for the 

form 

(Debye) 

h this 

factor wit 

form 

the 

evaluate 


and 

r 

small 

at 

V)] 

A/(2 

a 

[with 

.. 

br 

ar 

- 

1 

G(r) 

expand 


is 

it 

obtain 

y to 

Another wa 

smooth. 

is 

surface 

particle 

the 

provided 

shape 

particle 

any 

for 

Q 

as 

holds 

and 

law 

s 

Porod' 

is 

This 

surface. 

s 

sphere' 

the 

of 

area 

the 

is 

A 

where 

2 

) 

( 

2 

9 

Thus 

) 

resolution 

by 

out 

smeared 

be 

will 

ns 

oscillatio 

(the 

average 

on 

2 

/ 

9 

Q 

as 

cos 

9 

cos 

sin 

9 

) 

( 

) 

( 

cos 

. 

sin 

9 

radius) 

sphere 

the 

is 

R 

where 

1/R 

Q 

(i.e. 

particle 

spherical 

a 

for 

Q 

of 

values 

large 

at 

) 

( 

F(Q) 

of 

behavior 

the 

examine 

us 

Let 

2 

4 

4 

2 

2 

2 

2 

2 

2 

2 

4 

2 

3 

2 

4 

2 

4 

2 

π 

π 

= 

+ 

+ 

= 

∞ 

→ 

= 

→ 

= 

∞ 

→ 

→ 

⎥ 

⎦ 

⎤ 

⎢ 

⎣ 

⎡ 

− 

= 

⎥ 

⎦ 

⎤ 

⎢ 

⎣ 

⎡ − 

= 

>> 

Q 

A 

QR 

V 

F(Q) 

V 

QR 

V 

QR 

QR 

QR 

V 

QR 

QR 

QR 

QR 

QR 

V 

(QR) 

F(Q) 

QR

Scattering From Fractal Systems 

• Fractals are systems that are “self-similar” under a change of scale I.e. R -> CR 

• For a mass fractal the number of particles within a sphere of radius R is 

proportional to R D where D is the fractal dimension 

Thus 

4pR 

2 

dR. 

G( 

R) 

= number of particles between distance R and R + dR = cR 

D−3 

∴ G( 

R) 

= ( c / 4π 

) R 

r 

r r r 

iQ. 

R 2π 

and S( 

Q) 

= ∫ dR. 

e G( 

R) 

= dR. 

R. 

sin QR.( 

c / 4 ) R 

Q ∫ π 

c 1 

= D 

2 Q 

dx. 

x 

const 

For a surface fractal, onecan 

prove that S( 

Q) 

∝ which reduces to the Porod 

6− 

Ds 

Q 

form for smooth surfaces 

∫ 

D−2 

const 

. sin x = D 

Q 

of dimension 2. 

D−3 

D−1 

dR

ln(I) 

Typical Intensity Plot for SANS From Disordered 

Systems 

Zero Q intercept - gives particle volume if 

concentration is known 

Guinier region (slope = -r g 2 /3 gives particle “size”) 

ln(Q) 

Mass fractal dimension (slope = -D) 

Porod region - gives surface area and 

surface fractal dimension 

{slope = -(6-D s )}

Sedimentary Rocks Are One of the Most 

Extensive Fractal Systems* 

Variation of the average number of SEM 

features per unit length with feature size. 

Note the breakdown of fractality (D s =2.8 

to 2.9) for lengths larger than 4 microns 

*A. P. Radlinski (Austr. Geo. Survey) 

SANS & USANS data from sedimentary rock 

showing that the pore-rock interface is a surface 

fractal (D s =2.82) over 3 orders of magnitude in 

length scale

References 

• Viewgraphs describing the NIST 30-m SANS instrument 

– http://www.ncnr.nist.gov/programs/sans/tutorials/30mSANS_desc.pdf

LECTURE 6: Inelastic Scattering 

by 

Roger Pynn 

Los Alamos 


We Have Seen How Neutron Scattering 

Can Determine a Variety of Structures 

crystals 

X-Ray 

β = 3.9(3) 

β = 5.2(3) 

β = 4.4(2) 

β = 3.1(3) 

0.20(1)nm 


1.9(2)nm a-FeOOH β = 4.4(8) 

10.7(2)nm Fe 

2.1(2)nm 

0.44(2)nm 

0.07(1)nm 

interface 

0.03(1)nm 

MgO 

but what happens when the atoms are moving? 

Can we determine the directions and 

time-dependence of atomic motions? 

Can well tell whether motions are periodic? 

Etc. 

β = 8.6(4) 

β = 6.9(4) 

β = 6.2(5) 

surfaces & interfaces disordered/fractals biomachines 

These are the types of questions answered 

by inelastic neutron scattering

The Neutron Changes Both Energy & Momentum 

When Inelastically Scattered by Moving Nuclei

The Elastic & Inelastic Scattering Cross 

Sections Have an Intuitive Similarity 

• The intensity of elastic, coherent neutron scattering is proportional to the 

spatial Fourier Transform of the Pair Correlation Function, G(r) I.e. the 

probability of finding a particle at position r if there is simultaneously a 

particle at r=0 

• The intensity of inelastic coherent neutron scattering is proportional to 

the space and time Fourier Transforms of the time-dependent pair 

correlation function function, G(r,t) = probability of finding a particle at 

position r at time t when there is a particle at r=0 and t=0. 

• For inelastic incoherent scattering, the intensity is proportional to the 

space and time Fourier Transforms of the self-correlation function, G s (r,t) 

I.e. the probability of finding a particle at position r at time t when the 

same particle was at r=0 at t=0

The Inelastic Scattering Cross Section 

Lovesey. 

and 

Marshal 

or 

Squires 

by 

books 

in the 

example, 

for 

found, 

be 

can 

Details 

us. 

ally tedio 

mathematic 

is 

operators) 

mechanical 

quantum 

commuting 

- 

non 

as 

treated 

be 


have 

functions 

- 

and 

s 

' 

the 

(in which 

functions 

n 

correlatio 

the 

of 

evaluation 

The 

)) 

( 

( 

)) 

0 

( 

( 

1 

) 

, 

( 

and 

) 

, 

( 

) 

0 

, 

( 

1 

) 

, 

( 

: 

case 

scattering 

elastic 

for the 

those 

similar to 

y 

intuitivel 

are 

that 

functions 

n 

correlatio 

the 

and 

) 

, 

( 

2 

1 

) 

, 

( 

and 

) 

, 

( 

2 

1 

) 

, 

( 

where 

) 

, 

( 

' 

. 

and 

) 

, 

( 

' 

. 

that 

Recall 

) 

. 

( 

) 

. 

( 

2 

2 

2 

2 

δ 

ρ 

δ 

δ 

ρ 

ρ 

π 

ω 

π 

ω 

ω 

σ 

ω 

σ 

ω 

ω 

r 

d 

t 

R 

R 

r 

R 

r 

N 

t 

r 

G 

r 

d 

t 

R 

r 

r 

N 

t 

r 

G 

dt 

r 

d 

e 

t 

r 

G 

Q 

S 

dt 

r 

d 

e 

t 

r 

G 

Q 

S 

Q 

NS 

k 

k 

b 

dE 

d 

d 

Q 

NS 

k 

k 

b 

dE 

d 

d 

j 

j 

j 

s 

N 

N 

t 

r 

Q 

i 

s 

i 

t 

r 

Q 

i 

i 

inc 

inc 

coh 

coh 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

r 

h 

r 

r 

r 

h 

r 

r 

r 

r 

r 

r 

r 

∑∫ 

∫ 

∫∫ 

∫∫ 

− 

+ 

− 

= 

+ 

= 

= 

= 

= 

⎟ 

⎟ 

⎠ 

⎞ 

⎜ 

⎜ 

⎝ 

⎛ 

Ω 

= 

⎟ 

⎟ 

⎠ 

⎞ 

⎜ 

⎜ 

⎝ 

⎛ 

Ω 

− 

−

Examples of S(Q,ω) and S s(Q,ω) 

• Expressions for S(Q,ω) and S s (Q,ω) can be worked out for a 

number of cases e.g: 

– Excitation or absorption of one quantum of lattice vibrational 

energy (phonon) 

– Various models for atomic motions in liquids and glasses 

– Various models of atomic & molecular translational & rotational 

diffusion 

– Rotational tunneling of molecules 

– Single particle motions at high momentum transfers 

– Transitions between crystal field levels 

– Magnons and other magnetic excitations such as spinons 

• Inelastic neutron scattering reveals details of the shapes of 

interaction potentials in materials

A Phonon is a Quantized Lattice Vibration 

• Consider linear chain of particles of mass M coupled by 

springs. Force on n’th particle is 

Fn = α 0 un 

+ α1( 

un−1 

+ un+ 

1) 

+ α 2( 

un−2 

+ un+ 

2) 

+ ... 

• Equation of motion is 

• Solution is: 

a 

First neighbor force constant displacements 

2π 

4π 

N 2π 

q = 

0 , ± , ± ,...... ± 

L L 2 L 

F = Mu& 

& 

i ( qna−ωt 

) 

2 

un ( t) 

= Aqe 

with ωq 

= 

Phonon Dispersion Relation: 

Measurable by inelastic neutron scattering 

n 

u 

n 

4 

M 

∑ 

ν 

α 

ν 

sin 

2 

1 

( νqa) 

2 

1.4 

1.2 

1 

0.8 

0.6 

0.4 

0.2 

w 

-1 -0.5 0.5 1 

qa/2π

Inelastic Magnetic Scattering of Neutrons 

• In the simplest case, atomic spins in a ferromagnet precess 

about the direction of mean magnetization 

r r r r 

+ 

H = J ( l − l ') 

S . S = H + hω 

b b 

with 

hω 

q 

∑ 

l, 

l' 

= 

2 

q Dq = ω h 

exchange coupling 

2S( 

J 

0 

− 

J 

q 

l 

) 

l' 

is the dispersion 

0 

where 

∑ 

q 

ground state energy 

J 

q 

= 

q 

∑ 

l 

q 

q 

r 

J ( l ) e 

relation for a ferromagnet 

Spin wave animation courtesy of A. Zheludev (ORNL) 

r r 

iq. 

l 

spin waves (magnons) 

Fluctuating spin is 

perpendicular to mean spin 

direction => spin-flip 

neutron scattering

Measured Inelastic Neutron Scattering Signals in Crystalline 

Solids Show Both Collective & Local Fluctuations* 

Spin waves – collective 

excitations 

* Courtesy of Dan Neumann, NIST 

Local spin resonances (e.g. ZnCr 2 O 4 ) 

Crystal Field splittings 

(HoPd 2 Sn) – local excitations

Measured Inelastic Neutron Scattering Signals in Liquids 

Generally Show Diffusive Behavior 

“Simple” liquids (e.g. water) 

Quantum Fluids (e.g. He in porous silica) 

Complex Fluids (e.g. SDS)

Measured Inelastic Neutron Scattering in Molecular 

Systems Span Large Ranges of Energy 

Vibrational spectroscopy 

(e.g. C 60 ) 

Molecular reorientation 

(e.g. pyrazine) 

Polymers Proteins 

Rotational tunneling 

(e.g. CH 3 I)

Atomic Motions for Longitudinal & Transverse Phonons 

Q Q 

Transverse phonon Longitudinal phonon 

r 

r 

= ( 0, 

0. 

1, 

0) 

a 

= ( 0. 

1, 

0, 

0) 

a 

e T 

r 

R 

l 

r 

Q = 

= 

r 

R 

l 0 

2π 

a 

a 

r 

+ e 

( 0. 

1, 

0, 

0) 

s 

e 

r r 

i( 

Q. 

R −ωt 

) 

l 

e L

Transverse Optic and Acoustic Phonons 

) 

. 

( 

0 t 

R 

Q 

i 

s 

lk 

lk 

l 

e 

e 

R 

R 

ω 

− 

+ 

= 

r 

r 

r 

r 

r 

a 

e 

a 

e 

blue 

red 

) 

0 

, 

14 

. 

0 

, 

0 

( 

) 

0 

, 

1 

. 

0 

, 

0 

( 

= 

= 

r 

r 

a 

e 

a 

e 

blue 

red 

) 

0 

, 

14 

. 

0 

, 

0 

( 

) 

0 

, 

1 

. 

0 

, 

0 

( 

− 

= 

= 

r 

r 

Acoustic Optic 

Q

Phonons – the Classical Use for Inelastic Neutron Scattering 

• Coherent scattering measures scattering from single phonons 

2 ⎛ d σ 

⎞ 

⎜ 

d dE ⎟ 

⎝ Ω ⎠ 

coh± 

1 

= σ 

coh 

• Note the following features: 

k 

k 

– Energy & momentum delta functions => see single phonons (labeled s) 

– Different thermal factors for phonon creation (n s +1) & annihilation (n s ) 

– Can see phonons in different Brillouin zones (different recip. lattice vectors, G) 

– Cross section depends on relative orientation of Q & atomic motions (e s ) 

– Cross section depends on phonon frequency (ω s ) and atomic mass (M) 

– In general, scattering by multiple 

excitations is either insignificant 

or a small correction (the presence of 

other phonons appears in the Debye- 

Waller factor, W) 

' 

2 

π 

MV 

0 

e 

− 2W 

∑ ∑ 

r r 

( Q. 

e 

ω 

s G s 

s 

) 

2 

( n 

s 

1 ± 1 

+ ) δ ( ω m ω 

2 

s 

r r r 

) δ ( Q − q − G )

The Workhorse of Inelastic Scattering Instrumentation at 

Reactors Is the Three-axis Spectrometer 

k F 

Q 

k I 

“scattering triangle”

The Accessible Energy and Wavevector 

Transfers Are Limited by Conservation Laws 

• Neutron cannot lose more than its initial kinetic energy & 

momentum must be conserved

Triple Axis Spectrometers Have Mapped Phonons 

Dispersion Relations in Many Materials 

• Point by point measurement in (Q,E) 

space 

• Usually keep either k I or k F fixed 

• Choose Brillouin zone (I.e. G) to maximize 

scattering cross section for phonons 

• Scan usually either at constant-Q 

(Brockhouse invention) or constant-E 

Phonon dispersion of 36 Ar

What Use Have Phonon Measurements Been? 

• Quantifying interatomic potentials in metals, rare gas solids, ionic 

crystals, covalently bonded materials etc 

• Quantifying anharmonicity (I.e. phonon-phonon interactions) 

• Measuring soft modes at 2 nd order structural phase transitions 

• Electron-phonon interactions including Kohn anomalies 

• Roton dispersion in liquid He 

• Relating phonons to other properties such as superconductivity, 

anomalous lattice expansion etc

Examples of Phonon Measurements 

Phonons in 36 Ar – validation 

of LJ potential 

Phonons in 110 Cd 

Roton dispersion in 4 He 

Kohn anomalies 

in 110 Cd 

Soft mode

Time-of-flight Methods Can Give Complete Dispersion Curves at a 

Single Instrument Setting in Favorable Circumstances 

CuGeO 3 is a 1-d magnet. With the unique axis parallel to the incident 

neutron beam, the complete magnon dispersion can be obtained

Much of the Scientific Impact of Neutron Scattering Has Involved 

the Measurement of Inelastic Scattering 

Ene rgy Trans fe r (me V) 

10000 

100 

1 

0.01 

10-4 

10-6 

Neutrons in Condens ed Matter Res earch 

Spallation SPSS - Chopper - Chopper Spectrometer 

ILL - without s pin-echo 

ILL - with s pin-e cho 





Viruses 

Metallurgical 

Systems 

Colloids 

Aggregate 

Motion 

Polymers 

and 

Biological 

Systems 

Coherent 

Modes in 

Glasses 

and Liquids 

Molecular 

Motions 

Membranes 

Proteins 

Critical 


Diffusive 

Modes 

Spin 

Waves 

Electron- 

phonon 

Interactions 

Crystal and 

Magnetic 

Structures 

Crystal 

Fields 

Slower 

Motions 

Resolved 

Molecular 

Reorientation 

Tunneling 

Spectroscopy 

Surface 

Effects? 

Itinerant 

Magnets 

Hydrogen 

Modes 

Molecular 

Vibrations 

Lattice 

Vibrations 

and 


Amorphous 

Systems 


Induced 

Excitations 

Momentum 

Distributions 

Accurate Liquid Structures 



0.001 0.01 0.1 1 10 100 

Q (Å -1 ) 

Energy & Wavevector Transfers accessible to Neutron Scattering

Neutron Spin Echo 

By 

Roger Pynn* 

Los Alamos National Laboratory 

*With contributions from B. Farago (ILL), S. Longeville (Saclay), 

and T. Keller (Stuttgart)

What Do We Need for a Basic Neutron Scattering Experiment? 

• A source of neutrons 

• A method to prescribe the wavevector of the neutrons incident on the sample 

• (An interesting sample) 

• A method to determine the wavevector of the scattered neutrons 

• A neutron detector 


Incident neutrons of wavevector k i Scattered neutrons of 

wavevector, k f 

Q 

k i 

k f 

Sample 

2 

h 2 

E = hω 

= ( ki 

− k 

2m 

We usually measure scattering 

as a function of energy (E) and 

wavevector (Q) transfer 

2 

f 

)

• Uncertainties in the neutron 

wavelength & direction of travel 

imply that Q and E can only be 

defined with a certain precision 

Instrumental Resolution 

• When the box-like resolution 

volumes in the figure are convolved, 

the overall resolution is Gaussian 

(central limit theorem) and has an 

elliptical shape in (Q,E) space 

• The total signal in a scattering 

experiment is proportional to the 

phase space volume within the 

elliptical resolution volume – the better the resolution, the smaller the 

resolution volume and the lower the count rate

The Goal of Neutron Spin Echo is to Break the Inverse 

Relationship between Intensity & Resolution 

• Traditional – define both incident & scattered wavevectors in order to 

define E and Q accurately 

• Traditional – use collimators, monochromators, choppers etc to define 

both k i and k f 

• NSE – measure as a function of the difference between appropriate 

components of k i and k f (original use: measure k i –k f i.e. energy change) 

• NSE – use the neutron’s spin polarization to encode the difference 

between components of k i and k f 

• NSE – can use large beam divergence &/or poor monochromatization to 

increase signal intensity, while maintaining very good resolution

The Underlying Physics of Neutron Spin Echo (NSE) 

Technology is Larmor Precession of the Neutron’s Spin 

• The time evolution of the expectation value of the spin of a 

spin-1/2 particle in a magnetic field can be B 

determined classically as: 

r 

ds 

r r 

= γs 

∧ B ⇒ ω L 

dt 

= γ B 

s 

−1 

−1 

γ = −2913 

* 2π 

Gauss . s 

• The total precession angle of the spin, φ, depends on the time 

the neutron spends in the field: φ = ω t 

B(Gauss) 

10 

ω L (10 3 rad.s -1 ) 

183 

L 

N (msec -1 ) 

29 

Turns/m for 

4 Å neutrons 

~29

Larmor Precession allows the Neutron Spin to be Manipulated 

using π or π/2 Spin-Turn Coils: Both are Needed for NSE 


the neutron spends in the B field 

φ = ω t = γBd 

/ v 

L 

d 

B 

Neutron velocity, v 

1 

Number of turns = 

. B[ Gauss]. 

d[ 

cm]. 

λ[ 

Angstroms] 

135.65

How does a Neutron Spin Behave when the 

Magnetic Field Changes Direction? 

When H rotates with 

frequency ω, H 0 ->H 1 

->H 2 …, and the spin 

trajectory is described 

by a cone rolling on the 

plane in which H moves 

• Distinguish two cases: adiabatic and sudden 

• Adiabatic –tan(δ) > 1 – large ω – spin precesses around new 

field direction – this limit is used to design spin-turn devices

Neutron Spin Echo (NSE) uses Larmor Precession to 

“Code” Neutron Velocities 

• A neutron spin precesses at the Larmor frequency in a 

magnetic field, B. ω L = γB 


the neutron spends in the field 

σ d r r 

H, 

φ = ω t = γBd 

L 

/ 

v 

B 

Neutron velocity, v 

1 

Number of turns = . B[ Gauss]. 

d[ 

cm]. 

λ[ 

Angstroms] 

135.65 

The precession angle φ is a measure of the neutron’s speed v

The Principles of NSE are Very Simple 

• If a spin rotates anticlockwise & then clockwise by the same 

amount it comes back to the same orientation 

– Need to reverse the direction of the applied field 

– Independent of neutron speed provided the speed is constant 

• The same effect can be obtained by reversing the precession 

angle at the mid-point and continuing the precession in the 

same sense 

– Use a π rotation 

• If the neutron’s velocity, v, is changed by the sample, its spin 

will not come back to the same orientation 

– The difference will be a measure of the change in the neutron’s speed or 

energy

In NSE*, Neutron Spins Precess Before and After Scattering & a 

y Polarization Echo is Obtained if Scattering is Elastic 

z x π/2 

F S 

π/2 

π 

Initially, 

neutrons 

are polarized 

along z 

F 

Rotate spins into 

x-y precession plane 

Final Polarization, 

P 

S 

Allow spins to 

precess around z: 

slower neutrons 

precess further over 

a fixed path-length 

= 

cos( φ − 

Rotate spins 

through π about 

x axis 

1 φ2 

) 

Elastic 


Event 

Rotate spins 

to z and 

measure 

polarization 

Allow spins to precess 

around z: all spins are in 

the same direction at the 

echo point if ∆E = 0 

* F. Mezei, Z. Physik, 255 (1972) 145

For Quasi-elastic Scattering, the Echo 

Polarization depends on Energy Transfer 

• If the neutron changes energy when it scatters, the precession 

phases before & after scattering, φ 1 & φ 2, will be different: 

1 2 2 

using hω 

= m( 

v1 

− v2 

) ≈ mvδv 

2 

2 3 

⎛ 1 1 ⎞ γBd 

γBdhω 

γBdm 

λ ω 

φ1 

−φ 

2 = γBd 

⎜ − ≈ δv 

≈ = 

2 

3 

2 

v1 

v ⎟ 

⎝ 2 ⎠ v mv 2πh 

• To lowest order, the difference between φ 1 & φ 2 depends only 

on ω (I.e. v 1 – v 2) & not on v 1 & v 2 separately 

• The measured polarization, , is the average of cos(φ 1 - φ 2) 

over all transmitted neutrons I.e. 

P 

= 

∫∫ 

I 

r 

λ) 

S( 

Q, 

ω) 

cos( φ1 

−φ 

) dλdω 

r 

I( 

λ) 

S( 

Q, 

ω) 

dλdω 

( 2 

∫∫

Neutron Polarization at the Echo Point is a 

Measure of the Intermediate Scattering Function 

r 

∫∫ I( 

λ) 

S( 

Q, 

ω) 

cos( φ1 

−φ 

2 ) dλdω 

r 

r 

P = r 

≈ ∫ S( 

Q, 

ω) 

cos( ωτ ) dω 

= I( Q,τ) 

∫∫ I( 

λ) 

S( 

Q, 

ω) 

dλdω 

Bd λ τ 

2 

m 3 

where the " spin echo time" τ = γBd 

λ 

(T.m) 

(nm) (ns) 

2 

2πh 

1 0.4 12 

• I(Q,t) is called the intermediate scattering function 

– Time Fourier transform of S(Q,ω) or the Q Fourier transform of G(r,t), the two 

particle correlation function 

• NSE probes the sample dynamics as a function of time rather than 

as a function of ω 

• The spin echo time, τ, is the “correlation time” 

1 

1 

0.6 

1.0 

40 

186

Neutron counts ( per 50 sec) 

Neutron Polarization is Measured using an 

Asymmetric Scan around the Echo Point 

2000 

1500 

1000 

500 

0 

0 1 2 3 4 5 6 7 

Bd Current in coil 3 

2 (Arbitrary Units) 

Echo Point 

The echo amplitude decreases 

when (Bd) 1 differs from (Bd) 2 

because the incident neutron 

beam is not monochromatic. 

For elastic scattering: 

⎡γm 

⎤ 

P ~ ∫ I( 

λ) cos⎢ 

1 2 . d 

h 

⎥ 

⎣ 

⎦ 

{ ( Bd) 

− ( Bd) 

} λ λ 

Because the echo point is the same for all neutron wavelengths, 

we can use a broad wavelength band and enhance the signal intensity

What does a NSE Spectrometer Look Like? 

IN11 at ILL was the First

Field-Integral Inhomogeneities cause τ to vary 

over the Neutron Beam: They can be Corrected 

• Solenoids (used as main precession fields) have fields that vary as r 2 

away from the axis of symmetry because of end effects (div B = 0) 

solenoid 

B 

• According to Ampere’s law, a current 

distribution that varies as r 2 can 

correct the field-integral inhomogeneities 

for parallel paths 

• Similar devices can be used 

to correct the integral along 

divergent paths 

Fresnel correction coil for IN15

Energy Transfer (meV) 

10000 

100 

1 

0.01 

10 -4 

10 -6 

Neutrons in Condens ed Matter Res earch 

Spallation SPSS - Chopper - Chopper Spectrometer 

ILL - w ithou t s pin -ec ho 

ILL - with s pin-ech o 





Viruses 

Metallurgical 

Systems 

Colloids 

Aggregate 

Motion 

Polymers 

and 

Biological 

Systems 

Coherent 

Modes in 

Glasses 

and Liquids 

Molecular 

Motions 

Membranes 

Proteins 

Critical 


Diffusive 

Modes 

Spin 

Waves 

Electron- 

phonon 

Interactio ns 

Crystal and 

Magnetic 

Structures 

Crystal 

Fields 

Slower 

Motions 

Resolved 

Molecular 

Reorientation 

Tunneling 

Spectroscopy 

Surface 

Effects? 

Itinerant 

Magnets 

Hydrogen 

Modes 

Mo lecu lar 

Vibrations 

Lattice 

Vibrations 

and 


Amorphous 

Systems 


Induced 

Excitations 

Momentum 

Distributions 

Accurate Liquid Structures 



0.001 0.01 0.1 1 10 100 

Q (Å -1 ) 

Neutron Spin Echo has significantly extended the (Q,E) range to which 

neutron scattering can be applied

Neutron Spin Echo study of Deformations 

of Spherical Droplets* 

* Courtesy of B. Farago

The Principle of Neutron Resonant Spin Echo 

• Within a coil, the neutron is subjected to a 

steady, strong field, B0, and a weak rf field 

B1cos(ωt) with a frequency ω = ω0 = γ B0 – Typically, B0 ~ 100 G and B1 ~ 1 G 

• In a frame rotating with frequency ω 0, the neutron spin sees a constant field 

of magnitude B 1 

• The length of the coil region is chosen so that the neutron spin precesses 

around B1 thru an angle π. 

• The neutron precession phase is: 

φ 

exit 

neutron 

= φ 

= 2φ 

exit 

RF 

entry 

RF 

+ ( φ 

−φ 

entry 

RF 

−φ 

entry 

neutron 

entry 

neutron 

) 

+ ω d / v 

0 

S 

entry 

S 

B 0 

exit 

S 

B 1 (t) 

rotated π 

exit 

B 1 

entry 

B 1

Neutron Spin Phases NRSE spectrometer in an NRSE Spectrometer* 

n 

B= B=0 

ÌÐ ËÔÒ ÓÖÒØØÓÒ 

ÌÑ Ø È× ¬Ð Ö ÒÙØÖÓÒ ËÔÒ Ô× Ë 

Ø Ø 

A B C D 

B 0 

tA tA’ t tB 

Sample 

B=0 

+ + + + 

+ + + + 

d lAB d d lCD d 

Ø Ø 

 

Ú 

Ø Ø 

Ú 

Ø Ø 

Ð 

Ú 

 

Ø Ø 

Ú 

Ø Ø 

Ð 

Ú 

 

Ø Ð 

Ú 

 

Ø Ø Ð 

Ú 

 

Ø Ø 

 

Ú 

Ø 

Ú Ø Ð 

Ú 

 

Ø Ø 

Ð 

Ú 

 

Ø 

Ú Ø Ð 

Ú 

 

Ø Ø 

Ð 

Ú 

 

Ø Ð 

Ú 

Ð 

Ú 

 

B 0 

B0 

B=0 

B0 

tC tC’ t tD 

Echo occurs for elastic 

scattering when 

l AB + d = l CD + d 

* Courtesy of S. Longeville

H = 0 

Just as for traditional NSE, if the 

scattering is elastic, all neutron spins 

arrive at the analyzer with unchanged 

polarization, regardless of neutron 

velocity. 

If the neutron velocity changes, the 

neutron beam is depolarized

The Measured Polarization for NRSE is given by 

an Expression Similar to that for Classical NSE 

• Assume that v’ = v + δv with δv small and expand to lowest order, giving: 

P 

= 

where 

r 

I( 

λ) 

S( 

Q, 

ω) 

cos( ωτ 

∫∫ r 

∫∫ 

the " spin 

I( 

λ) 

S( 

Q, 

ω) 

dλdω 

echo 

time" 

τ 

NRSE 

NRSE 

) dλdω 

= 2γB 

• Note the additional factor of 2 in the echo time compared with classical NSE 

(a factor of 4 is obtained with “bootstrap” rf coils) 

• The echo is obtained by varying the distance, l, between rf coils 

• In NRSE, we measure neutron velocity using fixed “clocks” (the rf coils) 

whereas in NSE each neutron “carries its own clock” whose (Larmor) rate is 

set by the local magnetic field 

0 

2 

m 

( l + d) 

2πh 

2 

λ 

3

An NRSE Triple Axis Spectrometer at HMI: 

Note the Tilted Coils

Measuring Line Shapes for Inelastic Scattering 

• Spin echo polarization is the FT of 

scattering within the spectrometer 

transmission function 

• An echo is obtained when 

d 

dk 

1 

⎡( 

Bd) 

⎢ 

⎣ k1 

1 

2 

• Normally the lines of constant spinecho 

phase have no gradient in Q,ω 

space because the phase depends 

only on 

k 

( Bd) 

− 

k 

• The phase lines can be tilted by using 

“tilted” precession magnets 

2 

⎤ 

⎥ 

⎦ 

= 

0 

⇒ 

N 

k 

1 

2 

1 

= 

N 

k 

2 

2 

2 

ω 

ω 

Q 

Q

By “Tilting” the Precession-Field Region, Spin Precession Can Be 

Used to Code a Specific Component of the Neutron Wavevector 

If a neutron passes through a 

rectangular field region at an 

angle, its total precession phase 

will depend only on k⊥ . 

ω 

L 

= γB 

φ = ω t 

L 

= γB 

d 

= 

vsin 

χ 

KBd 

k 

with K = 0.291 (Gauss.cm.Å) -1 

⊥ 

d 

B 

χ 

k // 

Stop precession here 

Start precession here 

k 

k 

⊥

“Phonon Focusing” 

• For a single incident neutron wavevector, k I , neutrons are 

scattered to k F by a phonon of frequency ω o and to k f by 

neighboring phonons lying on the “scattering surface”. 

– The topology of the scattering surface is related to that of the phonon dispersion 

surface and it is locally flat 

• Provided the edges of the NSE precession field region are 

parallel to the scattering surface, all neutrons with scattering 

wavevectors on the scattering surface will have equal spin-echo 

phase 

k I 

Q 0 

k F k f 

k f⊥ 

scattering surface 

Precession field region

“Tilted Fields” 

• Phonon focusing using tilted fields is available at ILL and in 

Japan (JAERI)….however, 

• The technique is more easily implemented using the NRSE 

method and is installed as an option on a 3-axis spectrometers 

at HMI and at Munich 

• Tilted fields can also be used can also be used for elastic 

scattering and may be used in future to: 

– Increase the length scale accessible to SANS 

– Separate diffuse scattering from specular scattering in reflectometry 

– Measure in-plane order in thin films 

– Improve Q resolution for diffraction

An NRSE Triple Axis Spectrometer at HMI: 

Note the Tilted Coils

Nanoscience & Biology Need Structural 

Probes for 1-100 nm 

10 nm holes in PMMA 

Actin 

1µ 

Si colloidal crystal 

CdSe nanoparticles 

2µ 

10µ 

Structures over many length 

scales in self-assembly of 

ZnS and cloned viruses 

Peptide-amphiphile nanofiber 

Thin copolymer films

“Tilted” Fields for Diffraction: SANS 

d 

B 

χ 

χ 

B 

θ 

π/2 π 

π/2 

• Any unscattered neutron (θ=0) experiences the same precession angles 

(φ 1 and φ 2) before and after scattering, whatever its angle of incidence 

• Precession angles are different for scattered neutrons 

KBd 

φ1 

= and 

k sin χ 

P 

= 

∫ 

φ 1 

φ 

2 

k sin( χ + θ ) 

⎡ KBd cos χ ⎤ 

dQ. 

S( 

Q). 

cos⎢ 

Q 

2 2 ⎥ 

⎢⎣ 

k sin χ ⎥⎦ 

= 

KBd 

Spin Echo Length, 

φ 2 

⎡ KBd cos χ ⎤ 

⇒ cos( φ1 

−φ 

2) 

≈ cos⎢ 

θ 

2 ⎥ 

⎢⎣ 

k sin χ ⎥⎦ 

Polarization proportional to 

Fourier Transform of S(Q) 

r = 

KBd cos χ /( k sin χ) 

2

How Large is the Spin Echo Length for SANS? 

Bd/sinχ 

(Gauss.cm) 

3,000 

5,000 

5,000 

5,000 

λ 

(Angstroms) 

4 

4 

6 

6 

χ 

(degrees) 

20 

20 

20 

10 

r 

(Angstroms) 

1,000 

1,500 

3,500 

7,500 

It is relatively straightforward to probe length scales of ~ 1 micron

Spin Echo SANS using Magnetic Films: 

200 nm Correlation Distance Measured

Close up of SESAME (Spin Echo Scattering 

Angle Measurement) Apparatus

Conclusion: 

NSE Provides a Way to Separate Resolution from 

Monochromatization & Beam Divergence 

• The method currently provides the best energy resolution for 

inelastic neutron scattering (~ neV) 

– Both classical NSE and NRSE achieve similar energy resolution 

– NRSE is more easily adapted to “phonon focusing” I.e. measuring the energy 

line-widths of phonon excitations 

• The method is likely to be used in future to improve (Q) 

resolution for elastic scattering 

– Extend size range for SANS (SESANS) 

– May allow 100 – 1000 x gain in measurement speed for some SANS exps 

– Separate specular and diffuse scattering in reflectometry 

– Measure in-plane ordering in thin films (SERGIS)

An Introduction to Neutron Scattering - Spallation Neutron Source

Create successful ePaper yourself

Delete template?

Save as template?