PHYS01200804001 Sohrab Abbas - Homi Bhabha National Institute

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factor exp(-W). The centre, θ C , of the total reflectivity range differs from θ B by LI 1 0 C B (1 ) k b 2sin 2 , (24) 0 B a deviation brought about by the average refractive index, (1–χ O /2). As mentioned, K O and K H have to be complex even for a nonabsorbing crystal, over the total reflectivity regime. Since the incident wave vector k O in vacuum is real, the boundary condition I) dictates that the imaginary component K O of K O must necessarily lie along the surface normal. Moreover K H = K O , since K H and K O differ by a real vector H. Thus both the incident and the diffracted wave intensities must attenuate in a direction perpendicular to the crystal surface as exp (-2K O Z), Z being the depth in the crystal, with K sin sin( )sin( ) " H 0 B S B S . (25) The attenuation (and not absorption) is a direct consequence of the diversion of the incident wave intensity into the diffracted wave at the expense of penetration into the crystal. This phenomenon is known as primary extinction, to distinguish it from the secondary extinction caused by overlying grains in polycrystalline samples. The primary extinction is the strongest for =/2, i.e. at θ C . The effect is the most pronounced for the two extreme Bragg configurations and weakest for the symmetric Bragg case. For neutron incidence outside the total reflectivity domain, i.e., y 1 , the fraction of incident intensity diffracted at the front face of a thick single crystal is given by D 2 2 2 2 I ( y y 1) 1 1y / 1 1 y . (26) Here, y denotes scaled deviation of incidence angle from centre of Bragg diffraction C i.e., C y 2 / w . (27) i 22
However, for a thin crystal, the reflectivity remains unity over y 1 , whereas for y 1 , we obtain the Ewald reflectivity D 2 I (1 1 y ) . (28) 2.2.2.1 Neutron collimation Experimenters yearn for monochromatic neutron beams with δ-function angular profiles. Such beams allow measurements of ultra-small beam deflections and determination of micrometre-sized agglomerate distributions within samples through their USANS (ultra-small angle neutron scattering) studies. Early attempts at collimation used a pair of ~ mm wide slits set at a distance ~ m apart to limit the angular divergence of the beam. An advanced collimator introduced by Soller [133] consists of a number of uniformly spaced thin cadmium-coated metal sheets (steel or copper) forming a series of long (~ m) and narrow (~ mm) rectangular channels between the sheets. Each channel allows a narrow parallel beam of neutrons to pass through since highly divergent neutrons get absorbed in the cadmium coating [134-135]. Soller slits afforded large beam cross-sections without sacrificing the angular collimation. The slit collimation for thermal and cold neutrons however is constrained to the minute of arc regime; efforts at better collimation are defeated by neutron diffraction at the slit edges apart from a heavy loss in the beam intensity. Tighter collimation ~ arcsec is achievable through neutron diffraction from a single crystal. In the symmetric Bragg case (θ S =0 and b=–1), the diffracted wave exits at exactly the same angle to the planes as the incident wave and both w i and w e equal w 2 H / sin 2B for a centrosymmetric crystal. For 2Å neutrons exciting a {220} reflection in silicon, w = 1.2 arcsec. Neutron double-crystal diffractometers employ single Bragg reflections ~ arcsec, both as a monochromator and analyser albeit with long tails. However, the tails were suppressed by 23
Page 1: Studying neutron dynamical diffract
Page 4 and 5: DECLARATION I, hereby declare that
Page 6 and 7: ACKNOWLEDGEMENTS I sincerely, thank
Page 8 and 9: Fig.10 Boundary conditions on wave
Page 10 and 11: Fig.28 Theoretical curves for Si {1
Page 12 and 13: sample. Least-squares fit to the sa
Page 14 and 15: Fig.62 Interference contrast and av
Page 16 and 17: 3.4 Experimental 49 3.5 Results and
Page 18 and 19: SYNOPSIS The present thesis aims to
Page 20 and 21: analytic expressions for intensity
Page 22 and 23: Research A, 634 (2011) S41. [9] S.
Page 24 and 25: spallation sources yield peak neutr
Page 26 and 27: SU(2) phase, and separation of geom
Page 28 and 29: [33-34,48]. The bi-refracting natur
Page 30 and 31: application of the dynamical theory
Page 32 and 33: I. One of the major disadvantages o
Page 34 and 35: nuclear physics very important as i
Page 36 and 37: potential for most nuclei in spite
Page 38 and 39: Due to the rapid strides made latel
Page 40 and 41: for a positive value of b c i.e. fo
Page 42 and 43: k b k O H . ni sin( B S) . (18)
Page 46 and 47: exploiting multiple successive iden
Page 48 and 49: We shall confine our discussion hen
Page 50 and 51: The depth Z is measured from the in
Page 52 and 53: wings (large-y regions). The freque
Page 54 and 55: expressed as T(y, t )R(y, t )R(
Page 56 and 57: studies. With such beams, the and
Page 58 and 59: CHAPTER 3 Neutron forward diffracti
Page 60 and 61: eing the separation of points M and
Page 62 and 63: 3.1 Neutron forward diffraction Con
Page 64 and 65: excited by the incident wave (Fig.1
Page 66 and 67: Therefore, the forward diffracted i
Page 68 and 69: Fig.15 Calculated cr and I O for a
Page 70 and 71: (1 I ) D tan b . ID cot( B S)
Page 72 and 73: Fig.18 Experimental layout to measu
Page 74 and 75: Fig.20 Bragg reflection intensity f
Page 76 and 77: fraction I O through the Bragg pris
Page 78 and 79: Fig.22 Experimental (points) and th
Page 80 and 81: Table 2: Observed deflection sensit
Page 82 and 83: total reflectivity domain. These ar
Page 84 and 85: Here the term C() accounts for the
Page 86 and 87: Fig.28 Theoretical curves for Si {1
Page 88 and 89: I H (θ H ) peaks of 0.18 height an
Page 90 and 91: other with a compression factor lim
Page 92 and 93: an overall compression factor of 14
Page 94 and 95:
4.3 Experimental 4.3.1 Bragg prism
Page 96 and 97:
4.3.2 Experiment The experiment was
Page 98 and 99:
Fig.38 Bragg reflection rocking cur
Page 100 and 101:
4.4 Applications of novel SUSANS se
Page 102 and 103:
Fig.42 First Q ~ 10 -6 Å -1 SUSANS
Page 104 and 105:
2 (1) -(δk i .Δ i ) /2 Γ (Δ) =
Page 106 and 107:
2 I( ) A( ) . (76) The least squa
Page 108 and 109:
Fig.46 Analyser rocking curve for a
Page 110 and 111:
This instrument was also employed t
Page 112 and 113:
Fig.48 Theoretical flat-topped angu
Page 114 and 115:
difference between the two neutron
Page 116 and 117:
Fig.52 Phase dispersion in a 26.5 m
Page 118 and 119:
the beam and remains visible up to
Page 120 and 121:
cos bc 4NDd1 1 2cot 2 III 2 2
Page 122 and 123:
yield about an order of magnitude r
Page 124 and 125:
The refractive index, n = (1-Nb c
Page 126 and 127:
with a white, and hence high-intens
Page 128 and 129:
proposed dual sample. The dual samp
Page 130 and 131:
The experiment was performed with =
Page 132 and 133:
Fig.59 Typical interferograms for p
Page 134 and 135:
Fig.63 Path I, II and I-II phases d
Page 136 and 137:
Fig.65 Average intensity with the s
Page 138 and 139:
Fig.69 New metrological mapping of
Page 140 and 141:
Si at 26.2 o C, applying a correcti
Page 142 and 143:
selected asymmetric Bragg prism may
Page 144 and 145:
III. High precision determination o
Page 146 and 147:
REFERENCES [1] H. Rauch and S. Wern
Page 148 and 149:
[36] J. S. Nico and W. M. Snow, Ann
Page 150 and 151:
[67] V. F. Sears, Can. J. Phys., 56
Page 152 and 153:
[97] A. Cabello, S. Filipp, H. Rauc
Page 154 and 155:
[127] A. Ioffe, D. L. Jacobson, M.
Page 156:
2009, (Grenoble, France, 2010) 3-15
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PHYS01200804001 Sohrab Abbas - Homi Bhabha National Institute

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