Tables for
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

International Tables for Crystallography (2006). Vol. C, ch. 6.2, pp. 597-598


H. Lipson,a J. I. Langforda and H.-C. Hub

aSchool of Physics & Astronomy, University of Birmingham, Birmingham B15 2TT, England, and bChina Institute of Atomic Energy, PO Box 275 (18), Beijing 102413, People's Republic of China

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Summary of formulae for integrated powers of reflection

Absorption is neglected in both (g) and (h).

(a) Crystal element [\let\normalbaselines\relax\openup4pt\matrix{\rho =Q\delta V\hfill&\cr Q ={\displaystyle{{N^{2}e^{4}\lambda^{3}}\over{2m^{2}c^{4}}}{{1+\cos^{2}2\theta}\over{\sin 2\theta}}|F|^{2}}\hfill&\hfill\hbox{for non-polarized X-rays}\cr {\displaystyle{Q={{N^{2}\lambda^{3}}\over{\sin 2\theta}}|F|^{2}}}\hfill&\hfill\hbox{for neutrons}\cr}]
(b) Reflection from a crystal slab of thickness t[\eqalign{\sigma&=QW(\Delta\theta_{0})\cr\xi&=\mu/\sigma}]
1. Symmetrical Bragg geometry[\eqalignno{P_{H}/P_{0}&=(\sigma t\,{\rm cosec} \,\theta) /(1+\sigma t\,{\rm cosec}\, \theta)&\hbox{for }\mu =0\cr P_{H}/P_{0}& =\{1-\exp [-2(\xi ^{2}+2\xi)^{1/2}\sigma t\,{\rm cosec}\, \theta ]\} \{ (\xi ^{2}+2\xi )^{1/2}&\cr&\quad + \xi +1+[(\xi ^{2}+2\xi )^{1/2}-(\xi +1)]&\cr&\quad\times \exp [-2(\xi ^{2}+2\xi )^{1/2}\sigma t\,{\rm cosec}\,\theta ]\}^{-1}&\hbox{for }\mu \ne 0\cr \rho ^{\prime }&= {{Q[ 1-\exp (-2\mu t\,{\rm cosec}\,\theta )] }\over{2\mu }}&\hbox{for }\mu/\sigma_{0} >> 1}]
2. Asymmetrical Bragg geometry, when the reflecting planes are inclined at an angle [\varphi] to the crystal surface, and the surface normal is in the plane of the incident and reflected beams.[\eqalign{\tau &=\sigma t\,{\rm cosec} ( \theta +\varphi )\cr b&=-\sin (\theta +\varphi )/\sin (\theta -\varphi )}]angle of incidence [(\theta+\varphi)] and angle of emergence [(\theta-\varphi)] to the crystal surface[\eqalignno{{{P_{H}}\over{P_{0}}}&={{1-\exp [-| 1 + b| \,{\rm cosec} (\theta +\varphi )\sigma t]}\over{1-|b|\exp [-|1+b| \,{\rm cosec} (\theta +\varphi )\sigma t]}}&\cr&&\hbox{for } |b| \lt 1, \varphi \lt 0{^{\circ }} \hbox{ and } \mu =0\cr {{P_{H}}\over{P_{0}}}&={{1-\exp [-| 1+b|\,{\rm cosec} (\theta +\varphi )\sigma t]}\over{|b|-\exp [-|1+b|\,{\rm cosec} (\theta +\varphi)\sigma t]}}&\cr& &\hfill \hbox{for } |b| > 1, \varphi > 0{^{\circ }} \hbox{ and } \mu =0}]Define [u=[(1-b)^{2}(\xi +1)^{2}+4b]{}^{1/2}] and [v=(1-b)(\xi +1)]:[\eqalignno{{{P_{H}}\over{P_{0}}}&={{2[1-\exp (-u\tau )]}\over{(u+v)+(u-v)\exp (-u\tau )}}&\hbox{for } \mu \ge 0\cr \rho ^{\prime }&={{Q\{ 1-\exp [-(1-b)\mu t\,{\rm cosec}(\theta +\varphi )]\} }{[(1-b)\mu ]^{-1}}}&\cr &={{Q}\over{2\mu}}(1-\cot \theta \tan \varphi )\left\{ 1-\exp \left[-{{2\mu t\,{\rm cosec} (\theta +\varphi )}\over{1-\cot \theta \tan \varphi }}\right]\right\}\cr&&\hbox{for } \mu /\sigma _{0}>>1}]
(c) Transmission from a crystal slab of thickness t
1. Symmetrical Laue geometry[\eqalignno{P_{H}/P_{0}&=\exp (-\xi \sigma t\sec \theta )[1-\exp (-2\sigma t\sec \theta)]/2 &\hbox{for } \mu \ge 0\cr\rho ^{\prime }&=Qt\sec \theta \exp (-\mu t\sec \theta)&\hbox{for } \mu /\sigma _{0}>>1}]
2. Asymmetrical Laue geometry, when the reflecting planes are at [\pi {}/2-\varphi] to the crystal surface, with the normal in the plane of the incident and reflected beams.[\eqalign{\tau &=\sigma t\,{\rm sec} ( \theta -\varphi )\cr b&=\sec (\theta +\varphi )/\sec (\theta -\varphi )}]angle of incidence [(\theta+\varphi)] and angle of emergence [(\theta-\varphi)] to the normal to the crystal surface[\eqalignno{{{P_{H}}/{P_{0}}}&=(1-\tan \theta \tan \varphi )\{ {1-\exp [-\sec (\theta +\varphi )\sigma t-\sec (\theta -\varphi )\sigma t]}\} /2 &\cr&&\hbox{for } \mu = 0}]Define [u=[(1-b){}^{2}(\xi +1){}^{2}+4b]{}^{1/2} ] and [w=(1+b)(\xi +1)][\eqalignno{{{P_{H}}\over{P_{0}}}&=\{ \exp[-(w-u)\tau /2]-\exp [-(w+u)\tau /2]\} /u&\cr&&\hbox{for } \mu \ge 0\cr \rho ^{\prime} &={{Q}\over{(1-b)\mu }}\{\exp[-\mu t\sec (\theta +\varphi )]-\exp [-\mu t\sec (\theta -\varphi )] \}&\cr &={{Q}\over{\left[{\displaystyle1-{{\sec(\theta \!+\!\varphi )}\over{\sec (\theta\! -\!\varphi )}}}\right]\mu }}\{\exp [-\mu t\sec (\theta +\varphi )]-\exp [-\mu t\sec (\theta -\varphi )]\}&\cr&&\hbox{for } \mu /\sigma _{0}>>1}]
(d) Powder halo: no absorption correction included[{P\over I_0} = {Q\,p''V\cos\theta\over 2}={N{}^2e{}^4\lambda{}^3 V \over 8m{}^2c{}^4}\;{1+\cos{}^22\theta \over \sin\theta}\;p''|F|{}^2,]where P is the diffracted power.
(e) Debye–Scherrer lines on cylindrical film: no absorption correction included[{P_l\over I_0} = {Q\,p''lV \over 8\pi r\sin\theta} = {N{}^2e{}^4\lambda{}^3l V \over32\pi m{}^2c{}^4r}\; {1+\cos{}^2 2\theta\over \sin{}^2\theta\cos\theta}\;p''|F|{}^2,]where l is the length of line measured and r is the radius of the camera. Pl is the power reflected into length l.
(f) Reflection from a thick block of powdered crystal of negligible transmission [{P_l\over I_0} = {Q\,p''l \over 16\pi\mu r\sin\theta} = {N{}^2e{}^4\lambda{}^3l \over 64\pi m{}^2c{}^4\mu r} \; {1+\cos{}^22\theta \over\sin{}^2\theta\cos\theta}\; p''|F|{}^2]
(g) Transmission through block of powdered crystal of thickness t[\eqalign{ {P\over I_0}&= {Q\,p''lt \over 4\pi r\sin2\theta}\;{\delta'\over \delta} \cr&= {N{}^2e{}^4\lambda{}^3 \over 8\pi m{}^2c{}^4r}\; {lt\delta' \over \delta}\; {1+\cos{}^22\theta \over \sin{}^22\theta}\; p''|F|{}^2,}]where δ′, δ are the densities of the block of powder and of the crystal in bulk, respectively.
(h) Rotation photograph of small crystal, volume V
1. Beam normal to axis[\eqalign{ \rho &={QVp'\over 2\pi(\cos{}^2\varphi-\sin{}^2\theta){}^{1/2}} \cr &={N{}^2e{}^4\lambda{}^3V \over 4\pi m{}^2c{}^4} \; {1+\cos{}^2\theta \over \sin2\theta}\; {\cos\theta\over (\cos{}^2\varphi-\sin{}^2\theta){}^{1/2}}\; p'|F|{}^2}]
2. Equi-inclination Weissenberg photograph[\rho = {QV\over 2\pi\xi\cos\theta}\; {N{}^2e{}^4\lambda{}^3V \over 4\pi m{}^2c{}^4}\; {1+\cos{}^22\theta \over \xi\cos\theta}\;|F|{}^2]

Q Integrated reflection from a crystal of unit volume
[\delta V] Volume of crystal element
e, m Electronic charge and mass
c Speed of light
λ Wavelength of radiation
μ Linear absorption coefficient for X-rays or total attenuation coefficient for neutrons
Angle between incident and diffracted beams
[\varphi] In (b) and (c), as defined; in (h), latitude of reciprocal-lattice point relative to axis of rotation
V Volume of crystal, or of irradiated part of powder sample
N Number of unit cells per unit volume
ξ In (b) and (c), as defined; in (h), radial coordinate xi used in interpreting Weissenberg photographs
I0 Energy of radiation falling normally on unit area per second
hkl Indices of reflection
F Structure factor of hkl reflection
[W(\Delta\theta_0)] Distribution function of the mosaic blocks at angular deviation [\Delta\theta_0] from the average reflecting plane
σ Diffraction cross section per unit volume
σ0 Diffraction cross section per unit volume at [\Delta\theta_0=0]
b Asymmetry parameter
τ Reduced thickness of the crystal slab
PH/P0 Reflection power ratio, i.e. the ratio of the diffracted power to the incident power
ρ Integrated reflection power ratio from a crystal element
ρ′ Integrated reflection power ratio, angular integration of reflection power ratio
p Multiplicity factor for single-crystal methods
p′′ Multiplicity factor for powder methods