International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

International Tables for Crystallography (2006). Vol. B, ch. 5.1, pp. 541-542   | 1 | 2 |

Section 5.1.6.1. Absorption coefficient

A. Authiera*

aLaboratoire de Minéralogie-Cristallographie, Université P. et M. Curie, 4 Place Jussieu, F-75252 Paris CEDEX 05, France
Correspondence e-mail: authier@lmcp.jussieu.fr

5.1.6.1. Absorption coefficient

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In transmission geometry, the imaginary part of [X_{o}] is small and, using a first-order approximation for the expansion of [(\eta^{2} + 1)^{1/2}], (5.1.5.1)[link] and (5.1.5.2)[link], the effective absorption coefficient in the absorption case is [\eqalignno{\mu_{j} &= \mu_{o} \bigg[\textstyle{{1 \over 2}} (1 + \gamma^{-1}) \cr &\quad \mp {(\eta_{r}/2) (1 - \gamma^{-1}) + |C|(\gamma^{-1})^{1/2} |F_{ih}/F_{io}| \cos \varphi \over (\eta_{r}^{2} + 1)^{1/2}}\bigg], &(5.1.6.1)}] where [\varphi = \varphi_{rh} - \varphi_{ih}] is the phase difference between [F_{rh}] and [F_{ih}] [equation (5.1.2.10)[link]], the upper sign (−) for the ∓ term corresponds to branch 1 and the lower sign (+) corresponds to branch 2 of the dispersion surface. In the symmetric Laue case ([\gamma = 1], reflecting planes normal to the crystal surface), equation (5.1.6.1)[link] reduces to [\mu_{j} = \mu_{o} \left[1 \mp {|C| |F_{ih}/F_{io}| \cos \varphi \over (\eta_{r}^{2} + 1)^{1/2}}\right].]

Fig. 5.1.6.1[link] shows the variations of the effective absorption coefficient [\mu_{j}] with [\eta_{r}] for wavefields belonging to branches 1 and 2 in the case of the 400 reflection of GaAs with Cu Kα radiation. It can be seen that for [\eta_{r} = 0] the absorption coefficient for branch 1 becomes significantly smaller than the normal absorption coefficient, [\mu_{o}]. The minimum absorption coefficient, [\mu_{o} (1 - |CF_{ih}/F_{io}| \cos \varphi)], depends on the nature of the reflection through the structure factor and on the temperature through the Debye–Waller factor included in [F_{ih}] [equation (5.1.2.10b)[link]] (Ohtsuki, 1964[link], 1965[link]). For instance, in diamond-type structures, it is smaller for reflections with even indices than for reflections with odd indices. The influence of temperature is very important when [|F_{ih}/F_{io}|] is close to one; for example, for germanium 220 and Mo Kα radiation, the minimum absorption coefficient at 5 K is reduced to about 1% of its normal value, [\mu_{o}] (Ludewig, 1969[link]).

[Figure 5.1.6.1]

Figure 5.1.6.1 | top | pdf |

Variation of the effective absorption with the deviation parameter in the transmission case for the 400 reflection of GaAs using Cu Kα radiation. Solid curve: branch 1; broken curve: branch 2.

References

Ludewig, J. (1969). Debye–Waller factor and anomalous absorption (Ge; 293–5 K). Acta Cryst. A25, 116–118.
Ohtsuki, Y. H. (1964). Temperature dependence of X-ray absorption by crystals. I. Photo-electric absorption. J. Phys. Soc. Jpn, 19, 2285–2292.
Ohtsuki, Y. H. (1965). Temperature dependence of X-ray absorption by crystals. II. Direct phonon absorption. J. Phys. Soc. Jpn, 20, 374–380.








































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