Definitions

Creagh, D. C.

doi:10.1107/97809553602060000592

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

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International Tables for Crystallography (2006). Vol. C. ch. 4.2, p. 213

Section 4.2.3.1.1. Definitions

D. C. Creagh^b

4.2.3.1.1. Definitions

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This section deals with the manner in which the photon scattering and absorption cross sections of an atom varies with the energy of the incident photon. Further information concerning these cross sections and tables of the X-ray attenuation coefficients are given in Section 4.2.4. Information concerning the anomalous-dispersion corrections is given in Section 4.2.6.

When a highly collimated beam of monoenergetic photons passes through a medium of thickness t, it suffers a decrease in intensity according to the relation $[I=I_0\exp(-\mu_lt), \eqno (4.2.3.1)]$ where μ_l is the linear attenuation coefficient. Most tabulations express μ_l in c.g.s. units, μ_l having the units cm⁻¹.

An alternative, often more convenient, way of expressing the decrease in intensity involves the measurement of the mass per unit area m_A of the specimen rather than the specimen thickness, in which case equation (4.2.3.1) takes the form $[I=I_0\exp[-(\mu/\rho)m_A], \eqno (4.2.3.2)]$ where ρ is the density of the material and (μ/ρ) is the mass absorption coefficient. The linear attenuation coefficient of a medium comprising atoms of different types is related to the mass absorption coefficients by $[\mu_l=\rho\textstyle\sum\limits_ig_i(\mu/\rho)_i, \eqno (4.2.3.3)]$ where [g_i] is the mass fraction of the atoms of the ith species for which the mass absorption coefficient is $[(\mu/\rho)_i]$ . The summation extends over all the atoms comprising the medium. For a crystal having a unit-cell volume of [V_c] , $[\mu_l={1\over V_c}\sum\sigma_i, \eqno (4.2.3.4)]$ where $[\sigma_i]$ is the photon scattering and absorption cross section. If $[\sigma_i]$ is expressed in terms of barns/atom then [V_c] must be expressed in terms of Å³ and μ_l is in cm⁻¹. (1 barn = 10⁻²⁸ m².)

The mass attenuation coefficient μ/ρ is related to the total photon–atom scattering cross section σ according to $[\eqalignno{{\mu\over\rho}({\rm cm}^2/{\rm g}) &=(N_{\rm A}/M)\sigma ({\rm cm}^2/{\rm atom})\cr &=(N_{\rm A}/M)\times10^{-24}\sigma\,({\rm barns/atom}), &(4.2.3.5)}]$ where $[N_{\rm A}=]$ Avogadro's number = 6.0221367 (36) × 10²³ atoms/gram atom (Cohen & Taylor, 1987) and M = atomic weight relative to M(¹²C) = 12.0000.

References

Cohen, E. R. & Taylor, B. N. (1987). The 1986 adjustment of the fundamental physical constants. Rev. Mod. Phys. 89, 1121–1148.Google Scholar

International Tables for Crystallography (2006). Vol. C. ch. 4.2, p. 213