The classical approach

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, pp. 243-244

Section 4.2.6.2.1. The classical approach

D. C. Creagh^b

4.2.6.2.1. The classical approach

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In the classical approach, electrons are thought of as occupying energy levels within the atom characterized by an angular frequency $[\omega_n]$ and a damping factor $[\kappa_n]$ . The forced vibration of an electron gives rise to a dipolar radiation field, when the atomic scattering factor can be shown to be $[f={\omega^2\over\omega^2-\omega^2_n-i\kappa_n\omega}. \eqno (4.2.6.8)]$ If the probability that the electron is to be found in the nth orbit is [g_n] , the real part of the atomic scattering factor may be written as $[{\rm Re}(\,f)=\sum_ng_n+\sum_n\displaystyle{g_n\omega^2_n\over \omega^2-\omega^2_n}.\eqno (4.2.6.9)]$ The probability [ g_n] is referred to as the oscillator strength corresponding to the virtual oscillator having natural frequency $[\omega_n]$ . Equation (4.2.6.9) may be written as $[{\rm Re}(f) = f_0 + f^\prime, \eqno (4.2.6.10)]$ where [f_0] represents the sum of all the elements of the set of oscillator strengths and is unity for a single-electron atom. The second term may be written as $[f'={\int\limits^\infty_{\omega_{\kappa_i}}}{\omega'^2({\rm d}g_\kappa/{\rm d}\omega')\over\omega^2-\omega'^2}\,{\rm d}\omega' \eqno (4.2.6.11)]$ if the atom is assumed to have an infinite number of energy states. For an atom containing κ electrons, it is assumed that the overall value of f′ is the coherent sum of the contribution of each individual electron, whence $[f'={\sum_\kappa\int\limits^\infty_{\omega_\kappa}}{\omega^{\prime 2}({\rm d}g_\kappa/{\rm d}\omega')\over\omega^2-\omega^{\prime 2}}\,{\rm d}\omega' \eqno (4.2.6.12)]$ and the oscillator strength of the κth electron $[g_\kappa={\int\limits^\infty_{\omega_\kappa} }\bigg[{{\rm d}g\over{\rm d}\omega}\bigg]_\kappa\,{\rm d}\omega]$ is not unity, but the total oscillator strength for the atom must be equal to the total number of electrons in the atom.

The imaginary part of the dispersion correction f′′ is associated with the damping of the incident wave by the bound electrons. It is therefore functionally related to the linear absorption coefficient, $[\mu_l]$ , which can be determined from experimental measurement of the decrease in intensity of the photon beam as it passes through a medium containing the atoms under investigation. It can be shown that the attenuation coefficient per atom $[\mu_a]$ is related to the density of the oscillator states by $[\mu_a={2\pi^2e^2\over\varepsilon_0mc}\bigg[{{\rm d}g\over{\rm d}\omega}\bigg], \eqno (4.2.6.13)]$ whence $[f''={\pi\over2}\omega\bigg[{{\rm d}g_\kappa\over{\rm d}\omega}\bigg]. \eqno (4.2.6.14)]$

An expression linking the real and imaginary parts of the dispersion corrections can now be written: $[f'={2\over\pi}\sum_\kappa P\int\limits^\infty_{\omega_\kappa}\displaystyle{\omega'f''(\omega',0)\over\omega^2-\omega'^2}\,{\rm d}\omega'.\eqno (4.2.6.15)]$ This is referred to as the Kramers–Kronig transform. Note that the term involving the restoring force has been omitted from this equation.

Equations (4.2.6.12), (4.2.6.14), and (4.2.6.15) are the fundamental equations of the classical theory of photon scattering, and it is to these equations that the predictions of other theories are compared.

References

International Tables for Crystallography (2006). Vol. C. ch. 4.2, pp. 243-244