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The Atomic Scattering Factor. The Oscillator Strength

Assuming the simple case where k is very small and $\omega$ is quite different from $\omega_s$, then, we obtain approximately:

$$f^{\prime} = \frac{\omega^2}{\omega^2 - \omega^2_s}$$ (13)

and

$$f^{\prime\prime} = 0$$ (14)

$$\mu_a = 0$$

Under these conditions, we shall concentrate on the real part $f^{\prime}$ of the dipole scattering factor. Let us now consider an atom containing g(1),$\dots$, g(2),$\dots$, g(s),$\dots$ dipole oscillators of natural frequencies $\omega_1$,$\dots$, $\omega_2$,$\dots$,$\omega_s$,$\dots$respectively. Expression (13) can now be generalized by summing the contributions of all the oscillators contained in the atom so as to give the real part of the atomic scattering factor:

$$f^{\prime} = \sum_s\frac{g(s)\omega^2}{\omega^2 - \omega^2_s}$$ (15)

The number g(s) of dipole oscillators of natural frequency $\omega_s$existing in an atom is called the `oscillator strength' corresponding to that particular frequency. The calculation of the oscillator strength is the main difficulty in obtaining the resonance contributions to the atomic scattering factors.

We shall not discuss here either the implications or the validity of the generalization just made which has far reaching consequences. A comparison of these arguments with those from the quantum theory of atomic scattering is quite adequate, but will not be undertaken here, since we rather intend to concentrate on the applications of anomalous scattering. The reading is referred, for such a comparison, to James' book.⁶¹

If we now re-examine the foregoing arguments we note that the calculations were explicitly made for points located in the plane perpendicular to the electric vector of the incident wave, that is equivalent to taking the polarization factor equal to unity. That is why the expression (15) obtained for the atomic scattering factor is independent of the diffraction angle. In fact, as it is well known, this is not generally true, for instance for a spherically symmetric atom, f = f(sin $\theta$/$\lambda$).

f could be independent of $\theta$ if the incident wavelength was big with respect to the dimensions of the atom where the electron density is not negligible; notwithstanding the opposite is the case normally encountered in practice since the atomic dimensions and the wavelengths normally used in crystallography are of the same order of magnitude, namely, one to two ångströms.

However, our aim is to obtain only the dispersion terms and not the entire atomic scattering factor. The terms which correspond, for instance, to the K absorption edge are important only when the incident frequency $\omega$ is close to $\omega_K$. The relevant electron distribution in the case is only that of the K electrons. It is then easy to verify that $\lambda_K$ is much larger than the dimensions of the atomic region where the K-electron density is appreciable; this means that the phase difference in the scattered waves due to the difference in position of the K electrons within the atom will be small and their total contribution will thus be practically independent of the diffusion angle. The same arguments are valid, mutatis mutandis , to the case of L, M, etc., electrons. It follows that the resonant contribution to the atomic scattering should be nearly independent of the diffusion angle, which means that the hypothesis used in the previous calculations can be applied to a fair approximation.

In the quantum mechanical treatment the analogues of the classical oscillator strengths are magnitudes g(k, n) which are proportional to the transition probability of an electron passing from a state k to a state n. For an electron atom,

$$\sum_{n}g(k, n) = 1$$ (16)

i.e. the Thomas-Reiche-Kuhn sum rule, holds. This rule, extended to the case of a many-electron atom, states that the sum of the oscillator strength is equal to the atomic number Z.

Summing up, the atom scatters as if it was composed by dipole oscillators of given natural frequencies, identical to the Bohr frequencies, their number or oscillator strength being proportional to the transition probability of state k into state n. It is here important to note that the states k include all the discrete states of negative energy and the continuum of positive energy states.

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