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The Imaginary Component ${\Delta}f^{\prime\prime}$

The imaginary component of the atomic scattering factor corresponds to a component of the scattered radiation from the atom having a phase in the forward direction that lags ${\pi}$/2 behind that of the primary wave.

As is known, in the (Thomson) scattering by free electrons, or very approximately in the case of bound electrons and very high frequencies the phase-lag of the forward scattered wave is ${\pi}$. As the frequency of the incident radiation approaches an absorption edge the amplitude of the imaginary component increases so that the resultant phase may differ significantly from ${\pi}$.

In order to have an appreciable resonant effect, the incident frequency must lie very close to one of the natural oscillator frequencies. However, since the virtual oscillators in an atom have frequencies covering a continuous distribution, the resonance effect may be appreciable even for frequencies not so close to one of the absorption edges, for it still coincides with one of the continuous distribution. A system having a discrete set of oscillators would then behave in an entirely different way.

Let us consider, for instance, the case of K electrons. As we know ${\Delta}f^{\prime\prime}_K$ = 0 for $\omega_i$ < $\omega_K$. It is then only necessary to consider frequencies larger than $\omega_K$, since only in this case they may coincide with those of the continuum.

An extension of equation (12) gives the relationship between the K-contribution to the absorption coefficient and ${\Delta}f^{\prime\prime}_K$,the K-contribution to the imaginary component of the atomic scattering factor:

\begin{displaymath} {\mu}_K({\omega}_i) =
\frac{4{\pi}e^2}{mc{\omega}_i}\cdot{\Delta}f^{\prime\prime}_K\end{displaymath} (12')

From relation (23) between the oscillator density and the absorption coefficient one obtains:

\Big(\frac{\mbox{d}g}{\mbox{d}\omega}\Big)_K = \frac{mc}{2{\pi}^2e^2}\cdot{\mu}_K(\omega),\end{displaymath} (23')
and then:

{\Delta}f^{\prime\prime}_K = \frac{\pi}{2} {\omega}_i
\Big(\frac{\mbox{d}g}{\mbox{d}\omega}\Big)_K.\end{displaymath} (27)

Assuming a dependence of ${\mu}_K$ on $\omega$ according to (24),

{\Delta}f^{\prime\prime}_K = \frac{mc}{4{\pi}e^2} {\omega}_i
\Big(\frac{{\omega}_K}{{\omega}_i}\Big)^n \cdot{\mu}({\omega}_K),\end{displaymath} (28)
is obtained, giving the K-contribution as a function of the incident frequency.

It is interesting to express the relationship between ${\Delta}f^{\prime\prime}_K$ and the corresponding oscillator gK:

{\Delta}f^{\prime\prime}_K = \frac{\pi}{2}(n - 1)
\Big(\frac{{\omega}_K}{{\omega}_i}\Big)^{n-1}\cdot g_K\end{displaymath} (29)
valid for $\omega_i$ < $\omega_K$, which was obtained from equations (25) and (28).

Figure 2 represents ${{\Delta}f^{\prime\prime}_K}/{g_K}$ as a function of $x = {\omega}_i$/$\omega_K$, the effect of damping is indicated qualitatively by the dotted line.

The atomic scattering factor may be expressed as the sum of three terms:

f = f_0 + {\Delta}f^{\prime} + i{\Delta}f^{\prime\prime}\end{displaymath}

where f0 is a function of (sin $\theta$)/$\lambda$, whose values calculated for atoms with spherical symmetry are tabulated, for instance, in the International Tables for X-Ray Crystallography , Vol. 3.73 As was shown in this chapter the components, ${\Delta}f^{\prime}$ and ${\Delta}f^{\prime\prime}$ are not negligible when the incident frequency is slightly larger than that of the absorption edges of the atom. These two components, related to the photoelectric absorption coefficients as well as to the oscillator density and oscillator strengths associated to the atom, are given by:

{\Delta}f^{\prime} = \sum_s {\Delta}f^{\prime}_s = \sum_s
 ...omega}\Big)_s{\omega}^2\mbox{d}\omega}{{\omega}^2_i-{\omega}^2}\end{displaymath} (30)

{\Delta}f^{\prime\prime} = \sum_s {\Delta}f^{\prime\prime}_s...
 ..._s = \sum_s
\frac{mc}{4{\pi}e^2} {\omega}_i{\mu}_s({\omega}_i).\end{displaymath} (31)

s = K, L, M, N, \dots\end{displaymath}

It is possible by using quantum mechanical methods to determine the oscillator densities from the atomic wave functions. This method was first introduced by Hönl,62-63 who, of course, did not have at the time the computing facilities nowadays provided by modern computers, so that his main contribution was in developing the method and calculating some values for hydrogen-like K-electrons.

Figure 2: Imaginary dispersion correction as a function of incident frequency according to James.
\includegraphics {fig2.ps}

A method based on the utilization of the relationships between oscillator strengths and densities from one side and photoelectric absorption coefficients on the other, developed by Parratt and Hempstead,36 did not succeed initially in providing satisfactory results due mainly to the lack of good experimental values of the absorption coefficients and also, to my belief, to the lack of reliable experimental and theoretical atomic scattering factors.

Cromer67 used self-consistent field relativistic Dirac-Slater wave functions to calculate accurate oscillator strengths for elements 10 through 98. He calculated then the dispersion terms ${\Delta}f^{\prime}$ by using Parratt and Hempstead's36 solution of equation (26) and summing over the different absorption edges. The imaginary terms $f^{\prime\prime}$ were also computed for the same elements from equation (29). The values of the parameters n used by Cromer,67 Dauben and Templeton28 and other authors were taken from the discussion by Parratt and Hempstead,36 as n = 11/4 for the 1s1/2 edge, n = 7/3 for the 2s1/2 edge and n = 5/2 for all other edges.

The use of the dispersion terms for the solution of crystal structures, which will be discussed later, made it necessary to obtain the values of ${\Delta}f^{\prime}$ for a wide range of wavelengths for atoms where the anomalous dispersion effects are significant. This calculation was performed by Saravia and Caticha-Ellis33 for elements 20 through 83 for 32 different $K\alpha$ wavelengths ranging from Ti$K\alpha$ (2.75 Å) to I$K\alpha$ (0.435 Å), by using the method of Parratt and Hempstead,36 the absorption edges from Cauchois and Hulubei74 and the oscillator strengths from Cromer.67 Essentially the same calculations were later performed by Hazell34 for eleven $K\beta$radiations.

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Next: References Up: Anomalous Dispersion of X-rays in Crystallography Previous: Use of the Photoelectric Absorption Measurements

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