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There is, in principle, a simpler approach to the calculation of the dispersion
terms by using the following relationship between the oscillator density
functions (d*g*/d) and the photoelectric absorption coefficient
:^{61}

(23) |

Using experimental values of () one could obtain values for the
oscillator densities from equation (23). However, accurate values for
() are not presently available for all the elements in a useful
range of . Then, the usefulness of an empirical method to obtain
(d*g*/d)_{s}, *g*_{s}, and
based on the experimental values of
() is very limited.

A semi-empirical method has been used by taking the well known approximate functional dependence of ():

(24) |

By choosing the best experimental values for *n* and ()integration of equation (17) should provide reasonably good values of *g*_{s}.
One can substitute, in the general case, equations (23) and (24) in (17) to
obtain:

(25) |

(26) |

Equation (26) has been integrated in the general case by Parratt and
Hempstead.^{36} These authors have expressed their results in the form of
`universal anomalous dispersion curves', which are essentially the
representation of the integral in equation (26) with *n* as a parameter, as a
function of . When damping is neglected, these curves
are independent of the atomic number and of the electronic shell involved. To
obtain a particular value of , the value on the curve with
the correct value of *n* is multiplied by the oscillator strength calculated
from equation (25). is then obtained by summation through
all *s* = *K*, *L*, *M*, shells. The shape of the curves, reproduced in Fig.
1, is quite instructive. They are qualitatively correct and show that the
dispersion contributions from the various electron shells are rarely negligible,
or, in other words, as Parratt and Hempstead point out, there is practically no
region of normal dispersion.

Since the method used by Parratt and Hempstead is based on experimentally
determined values and uses exact integrations one should expect results in
better agreement with the independent measurements made of the atomic scattering
factor than is the case for the values obtained using Hönl's theory based on
hydrogen-like electron shells. The calculations made by Parratt and Hempstead
for the *K* region of copper and the *L* region of tungsten using only one term
in the oscillator distribution for each electron shell, actually showed a less
satisfactory agreement than Hönl's theory. This rather discouraging result was
attributed by Parratt and Hempstead to (a) the difficulties inherent in the
experimental measurements and (b) neglect of parts of the calculations in
previous comparisons.

In fact, the experimental differences (*f* - *f _{0}*) which they used to compare
their calculations were presumably of a rather low precision, since they were
based on values of

It would be interesting to remeasure the values of *f* using modern techniques
and subtract better values of *f _{0}* as are currently available nowadays in order
to make a definite comparison.

**Copyright © 1981, 1998 International Union of
Crystallography**