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For the continuum of positive energy states, the summations become integrals.
The natural frequency for them changes continuously, so that rather than the
discrete value *g*() it is necessary to define the oscillator density
(d*g*/d) at the frequency . The number of oscillators with
frequencies between and + d is (d*g*/d)d. This number is zero for < , where is
the frequency associated with the *s* absorption edge. The oscillator strength
due to all *K* electrons for instance, is obtained by integration in the whole
range of frequencies to , i.e. the interval where the number
of oscillators related to the continuum of positive energy states is different
from zero:

(17) |

(18) |

(19) |

Bethe^{65} calculated the oscillator strengths *g*(*s*, *m*) for hydrogen-like
atoms.

From equation (19), the generalized Thomas-Reiche-Kuhn rule can be obtained
observing that: *g*(*s*, *m*) = -*g*(*m*, *s*). The justification of this
argument is of a statistical nature, for both transitions should have the same
probability, since their net result must average zero.

Thus,

(20) |

(15') |

becomes

(21) |

Equation (21) only applies when , the incident frequency, corresponds
to a wavelength large in comparison with the atomic dimensions.
For frequencies higher than the natural frequencies of the atom and wavelengths
of the order of the atomic dimensions, we may substitute Z by *f _{0}*, the normal
scattering factor, with good approximation and write:

(22) |

Equations (21) and (22) are valid for any wavelength except for very short ones when relativistic corrections are not negligible. Damping has also been neglected: an approximation usually adopted in the calculation of and , which however does not introduce unduly large errors except in the intervals

around the absorption edges.
The `normal' scattering factor *f _{0}* has

The Hartree wave functions were tried by Cromer^{67} to obtain the oscillator
strengths, but he found them inadequate for this purpose, particularly for the
heavy elements. He tried then the relativistic wave functions without exchange,
calculated by Cohen^{68} in the test cases of tungsten and uranium, finding
better results. New relativistic wave functions computed by Lieberman, Waber
and Cromer^{69} became available for all atoms, which included Slater's^{70-
71} approximate exchange correction and Latter's^{72} self-interaction term.
Using these wave functions, Cromer^{67} calculated a set of oscillator
strengths which have been in use up to now. From them, he obtained a set of
dispersion corrections for elements 10 through 98 for five different
wavelengths.

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