Analytical approximation

The following tables give the coefficients of an analytic approximation to the $\langle j_0\rangle$ integrals for the d electrons in ions of the 3d and 4d series, and the f electrons of some rare earth and actinide ions. The approximation has the form

$$\langle j_0(s)\rangle = A \exp(-as^2) + B \exp(-bs^2) + C \exp(-cs^2) + D$$ (7)

For these expansions $s = \sin\theta/\lambda$ in units of Å^-1.


The integrals $\langle j_L\rangle$ with $L\ne0$ are zero when s=0 and have been fitted with the form

$$\langle j_0(s)\rangle = \left(A\exp(-as^2) + B\exp(-bs^2) + C\exp(-cs^2) + D\right)s^2$$ (8)

The second set of tables give the coefficients obtained for <j2> for 3d, and 4d transition metals, htmlrefrare earthsrej2 and actinides, <j4> for htmlref3d3dj4, and 4d transition metals, rare earths and actinides, and <j6> for rare earths and actinides. For the transition metal series the fits were made with form factor integrals calculated from Hartree-Fock wave-functions [2]. For the rare-earth and actinide series the fits were with Dirac-Fock form factors [3,4]. In the tables the number following the atom symbol indicates the ionisation state of the atom. Thus the coefficients following Fe0 are for a neutral iron atom and those following Fe2 are for Fe²⁺.