Definition of form factors

The magnetic form factor $f({\bf {q}})$ is obtained from the fourier transform of the magnetisation distribution of a single magnetic atom. Assuming that it has a unique magnetisation direction it can be written

$${\bf M}\int m({\bf r})e^{i{\bf {q}}\cdot{\bf r}}\,dr={\bf M}f({\bf {q}})$$ (1)

where M gives the magnitude and direction of the moment and $m({\bf r})$ is a normalised scalar function which describes how the intensity of magnetisation varies over the volume of the atom. When the magnetisation arises from electrons in a single open shell the magnetic form factor can be calculated from the radial distribution of the electrons in that shell. The integrals from which the form factors are obtained have the form

$$\left\langle j_L(q)\right\rangle= \int U^2(r)j_l(qr)4\pi r^2\,dr$$ (2)

The j_l are the spherical Bessel functions defined by

$$j_l(x)={\sqrt{\pi\over2x}}J_{l+\frac12}(x)$$ (3)

If the open shell has orbital quantum number l the form factor for spin moment is

$$f_s({\bf {q}})=\frac1{\mathrm M_S}\sum_{L=0}^{2l}i^L\left\lan... ...t\rangle \sum_{M=-L}^L S_{LM}Y^L_M\left({\hat{\bf {q}}}\right)$$ (4)

and that for orbital moment

$$f_o({\bf {q}})=\frac1{\mathrm M_L}\sum_{L=0,2,\ldots}^{2l} \l... ...q)\right\rangle\right) \sum_{M=-L}^LB_{LM}Y^L_M(\hat{\bf {q}})$$ (5)

The coefficients S_LM, B_LM have to be computed from the orbital wave-function [1]. The total spin moment M_S is given by S₀₀ and the orbital moment M_L by B₀₀. For small q the dipole approximation

$$f({\bf {q}})=({\mathrm L+2S)}\left\langle j_0(q)\right\rangle {\mathrm L}\left\langle j_2(q)\right\rangle$$ (6)

can be used.