Magnetic form factors

Magnetic form factors in the dipole approximation
("双極子近似による磁気形状因子" in Japanese)

fomulae

Within the dipole approximation,

When the orbital angular momentum ${\bf L}_i$ is quenched and can be replaced by $(g-2){\bf S}_i$ with g-factor and spin angular momentum in the each ion i, the Fourier-transformed magnetisation operator ${\bf Q}$ and the magnetic form factor $F({\bf \kappa})$ are,

${\bf Q}=\frac{1}{2}gF({\bf \kappa})\Sig_i \exp(i{\bf \kappa}\cdot{\bf R}_i){\bf S}_i$

$F(\bf{\kappa})=\bar{j_0}(\bf{\kappa}) + (1-2/g)\bar{j_2}(\bf{\kappa})$

When the magnetic moment of the ion i can be characterized by the total angular momentum ${\bf J_i}$ ,

$\bf{Q}=\frac{1}{2}g{F({\bf \kappa})\Sig_i \exp(i{\bf \kappa}\cdot{\bf R}_i){\bf J}_i$

$F(\bf{\kappa})=\bar{j_0}(\bf{\kappa}) + \bar{j_2}(\bf{\kappa}) \frac{J(J + 1) + L(L + 1) -S(S + 1)}{3J(J + 1) + S(S + 1)-L(L + 1)$

Reference

S. W. Lovesey, Theory of Neutron Scattering from Condensed Matter Vol.2, Oxford University Press (1984).

The momentum transfer $\hbar |{\bf \kappa}|$ is described as,

$|\bf{\kappa}|=2k\sin\theta=4\pi\sin\theta/\lambda$

The coefficients used in these calculations were taken from

International Tables for Crystallography (2006). Vol. C, Chapter 4.4.