Magnetic form factor

 In neutron scattering, an intensity of a magnetic Bragg reflection is proportional to Fourier-transformed magnetization distribution projected onto the plane perpendicular to the scattering vector Q. If the magnetic moments were treated as classical vectors, S(r), localized at the center positions of the magnetic atoms, the intensity would be proportional to S_perp(Q)^2. ("_perp" means the projection onto the plane perpendicular to Q. S(Q) is the Fourier-transform of S(r))

However, a magnetic moment of an atom is not the quantity localized at a specific position r. Instead, it originates from the unpaired electrons, which have broad distributions in the outer open shell.  Thus, the intensity of a magnetic reflection depends not only on S_perp(Q)^2 but also on the Fourier transform of the normalized distribution function of the electrons contributing magnetic moments, which leads to the magnetic form factor, f(Q).

To obtain the magnetic form factors, we use integrals <jK> which include the radial wave function for the shell and the spherical Bessel function of Kth order[1].

In the dipole approximation, the magnetic form factor f(Q) is described by <j0> and <j2>. (This is valid when <j0> is larger than <j2>).

Analytic approximations to the <j_K(Q)> integrals are listed in the web site of ILL. By using the coefficients tabulated in the web site, we can reproduce the |Q| dependence of the form factor. 

For instance, <j_0(Q)> is calculated as follows:

Step 2

The approximation for the <j0> integral is given by 

The variable x is not the momentum transfer of neutron, Q, but sinθ/λ. 

Step 3

|Q| is given by 2ki sinθ(=2*2π/λ*sinθ), and therefore the difference between Q and x is the factor of 4π. 

The Q dependence of the <j0> integral for Fe3+ is obtained as below. 

In the dipole approximation[1], the magnetic form factor f(Q) is described by <j0(Q)> and <j2(Q)>. This is valid when <j0(Q)> is larger than <j2(Q)>.

i) When the orbital angular moment L can be replaced by (g-2)S, (this is the case for 3d transition metal ions),

ii) When the magnetic ion is characterized by a total angular momentum J, (this is the case for rare-earth ions),

where gJ is the Lande's g-factor defined as follows:

One can simulate the Q-dependence of the magnetic form factor using "Magnetic form factor viewer" in Fcal-n.js.

Reference 

[1] "Theory of Neutron Scattering from Condensed Matter vol. 2", S. W. Lovesey (Oxford science publications).