International
Tables for Crystallography Volume H Powder diffraction Edited by C. J. Gilmore, J. A. Kaduk and H. Schenk © International Union of Crystallography 2018 |
International Tables for Crystallography (2018). Vol. H, ch. 3.2, pp. 254-255
Section 3.2.2.2.3. Neutrons and magnetic scattering^{a}Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794–3800, USA |
Magnetic neutron scattering is also described through a structure factor which is, however, a vector. The magnetic moment of the neutron interacts with the magnetization density of unpaired electrons in the sample, which may possess spin and/or orbital angular momentum. The magnetic interaction is only sensitive to the component of magnetization perpendicular to the scattering vector. When discussing magnetic scattering, it is more common to use the scattering vector Q = 2πG. The magnetic structure factor is defined asHere γ = 1.9132 is the neutron gyromagnetic factor, f_{n}(Q) is the atomic magnetic form factor, m_{n} is the magnetization of the nth site in units of the Bohr magneton and is the unit vector in the direction of Q. The double cross product isolates the component of magnetization perpendicular to Q.
Note that many magnetically ordered materials have a magnetic cell which is larger than the chemical cell. Indeed, many magnetic phases are incommensurately modulated, i.e., the magnetic structure is not periodic with any combination of the chemical unit cell translation vectors. Such matters are beyond the scope of this introduction, and are handled in Chapter 7.13 of this volume.
For unpolarized neutron measurements (i.e., an average over all polarization states of the incoming and diffracted beam), the intensities of the nuclear and magnetic diffraction peaks may be computed separately and then added to determine the overall diffraction pattern. In cases where the chemical and magnetic cells are identical (e.g. simple ferromagnets) the nuclear and magnetic diffraction patterns overlap, and so one observes only intensity differences upon magnetic ordering. In the case of antiferromagnets, new magnetic diffraction peaks appear at positions not allowed for the chemical unit cell.
Note also that the magnetic form factor depends on the spin density in the magnetic orbitals, which are typically of greater spatial extent than either the total charge density or the nuclear density. Therefore, the intensity of magnetic neutron diffraction peaks falls off much more rapidly with (sin θ)/λ than do nuclear neutron diffraction peaks.