Diffraction of X-rays by magnetic materials. I. General formulae and measurements on ferro- and ferrimagnetic compounds

Abstract

The calculation of the amplitude of X-rays scattered by a magnetically ordered substance, carried out in the relativistic quantum theory (i.e. taking the spin into account), is detailed. The effect of the orbital momentum is described in an appendix. The practical formulae dealing with the polarization of the beams are given both in a simple form for the usual experiments and in a complete form, using the Stokes vectors, for the most general case. The experiments show a change in the intensity of the X-rays diffracted by a ferromagnetic (pure iron) or a ferrimagnetic (zinc-substituted magnetite) powder when the magnetization, perpendicular to the diffraction plane, is reversed. The relative values of these intensity changes range from 10 -4 to 5 x 10 -3 and agree in sign and magnitude with the predictions. They are proportional to the spin-density structure factor multiplied by the imaginary part of the chargedensity structure factor; the large anomalous scattering of the Cu Ka radiation in the iron-containing samples is used in the present experiments. moment associated with the spin does interact with the magnetic field of the radiation (Fig. 1). This effect can Driving force Reradiation-eE E -e E-dipol.-eE H -e ~ H-quadr. E grad/~.H --e~ffj ~C_.,~ ~ ~H $ E-dipol. IntroductionX-ray diffraction is usually interpreted through the Thomson scattering mechanism, i.e. the interaction between the electromagnetic radiation and the charge of the electrons. X-rays therefore seem to give information about the charge density only and not about the spin density. If one examines the phenomenon more thoroughly, it appears that the electronic spin also plays some role; the magnetic 0567-7394/81/030314-11 $01.00Tor qUeHx/ z ~; ~H H-dipol. be treated with relativistic quantum theory. The Klein & Nishina (1929) formula for the Compton effect takes this effect into account, but only implicitly since it concerns mean values taken over all spin states. In relativistic theory space and spin wave functions cannot be separated as they can be in the nonrelativistic limit. A perturbation on the movement of an electron has an effect which depends upon the spin. During the collision with a photon of momentum k(0 (k¢f) after collision) the electron is accelerated. One may admit that the effect of this acceleration depends upon the direction of the spin by some term of the order of I k(f) -k(0 I/mc, which expresses the relativistic character of the acceleration undergone by the electron (Fig. 2a). This is analogous to the Mott asymmetry (Tolhoek, 1956) observed in the scattering of an electron by the electric field of a nucleus, or to the Schwinger (1948) scattering (see a recent review by Felcher & Peterson, 1975), which is similar to the Mott assymetry but concerns neutrons instead of electrons (Fig. 2b). These effects can also be compared to the spin-orbit coupling (Fig. 2c), the magnitude of which is still determined by the ratio I pl/mc (p is the electron momentum). One should not forget that these comparisons do not account f...

Soft X-Ray Resonant Magnetic Diffraction

“…The use of x-ray scattering to probe the magnetic structure of crystalline solids started over 25 years ago with the study of NiO by de Bergevin and Brunel [3]. However, such scattering is an extremely weak phenomenon, typically 10 8 times weaker than charge reflections, with the result that most magnetic structure determination has employed neutron diffraction.…”

Measuring the Dzyaloshinskii–Moriya interaction in a weak ferromagnet

“…Unfortunately, the standard techniques for characterizing antiferromagnetic structures-polarized neutron or X-ray diffraction-do not help: the sign of the twist appears in the phase of the diffracted wave, which is lost in an intensity measurement-an aspect of the famous 'phase problem' of crystallography. Borrowing from the ideas behind holography, it was recently suggested by some of us 16 that the sign of the Dzyaloshinskii-Moriya vector could be measured with resonant X-ray diffraction by observing interference between the resonant 17 and magnetic 18 scattering amplitudes. The resonant scattering process adopted is a rather exotic one, involving pure electric quadrupole events (that is, beyond the usual dipole approximation).…”

Magnetic X-Ray Scattering

“…However, the complete relativistic treatment of the interaction of plotons with charged particles depends on the spin of these particles [9]. Because the space and spin components cannot be separated in the relativistic theory, a perturbation of the electron movement has an effect which depends on its spin state.…”

“…A series of articles [9][10][11][12][13] have described the approximations and the formalism involved in deriving expressions for the scattering amplitudes. Here, we will limit ourselves to a description of the processes.…”