Magnetic order, modulated

This result is an exact expression for the transition matrix element. Physically we have a dipole interaction with the vector potential and a dipole interaction with the magnetic field modulated by a phase factor. The problem is that this integral is difficult to compute. An approximation can be invoked. The wavevector has a magnitude equal to 1 /X. The position r is set to the position of an atom and is on the order of the radius of that atom. Thus K r a/X. So if the wavelength of the radiation is much larger than the radius of the atom, which is the case with optical radiation, we may then invoke the approximation e k r 1 + ik r. This is commonly known as the Bom approximation. This first-order term under this approximation is also seen to vanish in the first two terms as it multiplies the term p e. A further simplification occurs, since the term a (k x e) has only diagonal entries, and our transition matrix evaluates these over orthogonal states. Hence, the last term vanishes. We are then left with the simplified variant of the transition matrix ... 

In addition to one-dimensional modulations considered above, both two-and three-dimensional modulations are possible. Furthermore, atomic parameters affected by modulations may be one or several of the following positional (as shown in Figure 1.52 and Figure 1.53), occupancy, thermal displacement, and orientation of magnetic moments. The latter, i.e. commensurately or incommensurately modulated orientations of magnetic moments are quite common in various magnetically ordered structures (e.g. pure lanthanide metals such as Er and Ho), and both the value of the modulation vector and the amplitude of the modulation function often vary with temperature. 

TbGa. This compound displays complex magnetic order, which has been determined in the neutron work accompanying the [tSR studies. Below 7n = 20K a modulated incommensurate AFM phase is present. There are changes in modulation... 

Fig. 130. Left ZF xSR spectrum of Ceo87Lao, 3Ri2Si2 together with the theoretical depolarization fimetion based on incommensurate sinusoidally modulated spin wave magnetic order (see text for details). From Yamamoto et al. (1997). Right Temperature dependence of the mean field seen by muons in magnetic Ce(R% jsRho ] 5)2 12 A single crystal oriented with the c-axis perpendicular to the muon beam was used. From Murayama et al.

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