Magnetic wavevector

The Maxwell theory of X-ray scattering by stable systems, both solids and liquids, is described in many textbooks. A simple and compact presentation is given in Chapter 15 of Electrodynamics of Continuous Media [20]. The incident electric and magnetic X-ray helds are plane waves Ex(r, f) = Exo exp[i(q r — fixO] H(r, t) = H o exp[/(q r — fixO] with a spatially and temporally constant amplitude. The electric field Ex(r, t) induces a forced oscillation of the electrons in the body. They then act as elementary antennas emitting the scattered X-ray radiation. For many purposes, the electrons may be considered to be free. One then finds that the intensity /x(q) of the X-ray radiation scattered along the wavevector q is... 

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 equ. (8.51) the summation over the magnetic quantum numbers takes care of the unobserved substates, and the -function ensures energy conservation. The essential part which is of interest in the present context is the transition matrix element rfi(icams>, KbmSb ha) whose dependence on the wavevectors Ka and Kb of the emitted electrons with spin projections mSa and mSb and on the photon energy hot is indicated explicitly. Following the detailed discussion in [TAA87] this matrix... 

Formally, if one has the experimental values of the dielectric tensor e, the magnetic permeability tensor /jl, and the optical rotation tensors p and p for the substrate, one can construct first the optical matrix M, then the differential propagation matrix A, and C, which, to repeat, is the x component of the wavevector of the incident wave. Once A is known, the law of propagation (wave equation) for the generalized field vector ift (the components of E and H parallel to the x and y axes) is specified by Eq. (2.15.18). Experimentally, one travels this path backwards. 

The wavevector is a good quantum number e.g., the orbitals of the Kohn-Sham equations [21] can be rigorously labelled by k and spin. In three dimensions, four quantum numbers are required to characterize an eigenstate. In spherically symmetric atoms, the numbers correspond to n, I, m, s, the principal, angular momentum, azimuthal and spin quantum numbers, respectively. Bloch s theorem states that the equivalent quantum numbers in a crystal are k, k, k and spin. The spin index is usually dropped for non-magnetic materials. 

Since in this one-dimensional case one has complete freedom to define the jc and y directions in a convenient manner, it is simplest to consider the wavevector as lying in the Jc direction. One then has two cases (i) where the E field is perpendicular to the plane formed by the two vectors q and z — Le., E is in the y direction and (ii) where the magnetic field is perpendicular to this plane. In both of these cases, denoted as -waves and -waves respectively, both the wave equation and Eq. (5.34) are satisfied and the form of the solutions is simplified. One then constructs the solutions in all the regions and finds a system of equations of the form of Eq. (5.35). This system only has a solution when the determinant of the coefficients is equal to zero this condition yields two dispersion relations, one associated with the -waves and another with the H waves ... 

The traditional kinematic constraints on high-energy magnetic neutron scattering are severe. A large scalar difference between the incident and scattered neutron wavevectors, and respectively, is required,... 

Fig. 44. Transitions across hybridization gaps of the quasiparticle band structure (a) can, depending on the position of the chemical potential (assumed inside the pseudogap for this calculation), and on the other band-structure features, give rise to pronounced structures in the non-local interaction part Y = x K of the Stoner denominator [compare eqs. (19) and (20)] at small wavevectors (b). In (b) a = y/ir with y defined in eq. (20) is varied somewhat around its proper value of (Grewe and Welslau 1988) to exhibit the strong tendency towards a magnetic instability occurring for Y(9 it> i/ = 0) = l.

The anisotropy of the RKKY interaction usually leads to rather complex magnetie struetures. In particular, nesting of the Fermi surface can have dramatic effects. An exemplary system is CeAlj (Steglieh et al. 1979b) its spin arrangement displays a type-II antiferromagnetic structure in which the magnetic moments are parallel to [111] and point into opposite directions on the lattice sites (0,0,0) and (3,5,5). The moments are modulated by an ineommensurate wavevector (i + t, 5 — t, 5) with t = 0.11 (Barbara et al. 1977). 

Here, k is the change in the wavevector of the neutron and r, s and p are the electron position, spin and momentum operators, respectively. If jtr is the neutron spin operator, the magnetic interaction potential is... 

By way of introduction to a complete theory of scattering by atomic electrons we consider the limiting case where the wavevector transfer k is vanishingly small. In this instance, the magnetic interaction reduces to the form... 

Between 5 and 20 K, the heat capacity shows complicated behavior. From thermal expansion measurements, Zochowski and McEwen (1986) determined the upper Neel transition with ordering on hexagonal sites to be 20.0 K and a second transition at 19.1 K described as being due to magnetic satellites split transversely as the modulation wavevectors tilt away from the (100) directions. Both transitions were apparently first order although from X-ray diffraction measurements Bulatov and Dolzhenko... 

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