Magnetism: spherical harmonics

Transitions between states are subject to certain restrictions called selection rules. The conservation of angular momentum and the parity of the spherical harmonics limit transitions for hydrogen-like atoms to those for which A/ = 1 and for which Am = 0, 1. Thus, an observed spectral line vq in the absence of the magnetic field, given by equation (6.83), is split into three lines with wave numbers vq + (/ bB/he), vq, and vq — (HbB/he). 

Expressions for the electric and magnetic fields can likewise be obtained. These plane-wave solutions are then expanded in terms of spherical harmonics... 

Payne, 1990 Sacks and Noguera, 1991). All those authors used a spherical-harmonic expansion to represent tip wavefunctions in the gap region, which is a natural choice. The spherical-harmonic expansion is used extensively in solid-state physics as well as in quantum chemistry for describing and classifying electronic states. In problems without a magnetic field, the real spherical harmonics are preferred, as described in Appendix A. 

For magnet configurations in which coils are coaxial and symmetric about the illustrated xy-plane, such as the magnet configurations in Figure 2A and C, the spherical harmonic expansion results in the elimination of all even order terms within the expansion. To further reduce computational complexity, the strategy employed here considers only one quarter of the magnet domain, and thus, the constraints in Equation (5) simplify to ... 

In the MSE approach, the size and field homogeneity of the FOV are proportional to number of zeroed inner spherical harmonics, and number of vanished outer harmonics defines the size of the system footprint. We recall that the number of inner and outer spherical harmonics made to vanish refers to the order and degree of the design. Once the order and degree are specified based on the requirements of the FOV and the stray field, and the magnet domain has been established, then the MSE current density map is calculated. 

Suppose that a plane jc-polarized wave is incident on a homogeneous, isotropic sphere of radius a (Fig. 4.1). As we showed in the preceding section, the incident electric field may be expanded in an infinite series of vector spherical harmonics. The corresponding incident magnetic field is obtained from the curl of (4.37) ... 

Consider now the field scattered by an isotropic, optically active sphere of radius a, which is embedded in a nonactive medium with wave number k and illuminated by an x-polarized wave. Most of the groundwork for the solution to this problem has been laid in Chapter 4, where the expansions (4.37) and (4.38) of the incident electric and magnetic fields are given. Equation (8.11) requires that the expansion functions for Q be of the form M N therefore, the vector spherical harmonics expansions of the fields inside the sphere are... 

The applicability of Eq. (21) rests on the validity of the assumption that the averages over internal and external variables are uncorrelated and thus can be calculated separately. Furthermore, theexpression of Eq. (21) emphasizes the close similarity of the irreducible Cartesian representation to the expression of the problem in terms of polar angles and the normalized 2nd rank spherical harmonics Y (see Eq. (7)). The corresponding polar angles ( (1), (t)) and (C(t), (t)), shown in Fig. 2B, describe the orientation of the internuclear vector and the magnetic field relative to the arbitrary reference frame, respectively. The different representations are related according to the following relationships.37... 

In Eq. (4.323) notation of the type Y(a,b) means that a spherical harmonic is built on the components of a unit vector a in the coordinate system whose polar axis points along the unit vector b. The functions. + and, S4 in Eq. (4.324) are the equilibrium parameters of the magnetic order of the particle defined, in general, by Eqs. (4.80)-(4.83). 

Equations (4.329) for a solid assembly and (4.332) for a magnetic suspension are solved by expanding W with respect to the appropriate sets of functions. Convenient as such are the spherical harmonics defined by Eq. (4.318). In this context, the internal spherical harmonics used for solving Eq. (4.329) are written Xf (e, n). In the case of a magnetic fluid on this basis, a set of external harmonics is added, which are built on the angles of e with h as the polar axis. Application of a field couples [see the kinetic equation (4.332)] the internal and external degrees of freedom of the particle so that the dynamic variables become inseparable. With regard to this fact, the solution of equation (4.332) is constructed in the functional space that is a direct product of the internal and external harmonics ... 

The constant A = 3.96 cm-1 has been obtained from the free-molecule zero-field splitting (Mizushima, 1975) and C is a Racah spherical harmonic with 1 = 2. The tensor that describes the interaction between the magnetic dipole moments ge Sp, where ge equals 2.0023 and fxB is the Bohr magneton, can be written immediately as... 

The shim coils are constructed based on an expansion of the magnetic field inhomogeneity in terms of spherical harmonics. The fields produced by the coils are orthogonal and can be adjusted independent of each other. Following (2.1.18) the shim fields Bs m are characterized by the associated Legendre polynomials P m (cos 6) and the expansion... 

To enable us to calculate the relaxation time of the magnetization in a direction perpendicular to the applied field it is necessary to expand IV in spherical harmonics involving the two space coordinates i and . For this we use the form given in the introduction to this section which is... 

The spherical harmonic analysis so far presented for uniaxial anisotropy is mainly concerned with the relaxation in a direction parallel to the easy axis of the uniaxial anisotropy. We have not considered in detail the behavior resulting from the transverse application of an external field and the relaxation in that direction for uniaxial anisotropy. Thus we have only considered potentials of the form V(r, t) = V(i, t) where the azimuthal or dependence in Brown s equation is irrelevant to the calculation of the relaxation times. This has simplified the reduction of that equation to a set of differential-difference equations. In this section we consider the reduction when the azimuthal dependence is included. This is of importance in the transition of the system from magnetic relaxation to ferromagnetic resonance. The original study [17] was made using the method of separation of variables on Brown s equation which reduced the solution to an eigenvalue problem. We reconsider the solution by casting... 

To relate Dff to the anisotropic motion of a molecule in a liquid crystalline solvent, we employ the function P(d, < ), defined as the probability per unit solid angle of a molecular orientation specified by the angles 6 and <3>, the polar coordinates of the applied magnetic field direction relative to a molecule-fixed Cartesian coordinate system. We expand P(0, ) in real spherical harmonics ... 

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