Spin, z component

For opterators that work also on spin variables, the expectation values are given by equations analogous to (5) except that P is replaced by p and the integrations are over x (i.e. both r and s) instead of r alone. Thus, the total spin z component has an expectation value... 

The one-electron density matrix gives rise to others, for example P KK ri-,r i), whose diagonal element gives the probable number of electrons per unit volume (without reference to spin) in the spatial volume element dri at point ri and the spin density matrix Q KK ri r i), whose diagonal element gives the contribution to the expectation value of the total spin z-component, < Sz >, associated with the same volume element. These functions are related to p KK xi-,x i) as follows ... 

Because of the difference in the nature of holes and particles, we need to pay special attention to the possible spin states of the entire system. When the electron is removed from state " (r - R ), the many-body state has a total spin z-component = -Sh, since the original ground state had spin 0. Therefore, the new state created by adding a particle in state - R ) produces a many-body state with total spin z-component S = Sp — Sh. This reveals that when we deal with hole states, we must take their contribution to the spin as the opposite of what a normal particle would contribute. Taking into consideration the fact that the hole has opposite wave-vector of a particle in the same state, we conclude that the hole corresponds to the time-reversed particle state, since the effect of the time-reversal operator T on the energy and the wavefunction is ... 

The Hamiltonian (3.4) is a function of the usual spatial coordinates x, y, z or r, 0, (j)). Electrons possess the intrinsic property of spin, however, which is to be thought of as a property in an independent, or orthogonal, space (spin space). Spin is actually a consequence of the theory of relativity but we shall merely graft on the property in an ad hoc fashion. The spin, s, of an electron (don t confuse with s orbitals ) takes the value 1/2 only. The z component of spin, m, takes (25 + 1) values of ms, ranging 5, 5-l,...-s. Thus for the single electron, = +1/2 or -1/2, also labelled a or p, or indicated by t or i. 

The spin angular momentum of a term is labelled with S and may take integrally separated values based on 0 or 1/2 depending upon the d configuration viz. S = 0, 1, 2... or 5 = 1/2, 3/2, 5/2... Associated with each such S value are (2S + 1) values of Ms for the z components of spin angular momentum, with Ms taking the values S,... 

Triplet state. The spin eigenfunctions of the triplet state, with (z-component of the total spin) = 1, are written as [29]... 

The second important spin-angular operation is the 90° rotation where the polarization is transformed from the z to the x direction or vice versa. A Mezei coil in the x,z plane is adjusted such that the resultant field points exactly in the direction of the bisection of the angle between x and z. A 180° rotation around this axis transforms the z component of polarization to the x direction. At the same time, the sign of the y component is inverted (Fig. lc). 

If the external magnetic field B(r), and m(r) have only a nonvanishing Z-component, B(r) = (0,0, B(r)) and m(r) = (0,0, m(r)), the universal functional F[p, m] may then be considered as a functional of the spin densities ps(r) and p(r), F[ps(r), p(r)], because the spin density is proportional to the z-component of the magnetization m(r) = p-bPsW P-b is the electron Bohr magneton. It is of worth mentioning that it is possible to define two spin densities that are the diagonal elements of the density matrix introduced by von Barth and Hedin [3]. These correspond to the spin-up (alpha) electrons density pT(r), and the spin-down (beta) electrons density p (r). In terms of these quantities, the electron and spin densities can be written as... 

Figure 3.9 (A) Nuclear spin energy for a nucleus having 7=1 (e.g., N) plotted as a function of Bq. (B) Nuclear vectors relative to Bo with vector length h[7(7+l)] and z component hm so that cos0 =m[7(7-i-1)] ". (Adapted with permission of Nelson Thornes Ltd. from Figures la and lb of reference 21.)...

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