Spherical Symmetry and Spins

Abstract A brief excursion is made into the concept of continuous groups, with an example of the rotation groups. The purpose is to familiarize the reader with the concept of electron spin. The coupling of spins is discussed. Applications are taken from Crystal-Field Theory and Electron Spin Resonance. 

---

It has been pointed out above that two electrons in the Is orbital must have their spins opposed, and hence give rise to the singlet state So, with no spin or orbital angular momentum, and hence with no magnetic moment. Similarly it is found that a completed subshell of electrons, such as six electrons occupying the three 2p orbitals, must have S — 0 and L = 0, corresponding to the Russell-Saunders term symbol lS0 such a completed subshell has spherical symmetry and zero magnetic moment. The application of the Pauli exclusion prin-... 

We say that z forms a basis for A,or that z belongs to Ai, or that z transforms according to the totally symmetric representation Ai. The s orbitals have spherical symmetry and so always belong to IY This is taken to be understood and is not stated explicitly in character tables. Rx, Ry, Rz tell us how rotations about x, y, and z transform (see Section 4.6). Table 4.5 is in fact only a partial character table, which includes only the vector representations. When we allow for the existence of electron spin, the state function ip(x y z) is replaced by f(x y z)x(ms), where x(ms) describes the electron spin. There are two ways of dealing with this complication. In the first one, the introduction of a new... 

Next we can dispose of the matrix elements of the one-electron operator Hy,Eq. (3-3 b) the kinetic energy operator, the electron-nuclear attraction potential arising from the metal nucleus, and the spin-independent relativistic terms have spherical symmetry and can be treated through the definition of the basis orbitals ip, Eq. (3-11). The spin-... 

For spherical symmetry with spin-orbit included there are many more separate states to be considered. For f the number increases from 73 SL states to 198 SLJ states for f ds the numbers are 1166 and 3707. Table 15.4 shows the J distribution for these two configurations, with the selection rules for the spin-orbit coefficients. Because there are off-diagonal C matrix elements between adjacent S and L values, there is mixing between the states and S and L are no... 

As discussed in Sect. 6.2, the electronic states of a paramagnetic ion are determined by the spin Hamiltonian, (6.1). At finite temperamres, the crystal field is modulated because of thermal oscillations of the ligands. This results in spin-lattice relaxation, i.e. transitions between the electronic eigenstates induced by interactions between the ionic spin and the phonons [10, 11, 31, 32]. The spin-lattice relaxation frequency increases with increasing temperature because of the temperature dependence of the population of the phonon states. For high-spin Fe ", the coupling between the spin and the lattice is weak because of the spherical symmetry of the ground state. This... 

The no-pair DCB Hamiltonian (6) is used as a starting point for variational or many-body relativistic calculations [9], The procedure is similar to the nonrelativistic case, with the Hartree-Fock orbitals replaced by the four-component Dirac-Fock-Breit (DFB) functions. The spherical symmetry of atoms leads to the separation of the one-electron equation into radial and spin-angular parts [10], The radial four-spinor has the so-called large component the upper two places and the small component Q, in the lower two. The quantum number k (with k =j+ 1/2) comes from the spin-angular equation, and n is the principal quantum number, which counts the solutions of the radial equation with the same k. Defining... 

---