Green function theory of ferromagnetism used to examine implications of cube-root behavior of magnetization near Curie point

A spin system with unperturbed Hamiltonian ${\mathcal{H}}_{0}=\frac{1}{2}\ensuremath{\Sigma}\ensuremath{\Sigma}{B}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}\ensuremath{-}\ensuremath{\gamma}H\ensuremath{\Sigma}{{\ensuremath{\sigma}}_{x}}^{j}$ relaxing via the spin-lattice coupling $G=\frac{1}{2}\ensuremath{\Sigma}\ensuremath{\Sigma}{C}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}({{\ensuremath{\sigma}}_{y}}^{k}+{{\ensuremath{\sigma}}_{z}}^{k})$ is studied by means of the general density-matrix theory of magnetic relaxation. By making some assumptions about the magnitude and time constants of the lattice correlation functions $〈{C}_{\mathrm{ij}}(t){C}_{\mathrm{kl}}(0)〉$, a master equation is obtained. It agrees at high temperatures with a master equation previously suggested by Glauber for the one-dimensional nearest-neighbor case. At high temperatures the magnetic moment relaxes with a single relaxation time, and the spin pair-correlation functions satisfy a closed set of equations. At low temperatures, however, the equations for the magnetization and the correlation functions are coupled to higher-order moments.

A detailed investigation is made of the approach to equilibrium of system of spins with $I=\frac{1}{2}$. The spin-spin interaction ${\ensuremath{\Sigma}}_{k}{\ensuremath{\Sigma}}_{j<k}{B}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}$ with ${B}_{\mathrm{jk}}$ constant is treated exactly, while the additional interaction ${\ensuremath{\Sigma}}_{k}{\ensuremath{\Sigma}}_{j<k}{L}_{\mathrm{jk}}{{\ensuremath{\sigma}}_{x}}^{j}{{\ensuremath{\sigma}}_{x}}^{k}$ with ${L}_{\mathrm{jk}}$ depending on lattice vibrations is treated by means of the assumption of sufficiently short correlation times for the ${L}_{\mathrm{jk}}$ operators. All correlations between the ${L}_{\mathrm{jk}}$ are included. The effect of the correlations usually neglected is expressed in terms of a sum over states somewhat resembling an Ising-model partition function with the time replacing interaction strength. The oscillatory relaxation via the ${B}_{\mathrm{jk}}$ and the monotonic relaxation via the phonons compete with each other; interference effects between the two relaxation modes also occur; the origin and nature of the irreversibility are very different for the two relaxation modes.

Sets of equations for the moments of the vibrational energy distribution for a diatomic species in a gas are derived, considering simultaneous dissociation and vibrational relaxation. In first approximation a set of two coupled equations are obtained, one for the number of diatomic molecules and the other for the vibrational energy per molecule. The latter contains two correction terms to the Bethe—Teller equation, due to the coupling. The next higher approximation gives three coupled equations, in which the second moment (dispersion) of the distribution function is taken into account. In these equations the dissociation rate ``constant'' is a function of both the average vibrational energy and the second moment. The recombination process is described by three constants which define the over-all rate, the average vibrational energy, and the dispersion about the average energy of just recombined molecules. The starting point in the development is a gain—loss equation for occupation numbers of vibrational levels. The assumption of harmonic vibrations is used. In order to close the set of equations in any order of approximation, those terms involving higher moments are determined by application of a general method due to Jaynes which gives the least biased estimate in view of the information contained in the moments considered. Consideration of the steady state in first approximation leads to Onsager reciprocal relations and, in second approximation, to more complicated relations.

Time-dependent linear gyromagnetic effects are treated in terms of Kubo's theory of irreversible processes. Quantitative and qualitative relations between various possible experiments are pointed out. The time constants can give information about relative relaxation rates of spin and orbital angular momentum. If the hyperfine interaction is strong, the Einstein-de Haas effect may give information on electron-nuclear spin correlations; also nuclear polarization by rotation analogous to Rose-Gorter polarization is expected to be possible; a small inertial line shift due to rotation is noted.

A unified theory of atomic and macroscopic gyromagnetic phenomena and ordinary magnetic polarization is given. Nuclear gyromagnetic effects and nuclear magnetic-field polarization are also considered. The dynamics of spin systems are considered, in order to derive the quantummechanical Larmor equation. The equilibrium gyromagnetic-magnetic theory is treated in detail. Time- dependent behavior of nuclear spins in rotating crystals is examined. (T.F.H.)