Matrix Elements of the System of Moment Equations Governing the Kinetics of Superparamagnetic Particles

Abstract

An exact equation for the matrix elements of the system of moments [the expectation values of the spherical harmonics ͗Y n,m ͘ ͑t͒], which govern the kinetics of the magnetization M of a single domain particle, is derived. This derivation is based on the representation of the stochastic Gilbert equation augmented by a random field and the corresponding Fokker-Planck equation for the distribution function of the orientations of M in terms of the angular momentum operators and on the subsequent evaluation of the matrix elements in terms of the Clebsch-Gordan coefficients. [S0031-9007(99)08847-X]

Birefringence and dielectric relaxation in strong electric fields

“…(3.17). The derivation of the stochastic equation for Y/,m from Gilbert's equation (9.21) and the averaging of the equation so obtained over an ensemble of particles, which all have at time t the same magnetization M(t), was given in [132,133], where the following equation for the sharp values Y L ,~ was obtained, namely,…”

“…(9.25) can be expressed in terms of the angular momentum operators Lz, L 5 , L2 [57,60] Thus, one obtains [132,133] where V = U -(M . H ) = V+ + VBy making use of the theory of angular momentum, one may essentially simplify the solution of the problem under consideration because the action of the angular moment operators on Yi,m is given by [57] Furthermore, the functions Y<: and Y < l , may also be considered as operators, which act on YI,, in accordance with the relation [132] 81~( 21+ 1 where Here, it is assumed that s 2 0 and the summation is carried out over those values of indexes, for which the Clebsch-Gordan coefficients (11, ml ,12, rnzll, rn) do not vanish. The Y I ,~ in Eq.…”

Spin Relaxation in Phase Space

“…Thus, it is evident that it is relatively easy to generalize the phase space formalism to non-axially symmetric problems for an arbitrary spin Hamiltonian in a manner exactly analogous to that given for the classical Fokker-Planck equation (3) in Ref. 145.…”

“…Here our objective is to illustrate this method for the formal phase space master equation (252) pertaining to a Hamiltonian ˆS H as accomplished for the Fokker-Planck equation (3) for classical spins for arbitrary magnetocrystalline anisotropy-Zeeman energy potentials [5,145]. We shall illustrate how the magnetization and its relaxation times may be evaluated in the linear response approximation and how the solution of the corresponding classical problem [5,6] carries over into the quantum domain.…”

Precessional effects in the linear dynamic susceptibility of uniaxial superparamagnets: Dependence of the ac response on the dissipation parameter

“…In order to verify our conjectures concerning the ␣ dependence of the linear response to a small ac field H(t) ͑i.e., assuming ␤M s HӶ1͒, we have calculated using linear-response theory the complex magnetic susceptibility ͑ ͒ of the system. The susceptibility was calculated by using a matrix continuedfraction solution 20,21 of the system of moments ͓the expectation values of the spherical harmonics ͗Y l,m ͘(t)͔ governing the kinetics of the magnetization M ͑the moment system can be obtained either from the FPE or from the LLG equation 22 ͒. The details of the calculation can be found elsewhere 20,21 ; it is assumed that H(t) is directed along H 0 .…”