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Spin Dynamics in Disordered Solids

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Encyclopedia of Complexity and Systems Science

Definition of the Subject

The main specifics of the theory of spin dynamics in disordered solids result from the fact that calculation of observable values muststart from the solution of the equation of motion, and then they should be averaged over random distribution of spins in the sample. Nominallyany problem in statistical physics looks analogous, but the content of existing text books contains much more simple bypass methods to achievethe results. An important bypass class, determined in the preceding century, consists of problems that are equivalent to the motion of weak (orseldom) interacting (quasi)particles in translational invariant media. The basis of this solution is formed by the Boltzmann equation (devisedin the 19th century) and by methods of deriving hydrodynamic equations based on this foundation [1]. Other important advancements in physical kinetics are connected with the invention ofa projection technique byNakajima–Zwanzig for deriving...

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Abbreviations

Averaging:

Two kinds of averaging are necessary for any observable \( { \hat Q } \): a standard quantum mechanical one \( { Q=\langle \hat Q \rangle = \operatorname{Tr}(\hat{Q}\rho) } \), where ρ is the density matrix, and consequent averaging over all possible positions of particles in disordered media \( { \langle Q\rangle_c = \langle\langle\hat Q\rangle\rangle_c } \).

Continuum media approximation j:

Positions of particles forming the disordered solids can be considered as a subset (impurity sites) of crystal lattice sites, randomly distributed on the lattice with a small probability \( { c\ll 1 } \) to find a given site occupied. Many important results can be received in continuum media approximation (CMA) when prime cell volume \( { \Omega_\mathrm{c} \to 0 } \) together with impurity concentration \( { c\to 0 } \) at a fixed value of impurity density \( { n=c/\Omega_\mathrm{c} } \).

Disordered solids :

Statically disordered media are considered, this means that constituent particles (atoms, ions and so on) do not participate in significant translational motions during the relaxation time under discussion, and therefore their positions are fixed (frozen) in the main approximation.

High-temperature approximation :

High-temperature approximation (HTA) in spin dynamics consists of using the simplest density matrix \( \rho=1/\operatorname{Tr}1 \), corresponding to the limit of infinite temperature T of canonical Gibbs distribution \( \rho_\mathrm{G}=\exp(-H/T)/ \operatorname{Tr}\exp(-H/T) \); here H is the Hamiltonian. Spin dynamics remains nontrivial and rich in this limit. Sometimes HTA means application of \( \rho=(1-H/T)/\operatorname{Tr}1 \).

Local field :

See “Secular part of dipole–dipole interactions”.

Local frequency :

The frequency \( { \omega_\text{loc}=\gamma H_\text{loc} } \) of rotation of the spin in local field \( { H_{\text{loc}} } \).

Secular part of dipole–dipole interactions :

As a rule any spin is considered as subjected to an external static magnetic field \( { \mathbf{H}_0=H_0\mathbf{n}_z } \) (directed along the z‑axis) and local field, produced by dipole magnetic moments of surrounding spins. The Hamiltonian of the dipole–dipole interaction of spins \( { \mathbf{I}_i } \) and \( { \mathbf{I}_j } \) is of the form \( { H_d^{(0)}(i,j) = r_{ij}^{-3}((\mathbf{m}_i\mathbf{m}_j) - 3(\mathbf{n}_{ij}\mathbf{m}_i)(3\mathbf{n}_{ij}\mathbf{m}_j)) } \), where \( { \mathbf{r}_{ij} = \mathbf{n}_{ij}r_{ij} = \mathbf{r}_i-\mathbf{r}_j } \), and \( { \mathbf{r}_i } \) is the position of spin “i” having magnetic moment \( { \mathbf{m}_i = \hbar\gamma_iI_i } \). If \( { H_0\gg H_\text{loc} } \), where H \( { _{\text{loc}} } \) is the mean square field produced at any spin by surrounding spins (local field), then the Hamiltonian \( { H_d^{(0)}(i,j) } \) can be substituted by the so‐called secular dipole–dipole Hamiltonian \( { H_{\!d}(i,j) \!=\! \smash{\frac{1}{2r_{ij}^3}} (1\!-\!3(\mathbf{n}_{ij}\mathbf{n}_z)^2) (\mathbf{m}_i\mathbf{m}_j\!-\!3(\mathbf{m}_i\mathbf{n}_z) (\mathbf{m}_j\mathbf{n}_z)) } \). The accuracy of the substitution is not less than \( { \sim H_\text{loc}/H_0 } \).

Spin dynamics :

Spin dynamics is considered as a time evolution of correlation functions directly connected with measurable quantities of paramagnetic samples.

Units:

As a rule \( { \hslash=1 } \) is supposed. Part of the equations contains ℏ written explicitly.

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Authors and Affiliations

  1. Institute for Theoretical and Experimental Physics, Moscow, Russia

    Fridrikh S. Dzheparov

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  1. Fridrikh S. Dzheparov
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  1. RAMTECH LIMITED, 122 Escalle Lane, Larkspur, CA, 94939, USA

    Robert A. Meyers Ph. D. (Editor-in-Chief) (Editor-in-Chief)

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Dzheparov, F.S. (2009). Spin Dynamics in Disordered Solids. In: Meyers, R. (eds) Encyclopedia of Complexity and Systems Science. Springer, New York, NY. https://doi.org/10.1007/978-0-387-30440-3_513

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