Fokker-Planck and Landau-Lifshitz-Bloch Equations for Classical Ferromagnets

A macroscopic equation of motion for the magnetization of a ferromagnet at elevated temperatures should contain both transverse and longitudinal relaxation terms and interpolate between Landau-Lifshitz equation at low temperatures and the Bloch equation at high temperatures. It is shown that for the classical model where spin-bath interactions are described by stochastic Langevin fields and spin-spin interactions are treated within the mean-field approximation (MFA), such a ``Landau-Lifshitz-Bloch'' (LLB) equation can be derived exactly from the Fokker-Planck equation, if the external conditions change slowly enough. For weakly anisotropic ferromagnets within the MFA the LLB equation can be written in a macroscopic form based on the free-energy functional interpolating between the Landau free energy near T_C and the ``micromagnetic'' free energy, which neglects changes of the magnetization magnitude |{\bf M}|, at low temperatures.Comment: 9 pages, no figures, a small error correcte

Bloch-Wall Phase Transition in the Spherical Model

The temperature-induced second-order phase transition from Bloch to linear (Ising-like) domain walls in uniaxial ferromagnets is investigated for the model of D-component classical spin vectors in the limit D \to \infty. This exactly soluble model is equivalent to the standard spherical model in the homogeneous case, but deviates from it and is free from unphysical behavior in a general inhomogeneous situation. It is shown that the thermal fluctuations of the transverse magnetization in the wall (the Bloch-wall order parameter) result in the diminishing of the wall transition temperature T_B in comparison to its mean-field value, thus favouring the existence of linear walls. For finite values of T_B an additional anisotropy in the basis plane x,y is required; in purely uniaxial ferromagnets a domain wall behaves like a 2-dimensional system with a continuous spin symmetry and does not order into the Bloch one.Comment: 16 pages, 2 figure

Nonlinear response of superparamagnets with finite damping: an analytical approach

The strongly damping-dependent nonlinear dynamical response of classical superparamagnets is investigated by means of an analytical approach. Using rigorous balance equations for the spin occupation numbers a simple approximate expression is derived for the nonlinear susceptibility. The results are in good agreement with those obtained from the exact (continued-fraction) solution of the Fokker-Planck equation. The formula obtained could be of assistance in the modelling of the experimental data and the determination of the damping coefficient in superparamagnets.Comment: 7 PR pages, 2 figure

Quantum Nonlinear Switching Model

We present a method, the dynamical cumulant expansion, that allows to calculate quantum corrections for time-dependent quantities of interacting spin systems or single spins with anisotropy. This method is applied to the quantum-spin model \hat{H} = -H_z(t)S_z + V(\bf{S}) with H_z(\pm\infty) = \pm\infty and \Psi (-\infty)=|-S> we study the quantity P(t)=(1-_t/S)/2. The case V(\bf{S})=-H_x S_x corresponds to the standard Landau-Zener-Stueckelberg model of tunneling at avoided-level crossing for N=2S independent particles mapped onto a single-spin-S problem, P(t) being the staying probability. Here the solution does not depend on S and follows, e.g., from the classical Landau-Lifshitz equation. A term -DS_z^2 accounts for particles' interaction and it makes the model nonlinear and essentially quantum mechanical. The 1/S corrections obtained with our method are in a good accord with a full quantum-mechanical solution if the classical motion is regular, as for D>0.Comment: 4 Phys. Rev. pages 2 Fig

Landau-Zener-Stueckelberg effect in a model of interacting tunneling systems

The Landau-Zener-Stueckelberg (LZS) effect in a model system of interacting tunneling particles is studied numerically and analytically. Each of N tunneling particles interacts with each of the others with the same coupling J. This problem maps onto that of the LZS effect for a large spin S=N/2. The mean-field limit N=>\infty corresponds to the classical limit S=>\infty for the effective spin. It is shown that the ferromagnetic coupling J>0 tends to suppress the LZS transitions. For N=>\infty there is a critical value of J above which the staying probability P does not go to zero in the slow sweep limit, unlike the standard LZS effect. In the same limit for J>0 LZS transitions are boosted and P=0 for a set of finite values of the sweep rate. Various limiting cases such as strong and weak interaction, slow and fast sweep are considered analytically. It is shown that the mean-field approach works well for arbitrary N if the interaction J is weak.Comment: 13 PR pages, 15 Fig

Quantum statistical metastability for a finite spin

We study quantum-classical escape-rate transitions for uniaxial and biaxial models with finite spins S=10 (such as Mn_12Ac and Fe_8) and S=100 by a direct numerical approach. At second-order transitions the level making a dominant contribution into thermally assisted tunneling changes gradually with temperature whereas at first-order transitions a group of levels is skipped. For finite spins, the quasiclassical boundaries between first- and second-order transitions are shifted, favoring a second-order transition: For Fe_8 in zero field the transition should be first order according to a theory with S \to \infty, but we show that there are no skipped levels at the transition. Applying a field along the hard axis in Fe_8 makes transition the strongest first order. For the same model with S=100 we confirmed the existence of a region where a second-order transition is followed by a first-order transition [X. Martines Hidalgo and E. M. Chudnovsky, J. Phys.: Condensed Matter (in press)].Comment: 7 Phys. Rev. pages, 10 figures, submitted to PR

Phase transition between quantum and classical regimes for the escape rate of a biaxial spin system

Employing the method of mapping the spin problem onto a particle one, we have derived the particle Hamiltonian for a biaxial spin system with a transverse or longitudinal magnetic field. Using the Hamiltonian and introducing the parameter $p (\equiv (U_{max}-E)/(U_{max}-U_{min}))$ where $U_{max}$ (U_{min}) corresponds to the top (bottom) of the potential and $E$ is the energy of the particle, we have studied the first- or second-order transition around the crossover temperature between thermal and quantum regimes for the escape rate, depending on the anisotropy constant and the external magnetic field. It is shown that the phase boundary separating the first- and second-order transition and its crossover temperature are greatly influenced by the transverse anisotropy constant as well as the transverse or longitudinal magnetic field.Comment: 5 pages + 3 figures, to be published in Phys. Rev.

Quantum dynamics of a nanomagnet in a rotating field

Quantum dynamics of a two-state spin system in a rotating magnetic field has been studied. Analytical and numerical results for the transition probability have been obtained along the lines of the Landau-Zener-Stueckelberg theory. The effect of various kinds of noise on the evolution of the system has been analyzed.Comment: 7 pages, 7 figure

The 1/D Expansion for Classical Magnets: Low-Dimensional Models with Magnetic Field

The field-dependent magnetization m(H,T) of 1- and 2-dimensional classical magnets described by the $D$-component vector model is calculated analytically in the whole range of temperature and magnetic fields with the help of the 1/D expansion. In the 1-st order in 1/D the theory reproduces with a good accuracy the temperature dependence of the zero-field susceptibility of antiferromagnets \chi with the maximum at T \lsim |J_0|/D (J_0 is the Fourier component of the exchange interaction) and describes for the first time the singular behavior of \chi(H,T) at small temperatures and magnetic fields: \lim_{T\to 0}\lim_{H\to 0} \chi(H,T)=1/(2|J_0|)(1-1/D) and \lim_{H\to 0}\lim_{T\to 0} \chi(H,T)=1/(2|J_0|)

СПЕКТРОМЕТР ДЛЯ АНАЛИЗА МЕТАЛЛОВ «ГРАНД-ЭКСПЕРТ». СОВРЕМЕННОЕ СОСТОЯНИЕ И АНАЛИТИЧЕСКИЕ ВОЗМОЖНОСТИ

This paper presents the characteristics of the modern Grand-Expert spectrometer for the analysis of metals and alloys. The spectrometer has an updated optical scheme and a new spectrum analyzer to solve a wide range of analytical tasks. The analytical capabilities of the spectrometer were investigated for the analysis of steels and high-purity copper and aluminum as an example. For each of the bases, the updated optical scheme made it possible to realize new opportunities for controlling the homogeneity of the sample material and the presence of micro-inclusions on the sample surface and for determining low impurity contents in the pure metal bases. The spectrometer uses a modern semiconductor spark generator with adjustable frequency, current intensity, and duration of individual spark pulses. Spectra of metal samples for individual spark pulses were obtained in real time for the investigated sample. The operation of the spectrometer in different modes and with different exposure times was tested to select the optimal parameters of calibration characteristics. Computer control provides full synchronization of the generator mode setting, argon feeding, and spectrum registration. For steels, we selected sparking modes with high stability of spectral line intensities and analyte concentrations, and for pure metals (copper and aluminum), modes providing low detection limits of impurity elements and good stability of the results.Keywords: optical spectrometry, atomic-emission spectrometer, spectral analyzer, MAES, determination of metal composition DOI: http://dx.doi.org/10.15826/analitika.2021.25.4.008V.G. Garanin  VMK-Optoelektronika, ul. Akademika Koptyuga 1, Novosibirsk, 630090, Russian FederationВ статье приведены характеристики современного спектрометра для анализа металлов и сплавов «Гранд-Эксперт» с обновленной оптической схемой и новым анализатором спектров, которые позволили решить широкий круг аналитических задач. Рассмотрены аналитические возможности спектрометра на примере анализа сталей, высокочистых меди и алюминия. Для каждой основы обновленная оптическая схема позволила реализовать новые возможности в части контроля однородности материала пробы и наличия на поверхности анализируемых образцов микровключений, а также определения в чистых металлических основах низких содержаний примесей. В спектрометре применяется современный полупроводниковый искровой генератор с регулировкой частоты, силы тока и длительности отдельных искровых импульсов. Получены спектры металлических образцов для отдельных искровых импульсов в режиме реального времени для исследованной пробы. Для выбора оптимальных параметров градуировочных характеристик проверена работа спектрометра в разных режимах и с разным временем экспозиций. Компьютерное управление обеспечивает полную синхронизацию установки режимов генератора, подачи аргона и регистрации спектров. Для сталей выбраны режимы обыскривания с высокой стабильностью интенсивностей спектральных линий и концентраций основных определяемых элементов. Для чистых металлов (меди и алюминия) выбраны режимы обеспечивающие низкие пределы обнаружения элементов-примесей и хорошую стабильность получаемых результатов.Ключевые слова: оптическая спектрометрия, атомно-эмиссионный, спектрометр, анализатор спектров, МАЭС, определение состава металловDOI: http://dx.doi.org/10.15826/analitika.2021.25.4.00