Ferromagnetic Hysteresis by Heisenberg Partition Function and Its Processing Methods in Nanoparticle Magnetization Modeling

The Heisenberg {\it ab initio} theory of magnetization is developed to apply for multilayer nanoparticles. The theory is based on distribution and partition functions modification with account the difference between exchange integral and closest neighbour numbers, that change the system of resulting transcendental equation for magnetization and its reversal to form either a paramagnetic type curve or hysteresis loops patterns. The equations are obtained within the Heisenberg partition function construction by Heitler diagonalization of energy matrix via irreducible representations of permutation symmetry group. A combination with the Gauss distribution gives the explicit expression for the partition function in the asymptotic limit] at large spin range in terms of transcendent function. The exchange integral, as a parameter of the equation of state (material equation) is evaluated from Curie temperature value by means of a formula derived within the presented theory. Methods of data processing from the simultaneous solution of the material equation system are proposed. The multi-valued function of hysteresis loop is found by combination of graphical approach and special procedure for elimination of mistaken peaks and prolapses of the patterns. The theory and computation methods are applied to spherical particles with separate surface layers consideration. The contribution of the surface layers, that are specified by number of closest neighbors and exchange integrals into overall magnetization, is studied for two-layer and three-layer models, that are discussed and compared graphically.

Study of the equilibrium critical behavior of thin granular membranes by Monte Carlo methods

Simulation of thin granular films and twolayer structures is carried out by Monte Carlo methods. Temperature dependence of equilibrium macroscopic characteristics of granular films such as magnetization, energy, heat capacity, and magnetic susceptibility are calcu-lated for different linear sizes of granules. Influence of exchange integral values for inter-action between granules in film and interlayer interaction in twolayer structure is studied.

Enstrophy and circulation scaling for Navier–Stokes reconnection

As reconnection begins and the enstrophy $Z$ grows for two configurations, helical trefoil knots and anti-parallel vortices, two regimes of self-similar collapse are observed. First, during trefoil reconnection a new $\sqrt{\unicode[STIX]{x1D708}}Z$ scaling, where $\unicode[STIX]{x1D708}$ is viscosity, is identified before any $\unicode[STIX]{x1D716}=\unicode[STIX]{x1D708}Z$ dissipation scaling begins. Further rescaling shows linearly decreasing $B_{\unicode[STIX]{x1D708}}(t)=(\sqrt{\unicode[STIX]{x1D708}}Z)^{-1/2}$ at configuration-dependent crossing times $t_{x}$. Gaps in the vortex structures identify the $t_{x}$ as when reconnection ends and collapse onto $\unicode[STIX]{x1D708}$-independent curves can be obtained using $A_{\unicode[STIX]{x1D708}}(t)=(T_{c}(\unicode[STIX]{x1D708})-t_{x})(B_{\unicode[STIX]{x1D708}}(t)-B_{\unicode[STIX]{x1D708}}(t_{x}))$. The critical times $T_{c}(\unicode[STIX]{x1D708})$ are identified empirically by extrapolating the linear $B_{\unicode[STIX]{x1D708}}(t)$ regimes to $B_{\unicode[STIX]{x1D708}}^{{\sim}}(T_{c})=0$, yielding an $A_{\unicode[STIX]{x1D708}}(t)$ collapse that forms early as $\unicode[STIX]{x1D708}$ varies by 256. These solutions are regular or non-singular, as shown by decreasing cubic velocity norms $\Vert u\Vert _{L_{\ell }^{3}}^{}$. For the anti-parallel vortices, first there is an exchange of circulation, from $\unicode[STIX]{x1D6E4}_{y}(y=0)$ to $\unicode[STIX]{x1D6E4}_{z}(z=0)$, mediated by the viscous circulation exchange integral $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6E4}}(t)$, which is followed by a modified $B_{\unicode[STIX]{x1D708}}(t)$ collapse until the reconnection ends at $t_{x}$. Singular Leray scaling and mathematical bounds for higher-order Sobolev norms are used to help explain the origins of the new scaling and why the domain size $\ell$ has to increase to maintain the collapse of $A_{\unicode[STIX]{x1D708}}(t)$ and $\unicode[STIX]{x1D716}_{\unicode[STIX]{x1D6E4}}$ as $\unicode[STIX]{x1D708}$ decreases.

Влияние межатомного обменного взаимодействия на магнитные фазовые переходы в условиях спинового кроссовера при высоких давлениях

The phase diagram of an antiferromagnet with the spin crossover from the high-spin to low-spin state with S = 0 with increasing external pressure has been calculated with regard to the pressure dependence of the exchange integral. The calculated results are compared with the experimental data on ferropericlase Fe_ x Mg_1 – x O.

Фаза Имри-Ма в нанокристаллическом ферромагнетике

A possibility of appearing a disordered Imry–Ma state induced by fluctuations of the anisotropy easy axis direction in a nanocrystalline sample in the case of a weak exchange interaction between crystallites has been studied. A phase diagram of the system has been built in variables “the characteristic crystallite size–the exchange integral of the intercrystallite interaction.” The characteristic value of statistic fluctuations of the crystallographic anisotropy has been estimated, and the dependences of the coercive field on the crystallite size have been found for systems of various space dimensions.

Magnetic nanopantograph in the SrCu2(BO3)2 Shastry–Sutherland lattice

Magnetic materials having competing, i.e., frustrated, interactions can display magnetism prolific in intricate structures, discrete jumps, plateaus, and exotic spin states with increasing applied magnetic fields. When the associated elastic energy cost is not too expensive, this high potential can be enhanced by the existence of an omnipresent magnetoelastic coupling. Here we report experimental and theoretical evidence of a nonnegligible magnetoelastic coupling in one of these fascinating materials, SrCu2(BO3)2 (SCBO). First, using pulsed-field transversal and longitudinal magnetostriction measurements we show that its physical dimensions, indeed, mimic closely its unusually rich field-induced magnetism. Second, using density functional-based calculations we find that the driving force behind the magnetoelastic coupling is the CuOCu^ superexchange angle that, due to the orthogonal Cu2+ dimers acting as pantographs, can shrink significantly (0.44%) with minute (0.01%) variations in the lattice parameters. With this original approach we also find a reduction of ∼10% in the intradimer exchange integral J, enough to make predictions for the highly magnetized states and the effects of applied pressure on SCBO.