Heisenberg Exchange Coupling Research Articles

Valence bond description of antiferromagnetic coupling in transition metal dimers

A single configuration model containing nonorthogonal magnetic orbitals is developed to represent the important features of the antiferromagnetic state of a transition metal dimer. A state of mixed spin symmetry and lowered space symmetry is constructed which has both conceptual and practical computational value. Either unrestricted Hartree–Fock theory or spin polarized density functional theory, e.g., Xα theory, can be used to generate the mixed spin state wave function. The most important consequence of the theory is that the Heisenberg exchange coupling constant J can be calculated simply from the energies of the mixed spin state and the highest pure spin multiplet.

Effect of a cubic crystal field on the critical behavior of a 3D model with Heisenberg exchange coupling: A high-temperature series investigation

The effect of a cubic anisotropy on the critical behavior of systems with isotropic exchange coupling has been the subject of many approximate renormalization-group studies with conflicting results for the $n=3$, Heisenberg case. These studies disagree on the relevance of cubic crystal-field strength, i.e., whether this field induces a crossover from the $n=3$ isotropic behavior to a distinctive cubic behavior or not. We have derived and analyzed tenth-order, high-temperature susceptibility series for the spin-infinity Heisenberg model with a nearest-neighbor exchange and a crystal field of cubic symmetry for the fcc lattice. The fixed length of the spins significantly affects the form of the series causing many terms to be zero. Analysis of our tenth-order series for fixed values of the cubic field strength ${\ensuremath{\nu}}_{4}$ yields an effective susceptibility index which is indistinguishable from the Heisenberg value for ${\ensuremath{\nu}}_{4}\ensuremath{\lesssim}5J,J$ being the exchange constant. Naive analysis of our series for the cubic field derivatives of the susceptibility suggests that the cubic field strength is a relevant variable with an associated crossover exponent $\ensuremath{\varphi}=0.63\ifmmode\pm\else\textpm\fi{}0.10$, much larger than the value, $\ensuremath{\varphi}\ensuremath{\sim}0.1$, expected from the renormalization-group calculations.

Dynamics of mixed-spin paramagnet

Second and fourth frequency moments of the wave-vector dependent correlation functions of the A−A, B−B, and A−B spins are computed at high temperatures for a mixed spin paramagnet with nearest neighbour Heisenberg exchange couplings J AA v , J BB v , J AB v with unaxial symmetry, v ≡ 0, +. Expression for spin diffusion coefficient is given.

Magnetic excitations in rare-earth antiferromagnets with bilinear and biquadratic pair couplings: Application to DySb

A Green's-function formalism is constructed for the purpose of computing the elementary excitation energies in a type-II antiferromagnet with Heisenberg exchange and quadrupolar couplings in a cubic crystal field. Three types of excitation modes are found: a longitudinal mode ($L$ mode) associated with ${O}_{0}^{1}$ and ${O}_{0}^{2}$ operators ($\ensuremath{\Delta}m=0$), a transverse mode ($T1$ mode) associated with ${O}_{\ifmmode\pm\else\textpm\fi{}1}^{1}$ and ${O}_{\ifmmode\pm\else\textpm\fi{}1}^{2}$ operators ($\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}1$), and a second transverse mode ($T2$ mode) associated with ${O}_{\ifmmode\pm\else\textpm\fi{}2}^{2}$ operators ($\ensuremath{\Delta}m=\ifmmode\pm\else\textpm\fi{}2$). In the ordered phase the $L$-mode and $T1$-mode excitations are mixed magnetic dipolar and quadrupolar excitations. In the disordered phase, as a consequence of cubic symmetry, the magnetic dipolar modes decouple from the quadrupolar modes, giving rise to the possibility of observing a pure quadrupolar excitation. Cubic symmetry also demands that in the disordered phase certain of the excitation energies in the $L$, $T1$, and $T2$ modes have identical dispersion curves. In general, the dispersion in both the ordered and disordered phase is complicated owing to the inclusion of next-nearest-neighbor coupling. The theory is applied to DySb, a type-II antiferromagnet with strong evidences of quadrupolar coupling.

Effects of randomness on three-dimensional magnetic ordering of quasi-low-dimensional spin systems

Motivated by recent work of Hone and co-workers on three-dimensional (3D) ordering of impure Ising and classical Heisenberg linear-chain systems, we present a coherent-potential-approximation (CPA) analysis of quasi-one-dimensional and nearly-two-dimensional spin systems with Heisenberg exchange coupling for which 3D magnetic ordering, in the absence of spatial randomness, is assumed to be described by a random-phase decoupled Green's-function theory. The effects caused by the introduction of arbitrary concentration of nonmagnetic impurities, with quenched-in random distribution and with or without the simultaneous presence of exchange-bond randomness between magnetic ions, are analyzed within such a CPA procedure. Numerical results for the renormalization of spin-wave stiffness along various symmetry directions and for the 3D transition temperature are given as a function of the (arbitrary) impurity concentration. Qualitative comparison with the results given by Hone et al. can be carried out and it is found that in the regime of very small interchain coupling strengths and impurity concentrations, where the latter theory can be expected to be reliable, these results are in reasonable agreement with theirs. Similarly, in the large-impurity limit near the percolation threshold, we find the behavior of the present results to be in accord with that generally expected for such systems. Lastly, we note that our procedure indicates that spatial isotropy of the spin-wave dispersion is "effectively" achieved at some intermediate concentration, the magnitude of which depends on the relative size \ensuremath{\alpha} of the inter- and intrachain couplings.

Paramagnetic Susceptibilities of Metallic Samarium Compounds

It is shown that the influence of conduction-electron polarization effects upon the susceptibilities of metals containing the tripositive samarium ion is very much greater than upon metals containing normal rare earths. A theory of the susceptibility of metallic samarium materials is developed which takes account of these polarization effects and of interionic Heisenberg exchange couplings, the admixture of the $J=\frac{7}{2}$ state into the $J=\frac{5}{2}$ ground state, which is assumed to be the only one to be thermally populated, but which, however, does not take account of crystal-field splittings. The susceptibility is found to be of the unexpectedly simple form $\ensuremath{\chi}(T)={\ensuremath{\chi}}_{0}+\frac{D}{(T\ensuremath{-}\ensuremath{\theta})}$, in which the only dependence on temperature $T$ is that explicitly shown. This expression is found to fit the published data for the susceptibility of dhcp samarium to an accuracy of 1% in the temperature region 110-230 \ifmmode^\circ\else\textdegree\fi{}K, and the parameters extracted from the fit are found to be in excellent agreement with those obtained for the other light rare-earth metals. An expression is also derived for the susceptibilities of metals containing normal rare earths, which takes account of both conduction-electron polarization and crystal field effects.

Antiferromagnetic chain

For a chain of N atoms of spin 1 2 with nearest neighbour Heisenberg exchange coupling, the Hamiltonian is linearized and the resulting model is solved exactly. The qualitative thermodynamic behaviour is established for the model, which exhibits a long-range order and is shown to undergo a phase transition at a critical temperature T c .

Application of the Bethe-Weiss Method to the Theory of Antiferromagnetism

The P. R. Weiss method developed in the theory of ferromagnetism is applied to antiferromagnetism by introducing sublattices. Atomic lattices (spin \textonehalf{} per atom) with negative Heisenberg exchange coupling $J$ between the nearest neighbors are investigated. It has been found that two-dimensional lattices cannot sustain antiferromagnetic order. The Curie temperatures of the simple cubic and b.c.c. structure are, respectively, $\frac{2.004|J|}{k}$ and $\frac{3.18|J|}{k}$. That the f.c.c. lattice cannot be ordered by the interactions among the nearest neighbors is deduced from the disorder of a quadratic layer and the ineffectiveness of the interactions between layers in a f.c.c. lattice in producing order. This helps to understand why the ordering pattern of spins in Mn ions in MnO should be such as observed by Shull. Curves are obtained for reciprocal susceptibility and for short-range order vs temperature above the Curie point ${T}_{c}$. The experimental formula $\ensuremath{\chi}=\frac{\mathrm{const}}{(T+\ensuremath{\theta})}$ is compared with our theory. We obtain for the simple cubic and b.c.c. lattice $\ensuremath{\theta}=1.5{T}_{c} \mathrm{and} 1.25{T}_{c}$ respectively if we extrapolate our theoretical curve from extremely high temperatures, and $\ensuremath{\theta}$ is slightly higher than these values if we extrapolate from the temperature range at which experimental readings are taken. This compares more favorably with the experimental data than the prediction $\ensuremath{\theta}={T}_{c}$ of the molecular field theory. The general validity of our theory and its failure in the range of low temperatures are discussed.