Molecular Field Approach Research Articles

Electron-paramagnetic-resonance spectra of ions substituted into transition-metal ion lattices

Below 2 K it is possible to observe well-resolved electron paramagnetic resonance spectra for ${\mathrm{Cu}}^{2+}$ substituted into $\ensuremath{\alpha}\ensuremath{-}\mathrm{NiS}{\mathrm{O}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}6{\mathrm{H}}_{2}\mathrm{O}$. Comparison with the $g$ values observed in the diamagnetic isostructural ZnSe${\mathrm{O}}_{4}$ \ifmmode\cdot\else\textperiodcentered\fi{} 6${\mathrm{H}}_{2}$O shows a positive $g$ shift of about 0.2 units in the ${\mathrm{Ni}}^{2+}$ lattice due to ${\mathrm{Cu}}^{2+}$-${\mathrm{Ni}}^{2+}$ exchange interaction. Two equivalent theories for the incorporation of guest-host exchange interaction into the spin Hamiltonian of the guest ion are presented. One of these theories is an extension of the standard perturbation techniques and the other utilizes a molecular-field approach. Both give similar results and indicate a nearest-neighbor ferromagnetic ${\mathrm{Cu}}^{2+}$-${\mathrm{Ni}}^{2+}$ exchange with $\ensuremath{-}2J=\ensuremath{-}0.145\ifmmode\pm\else\textpm\fi{}0.01$ ${\mathrm{cm}}^{\ensuremath{-}1}$. Experiments are also reported for ${\mathrm{Co}}^{2+}$, ${\mathrm{Mn}}^{2+}$, and ${\mathrm{V}}^{2+}$ substituted into either $\ensuremath{\alpha}\ensuremath{-}\mathrm{NiS}{\mathrm{O}}_{4}\ifmmode\cdot\else\textperiodcentered\fi{}6{\mathrm{H}}_{2}\mathrm{O}$, NiSe${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}6${\mathrm{H}}_{2}$O, or ZnSe${\mathrm{O}}_{4}$\ifmmode\cdot\else\textperiodcentered\fi{}6${\mathrm{H}}_{2}$O. The ${\mathrm{Co}}^{2+}$-${\mathrm{Ni}}^{2+}$ exchange is only slightly ferromagnetic, but our theories cannot give quantitative agreement with experiment. The ${\mathrm{Mn}}^{2+}$-${\mathrm{Ni}}^{2+}$ exchange is too small to be measured, but the ${\mathrm{V}}^{2+}$-${\mathrm{Ni}}^{2+}$ exchange indicates $\ensuremath{-}2J=+0.06\ifmmode\pm\else\textpm\fi{}0.01$ ${\mathrm{cm}}^{\ensuremath{-}1}$. All of these exchange interactions are for hydrated ions and take place through hydrogen bonds. The trends that we observe for exchange in the ${\mathrm{Ni}}^{2+}$ lattice are discussed.

MAGNETIC ANISOTROPY IN THE Fe-BORACITES

Using the Mossbauer technique we have investigated the electric and magnetic hyperfine interaction in Fe-boracites Fe3B7013X (X = C1, Br, I). The analysis of the Mossbauer spectra below the NBel temperature leads to 3 lattice sites in the case of Fe-CIand Fe-Br-boracite and to at least 4 sites in the Fe-I-boracite, which are different with regard to the sizes of the magnetic hyperfine fields and their directions with respect to the electric field gradient tensor. The Nkel temperature turned out to be 33.2 + 0.2 K and 11.8 & 0.2 K for the Fe-Iand Fe-CI-boracite, respectively. The experimental values for the hyperfine fields are compared with calculations based on Ligand Field Theory including spin orbit coupling and molecular field approach. Within this model the strong orbital contribution to the hyperfine field and the magnetic anisotropy can be explained.

Hyperfine fields and molecular fields in metals

Data on the temperature dependence of the hyperfine fields on Fe sites in Fe 3Si and FeRh (35 at.%) are presented. The sublattice magnetizations are practically independent of the number of magnetic neighbours at each site, suggesting that the molecular field approach is inapplicable to metallic systems.

Les systèmes MFFeF 3 (M = Li, Na, K, Rb, Cs, Tl, NH 4)

The binary systems MFFeF 3 (M = Li, Na, K, Rb, Cs, Tl, NH 4) have been investigated. The structures of the fluoroferrites M 2Fe 5F 17 (M = K, Rb, Cs, Tl), MFeF 4 (M = Na, K, Rb, Cs, NH 4), Na 5Fe 3F 14, M 2FeF 5 (M = K, Rb, Cs) and M 3FeF 6 (M = Li, Na, K, Rb, Cs, Tl, NH 4) can be deduced from networks of (FeF 6) 3− octahedra. The Néel temperatures decrease with the number of nearest neighbours iron atoms; these temperatures can be predicted according to the molecular field approach.

Theory of Ferro- and Antiferromagnetic Resonance in Solids Containing Ions of Complicated Level Structure

The time-dependent molecular-field (TDMF) approach to magnetic resonance is described and illustrated with six examples. For the simplest models, the random-phase approximation of TDMF is equivalent to Tyablikov spin waves. However, whereas Green's function and spin-wave techniques are forced into same-site decorrelation approximations when confronted with complicated single-ion dynamics, TDMF is general enough to include any single-ion terms without difficulty. Ions appear as coupled oscillators whose allowed transitions are pulled in forming collective modes. The procedure, being quite general, is readily automated and is thus convenient in actual calculations for physical problems.

Functional Integrals Representing Distribution Functions in Statistical Mechanics

We show how to obtain formal solutions of the chain of equations for distribution functions in classical statistical mechanics. These solutions are in the form of complex functional integrals. They are not unique, which fact is a fundamental property of the equations, and the different solutions are recognized by different integration paths in the complex function space. The different manners of integration correspond to different phases, of which some can be identified with the possible physical states. The treatment of the integrals in some cases is also discussed. They are closely related to generalizations of the molecular field approach to the problem. It is also shown that the functional integrals can be written as averages over an external field and that essentially the same form is valid in the quantum‐mechanical case.

Magnetic and crystallographic characteristics of (Pr,La)Al 3, (Pr,Y)Al 3, (Pr,La)Al 2 and (Pr,Y)Al 2

Magnetization data are reported for PrAl 2 (cubic) and PrAl 3 (hexagonal) and ternary alloys in which Pr is partially replaced by Y and La. Between 50 and 300°K Curie-Weiss behavior is observed with slopes as expected for tripositive Pr. The cubic alloys exhibit Curie temperatures ranging from 38.5°K for PrAl 2 to 5°K for Pr 0.05La 0.95Al 2. Pr 0.95Y 0.95Al 2 is the single exception; it remains paramagnetic to 4°K. The hexagonal alloys all develop Van Vleck paramagnetism at low temperatures. The operator equivalent method is employed to obtain eigenfunetions, eigenvalues and magnetic moments for the crystal field states for Pr +3 in both the cubic and hexagonal environments. Fourth and sixth order interactions are included in both instances and the second order term is also included for PrAl 3 since its axial ratio is not ideal. Susceptibility-temperature behavior of PrAl 3 and the hexagonal ternaries can be accounted for with two sets of parameters which concern the relative importance of the second, fourth and sixth order interactions; in each case the overall splitting is about 120°K. The maximal value of the overall splitting for the cubic alloys is established to be 25°K. Using a molecular field approach it is possible to understand the development of ferromagnetism in PrAl 2 which occurs despite the lack of a permanent moment for Pr in its ground state. With this approach T c is calculated to be 39°K, which is in excellent agreement with experiment, 38.5°K. The same approach gives a reasonably satisfactory accounting for the composition dependence of T c and the saturation moment.

Ising chain as the basic cluster in effective-field theories of magnetism

A linear chain is used as the basic cluster in effective-field theories of Ising systems. The chain gives limited improvement over finite clusters in a molecular-field approach. Results are independent of cluster size in a “constant coupling” approach.

Static Phenomena Near Critical Points: Theory and Experiment

This paper compares theory and experiment for behavior very near critical points. The primary experimental results are the "critical indices" which describe singularities in various thermodynamic derivatives and correlation functions. These indices are tabulated and compared with theory. The basic theoretical ideas are introduced via the molecular field approach, which brings in the concept of an order parameter and suggests that there are close relations among different phase transition problems. Although this theory is qualitatively correct it is quantitatively wrong, it predicts the wrong values of the critical indices. Another theoretical approach, the "scaling law" concept, which predicts relations among these indices, is described. The experimental evidence for and against the scaling laws is assessed. It is suggested that the scaling laws provide a promising approach to understanding phenomena near the critical point, but that they are by no means proved or disproved by the existing experimental data.

Magnetic Structure Transitions

The problem of magnetic phase transitions between ferromagnetism and antiferromagnetism, or between two different kinds of antiferromagnetic arrangement, is treated by a molecular field approach. It is shown that such transitions may occur if the molecular field coefficients vary with temperature; if they do occur, they will be of first order. The theoretical results agree qualitatively with the experimental results of Guillaud and Serres on MnAs.