Cluster variation method and Möbius inversion formula

The cluster variation method is reformulated with the aid of the Möbius inversion formula.     

Möbius function for the set of acyclic directed backbone graphs

Consider a two-rooted graphG, the edges of which are directed in such a way that there are no cycles and every edge belongs to some self-avoiding walk from rootu to rootv which follows the direction of the edges. Au-v backbone ofG is a subgraph formed by taking the union of any subset of directed self-avoiding walks fromu tov. Let _uv be the set of all such backbones ofG partially ordered by set-inclusion. We prove the conjecture of Bhatti and Essam that the Möbius function of this set is given, for acyclicb,b _uv withbb, by(b,b)=(–1) ^c-c , wherec andc are the respective cycle ranks ofb andb. The significance of this result in percolation theory is reviewed together with previous results for other sets of subgraphs.     

Finitized Conformal Spectra of the Ising Model on the Klein Bottle and Möbius Strip

We study the conformal spectra of the critical square lattice Ising model on the Klein bottle and Möbius strip using Yang–Baxter techniques and the solution of functional equations. In particular, we obtain expressions for the finitized conformal partition functions in terms of finitized Virasoro characters. This demonstrates that Yang–Baxter techniques and functional equations can be used to study the conformal spectra of more general exactly solvable lattice models in these topologies. The results rely on certain properties of the eigenvalues which are confirmed numerically.     

Phase diagram of the one-state potts model on the Cayley tree

The phase diagram of the one-state Potts model on the closed asymmetric Cayley tree with branching ratior=2 is obtained from the Bethe-Peierls map. The route to chaos, via the period doubling cascade, is obtained by considering the antiferromagnetic coupling limit. The connection of the Potts model with the percolation problem is shown by calculating the order parameter, its susceptibility, the internal energy, and the specific heat as well as their asymptotic behavior at the paramagnetic-ferromagnetic critical point. Due to the type of the lattice and to the polynomial character of the map, this is the simplest known example of a McKay-Berker-Kirkpatrick spin-glass.     

Analyticity of a hard-core multicomponent lattice gas

A multicomponent anti-Widom-Rowlinson lattice gas is introduced. An arbitrary numberM of particle types is permitted, all having the same activity. The only interactions are nearest-neighbor exclusions oflike particles (analogous to map-coloring problems). For any lattice it is shown that there is a finite numberM ₀ (depending only on the coordination number of the lattice) such that for allMM ₀ the infinite volume correlation functions exist and are analytic functions of the activity, for all positive values of the common activity.Research supported in part by NSF Grant No. GP-33535X, AFOSR Grant No. 73-2430B, and NSF Grant No. MPS75-20638.     

Tiling problems and undecidability in the cluster variation method

In cluster approximations for lattice systems the thermodynamic behavior of the infinite system is inferred from that of a relatively small, finite subsystem (cluster), approximations being made for the influence of the surrounding system. In this context we study, for translation-invariant classical lattice systems, the conditions under which a state for a cluster admits an extension to a global translation-invariant state. This extension problem is related to undecidable tiling problems. The implication is that restrictions of global translation-invariant states cannot be characterized purely locally in general. This means that there is an unavoidable element of uncertainty in the application of a cluster approximation.     

Consistent relations in the method of reducibility in the cluster variation method

A redundancy is noted in the set of Lagrange multipliers—effective fields and effective interactions—used in the expressions of the reduced density matrices given by Morita⁽⁴⁾ in some problems with less symmetry⁽⁶⁾. The expressions of the reduced density matrices without this redundancy are given. An argument showing that a certain expression of the free energy in terms of the Lagrange multipliers is stationary with respect to the variations of these is presented.     

A cluster variation method for lattice gas models

Czechoslovak Journal of Physics -     

A simple model for the magnetocrystalline anisotropy in mixed ferrite nanoparticles

A simple model, based on the relative occupancy of tetrahedral and octahedral sites by different cations, is proposed for the magnetocrystalline anisotropy of mixed ferrite nanoparticles. According to this model, the total magnetocrystalline anisotropy is the weighted average of the contributions of the anisotropies of Fe³⁺ and M²⁺ ions in A and B sites. The model predictions are confirmed in the case of cobalt-zinc ferrite.     

A note on the derivation of the Schrödinger-Langevin equation

It is shown that the dynamics of open systems interacting with a chaotic environment can be formulated in a quantum mechanical scheme by means of Nelson's stochastic quantization procedures. As a result, a wave equation for a particle in the chaotic environment is found to be of the Schrödinger-Langevin type associated with an additional nonlinear and random potential.     

Tuning of the periodicity of stable self-organized metallic templates

The atomic and electronic structures of Pb bilayer/Pt(111) are investigated with two theoretical calculations.We find that the stable (2 × 2)/(3 × 3) Pb/Pt(111) structure is a promising candidate for being used as a template with self-organized ordered Pb semi-cluster array on the first Pb monolayer.This stable structure can realize the ordered Au single-atom array around the Pb semi-clusters that can cause selective adsorption of noble atoms.The size of Pb magic number semi-cluster plays a more important role in determining the periodicity of the template than the lattice constant misfit between the substrate and the overlayer.This leads to quite a different periodicity between the two stable templates,which are (2 × 2)/(3 × 3) Pb/Pt(111) and Pb/Cu(111).Therefore,by considering the size of the stable semi-clusters and carefully selecting different substrate materials,we can tune the density of Pb semi-clusters as the nucleation points and then tune the periodicity of the stable template.     

Oxygen-vacancy ordering in the YBa2Cu3Oz basal plane studied by the cluster variation method

The cluster variation method is used to calculate a phase diagram for a two-dimensional Ising model representing the Cu₂O plane of the high-T _c superconductor YBa₂Cu₃O_z. Both first (V ₁ and second-neighbor (V ₂) interactions are considered, withV ₂/V ₁= –1/2; At high temperatures, the transition from the disordered (tetragonal) to the ordered (orthorhombic) phase is second-order. A tricritical point is found below which phase separation occurs. Fractional site occupancy and second-neighbor pair correlations are calculated as a function of temperature. The relevance of the model to the thermodynamics of ordering in the high-T _c compound is discussed.     

Concerning the cation distribution in MnFe2O4 synthesized through the thermal decomposition of oxalates

A single phase manganese ferrite powder have been synthesized through the thermal decomposition reaction of MnC₂O₄·2H₂O-FeC₂O₄·2H₂O (1:2 mole ratio) mixture in air. DTA-TG, XRD, Mössbauer spectroscopy, FT-IR and SEM techniques were used to investigate the effect of calcination temperature on the mixture. Firing of the mixture in the range 300-500 °C produce ultra-fine particles of α-Fe₂O₃ having paramagnetic properties. XRD, Mössbauer spectroscopy as well as SEM experiments showed the progressive increase in the particle size of α-Fe₂O₃ up to 500 °C. DTA study reveals an exothermic phase transition at 550 °C attributed to the formation of a Fe₂O₃-Mn₂O₃ solid solution which persists to appear up to 1000 °C. At 1100 °C, the single phase MnFe₂O₄ with a cubic structure predominated. The Mössbauer effect spectrum of the produced ferrite exhibits normal Zeeman split sextets due to Fe³⁺ions at tetrahedral (A) and octahedral (B) sites. The obtained cation distribution from Mössbauer spectroscopy is (Fe_0.92Mn_0.08)[Fe_1.08Mn_0.92]O₄.     

Effect of indium addition on the structure and magnetic properties of YIG

In this work we report the structure and magnetic properties of a series of single-phase indium-substituted yttrium iron garnet (In-YIG) nanoparticles with nominal composition of Y₃In_xFe_5−xO₁₂ (x=0.1, 0.2, 0.3 and 0.4) prepared by conventional mixed oxide route. Based on XRD results, the lattice parameters of the samples increased with increase in In³⁺ content due to its larger ionic radius. Mössbauer results confirmed the substitution of In³⁺ for Fe³⁺ in [a] site of YIG structure. Further, the magnitudes of the magnetic hyperfine field (MHF) were seen to reduce due to indium substitution. Moreover, a rising trend was observed for saturation magnetization (M_S) of the samples with x>0.2 owing to the substitution of non-magnetic In³⁺ for Fe³⁺. However, the observed initial drop of M_S for the sample with x=0.2 compared to that with x=0.1 is possibly attributed to the dominance of spin canting over the net magnetization rise caused by In³⁺ in [a] sites.     

A note on the homogeneous balance method

Based on the idea of the homogeneous balance method, a simple and efficient method is proposed for obtaining exact solutions of nonlinear partial differential equations. Some equations are investigated by this means and new solitary wave solutions or singular traveling wave solutions are found.     

A Mössbauer Effect Powder Diffractometer

An automated, high-sensitivity Mössbauer powder diffractometer is described, and its performance demonstrated with a bcc ⁵⁷Fe foil sample. Useful diffraction patterns are acquired in about 10 hours. The incoherent and coherent parts of the scattering are readily identified. Both are affected by thickness distortion, but the incoherent part is affected more strongly.     

Vibrational dynamics of biological molecules: Multi-quantum contributions

High-resolution X-ray measurements near a nuclear resonance reveal the complete vibrational spectrum of the probe nucleus. Because of this, nuclear resonance vibrational spectroscopy (NRVS) is a uniquely quantitative probe of the vibrational dynamics of reactive iron sites in proteins and other complex molecules. Our measurements of vibrational fundamentals have revealed both frequencies and amplitudes of ⁵⁷Fe vibrations in proteins and model compounds. Information on the direction of Fe motion has also been obtained from measurements on oriented single crystals, and provides an essential test of normal mode predictions. Here, we report the observation of weaker two-quantum vibrational excitations (overtones and combinations) for compounds that mimic the active site of heme proteins. The predicted intensities depend strongly on the direction of Fe motion. We compare the observed features with predictions based on the observed fundamentals, using information on the direction of Fe motion obtained either from DFT predictions or from single crystal measurements. Two-quantum excitations may become a useful tool to identify the directions of the Fe oscillations when single crystals are not available.     

Mössbauer studies and magnetic studies of Ni doped orthoferrites PrFe1−xNixO3 (x≤0.3)

The magnetic properties of polycrystalline PrFe_1−xNi_xO₃ (x≤0.3) system were studied using Mössbauer spectroscopy and magnetization measurements. The Mossbauer spectra exhibit six line spectra which loses its sharpness as the Ni substitution increases within the system. As the Ni concentration in the system increases, the hyperfine field and isomer shift shows decrease, which is vivid from the sluggish nature of the sextets. The small value of quadrupole splitting confirms the octahedral environment of the Fe⁺³ ions. The magnetization curves show the reversible behavior and represent the fall in negative molecular field leading to AFM frustration. From these results, we conclude that sagging in the spectra reveals the change from antiferromagnetic state to ferromagnetic state, which can be attributed to mixed state of Fe⁺³ ions i.e. high spin (HS) and low spin (LS) which is a consequence of progressive collapse of Hund’s rule due to HS→LS transition. These results confirm the weak ferromagnetic component due to canted-AFM spin arrangement of Fe³⁺ magnetic moments.