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Physica Scripta

Thermodynamic properties of a quasi one-dimensional quadrupolar coupled system

Carla Buzano

Published Feb 1, 1990• 7 citations
Publisher

Abstract

The critical properties of a three-dimensional spin-1 system characterized by quadrupolar interactions are analyzed under the assumption of weakly coupled chains. The problem is approached introducing first a trial one-dimensional Hamiltonian whose free energy can be evaluated exactly. Then using a variational procedure the free energy of the model is determined. The phase transitions are examined and the phase diagram derived. In addition some characteristic thermodynamic quantities such as average quadrupolar momentum, quadrupolar correlation function and specific heat are evaluated.

DOI
10.1088/0031-8949/41/2/015

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Journal of Physical Studies•Jan 1, 2001•0 citations
The critical properties of a three-dimensional spin-1 system characterized by quadrupolar interactions are analyzed under the assumption of weakly coupled chains. The problem is approached introducing first a trial one-dimensional Hamiltonian whose free energy can be evaluated exactly. Then using a variational procedure the free energy of the model is determined. The phase transitions are examined and the phase diagram derived. In addition some characteristic thermodynamic quantities such as average quadrupolar momentum, quadrupolar correlation function and specific heat are evaluated.
The spin-1 Ising model with bilinear and biquadratic interactions is investigated using a generalized constant-coupling approximation with two parameters. Equations to determine these parameters are derived by minimizing the variational free energy. The averages of the dipolar and quadrupolar moments expressed in terms of these parameters are first ascertained to give exact results for a linear chain, then investigated in detail for the quadrupolar-coupled system in comparison with the molecular-field results. The pair-correlation and the heat capacity of this system are also investigated in connection with the phase transition. For the system with the bilinear and biquadratic interactions the effect of the anisotropy constant to the phase transition is discussed.
The equilibrium properties of the spin-1 Ising system Hamiltonian with arbitrary bilinear (J), biquadratic (K) and odd (L), which is also called dipolar-quadrupolar, interactions is studied for zero magnetic field in the lowest approximation of the cluster variation method. The odd interaction is combined with the bilinear (dipolar) and biquadratic (quadrupolar) exchange interactions by the geometric mean. In this system, phase transitions depend on the ratio of the coupling parameters, alpha = J/K; therefore, the dependence of the nature of the phase transition on alpha is investigated extensively and it is found that for alpha less than or equal to 1 and alpha greater than or equal to 2000 a second-order phase transition occurs, and for 1 < alpha < 2000 a first-order phase transition occurs. The critical temperatures in the case of a second-order phase transition and the upper and lower limits of stability temperature in the case of a first-order phase transition are obtained for different values of alpha calculated using the Hessian determinant. The first-order phase transition temperatures are found by using the free energy values while increasing and decreasing the temperature. Besides the stable branches of the order parameters, we establish also the metastable and unstable parts of these curves and the thermal variations of these solutions as a function of the reduced temperature are investigated. The unstable solutions for the first-order phase transitions are obtained by displaying the free energy surfaces in the form of a contour map. Results are compared with the spin-1 Ising system Hamiltonian with the bilinear and biquadratic interactions and it is found that the odd interaction greatly influences the phase transitions.
The linear renormalization-group transformation is proposed to study critical temperatures and temperature dependence of the thermodynamic values (free energy, specific heat) of the weakly interacting spin chains. The method is examined in two classical systems: the standard Ising model on a rectangular lattice and the Ising spin chains coupled by pair-pair interactions. In the latter case the transformation does not exhibit any nontrivial fixed point. This result agrees with the exactly determined free energy of the system under consideration. The method is also applied to the weakly interacting quantum XY spin chains. The critical temperature as a function of the interchain interactions and the temperature dependence of the specific heat are found.
State space Riemannian geometry is obtained for the one-dimensional Blume-Emery-Griffiths model and its Blume-Capel and Griffiths model limits, and its (pseudo)critical as well as noncritical parameter regimes are extensively investigated. Two codimension one geometries are obtained by taking suitable hypersurfaces in the three-dimensional state space manifold, and the induced thermal metrics are accordingly interpreted in terms of constrained fluctuations. The three-dimensional scalar curvature and the two two-dimensional curvatures are shown to be consistent with Ruppeiner's conjecture relating the inverse of the singular free energy to the thermodynamic scalar curvature. Moreover, they are found to be in an excellent agreement over a greater part of the noncritical region with the corresponding correlation lengths for the spin and the quadrupolar order parameters. The scaling function for the free energy near the pseudocritical and tricritical points is obtained thermodynamically by using Ruppeiner's conjecture. A connection is made between the sign change in the curvatures and the change in fluctuation patterns of the order parameters. In the accompanying paper we shall analyze the geometry of the spin-one model in its mean field approximation.
The thermodynamic properties of the weakly interacting spin-1/2 and spin-1 chains with the intrachain coupling J and interchain coupling $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$ are studied by using the linear renormalization-group method. The magnetic susceptibility of the weakly interacting s=1/2s=1/2s=1/2 quantum anisotropic Heisenberg chains in the longitudinal field (h) is investigated. The scaling shifts of the parallel susceptibility maxima in the external magnetic field are considered. The same method is used to study the specific heat and susceptibility of the weakly interacting S=1S=1S=1 Ising models. In the special case (Blume-Capel model) the phase diagram with critical and first-order transition lines is found. The location of the tricritical point for $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}/J=0.2$ is estimated.
The microscopic approach to calculating and investigating the three-dimensional (3D) Ising-like system thermodynamics above and below the phase transition temperature Tc is developed using the collective variables method.Mathematical description is performed within the framework of the higher non-Gaussian approximation ( 6 model) taking into account the corrections to scaling.A nonuniversal factor depending on microscopic parameters of the system is singled out in the expressions for leading critical amplitudes and correction-to-scaling amplitudes.Numerical estimates of the critical region size, phase transition temperature as well as the graphs of temperature dependences of entropy and other thermodynamic characteristics near Tc are given for various values of eective radius of the exponentially decreasing interaction potential.The variation of the free energy of the system at the phase transition point, average spin moment, and speci c heat with increasing ratio of the potential eective radius to the simple cubic lattice constant is traced.The results of calculations and their comparison with the other authors' data show that the 6 model provides a more adequate quantitative description of the critical behaviour of a 3D Ising-like system than the 4 model.