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Physical review. B, Solid state

Specific-Heat Scaling Functions for Ising and Heisenberg Models and Comparison with Experiments on Nickel

Richard Krasnow, Eugene Stanley

Published Jul 1, 1973• 20 citations
Publisher

Abstract

We calculate the scaled constant-magnetization and constant magnetic field specific heats $\frac{[{C}_{M}(\ensuremath{\epsilon},M)\ensuremath{-}{C}_{M}(\ensuremath{\epsilon},0)]}{{M}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}}}}$ and $\frac{[{C}_{H}(\ensuremath{\epsilon},H)\ensuremath{-}{C}_{H}(\ensuremath{\epsilon},0)]}{{H}^{\ensuremath{-}\frac{\ensuremath{\alpha}}{\ensuremath{\beta}\ensuremath{\delta}}}}$ as functions of the scaled variable $x\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{M}^{\frac{1}{\ensuremath{\beta}}}}$ and $y\ensuremath{\equiv}\frac{\ensuremath{\epsilon}}{{H}^{\frac{1}{\ensuremath{\beta}\ensuremath{\delta}}}}$, respectively [where $\ensuremath{\epsilon}\ensuremath{\equiv}\frac{(T\ensuremath{-}{T}_{c})}{{T}_{c}}$ and ${T}_{c}$ is the critical temperature],for the spin-\textonehalf{} Ising model (bcc lattice) and the spin-\textonehalf{}, $\ensuremath{\infty}$ Heisenberg models (fcc lattice). Our calculations are based on previously calculated scaling functions for the $M(H,T)$ equation of state. We also calculate the amplitudes of the zero-field specific heat for $T>{T}_{c}$ and $T<{T}_{c}$. Thus, we obtain the functions ${C}_{M}(\ensuremath{\epsilon},M)$ and ${C}_{H}(\ensuremath{\epsilon},H)$, though for the Heisenberg models we cannot obtain the finite nonzero constant ${C}_{M}(0,0)={C}_{H}(0,0)$. We compare our calculated functions with the data of Connelly, Loomis, and Mapother on nickel.

DOI
10.1103/physrevb.8.332

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In recent years there have been many measurements of the scaling-law equation of state for different materials, and the "scaling function" so obtained has generally been fit by an empirical equation involving the selection of several adjustable parameters. We propose a method for calculating, directly from high-temperature series expansions, the function h(x)h(x)h(x) that determines the scaling-law equation of state $H={M}^{\ensuremath{\delta}}h(x)$. Previously, h(x)h(x)h(x) has been calculated only for the S=12S=\frac{1}{2}S=21​ Ising model, but the method is not generalizable to the case of the Heisenberg model because it relies upon the use of low-temperature expansions as well, and these are not known for the Heisenberg model. First we calculate h(x)h(x)h(x) for the Ising model (bcc, fcc, and simple cubic lattices) in order to assess the utility and credibility of our method. Our Ising model h(x)h(x)h(x) agrees well with the previous calculation that used both high-and low-temperature expansions. Next we calculate h(x)h(x)h(x) in its entire region of definition for the S=12S=\frac{1}{2}S=21​ Heisenberg model (fcc and bcc lattices) and the $S=\ensuremath{\infty}$ Heisenberg models (fcc lattice), where SSS denotes the spin quantum number. The accuracy of our resulting expressions is limited by the finite number of known terms in the corresponding high-temperature series expansions, but it is generally of the order of a few percent. In Paper II the scaling functions calculated here are compared with experiment and with the predictions of the universality hypothesis.
High-temperature series expansions of the partition functions for the Ising and Heisenberg models are analyzed for various values of the spin sss. The fcc lattice is used for which successive coefficients are sufficiently regular for estimates of critical behavior to be made with confidence. It is suggested that the magnetic susceptibility above the Curie temperature is of the form $A{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{\ensuremath{-}\frac{4}{3}}$ for the Heisenberg model for all sss, instead of ${A}^{\ensuremath{'}}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{\ensuremath{-}\frac{5}{4}}$ for the Ising model. Critical estimates of energy and entropy show that the magnitude of the "tail" of the specific heat anomaly is insensitive to the value of sss, and is about 2.5 times larger for the Heisenberg than for the Ising model. The sharpness of the anomaly at the Curie point increases as sss increases, and on passing from the Heisenberg to the Ising model. A brief reference is made to experimental results.
The study of the equation of state of the two-dimensional Ising model is initiated by use of the recently calculated nnn-point spin-correlation functions. In the scaling region where $T\ensuremath{\rightarrow}{T}_{c}$ and $h=B{|1\ensuremath{-}\frac{{T}_{c}}{T}|}^{\ensuremath{-}\frac{15}{8}}(\frac{H}{k{T}_{c}})$ is of order 1 (BBB is a lattice-dependent constant) the free energy below Tc{T}_{c}Tc​ is of the form $f(T, h)\ensuremath{-}f(T, 0)=\mathrm{const}{(1\ensuremath{-}\frac{{T}_{c}}{T})}^{2}$ $\ensuremath{\Sigma}{n=1}^{\ensuremath{\infty}}(\frac{1}{n!}){C}_{n}^{\ensuremath{-}}{h}^{n}$ where the lattice-dependent constant is chosen so that ${C}_{1}^{\ensuremath{-}}=1$. The constant ${C}_{2}^{\ensuremath{-}}$ has been studied previously. In this paper we study ${C}_{3}^{\ensuremath{-}}$ and ${C}_{4}^{\ensuremath{-}}$ and find as a first approximation $\frac{{C}_{3}^{\ensuremath{-}}}{3!}\ensuremath{\sim}\ensuremath{-}\frac{11}{72{\ensuremath{\pi}}^{2}}$ and $\frac{{C}_{4}^{\ensuremath{-}}}{4!}\ensuremath{\sim}\frac{5}{189\ensuremath{\pi}}+\frac{59}{162{\ensuremath{\pi}}^{3}}$. These are in close agreement with the low-temperature-series values of -0.01538 and 0.0195.
The spin-spin correlation functions and the critical-scattering intensity for Heisenberg models of general spin, S=12S=\frac{1}{2}S=21​ to $\ensuremath{\infty}$, on the sc, bcc, and fcc lattices are studied on the basis of high-temperature series expansions along the lines developed in Paper I [M. E. Fisher and R. J. Burford, Phys. Rev. 156, 583 (1967)]. Subject to increased uncertainties for low spin, it is concluded that the exponents $\ensuremath{\gamma}={1.375}_{\ensuremath{-}0.01}^{+0.02}$, $2\ensuremath{\nu}={1.405}_{\ensuremath{-}0.01}^{+0.02}$, and $\ensuremath{\eta}=0.043\ifmmode\pm\else\textpm\fi{}0.014$ describe all lattices and all spin. Explicit formulas are presented for the susceptibility/zero-angle scattering ${\ensuremath{\chi}}_{0}(T)$, for the inverse correlation length ${\ensuremath{\kappa}}_{1}(T)$, for the effective interaction range r1(T){r}_{1}(T)r1​(T), and using the Fisher-Burford approximant, for the total scattering $\stackrel{^}{\ensuremath{\chi}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}},T)$. The shape parameter ${\ensuremath{\varphi}}_{c}$ attains the "universal" value ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 11 for large spin but shows signs of spin dependence (and lattice dependence) for low spin. At fixed k the scattering is predicted to display a maximum above Tc{T}_{c}Tc​ determined by $\frac{{\ensuremath{\kappa}}_{1}({T}_{max})}{k}\ensuremath{\simeq}0.10 (\mathrm{for} S\ensuremath{\gtrsim}2)$ to 0.15. A detailed study is made of the structure dependence of the critical-point correlations ${〈{S}_{0}^{z}{S}_{\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}}^{z}〉}_{c}$ for various models. This leads to the revised, universal estimate ${\ensuremath{\varphi}}_{c}\ensuremath{\simeq}0$. 15 for all three cubic lattice, spin-\textonehalf{} Ising models. The results are compared d briefly with various experiments which support $\ensuremath{\eta}\ensuremath{\gtrsim}0.05$.
We report a detailed study of twelve-term, high-temperature series for the second moment of spin-pair correlations ${\ensuremath{\mu}}_{2}(t)$ and the specific heat cH(t){c}_{H}(t)cH​(t) of the nearest-neighbor spin-sss Ising model in zero magnetic field on the fcc lattice. Near criticality we find ${\ensuremath{\mu}}_{2}(t)={A}_{2}(s)t{(s)}^{\ensuremath{-}(\ensuremath{\gamma}+2\ensuremath{\nu})}[1 + {B}_{2}(s)t{(s)}^{{\ensuremath{\Delta}}_{1}}+ \ensuremath{\cdots}]$, {$t(s)=\frac{[T\ensuremath{-}{T}_{c}(s)]}{T}$}, showing a confluent correction to the dominant scaling singularity. To within uncertainties the exponents have the universal (i.e., spin-independent) values $\ensuremath{\nu}={0.638}_{\ensuremath{-}0.008}^{+0.002}$ (with $\ensuremath{\gamma}={1.250}_{\ensuremath{-}0.007}^{+0.003}$) and ${\ensuremath{\Delta}}_{1}=0.6\ifmmode\pm\else\textpm\fi{}0.1$. The confluent exponent ${\ensuremath{\Delta}}_{1}$ is in reasonable agreement with the correction-to-scaling index derived from earlier analysis of the susceptibility, as predicted by renormalization-group arguments. A similar analysis of the specific heat cH{c}_{H}cH​ for the same model finds no detectable confluent singularities in rather noisy, high-temperature series and gives $\ensuremath{\alpha}=0.125\ifmmode\pm\else\textpm\fi{}0.020$ in general confirmation of earlier s=12s=\frac{1}{2}s=21​ estimates. With $\ensuremath{\nu}$ as quoted above the hyperscaling relation $d\ensuremath{\nu}=2\ensuremath{-}\ensuremath{\alpha}$ at d=3d=3d=3 requires $\ensuremath{\alpha}={0.086}_{\ensuremath{-}0.006}^{+0.024}$ so the validity of hyperscaling remains problematical.
Using a Monte Carlo procedure with a self-consistent-field boundary condition, the magnetization of a simple cubic classical Heisenberg ferromagnet with nearest-neighbor interactions only is calculated. For reduced temperatures and reduced fields in the range $0.02<~|\frac{T}{{T}_{c}}\ensuremath{-}1|$, HJ< 0.5\frac{H}{J}<~0.5JH​< 0.5 the data of this computer experiment can be described in terms of the "effective" critical exponents $0.31<~\ensuremath{\beta}<~0.35$, $1.33<~\ensuremath{\gamma}<~1.39$, and $4.9<~\ensuremath{\delta}<~5.3$. Although the present data are not very close to the critical point they do obey the homogeneity requirement and determine the scaling function rather precisely. This function agrees very well with the scaling function for the face-centered-cubic classical Heisenberg magnet derived recently by Milo\ifmmode \check{s}\else \v{s}\fi{}evi\ifmmode \acute{c}\else \'{c}\fi{} and Stanley using high-temperature---series-expansion techniques. This agreement supports their hypothesis that neither critical exponents nor the scaling function depend on the lattice structure in the Heisenberg model.