Ising Value Research Articles

Thermal expansion of methanol + cyclohexane near the critical solution point

We report precise measurement of volume changes as function of temperature in the one phase region of methanol + cyclohexane near the critical solution point. We find a value for the critical exponent α of 0.11 ± 0.07 in the region 11 Mk < T – T c < 6.4 K. This value agrees with the Ising value of 1 8 . For temperatures closer to Tc we observe a stronger divergence. In addition we observe unusual behavior over a small region of temperature centered at T – T c = 15 mK.

Critical behavior of a 3-D Ising model in a random field

A Monte Carlo method has been used to study a simple cubic Ising ferromagnet in a random quenched magnetic field. The Hamiltonian for this model is ℋ==JΣ(ij)σiσj−ΣiHiσi, where σi,σj=±1, J is the nearest-neighbor interaction constant, and the field Hi=tH is fixed at each site with ti=±1 at random and Σti=0. L×L×L lattices with periodic boundary conditions have been studied for a range of H and T. As expected we find a ferromagnetic ordered state which for small H undergoes a second order phase change to the paramagnetic state with increasing temperature. A finite size scaling analysis of the preliminary data suggests that the critical exponent β is substantially smaller (β∼0.2) than the usual 3-dim Ising value of 0.31. Results obtained for small lattices indicate that below kT/J-2 the transition becomes first order suggesting that a tricritical point appears on the critcal field curve.

Heat capacity of the binary liquid system triethylamine–water near the critical solution point

The specific heat Cpx has been measured for the binary system triethylamine–water as a function of temperature near the critical solution temperature at the critical concentration. Simple power law fits of the data resulted in rather large values (around 0.25) for the critical exponent α. For a fixed value of α equal to the Ising value αI=0.125 good fits could only be obtained if more complex expressions were used in fitting the data. For this case the best result was obtained by including a correction-to-scaling term. Combining our result for the amplitude of the specific heat singularity with the correlation length amplitude allowed us to test two-scale-factor universality. Our result strongly indicates that, in contrast to earlier findings, two-scale-factor universality is also valid for binary systems.

Critical behavior of the heat capacity in SrTiO3

Using an ac calorimetric method, heat-capacity measurements were made for a "monodomain" and a "polydomain" sample. The former exhibited a clear critical behavior which was analyzed with an expression containing a critical-exponent term. The exponent has an Ising value within a standard deviation. On the other hand, the polydomain sample showed a smaller and different anomaly. Using a special sample-setting method, the above behaviors were verified directly by measuring the heat capacity under externally applied stress conditions. The differences in heat capacities observed leads us to imagine that crossover from Heisenberg to Ising behavior with external stress takes place due to the existence of the bicritical point.

Thermal expansion of isobutyric acid+water near the upper critical solution temperature

We report the precise measurement of volume changes as a function of temperature near the critical solution point of isobutyric acid–water. We find not only evidence of a critical anomaly in the thermal expansion but also a small range of temperature in which it is negative, a requirement for thermodynamic consistency. Although the uncertainty in α, the critical exponent determined from these measurements, is on the same order as α itself, we find that α is consistently in the range 0.08–0.14; that is comparable to the Ising value of 1/8.

Observation of lattice- and spin-dimensionality crossovers in the susceptibility of quasi 2-dimensional Heisenberg ferromagnets

We have studied the magnetic susceptibility (χ) for T → T + c in the quasi 2-dimensional (2-d) Heisenberg ferromagnets ( C n H 2 n + 1 NH 3) 2 CuCl 4 ( n = 1, 2, 3, 10). Through the series of compounds the ratio | R| = | J′/ J| of inter- to intralayer interactions varies from 10 −3 − 10 −6. In comparing χ curves for different R, clear evidence is obtained for crossover from 2-d to 3-d behaviour as T → T + c . In agreement with theoretical predictions, the quantity χ| J′| determines the ‘crossover temperature’, the sign of J′ determining whether χ deviates upward or downward from 2-d behavior. In addition, spin-dimensionality crossovers are observed, due to the presence of small XY anisotropy (0.3%) and Ising anisotropy (0.02%) in the interactions. For compounds with | R| < 10 −4 the lattice-dimensionality crossover is preceded by spin-dimensionality crossovers, from 2-d Heisenberg to 2-d XY behavior ( at ϵ ≏ 1 − T c/T ≅ 0.5) and subsequently to 2-d Ising behavior (at ϵ ≅ 0.2). The critical behavior of χ (for ϵ < 0.2) is thus of Ising character, and the lattice-dimensionality crossovers (at ϵ = 0.02−0.05) entail a change in the critical exponent γ from the 2-d Ising to the 3-d Ising values (1.75 and 1.25, resp.), the crossover revealing itself as a “kink” in the log χ versus log ϵ plot. In comparing χ curves for materials with different amounts of spin-space anisotropy, extrapolation to the case of zero anisotropy is made, and an estimate of the χ behavior for the 2-d Heisenberg ferromagnet is obtained down to kT/ J = 0.5. This temperature is a factor of three lower than that at which the high-temperature-series prediction for this model becomes unreliable.

Neutron-diffraction study of ND4Cl in the tricritical region

A detailed neutron scattering investigation has been made of the order-disorder transition in a crystal of N${({\mathrm{D}}_{0.93}{\mathrm{H}}_{0.07})}_{4}$Cl as a function of both temperature and pressure, with emphasis on the region close to the tricritical point (TCP). At atmospheric pressure, the temperature dependence of the order parameter $\ensuremath{\Psi}$ was obtained by means of a detailed crystal structural analysis, which revealed that the transition was first order. It was also found that $\ensuremath{\Psi}$ could be determined with suitable accuracy simply by measuring the intensity of the (221) reflection, which is negligibly small in the disordered region. Intensity data for this reflection were obtained at elevated pressures up to a maximum of 6 kbar both as a function of pressure and temperature, and the phase line was accurately located. This was approximately linear, with an average slope of about 8 \ifmmode^\circ\else\textdegree\fi{}K/kbar. From the behavior of the critical scattering, which showed quite abrupt changes in the disordered region close to the TCP, the latter was determined to lie at 128 \ifmmode\pm\else\textpm\fi{} 10 bar and (250.2 \ifmmode\pm\else\textpm\fi{} 0.1) \ifmmode^\circ\else\textdegree\fi{}K. The critical exponents at the TCP were determined for scans made at constant pressure and temperature, and were found to have significantly different values, $2{\ensuremath{\beta}}_{t}=0.36\ifmmode\pm\else\textpm\fi{}0.01 \mathrm{and} 0.28\ifmmode\pm\else\textpm\fi{}0.01$, respectively. In the second-order region a similar tendency was observed for scans made through a given transition point in these directions. At 280 \ifmmode^\circ\else\textdegree\fi{}K, $2\ensuremath{\beta}$ for constant pressure and temperature scans, respectively, had values 0.61 \ifmmode\pm\else\textpm\fi{} 0.04 and 0.50 \ifmmode\pm\else\textpm\fi{} 0.04, the former being close to the three-dimensional Ising value.

Magnetic Properties of FeCl2in Zero Field. II. Long-Range Order

In Paper I it has been shown that the critical behavior of Fe${\mathrm{Cl}}_{2}$ in zero field should be that appropriate to the three-dimensional $S=1$ Ising model. This paper reports a detailed study of the critical behavior of the sublattice magnetization below ${T}_{N}$ together with a qualitative survey of the wave-vector-dependent susceptibility above ${T}_{N}$. The sublattice magnetization is found to follow the power law $\frac{{M}_{T}}{{M}_{0}}=D{(1\ensuremath{-}\frac{T}{{T}_{N}})}^{\ensuremath{\beta}}$, with $D=1.47\ifmmode\pm\else\textpm\fi{}0.02$, ${T}_{N}=23.553\ifmmode\pm\else\textpm\fi{}0.01$ \ifmmode^\circ\else\textdegree\fi{}K, $\ensuremath{\beta}=0.29\ifmmode\pm\else\textpm\fi{}0.01$, over the range 0.1 to 0.001 in reduced temperature. The exponent $\ensuremath{\beta}$ is in reasonable agreement with the rigid-lattice Ising value of 0.313 \ifmmode\pm\else\textpm\fi{} 0.003. The regular behavior of ${M}_{T}$ together with the unusual metamagnetic properties of Fe${\mathrm{Cl}}_{2}$ make this an ideal system for a study of the critical behavior of the order parameter around a tricritical point. The wave-vector-dependent susceptibility ${\ensuremath{\chi}}^{\ensuremath{\alpha}\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})$ above ${T}_{N}$ is found to exhibit typical behavior for a three-dimensional anisotropic antiferromagnet. Only ${\ensuremath{\chi}}^{\mathrm{zz}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{\ensuremath{\tau}}})$, the component along the anisotropy axis, diverges at the phase transition. Furthermore ${\ensuremath{\chi}}^{\mathrm{zz}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{Q}})$ is found to be fully three dimensional; that is, it has the form of a peak rather than a ridge, in spite of the fact that the spin waves are essentially two dimensional in form.