High temperature Expansion

To simplify its derivation we begin by transforming variables according to 

From Eq. (4.53) we obtain the corresponding partition function of the Ising magnet by replacing H by its counterpart H given in Eq. (4.57), where, 

In the limit of sufficiently high temperature p — 0) this expression may be evaluated approximately using the expansion 

Transforming back to the original lattice-fluid parameters eit, fx, and we may write the last expression as 

Together, Eqs. (4.60) and (4.68) provide a means to calculate in the limit T — 00. At sufficiently high chemical potentials one may also expect all lattice. sites to be fully occupied such that Sj = (si) = 1 so that in the limit of sufficiently high temperatures one has access to the internal energy of the lattice fluid through the expression 

The density of TD is not known. We may, however, use Fig. 18.1.2 to obtain this volume by interpolation. For T = 0.54 the reduced volume may be accurately represented by 

This corresponds to 0.3 % of the total volume. Hence the total experimental excess volume (19.9.3) is about 0.8% of the molar volume. 

Now we may calculate the excess volume (19.9.3) using the rule (19.8.5). We obtain 

The sign is correctly given but the calculated effect is too small. 

We may use the method we have summarized in 2 of Ch. XVIII to obtain the order of magnitude of small quantum corrections for 

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The first tenn in the high-temperature expansion, is essentially the mean value of the perturbation averaged over the reference system. It provides a strict upper bound for the free energy called the Gibbs-Bogoliubov inequality. It follows from the observation that exp(-v)l-v which implies that ln(exp(-v)) hi(l -x) - (x). Hence... 

More generally, for other lattiees and dimensions, nmnerioal analysis of the high-temperature expansion provides infonnation on tire eritieal exponents and temperature. The high-temperature expansion of the suseeptibility may be written in powers of = p J as... 

Figure A2.3.29 Calculation of the critical temperature and the critical exponent y for the magnetic susceptibility of Ising lattices in different dimensions from high-temperature expansions.

Note that in the limit of isotropic spins (where Si 0), the results for coherent axes and for random anisotropy duly coincide and agree with ordinary high-temperature expansions. 

This is the same correction term as that obtained for isotropic spins by Waller [29] and van Vleck [30] using ordinary high-temperature expansions. 

Figure 3.3. Equilibrium linear susceptibility (x/Xiso) versus temperature for an infinite spherical sample on a simple cubic lattice. The dotted lines are the results for independent spins, while the solid lines show the results for parallel and random anisotropy calculated with thermodynamic perturbation theory, as well as for Ising spins calculated with an ordinary high-temperature expansions. We notice in this case that the linear susceptibility for systems with random anisotropy is the same as for isotropic spins calculated with an ordinary high-temperature expansion. The dipolar interaction strength is hj = a/2a = 0.004.

The magnetic susceptibility in zero field, x0, can likewise be expressed as a high temperature expansion... 

In the forward (clockwise) case, heat is taken in from a hot source and work is done by the hot substance during the high-temperature expansion ab also additional work is done at the expense of the thermal energy of... 

The sum runs over an infinite amount of terms, of which only a limited number can be calculated due to the rapidly increasing computational demand. However, a sufficiently high order of the series is a requirement in order to describe the susceptibility in proximity to a critical temperature and determine the exchange parameters. If the series is cut short to zeroth order, a Curie expression follows for the susceptibility if the series is developed to first order, the Curie-Weiss law follows. Thus, both Curie and Curie-Weiss expressions can be regarded as development stages in the high-temperature expansion, which also implies that these expressions should be valid only for high temperatures. 

The value of the exchange integral in these systems can be obtained by using Green functions, the spin wave theory, the high-temperature expansion series, etc.. .. The application of the expansion series is perhaps the simplest way of obtaining this value. 

At high temperatures (70 < T < 300 K), the magnetic susceptibility follows a Curie-Weiss law with 6p = +22 K. Applying the high-temperature expansion series, the exchange constant is obtained160,161) (Fig. 25). 

Fluorides are suitable 2-D and 1-D magnetic models. In the K2NiF4 series, for instance, the antiferromagnetic behavior of fluorinated compounds can perfectly be explained by means of high-temperature expansion series which is not always the case with the homologous oxides9). 

Wang-Landau Sampling for Quantum Systems High Temperature Expansion... 

To normalize g n ) there are two options. If Ho can be solved exactly, (0) can be determined directly. Otherwise, the normalization can be fixed using the high temperature expansion version of the algorithm to calculate Z j3) at any fixed value of A. Even without normalization we can still obtain entropy and energy differences. 

High-Temperature Expansion for Vibrational Free Energy Obtain a high-temperature expansion for the free energy due to thermal vibrations given in eqn (3.110). 

The high-temperature expansion, truncated at first order, reduces to van der Waals equation, when the reference system is a fluid of hard spheres. 

The high-temperature expansion could also be derived as a Taylor expansion of the free energy in powers of X about 7 = 0 ... 

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