High-temperature series expansion

Here, the Weiss temperature 9 is a function of the exchange energies /ex that can be obtained from a high-temperature series expansion ... 

Exact solutions do not exist for the exchange Hamiltonian of two-dimensional layers and three-dimensional networks. These situations can, however, be described using approximation methods such as molecular field approximation and high-temperature series expansion. 

For the case of ferromagnetic coupling, a high-temperature series expansion has been proposed ... 

Generally the critical temperature is lower in the more refined approximations than it is in the mean field approximation. It is not very different from the one obtained using high temperatures series expansions (Elliott et al., 1971). A numerical comparison of different approximations is shown in fig. 17.28. Somewhat subtle is the question of the order of the phase transition. The decoupling approximations going beyond the RPA-MF approximation lead to first order phase transitions for some or even all values of the relevant parameters. 

Fig. 238. Cu(HCOO)2 4H2O. The experimental paramagnetic susceptibility is compared with the calculated one resulting from the use of high temperature series expansions (HTS),...

Fig. 240. Cu(HCOO)2 -4H20. Temperature dependence of l/fC, and 1/ 2- Measuring points experimental solid lines high temperature series expansion solution to the Heisenberg two-dimensional antiferromagnetic lattice with J/hc=-54cm, g =2.34, gj =2.04. The calculated X T and values have been corrected for TIP=120 10" ...

Whereas the latter expression must be solved numerically for low temperatures, the entropy at high temperatures can be derived by a series expansion [4], For the Debye or Einstein models the entropy is essentially given in terms of a single parameter at high temperature ... 

The high temperature susceptibility data have been fitted to a series expansion formula suitable for a layer ferromagnetic (see below for structures) with 5 = 2, to give estimates of the nearest neighbour exchange integrals J, and some representative results for J and examples of Tc are in Table 33. 

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 first domain corresponds to high-incident heat fluxes, where the pyrolysis temperature (TP) is attained very fast, thus t Application of the first-order Taylor Series expansion to Equation 3.13 around tp/tc —> 0 yields the following formulation for the pyrolysis time (lp) ... 

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). 

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