Low-temperature expansion

For a 2D square lattice q = 4, and the high- and low-temperature expansions are related in a simple way... 

The Ising model has been solved exactly in one and two dimensions Onsager s solution of the model in two dimensions is only at zero field. Infomiation about the Ising model in tliree dunensions comes from high- and low-temperature expansions pioneered by Domb and Sykes [104] and others. We will discuss tire solution to the 1D Ising model in the presence of a magnetic field and the results of the solution to the 2D Ising model at zero field. 

Thermal Expansion. Most manufacturers literature (87,119,136—138) quotes a linear expansion coefficient within the 0—300°C range of 5.4 x 10"7 to 5.6 x 10 7 /°C. The effect of thermal history on low temperature expansion of Homosil (Heraeus Schott Quarzschmelze GmbH) and Osram s vitreous silicas is shown in Figure 4. The 1000, 1300, and 1720°C curves are for samples that were held at these temperatures until equilibrium density was achieved and then quenched in water. The effect of temperature on linear expansion of vitreous silica is compared with that of typical soda—lime and borosilicate glasses in Figure 5. The low thermal expansion of vitreous silica is the main reason that it has a high thermal shock resistance compared to other glasses. 

Run, L. C. and Ranov, T., "Efficiency of Low Temperature Expansion Machines," in Advances in Cryogenic Engineering, Vol. 10, K. D. Timmerhaus, Editor, Plenum Press, New York (1965). 

Free Electron Model for Electronic Entropy Using the free electron model of chap. 3, estimate the entropy of metals. Begin by obtaining an expression for the electronic density of states for the three-dimensional electron gas. Then, make a low-temperature expansion for the free energy of the electron gas. Use your results to derive eqn (6.12). 

When speaking of kinematic interaction, it should be noted that the problem of its separation in connection with the transition from Pauli operators to Bose operators is far from new. This problem arises, in particular, for the Heisenberg Hamiltonian, which corresponds, for example, to an isotropic ferromagnet with spin a = 1/2 when spin waves whose creation and annihilation operators obey Bose commutation relations are introduced. This problem was dealt with by many people, including Dyson (6), who obtained the low-temperature expansion for the magnetization. However, even before Dyson s paper, Van Kranendonk (7) proposed to take into account of the kinetic interaction by starting from a picture where one spin wave produces an obstacle for the passage of another spin wave, since two flipped spins cannot be located at the same site (for Frenkel excitons this means that two excitons cannot be localized simultaneously on one and the same molecule). 

Fig. 1. Cross section of gas-lubricated low-temperature expansion engine.

We note that the fractal dimensions found in the (low-temperature) expansion simulations of Nakano et al. [54] and Bhattacharya and Kieffer [50] are not in perfect agreement, again suggesting that either or both of the potential and the simulation protocol can also affect the final network morphology. 

Keeping the first two terms in the low-temperature expansion of hsCz) we obtain the next term in the susceptibility per particle ... 

A Simon liquefier is designed to permit the final helium expansion to take place from 0.808 MPa and 12 K to 0.101 MPa. If the heavy-walled container, constructed of 304 stainless steel with a mass of 70 kg, has an internal volume of 0.04 m, determine the liquid yield and the fraction of the container that is filled with liquid helium after the final low-temperature expansion step. This is an iterative calculation (see Barron, Ref. 19, p. 108). 

These exact treatments may be classified in two groups the first comprises methods which represent the lattice partition function (3.3.10) either as an expansion in powers of (high temperature expansion) or in powers of kTjw (low temperature expansion). Unfortunately such series converge very slowly in the region of the critical temperature (Kirkwood [1938] for a recent account of this theory c.f. Rushbrooke [1953] and Ter Haar [1954]). 

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