Zero-field

At r = 0, all the spins are either aligned up or down. The magnetization per site is an order parameter whieh vanishes at the eritieal point. Along the eoexistenee eiirve at zero field... 

An essential feature of mean-field theories is that the free energy is an analytical fiinction at the critical point. Landau [100] used this assumption, and the up-down symmetry of magnetic systems at zero field, to analyse their phase behaviour and detennine the mean-field critical exponents. It also suggests a way in which mean-field theory might be modified to confonn with experiment near the critical point, leading to a scaling law, first proposed by Widom [101], which has been experimentally verified. 

Assume that the free energy can be expanded in powers of the magnetization m which is the order parameter. At zero field, only even powers of m appear in the expansion, due to the up-down symmetry of the system, and... 

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. 

Both systems give the same results in the thennodynamic limit. We discuss the solution for the open chain at zero field and the closed chain for the more general case of H 0. 

This reduces to the results for the free energy at zero field H = 0)... 

The specific heat at zero field follows easily,... 

Onsager s solution to the 2D Ising model in zero field (H= 0) is one of the most celebrated results in theoretical chemistry [105] it is the first example of critical exponents. Also, the solution for the Ising model can be mapped onto the lattice gas, binary alloy and a host of other systems that have Hamiltonians that are isomorphic to the Ising model Hamiltonian. 

In 1962 Heller and Benedek made aeeiirate measurements of the zero-field magnetization of the antiferromagnet MnF2 as a fiinetion of temperature and reported a p of 0.335 0.005, a result supporting an experimental parallelism between fluids and magnets. 

That analyticity was the source of the problem should have been obvious from the work of Onsager (1944) [16] who obtained an exact solution for the two-dimensional Ising model in zero field and found that the heat capacity goes to infinity at the transition, a logarithmic singularity tiiat yields a = 0, but not the a = 0 of the analytic theory, which corresponds to a finite discontinuity. (Wliile diverging at the critical point, the heat capacity is synnnetrical without an actual discontinuity, so perhaps should be called third-order.)... 

Thus, for X = y, F = 0 which gives rise to planes of zero field strength (Figure 25.2b). At all other positions between the poles, the oscillating electric field (F) causes ions to be alternately attracted to and repelled by the pairs of rods (A, B Figure 25.2). 

Ebbesen[4] was the first to estimate a conductivity of the order of lO fim for the black core bulk material existing in two thirds of tubes and one third of nanoparticles. From this observation, it may naturally be inferred that the carbon arc deposit must contain material that is electrically conducting. An analysis of the temperature dependence of the zero-field resistivity of similar bulk materials[14,15] indicated that the absolute values of the conductivity were very sample dependent. 

Above 2 K, the temperature dependence of the zero-field resistivity of the microbundle measured by Langer et al. [9] was found to be governed by the temperature dependence of the carrier densities and well described by the simple two-band (STB) model derived by Klein [23] for electrons, , and hole, p, densities in semimetallic graphite ... 

In Fig.. I we present the temperature dependence of the conductance for one of the CNTs, measured by means of a three-probe technique, in respectively zero magnetic field, 7 T and 14 T. The zero-field results showed a logarithmic decrease of the conductance at higher temperature, followed by a saturation of the conductance at very low temperature. At zero magnetic field the saturation occurs at a critical temperature, = 0.2 K, which shifts to higher temperatures in the presence of a magnetic field. 

Optical detection of magnetic resonance (ODMR) was attempted for measurements of the pH effects on the triplet state of purine to investigate the protonation site of purine at low temperatures (78JA7131). The ODMR spectrum did not show the presence of more than one triplet state at liquid helium temperatures. Since the protonated tautomers 1H,9H (3a) and H,1H (3b) have similar bond structures, their triplets should have similar zero-field parameters and are thus not easy to distinguish by ODMR. 

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