Spin-wave behavior

From the spin wave (or magnon) model of magnetic systems, the heat capacity versus temperature behavior below the ordering temperature is given192 by the following equation ... 

Of course, the approximations made by the spin wave theory are reasonable at very low temperatures only, and thus it is plausible that this line of critical temperatures terminates at a transition point 7 Kt, the Kosterlitz-Thouless (1973) transition, while for T > 7kt one has a correlation function that decays exponentially at large distances. This behavior is recognized when singular spin configurations called vortices (fig. 33 Kawabata and Binder, 1977) are included in the treatment (Berezinskii, 1971, 1972). Because 0(x) is a multivalued function it is possible that a line integral such... 

Fig. 5.28 The 4/ shell of each Gd atom carries a local magnetic moment which remains constant with temperature. The relative axes of these magnetic moments fluctuate with rising temperature. This behavior can be modeled by spin waves being characterized by the angle a between nearest neighbors. From [127], copyright 1998, reproduced with permission from World Scientific Publishing Co. Pte. Ltd...

H H2, we know that it is synunetric across the symmetry hne which extends radially outward from the origin at = 0 (i.e. to the right of the conical intersection). It is also symmetric across the two symmetry lines which extend radially outward from the origin at = 120°. The geometric phase alters the symmetry of the real electronic wave function for H3, so that it is also antisymmetric across the symmetry line which extends radially outward from the origin at = 7t (i.e. to the left of the conical intersection). By symmetry, it is also antisymmetric across the two symmetry lines which extend radially outward from the origin at = 60°. The antisynunetric behavior is a direct consequence of the wave function s double-valuedness. In order to satisfy Fermi statistics for aU nuclear geometries, the product of the nuclear motion wave function and nuclear spin wave function must also be double-valued and be antisynunetric across the synunetry hnes at (f) = 0, (f> = 120°, and = —120°. The product must also be synunetric across the symmetry lines at = tt, = 60°, and 4> = —60°. 

As remarked in the introduction, section 1, the temperature dependence of the hyperfine field provides a measurement of the temperature dependence of the magnetization on the microscopic level. This point is illustrated by the data of Sweger et al. (1974) for Hen in Dy metal. In this case, Hen(T) is not well represented by a Brillouin function. Paralleling the bulk magnetization data, the behavior of Hen for T 100 K can be interpreted in terms of a spin-wave model of the form [Niira (I960)],... 

We wish next to analyze the behavior of the system in a spin-wave state. To this end, we define the transverse spin-spin correlation operator which measures the correlation between the non-z components of two spins at sites R, R from its definition, this operator is... 

The behavior of a multi-particle system with a symmetric wave function differs markedly from the behavior of a system with an antisymmetric wave function. Particles with integral spin and therefore symmetric wave functions satisfy Bose-Einstein statistics and are called bosons, while particles with antisymmetric wave functions satisfy Fermi-Dirac statistics and are called fermions. Systems of " He atoms (helium-4) and of He atoms (helium-3) provide an excellent illustration. The " He atom is a boson with spin 0 because the spins of the two protons and the two neutrons in the nucleus and of the two electrons are paired. The He atom is a fermion with spin because the single neutron in the nucleus is unpaired. Because these two atoms obey different statistics, the thermodynamic and other macroscopic properties of liquid helium-4 and liquid helium-3 are dramatically different. 

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