Spin wave theory

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. 

De Jongh showed that in 2-D systems, spin-reduction can also be deduced from the perpendicular magnetic susceptibility7,269. The experimental value of xi extrapoled to 0 K is in good agreement with the theoretical value obtained by means of the spin-wave theory (Xi (0). The difference between the latter value and that calculated via the molecular field approach Xkmf) is essentially due to the zero-point spin reduction. 

This model, which gives good results for high values of S, can neither be applied to S A nor to compounds with chain structures. More recently, improvements have been achieved by Ishikawa and Oguchi by including into the spin-wave theory kinematic interactions besides dynamic ones270) for CuF2 H20, S = A, / i(exp) = 0.63 and Xi calc. = 0.64. 

In A2FeFj compounds (A = K, Rb, Cs) an important spin reduction has been detected by neutron diffraction202 203 and Mossbauer resonance measurements230-232. The calculations are based only on the spin-wave theory disregardering the kinematic interactions since the spin value is important. Figure 36 describes the variation of the zero-point spin reduction with the anisotropy factor a. a is here (1 - coA/(oE) 2 where mA and a E are the Larmor frequencies corresponding to the anisotropy and exchange fields, respectively. 

Those curves that do not approach T = 0°K with zero slope are not realized in nature. The N6el model is a molecular field model, and is subject to the same criticisms as the Weiss field model for ferromagnets. Kaplan (325) has applied spin wave theory to ferri-magnets and worked out a Bloch Tz/2 law, similar to equation 98, for low temperatures. In this approximation M /M% remains constant,... 

The heat capacity of EuS was measured to test the predictions of spin-wave theory from 1° to 38°K. by McCollum and Callaway (137) and independently from 10° to 35°K. by Moruzzi and Teaney (145). A sharp Neel peak was found at 16.2 °K. Magnetic and lattice contributions to the heat capacities were resolved on the assumption of a dependence for the lattice and a T dependence for the magnetic contribution at temperatures above the Neel point. A plot of CT vs. yields a straight line between 21° and 31 °K. and a Debye temperature of 208 °K. 

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... 

The molar heat capacities of NiCl2 from 2 to 30 K have been investigated. Smoothed values in this region were listed. The results were compared with the predictions of the spin-wave theory. 

A prototype model for orientational ordering of electrostatic quadrupoles on a triangular net in a static crystal field was devised and systematically investigated based on mean-field, Landau, and spin-wave theory [21, 141] see Section II.A for an extensive discussion. 

A wave can be propagated through the material by means of this mechanism, Fig. 4.45(c). In a similar way to phonons (quantised mechanical vibrations of the lattice), magnons can be thought of as particles, with all their associated properties and interactions. Spin wave theory (Bloch, 1930) provides an interpretation of the low-temperature behaviour of saturation magnetisation in ferromagnets. 

R 638 C. Yamamoto and Y. Okamoto, Optically Active Polymers for Chiral Separation , Bull. Chem. Soc. Jpn., 2004,77,227 R 639 S. Yamamoto, Recent Progress of the Low-Dimensional Spin-Wave Theory , Los Alamos National Laboratory, Preprint Archive, Condensed Matter, Avail. URL http //xxx. lanl. gov/pdf/cond-mat/0310004 R 640 N. Yamasaki and J. Masamoto, Novel Reaction between Cyclic Formal and Ethylene Oxide , J. Polym. ScL, A, 2004,42,520 R 641 C. Yang, G. S. Jas and K. Kuczera, Structure, Dynamics and Interactions with Kinase Targets Computer Simulations of Calmodulin , Biochim. Biophys. Acta, 2004,1697,289... 

The FMR technique can also be used as a convenient means of studying the temperature dependence of the magnetization. In spin wave theory this temperature dependence is given by... 

Deviations from spin wave theory at low temperatures in various amorphous alloys were found by Bhagat et al. (1980), and explained in terms of a model where proper account is taken of the presence of magnetic clusters. A spin-cluster model was also used by Bhagat and Paul (1975) to explain the FMR data obtained by these authors in several amorphous RFe alloys. Weissenberger et al. (1984) investigated FMR in amorphous Y-Co alloys and did not observe deviations from normal spin wave behaviour below T. ... 

Spin wave theory for a simple ferromagnet gives a quadratic magnon dispersion relation at very low temperatures in the long wavelength limit ... 

For an antiferromagnet spin wave theory gives a linear magnon dispersion relation ... 

Fig. 7.39. The spin-wave data for Er by Nicklow et al. (1976). The lower curve (open circles represents the branch hio(+q) and the upper curve (solid circles) the branch h

Well below T, in the zero-temperature limit, the excitations out of the antifer-romagnetically ordered ground state of eq. (2) are spin waves. Neglecting interlayer coupling, conventional spin-wave theory in the classical (large-5) limit predicts a dynamic susceptibility (as measured ly inelastic neutron scattering) of the form... 

Fig. 3. Thermal triple-axis constant-energy-transfer scans through the antiferromagnetic zone center, (100), at selected temperatures below Tf, for La2Cu04- The dashed lines show fits to conventional spin wave theory convolved with the spectrometer resolution. In all cases, the observed peaks are unable to resolve the counter-propagating spin wave modes. This type of measurement can only place a lower bound (650 meV A) on the spin-wave velocity. From Yamada et al. (1989).

Spin wave theory of antiferromagnets is a powerful method to study the ground state in these cases but goes beyond the scope of the book, the interested reader is referred to the monographs of Yosida [7] and BlundeU [8]. 

In a recent critical analysis Dietrich et al. (1975) compare these experiments and argue that specific heat and NMR at temperatures less than about jTc sample only spin waves with low wave vector. In that region the spin wave energy depends only on the stiffness constant, being proportional to the sum of (Ji + J2) and thus Ji and I2 cannot be determined individually. On the other hand, inelastic neutron scattering determines the dispersion of the spin waves over the entire Brillouin zone and the best fit of spin wave theory to these data yield /i and I2 separately and J2 is shown to be positive. The recommended values for 7i,2 of EuO and EuS are also collected in table 19.2. 

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