Volume 33, Issue 3, 01 March 1962
 MAGNETIZATION


Statistical Mechanics of Ferromagnetism
View Description Hide DescriptionThe fact that reliable statistical mechanical theories of ferromagnetism existed only in the very low temperature region (spin wavetheory) and in the very high temperature region (Kramers and Opechowski) has seriously hampered the development of magnetic theory. The Weiss molecular field theory neglects all spin‐spin correlations and therefore gives too high a Curie temperature, misrepresents the details of the phase transition, and is unreliable as a basis for analyzing any effect depending on spin interactions. Cluster methods attempt to introduce correlation, but they are both inadequate and sometimes run into severe difficulties, such as the appearance of an anti‐Curie temperature (in the Bethe‐Peierls‐Weiss theory). The source of such difficulties is demonstrated by the formalism of P. W. Kasteleijn and J. Van Kranendonck [Physica 22, 317 (1956)] who showed that the cluster theories correspond to the choice of a self‐inconsistent two‐particle density matrix. Furthermore the spin wavetheory indicates that the interactions which are dominant in establishing spin‐spin correlations are iterated interactions through enormously many long and complicated paths connecting the spins. Cluster methods are unable to include these long paths, and stress instead the direct short‐path interactions. The need for a many‐body theory is therefore indicated.
One such many‐body theory is the Green function treatment of Bogoliubov and Tyablikov. That treatment is equivalent to the random phase approximation. The simplest formulation starts with the spin wavetheory of Dyson. The destruction operator for a Dyson spin wave is.Similarly Dyson introduces the Fourier components of the z components of spin.Unfortunately the spin waves defined in terms of these operators are neither orthogonal nor are they eigenstates of the Hamiltonian. In fact the commutation relations areand,where ε_{ k } is the energy of a simple spin wave, expressed as a function of k; for small k it is proportional to k ^{2}. The random phase approximation ``decouples'' the awkward higher‐order terms in the right‐hand members of these commutation relations. In lowest order this decoupling consists of replacing S _{λ} ^{ z } by N〈S_{z} 〉δ_{λ0}, where 〈S_{z} 〉 is the average value of the z component of spin, to be evaluated self‐consistently. Then the commutation relations becomeand.The first of these states that the quasi‐particles (magnons) produced by the creation operators a_{k} ^{+} are bosons. The second commutation‐relation states that the energy of such a quasi‐particle is μH+ε_{ k }〈S_{z} 〉/S. Thus the average number of such magnons, of wave vector k, is,which is the result given by Tyablikov^{1} and by Englert. It differs from the simple spin wavetheory by reducing the energy of each spin wave proportionally to the average magnetization.
The Green function method (or RPA) gives a result which forms a useful interpolation throughout the entire temperature range. It agrees to second order in 1/T with the Kramers‐Opechowski series in the high temperature (paramagnetic) region. Near the Curie temperature the theory reduces to the Weiss theory, and in the very low temperature region it agrees with the simple spin wavetheory. The corrections to the simple spin wavetheory at slightly higher temperatures are incorrectly given, however, as the RPA approximation predicts a correction to the magnetization varying as T ^{3}, whereas Dyson has shown that the lowest order correction varies as T ^{4}. This error in the theory arises from the decoupling procedure, which ignores spin‐wave correlation effects in replacing S _{λ} ^{ z } by zero (if λ≠0).
A direct approach to a statistical theory consists in a series expansion of the partition sum, a diagrammatic representation of the terms, and a partial summation of those terms corresponding to the long, circuitous paths connecting and correlating two spins. Such a theory has been carried out by Brout and co‐workers and by Horwitz and Callen. The results have been published, or are in publication in detailed form elsewhere for the Ising model, and the extension to the Heisenberg model will be presented in a separate more extensive publication. Again the theory provides a useful interpolation between the low and high temperature region. Evaluation of the theory at low temperatures, by R. Stinchcomb, indicates that it properly gives the Dyson dynamical (T ^{4}) correction.

Magnetization of Localized States in Metals
View Description Hide DescriptionA study has been made of the magnetic properties of dilute solutions of iron in various nonmagnetic 4d series elements and alloys. In some cases the iron atoms possess a localized magnetic moment which manifests itself as an inverse temperature dependence in the susceptibility of the solution. The occurrence of the moment is determined by the valence electron concentration in the solute element (or alloy). As a function of this quantity one finds portions of the 4d series in which localized moments are strictly absent, interspersed by regions—one centered near Mo, the other in the vicinity of Rh and Pd—in which magnetization occurs and at whose edges the moments appear almost discontinuously.
This behavior may be understood in terms of a theoretical model in which the magnetization is ascribed to a virtual level, of the type proposed by Friedel, of the iron atom. Polarization occurs when the virtual level lies near the Fermi level and is sufficiently sharp in energy. A self‐consistent Hartree‐Fock calculation indicates that under these circumstances the impurity atom develops an exchange potential which splits the level—causing it to have different energies for spin‐up and spin‐down electrons and thus giving rise to magnetization. The model gives a qualitative description of the experiments and, in particular, is able to account for the fact that the impurity atoms sometimes carry a moment that is a fraction of a Bohr magneton.

Pyromagnetic Test of Spin Wave Theory in Metallic Nickel
View Description Hide DescriptionThe temperature dependence of the spontaneous magnetization of metallic nickel has been studied between 4.2° and 120°K by a pyromagnetic technique developed by the authors. Fractional changes in magnetization as small as a few parts per million could be detected near 4.2°K. The resultant data were fitted by the method of least squares to a theoretical equation containing terms descriptive of thermal excitation of spin waves in the presence of an effective magnetic field plus a T ^{2} term descriptive of collective electron behavior. The best fit of the data to this equation is obtained using the spin wave terms alone, provided an intrinsic energy gap is assumed in the spin wave dispersion law of 2.7°K for magnetization parallel to the [111] axis and 1.9°K parallel to the [100] axis. Enhancement of this gap by an externally applied field follows theoretical predictions. It is noted that the measured difference between the gap temperature along the two principal axes has the value theoretically predicted from previous measurements of magnetic anisotropy energy, however the isotropic contribution observed in this experiment has not been theoretically anticipated. A possible origin for the isotropic gap is proposed in terms of interaction of polarized s and d electrons. It is also pointed out, however, that the ``isotropic effective field'' may be a spurious result, originating in thermal expansion effects not included in the theoretical equation to which the data were fitted. Finally, a new type of pyromagnetic measurement is described which can be used to determine the temperature dependence of the magnetic anisotropy.

NMR in Domains and Walls in Ferromagnetic CrBr_{3}
View Description Hide DescriptionNuclear magnetic resonance absorptions have been observed in zero applied field in the insulating ferromagnet CrBr_{3}. Three lines are shown to be a quadrupole split triplet from nuclei in the bulk of the domains. A fourth, with a resonance frequency of 56.58 Mc at 1.34°K and a 250‐kc linewidth, is ascribed to nuclei in the domain walls. The nuclear resonance frequencies are proportional to the local magnetization, and thus measure the temperature dependence of the magnetization both in domain walls and in domains. This is the first measurement of the difference between wall and domain magnetizations, and shows that (ΔM/M)_{wall} − (ΔM/M)_{domain}=T/200 in the spin wave region.

Temperature Dependence of YIG Magnetization
View Description Hide DescriptionThe temperature dependence of the saturation magnetization, 4πM, has been measured for pure, polished YIG spheres in the temperature range from 4–50°K. The technique used involved measuring the magnetic field separation of the 210 and 220 magnetostatic modes as a function of temperature. The mode spacing is equal simply to 4πM/5. This method offers a high degree of precision since the linewidths are less than 1 gauss and the separation of the two modes is the order of 500 gauss. Data were taken for several samples and with the external magnetic field oriented in both the [100] and [111] directions. The experimental results have been fitted to the theoretical expression of Dyson. Only the first‐order correction term to Bloch's equation was necessary, the result being . A fit to the term alone is definitely excluded.

Ferromagnetic Resonance in CrBr_{3}
View Description Hide DescriptionFerromagnetic resonance has been observed in single crystals of anhydrous chromium tribromide using frequencies of 20 to 27 kMc. At 1.5°K a g value of 2.006 is found along with a uniaxial anisotropy,K=9.4×10^{5} ergs/cm^{3}. Linewidths as narrow as 3.5 oe have been observed at 1.5°K.
