Volume 37, Issue 3, 01 March 1966
 THEORY


Temperature‐Dependent Localized Spin‐Wave Modes in Impure Ferromagnets
View Description Hide DescriptionA spin impurity in a Heisenberg ferromagnet can produce a localized spin‐wave mode if its spin or its exchange is sufficiently large. By assuming the effect of the perturbation to be very localized, and the local mode frequency to be so high that its thermal excitation is small, we obtain an estimate of the temperature dependence of the local mode frequency. It is found to decrease with temperature, but more slowly than the host magnetization. Furthermore, a mode which is nonexistent at low temperatures can appear at higher temperatures.

Exchange Constants and Effective Masses in the Ferromagnetic Rare‐Earth Metals
View Description Hide DescriptionThe value of the term k_{f} ^{4}ΣF(2k_{f}R_{n} ) occurring in the RKY theory of indirect interaction has been calculated by summing over 1.5×10^{5} lattice sites for the six heavy rare‐earth metals which exhibit ferromagnetism. The sum ΣF(2k_{f}R_{n} ) has previously been assumed to be constant; however, these calculations show that it varies from −6.62×10^{−3} to −6.91×10^{−3} using free‐electron k_{f} values, or from −2.97×10^{−3} to −7.07×10^{−3} using the k_{f} values obtained by Yosida. Using these values and experimental results of θ_{ p } and ρ_{spin} the variation of Γ and m^{x} through the six elements has been found. In some instances these new values differ appreciably from those currently accepted. The variation of Γ and m^{x} with volume has been investigated using the results of high‐pressure studies, and the values of ∂logΓ/∂logV and ∂ logm^{x} /∂ logV have been found. The change in the g values resulting from these Γ and m^{x} values is compared with the experimental values.

Nature of Exchange Coupling in Transition Metals
View Description Hide DescriptionIt is now fairly well accepted that the magnetic electrons of the d‐shell transition metals must be viewed as itinerant. It appears, moreover, that they are only moderately correlated: although they occupy moderately narrow bands and have a spatial distribution similar to that of atomic d electrons, they fluctuate from atom to atom almost as randomly as the electrons of nontransition metals, thanks to the operation of various screening effects. In the exchange coupling of these electrons, their itinerancy plays a central role, and the concepts of the Heisenberg model are quite inapplicable. Significant contributions to the exchange coupling can arise both from polarity energy (the energy cost of having too many or too few electrons on an atom) and from the Hund's‐rule coupling of electrons on the same atom. Correlation effects, though not strong enough to prevent sizable polar fluctuations, are very important in that they give rise to magnon modes, critical fluctuations, etc. Details of band structure are important in determining whether a metal prefers ferromagnetism,antiferromagnetism, or neither.

Interband Mixing and Induced Local‐Moment—Conduction‐Band Spin Distributions
View Description Hide DescriptionThe spin distribution induced in free‐electron‐like conduction bands by local (rare‐earth) electronic magnetic moments has been investigated quantitatively. Both interband mixing effects and the more familiar Ruderman—Kittel—Kasuya—Yosida exchange‐polarization terms have been accounted for. The former include (1) effective exchange^{1,2} terms, arising from the interband mixing, which contribute to both the average induced spin density and its spatial distribution; and (2) direct interband admixture terms^{1} which affect only the spatial distribution (and affect neither the average density nor the density at the origin of the impurity moment). The results are found to be sensitive to the conduction electron Fermi wave vector k_{F} : in some cases, the admixture terms have little effect on the spin distribution outside of the impurity site; in others, they produce spin‐density oscillations which are large in amplitude compared with the RKKY result.

Application of the Rudermann—Kittel Interaction to Metals
View Description Hide DescriptionThe indirect exchange interaction of magnetic cores via the conduction electrons has been calculated on the basis of a simple model, and the original Rudermann—Kittel result was found provided that (3m/4πℏ ^{2}) × J Sk_{F} ≪1, where J is the exchange interaction between the conduction electrons and the magnetic core electrons, and k_{F} the radius of the Fermi sphere. For the usual values J = 5 eV·Å^{3}, k_{F} = 1 Å^{−1}, and the spin S = 3, this parameter is 0.5. The zero points of the oscillatory interaction from this analysis may be particularly unreliable.

Effect of the Band States Near the Fermi Surface on the Magnetic Properties of Anderson's Extra‐Orbital Model
View Description Hide DescriptionThe magnetic properties of Anderson's ``extra‐orbital'' model are investigated. An exact ground‐state solution is obtained for the simple case in which the conduction band is replaced by a single effective band state. Using this solution, a criterion for a magnetic ground state is determined. The possible existence of a magnetic ground state is found to be critically dependent upon whether the energy level of the effective band state is on the Fermi surface or not. To study this critical behavior associated with band states near the FS, the full Anderson (continuum) model is investigated. It is shown that the ``extra‐orbital'' spin—flip scattering processes associated with band states near the FS produce a lnk T/Γ damping which prohibits the occurrence of a Hartree—Fock magnetic ground state.

Magnetic Form Factor of Nickel
View Description Hide DescriptionThe magnetic form factor of Ni is calculated using the pseudopotential method, the ferromagnetic interaction being supplied by a parametrized Hubbard—Kanamori Hamiltonian. This approach is useful in the present instance because the form factor is primarily determined by uncompensated majority spin electrons near the Fermi surface where LCAO d‐band functions represent a reasonable approximation for the wave functions. The asphericity of the d‐electron spin distribution as well as the constant negative polarization of electrons in the outer regions of the unit cell observed by Mook and Shull are reproduced. The latter effect is accounted for by a reversal of the s‐spin polarization in that region of the unit cell. A possible contribution due to a slow spatial variation of the spin polarization over the unit cell associated with compensated d electrons throughout the d band is also discussed.

Wick's Theorem for Spin Operators and Its Relation to the Coupled‐Fermion Representation
View Description Hide DescriptionIt is shown that Wick's theorem and the linked diagram expansion theorem for S = ½ operators (applied elsewhere by the authors to discuss antiferromagneticspin waves) can be derived by a direct application of conventional theorems to the coupled‐fermion representation of spin operators.

Perturbation Expansion for the Magnetization of the Spin ½ Antiferromagnet
View Description Hide DescriptionA perturbation expansion of the sublattice magnetization of a spin ½ antiferromagnet is developed. The theory proceeds from a Fermion representation of the spin operators for spin ½, in which an auxiliary (drone) set of Fermi operators is introduced to account for the boson character of spin operators attached to different lattice sites. The development of the present theory is parallel to that derived earlier by Mills, Kenan, and Korringa, with the role of the ``contracted lines'' of the earlier theory being played by the drone lines of the present theory.Integral equations are derived for the single‐particle spin propagator. A resummation over a class of graphs contributing to the kernel, which gives the spin‐wave result at low temperatures, is outlined.

Mechanism for the Direct Exchange Interaction of Europium in Compounds with Rocksalt Structure
View Description Hide DescriptionFrom the strong decrease in the nearest‐neighbor exchange interaction going from EuO to EuS, EuSe, EuTe, it has been concluded that this positive interaction must be caused by direct orbital overlap. Since the 4f cores are very small, this has to occur via partial occupation of outer orbits. The excitation to the 5dwave function by zero point lattice vibrations has been considered and is thought to be responsible for the ferromagnetism of these compounds. The order of magnitude has been estimated and agrees with experiment.

Space Group Theory for Spin Waves
View Description Hide DescriptionThe symmetry properties of spin waves in ferro‐, antiferro‐, and ferrimagnetic crystals can be conveniently discussed in terms of an appropriate space group as are electron energy states in band theory and normal modes in the theory of lattice vibrations. Dimmock and Wheeler have pointed out that the magnetic space group, because of the directional properties of the moments, has fewer elements than the lattice space group, and have discussed the spin‐wave properties in terms of this group. While this is strictly correct, the main part of the Hamiltonian usually used in such systems often has much more symmetry than this. For Heisenberg interactions the spins can be rotated independently so that the total group is a product of a lattice space group and a set of spin—space rotations which leave the spin pattern unchanged. This lattice space group is obtained by considering atoms with different spin orientations as inequivalent. The single ion anisotropy restricts the spin rotations. This leads to considerably more degeneracy in the spin‐wave spectrum than is predicted by the magnetic space group, the degeneracy being removed in real crystals by small interactions like dipole—dipole. Such groups have been constructed for common systems for use in evaluating the form of spin‐wave spectra, selection rules, etc.

Variation of the Critical Temperatures of Ising Antiferromagnets with Applied Magnetic Field
View Description Hide DescriptionHigh‐temperature expansions of the free energy of Ising ferro‐ and antiferromagnets on the square, simple cubic, and body‐centered cubic lattices have been derived directly from the corresponding low‐temperature expansions using a generalization of the technique proposed by Domb. These lattices, being loose‐packed, can be divided into two identical sublattices, α and β, such that every α site has only β sites as nearest neighbors. The high‐temperature expansions treat the finite magnetic fields on the α and β sites as independent quantities. With these series, which extend to eleventh order in the appropriate energies divided by kT, the high‐temperature expansions of the staggered susceptibility (the susceptibility with respect to a magnetic field whose sign on the α sites is the negative of its sign on the β sites) has been evaluated for finite applied constant magnetic field. This staggered susceptibility shows a strong singularity at the antiferromagnetic critical temperature, and is analagous to the normal susceptibility of an Ising ferromagnet. The variation of the critical temperature has been obtained numerically for various ratios of the applied magnetic‐field energy to the coupling energy using the method of Padé approximants. The results are well described by an expression of the form T_{c} (H)/T_{c} (0) = [1 — (H/H_{c} )^{2}]^{ξ}, with ξ = 0.87, 0.35, and 0.36 for the square, simple cubic, and bcc lattices, respectively. Here, H_{c} = —zJ/m, where z is the coordination number, J is the spin coupling energy, and m is the magnetic moment.

Thermal Properties of Spin‐Wave Impurity States
View Description Hide DescriptionThe effect of dilute magnetic impurities on the thermal properties of an ideal simple cubic spin ½ Heisenberg ferromagnet has been investigated. It is shown that for a small ratio ε of impurity—host to host—host exchange low‐lying ``s‐type'' virtual spin‐wave states result which cause a large density of states to occur at low energies. These low‐energy states lead to an accumulation of spin disorder at and near the impurity site. Consequently the impurity magnetization decreases far more rapidly than that of the host. This effect is accompanied by a large increase in the low‐temperature spin‐wave specific heat. Analytic solutions to the Green's function equations are calculated for temperatures near T = 0 and near T = T_{c} , the Curie temperature. Self‐consistent numerical solutions are presented for both the magnetization and spin‐wave specific heat as a function of temperature. For small ε the impurity magnetization is approximated by the Brillouin functionwhere J′ is the host‐impurity exchange and 〈S _{1} ^{ z }〉 is the thermal average for the impurity nearest‐neighbor spins. 〈S _{1} ^{ z }〉 is found to be depressed from the bulk value by an amount which increases with temperature and is about 0.84 of the bulk value as T→T_{c} .
A detailed account of these calculations will be published elsewhere.
