Volume 39, Issue 2, 01 February 1968

Magnetic Case History of the Eu^{3+} Ion
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Development of the Theory of Helical Spin Ordering
View Description Hide DescriptionTheory of helical spin ordering in a lattice having one magnetic atom per unit cell and its extension to the case of several atoms per unit cell have brought about deeper understanding of magnetic structures than before. These theories are reformulated concisely in the present paper and their applications to solid solutions,, (spiraling triangular spins), and spinels are discussed. The role of uniaxial anisotropy energy in spin ordering and a particular spin order of an integral period of seven layers in thulium, as well as the pairing of spins in holmium due to sixfold anisotropy energy, are discussed. Transition between a fan structure and the parallel alignment in magnetic field and its example in are mentioned. Finally, the existence of a spin‐wave mode of zero frequency in a general modified‐helical spin arrangement is pointed out.

Interpretation of the Magnetic Properties of the Rare Earth Chromites and the Rare Earth Manganites
View Description Hide DescriptionThe magnetostatic properties of the rare earth chromites, , and those of the rare earth manganites, , have been studied between 1.6° and 1500°K, in fields up to 28 000 Oe. These compounds are antiferromagnetic; they present two Néel points.
For the rare earth chromites, the first Néel point varies between 112°K for and 282°K for ; it corresponds to the ordering of the spins of the Cr^{3+}ions; below this transition temperature, a ferromagnetism is superimposed upon the antiferromagnetism. The second Néel point is situated in the neighborhood of the temperature of liquid helium; it corresponds to the ordering of the moments of the rare earth ions.
These properties are interpreted in terms of a molecular‐field model. The lattice of the Cr^{3+}ions and that of the R^{3+}ions are decomposed into two identical sublattices. The Cr‐Cr exchange interaction is negative; it is dominant; it defines principally the first Néel point. The R‐R interaction is negative and weaker; it defines the second Néel point. The R‐Cr interaction can be positive or negative, and its magnitude is small. The configuration of the moments of the chromiumions can be weakly ferromagnetic; this would induce a ferromagneticpolarization of the rare earth ion, and can give rise to a compensation temperature.
With the perovskite rare earth manganites, the first Néel point is between 45°K for and approximatively 131°K for ; a ferromagnetism is superimposed upon the antiferromagnetism. These substances behave like the rare earth chromites.
With the hexagonal rare earth manganites, the first Néel point is characterized only by a change in the slope of the susceptibility curve; there is no superimposed ferromagnetism. Near 10°K, it appears a ferromagnetic term which can be induced by the ordering of the spins of the rare earth ions. The rare earth‐manganese interaction is determined in the molecular‐field approximation.
The interpretation of the magnetic properties of the rare earth chromites has been published in the “Compt. Rend, 262, 799, 866 (1966); a full report on the interpretation of the magnetic properties of the rare earth maganites will be published in “J. Phys.”.

Spin‐Wave Interactions in a Heisenberg Ferromagnet
View Description Hide DescriptionThe theory of spin‐wave interactions is discussed using the double‐time temperature‐dependent Green's function technique. It is shown how the important parts of the third‐order Green's functions can be deduced rigorously. By neglecting the remaining parts of these third‐order functions it is then possible to obtain closed expressions for the first‐order Green's functions and the spin correlation functions. One further approximation, which however is rigorous in the two limits and , is used. In agreement with Dyson's spin‐wave theory it is found that ‘kinematic’ interactions produce no effects on the magnetization at low temperatures. However, it is shown that the kinematic interactions do affect the neutron cross sections for creation or annihalation of spin waves and the transverse magnetic susceptibility, and therefore, may be observed directly.

Spin Waves in 3d Metals
View Description Hide DescriptionA review is given of recent measurements of spin‐wave dispersion relations of the 3d metals by the Brookhaven neutron diffraction group using the diffraction technique and triple‐axis spectrometry. The parameters and in the relation have been determined at 295°K for Fe, Co, Ni, and some of their alloys. These values are compared with those obtained by thin‐film resonance and small‐angle scattering.
The most extensive measurements were carried out on Fe using a triple‐axis spectrometer. The dispersion relation was measured along the three principal symmetry directions for wavevectors up to . The stiffness constant as well as the linewidth of selected spin waves were studied for the temperature range between 77°K and the Curie temperature, 1042°K. Well‐defined magnons were observed up to a reduced temperature, but not above .

Nuclear‐Spin Exchange Interaction in Solid ^{3}He
View Description Hide DescriptionSolid ^{3}He has an unusually large nuclear spin‐spin exchange interaction, which is due to the large zero‐point motion of the atoms. This exchange can be described by a Hamiltonian of the form . The parameter is strongly dependent on the density, decreasing from at to at in the bcc phase. Hence the exchange is large in comparison with the energy of dipolar interaction . There is some evidence that the exchange is antiferromagnetic. One can calculate that at the highest stable molar volume, the transition to the ordered state should occur at about . In the still more dense hcp phase, the exchange continues to decrease with molar volume.
The experimental evidence for this exchange comes from nuclear magnetic relaxation measurements taken in several laboratories. This paper presents a review of longitudinal and transverse relaxation data as well as diffusion data and their interpretation in terms of exchange. The consequences of exchange on some thermodynamic properties is then discussed.

Spin Waves in Sodium and Potassium
View Description Hide DescriptionThe observation of spin‐wave excitations originating in the conduction electrons of sodium and potassium metal is presented. The qualitative behavior of these spin‐wave signals is given along with a detailed comparison of the experimental data with the theory (of spin waves) as derived by Platzman and Wolff. It is shown that with the aid of the theory, it is possible to extract information from the data which leads to the first two Legendre coefficients of the spin part of the Landau correlation function for a Fermi liquid. A discussion is given on how present uncertainties in these two coefficients can be substantially reduced in future work.

Localized vs Collective Descriptions of Magnetic Electrons
View Description Hide DescriptionSpontaneous band magnetism occurs as a transitional electron state, in the thermodynamic sense, between a localized‐electron (or small‐polaron) state and a conventional collective‐electron state. The localized‐electron state is well described by crystal‐field theory, together with superexchange and double‐exchange theories. Conventional band theory neglects electron correlations, except in the superconducting state. In the transitional state, electron correlations introduce not only the exchange interactions responsible for spontaneous magnetism, but also a deep minimum, if not an energy gap, in the density‐of‐states vs energy for half‐filled bands. Since Umklapp processes stabilize antiferromagnetic vs ferromagnetic order as the Fermi surface approaches a Brillouin‐zone boundary, it is possible to construct a semiempirical phase diagram for various electronic states in the space , , , where is the number of electrons per orbital per atom, is the transfer energy, and is the temperature. Magnetic data that illustrate a few significant features of this diagram are discussed briefly, and it is pointed out that spontaneous band magnetism is a relatively rare phenomenon because the transitional state occurs over only a small range of .

Magnetism in High Magnetic Fields
View Description Hide DescriptionA brief summary of several recent high‐field studies of magnetism in solids at the National Magnet Laboratory is presented. Topics include high field superconductors,magnetic phase transitions in simple antiferromagnets, capabilities of high field Mössbauer measurements for microscopic studies of magnetic structures, rare earth metals, collective electron ferromagnetism, and dilute alloys.

Magnetic Properties and Electrical Conduction of Copper‐Containing Sulfo‐ and Selenospinels
View Description Hide Descriptionwith and with show p‐type metallic conduction and are ferromagnetic, while with show n‐type metallic conduction and no magnetic ordering at 4°K. From the magnetic and electrical properties, which indicate that Cu is monovalent in these compounds, a model for the electronic structure of copper‐containing sulfospinels is deduced. Ferromagnetism and p‐type conduction are attributed to holes in a broad valence band,n‐type conduction to electrons in a broad conduction band.Susceptibility measurements on , , , , and are reported. The main contribution to the temperature‐independent susceptibility observed is attributed to the Van Vleck susceptibility of Co^{3+} or Rh^{3} ions in the low spin state .

Model Calculations for the Study of Direct and Superexchange Interactions
View Description Hide DescriptionInvestigations of the two‐atom and the three‐atom systems have been made in order to study both direct and superexchange mechanisms. In these model calculations all electrons are treated explicitly within the framework of the unrestricted Hartree‐Fock procedure; the Fock matrix is evaluated directly by Monte Carlo integration techniques. The effective exchange parameter is determined at several internuclear distances, and spin densities are exhibited for the ferromagnetic and anti‐ferromagnetic states of the three‐atom system. The AF state lies lower in energy beyond a calculated critical bond length. Localization of the uhf eigenfunctions and the significance of the Mott transition for magnetic ordering are discussed.

Systematics of the Right‐Angle Nearest‐Neighbor Exchange Interaction for Ions
View Description Hide DescriptionIn the chromiumchalcogenide spinels the strength of the ferromagnetic nearest‐neighbor exchange interaction increases as the anion is changed from . This tendancy was first revealed by the experiments of Baltzer et al. In order to explain this behavior, various mechanisms contributing to the right‐angle exchange coupling are examined. Both the superexchange and the direct exchange interactions between the magnetic ions are taken into account. For the four‐electron, three‐center model in which a ligand orbital forms a π‐bond with one magnetic ion, but is orthogonal to the other, it is demonstrated that: (1) An increase in the Cr^{3+} ligand covalency along the series leads to an enhancement of the ferromagnetic interaction. (2) There is a reduction in the antiferromagnetic direct cation‐cation interactions, because the Cr^{3+} ions are further apart. (3) For those ligand orbitals which form π‐bonds with both magnetic ions, the antiferromagnetic interaction tends to become stronger in going from . The last effect is shown to be overcome by a corresponding increase in the first (ferromagnetic term) described above. Larger covalency, leading to a reduction in the magnitude of the electron energy, also reduces the third (antiferromagnetic) mechanism.

Electric Multipole Interactions between Rare Earth Ions
View Description Hide DescriptionElectrostatic coupling between rare earth ions is important for nonradiative optical‐energy transfer processes and for effective spin‐spin interactions in ionic salts. In calculating such effects is it generally assumed that the electric quadrupole‐quadrupole term is much larger than the corresponding interaction between higher degree multipole moments because successive terms of a given parity are each a factor smaller than the preceding one. ( is the radius of 4f‐electron orbit and is the ionic separation). In this paper we point out that the relative importance of the higher‐degree terms may be enhanced by electrostatic shielding and induced‐moment effects similar to those affecting single ion crystal field terms, and we derive the Hamiltonian operators for the l‐l′ multipole interactions up to sixth degree. Comparison with the observed single ion crystal fields also suggests that electric multipole interactions might be relatively important throughout the rare earth series and not only for the larger light ions, as is often assumed.
For pairs of Kramers ions at low temperatures the various multipole interactions will contribute in second order to different terms of an effective spin Hamiltonian of the form , and the resulting interaction tensor will generally be quite anisotropic, subject only to symmetry. There is also a corresponding anisotropic contribution to the magnetic gtensor, whose principal values and axes may thus be different from those of the single ions. Detailed multipole calculations are prohibitively complicated in the general case, and unless it can be shown that the higher degree terms are in fact negligible the observable interaction parameters in any particular case must therefore be treated as strictly empirical quantities, restricted only by symmetry.
A more detailed discussion of these effects is being published elsewhere.

Some Properties of Ferromagnetoelectric Nickel Iodine Boracite
View Description Hide DescriptionSynthetic and very pure single crystals of nickel iodine boracite (, ), up to 10 mm in their largest dimension grown by a chemical transport process were investigated, using a self‐compensating recording balance under quasistatic conditions. Below 64°K ferromagnetic behavior in agreement with the results of Ascher, Rieder, Schmid, and Stössel was observed. Rotation diagrams showed a slight hexagonal component in the (111) plane of the crystal. In the (100) plane, however, large tetragonal torsion moments were recorded having frequent discontinuities and very large rotational hysteresis in fields of about 5 kOe. In a temporally and spatially constant magnetic field, a discontinuous but on the average linear decrease of the torsion moment was observed, taking about 30 min for it to vanish completely.
The above behavior can be only partly explained on the basis of ferromagnetic‐domain wall motion. The influence of ferroelectric‐domain processes coupled through ferromagnetoelectric interaction seems to be of considerable importance for the relaxation of magnetization and is therefore assessed and discussed.

Statistical Mechanics of Uniaxial Antiferromagnet with First‐ and Second‐Neighbor Exchanges
View Description Hide DescriptionLow‐temperature neutron‐scattering study together with the zero‐field AFMR in furnished a reliable set of values of first‐ and second‐neighbor exchange integrals and the uniaxial anisotropy energy. It is the purpose of this paper to see to what extent one can understand the high‐temperature properties such as the Néel temperature and the anisotropicsusceptibilities in terms of the known parameters obtained at low temperatures. For this purpose the cluster‐variation method is used in which up to the two‐spin correlation is taken into account. One of the main tasks of this method is to diagonalize effective one‐ and two‐spin Hamiltonians. It is noted here that the anisotropy energy could be as big, at least, as the second‐neighbor exchange energy. It is, therefore, not justifiable to treat the anisotropy energy as a small perturbation. In this paper diagonalization of the effective two‐spin Hamiltonian is achieved for the actual spin, , and the Néel temperature and the anisotropicsusceptibilities are calculated. Calculations of the sublattice magnetization and spin‐flop field are in progress.

Theory of Anomalous Magnetic and Transport Phenomena in Eu‐Chalcogenide Alloys
View Description Hide DescriptionAnomalous magnetic and transport properties of , in which is Gd or La, are calculated for the magnetic impurity state model. An excess electron is trapped on a central impurity atom, R, and forms a magnetic impurity state, extending mostly onto the nearest neighbor Eu^{2+} ions and aligning these spins through exchange interaction. Satisfactory agreement is obtained between calculated and experimental results for magnetic properties. For transport properties, a calculation based on the hopping‐type impurity conduction model in which the activation energy is determined by the exchange interaction agrees well with experiment for higher temperatures. The magnetic fieldeffect is also calculated. For low temperatures or high magnetic fields the impurity‐band‐conduction mechanism becomes dominant. The calculation on this model explains the experiment satisfactorily.

Gyrotropic Birefringence—Phenomenological Theory
View Description Hide DescriptionA phenomenological theory of gyrotropic birefringence is presented. Since the magnetic classes in which gyrotropic birefringence is allowed are among those in which the magnetoelectric effect may occur, the constitutive relations between the complex amplitudes of the fields are so written as to incorporate both these effects simultaneously. All induced effects are then combined in a suitably renormalized electric dipole moment. For the case of lossless media, the gyrotropic birefringence property tensor has 18 linearly independent components before crystalline symmetry considerations are introduced. It is shown that a physical basis for these 18 independent quantities may be found in electric quadrupole and magnetoelectric effects, with the former contributing ten independent quantities and the latter eight. In particular, the compounds and , in which an experimental observation of gyrotropic birefringence may be possible, are considered. Finally, the closely related effect of natural optical activity is discussed and correlated with the point of view presented here.

Domain‐Wall Mobility in Si‐Doped
View Description Hide DescriptionDomain‐wall mobility measurements have been carried out on picture frames made from single crystals of the composition in the temperature range from 77° to 300°K. The geometry of the picture frames was such that a wall parallel to the 110‐plane was favored. For and the mobility can be described by , with . The lowest mobility observed is 1 cm sec^{−1}Oe^{−1}. The values of and are consistent with loss measurements and can be understood as resulting from a diffusion aftereffect involving electron transitions between Fe^{2+} and Fe^{3+}ions. Below 220°K this heavily impeded wall motion could not be traced further, and a faster mode with was observed. The fast mode could also be observed above 220°K in applied fields surpassing a threshold field . We identify this fast mode with the one observed by Wanas. The transition between the two modes can be described by Janak's theory of diffusion‐damped wall motion.

Pressure Dependence of the Magnetization of Invar and Silectron from 30–450 kbar
View Description Hide DescriptionMeasurements of the change in saturation magnetization of Invar and Silectron subjected to shock‐wave compression from 30 to 450 kbar show that Invar exhibits a constant coefficient of saturation magnetization change with pressure, and that Silectron experiences a pressure‐induced transition to a nonferromagnetic phase. Shock waves are generated in tape‐wound core samples by projectile impact techniques which allow experiments in small pressure increments over a wide range in pressure. The Invar measurements give a value for of up to a magnetization change of 90% of the saturation magnetization. This value is the same as that obtained in previous static measurements to 5 kbar. Measurements on Silectron (grain‐oriented 3% Si‐97% Fe) cores show a change in magnetization beginning at 150 kbar which is the pressure at which a first‐order transition has been detected from previous shock‐wave pressure‐volume measurements. The present measurements indicate that the high‐pressure phase of Silectron is nonferromagnetic and show that a mixed‐phase region extends to a pressure of 225 kbar. These shock‐wave measurements cover a pressure range which is about two orders of magnitude greater than that used previously in static magnetization vs pressure measurements.