A characteristic class of quantities in nonequilibrium thermodynamics and a statistical justification of the local equilibrium approximation

A theory of bandwidth anisotropy in metallic ferromagnets developed previously is specialised to the case of 5d electrons in a hexagonal close-packed lattice. This theory is combined with a model for 5d electrons in the heavy rare earth metals to give a new theory for the low temperature values of the magnetic anisotropy coefficientsκ ₂ ⁰ andκ ₄ ⁰ in Tb, Dy, Ho, Er and Tm. In this theory the magnetic anisotropy is due to a combination of (i) crystal fields acting on 5d and 4f electrons and (ii) bandwidth anisotropy associated with a dependence of 5d bandwidths on magnetization direction. After use is made of empirical upper limits on the eighth order magnetic anisotropy in Gd, there remain four partially adjustable parameters of importance in the theory. These can be chosen to give a good fit to the six observed values forκ ₂ ⁰ andκ ₄ ⁰ in Tb, Dy and Ho. Crystal fields corresponding to negative point charges are seen by 5d electrons, but because of 4f – 5d interactions effective fields of larger magnitude and opposite sign act on 4f shells. Bandwidth anisotropy gives a significant contribution toκ ₄ ⁰ of opposite sign to that due to crystal fields, and dominates the latter in Tb and Er.     

Saturation magnetic moments,magnetic hyperfine fields and electric field gradients at nuclei in the heavy rare earth metals

The contributions of 4f, 5d and 6s electrons to the saturation magnetic moments and magnetic hyperfine fields in the heavy rare earth metals are calculated using the model described in the previous paper. It is found that 4f shell moments are reduced from their free ion values by amounts varying from 0.05µ _B in Gd to several tenths of a Bohr magneton in Tb and Dy, in qualitative agreement with a recent published analysis of neutron diffraction results in Tb, but that the calculated total saturation moments in Tb and Dy are slightly larger than commonly accepted experimental values. After 6s contributions to magnetic hyperfine fields are determined by fitting observations in Gd, the predicted differences between the fields for metals and those for free ions are such that the estimated uncertainty ranges of the theoretical values overlap the experimental ranges. The 5d contribution in the model is negative, varying from about –40 kOe in Tb to –200 kOe in Er. Electric field gradients are also analysed. Observed results can be fitted if the average effective Sternheimer screening factorR _d ^* for 5d electrons in the metals satisfies (1 —R _d ^* )0.7.     

The magnetic hyperfine field at Cd nuclei in the rare earth ferromagnets Gd,Tb, Dy,Ho, Er and Tm

The time differential perturbed angular correlation technique has been used to study the combined magnetic and electric hyperfine interactions at the site of a¹¹¹Cd impurity in the rare earth ferromagnets Gd, Tb, Dy, Ho, Er, and Tm at 4.2 °K. The following magnetic hyperfine fields at the site of¹¹¹Cd have been found: ¦H _hf ¦=340(7) kG in Gd, 275 (5) kG in Tb, 221 (4) kG in Dy, 116 (3) kG in Er and 60 (6) kG in Tm. In Ho two magnetically different sites were observed with magnetic fields of 159 (3) and 139 (3) kG. Both sites are equally populated. The coupling constantJ _5f of the conduction electron-4f interaction has been calculated for the different rare earth metals from the measured hyperfine fields by means of the RKKY theory.     

Revised model for 5d electrons in the heavy rare earth metals

A revised version of a recently published model for 5d electrons in the ferromagnetic state of the heavy rare earth metals is described. The model involves the broadening of local 5d states into overlapping bands with individual widthsW. In the new approach it is assumed that the local 5d wave functions lie at some point between those for atomic 4f _n 5d 6s ² configurations and those calculated for such configurations subject to the restriction that the 4f shell is kept with its moment rigidly fixed in some given direction. The admixture of non-aligned 4f states as in the atom lowers the local energy, but it also lowers the 5d bandwidth due to misfit of the 4f states which occur with and without the presence of a 5d electron. This second effect raises the energy of the low lying states in the band. The best local states are determined by minimising the total electronic energy of the system, using approximations which are most suitable for 4f shells with large excitation energies. Bandwidths are found by fitting the observed saturation magnetic moments in Gd and Tm, and satisfyW?1 eV.     

Models for the heavy rare earth metals and (rare earth) Fe2 compounds involving 5d and 6s electrons

A model involving 5d electrons is introduced to explain the differences between the observed saturation moments in the heavy rare earth metals and those of the corresponding tripositive ions. Atomic 5d states, whose energies are determined by 4f–5d and spin-orbit interactions, are assumed to be broadened into partly overlapping bands with individual widths of the order of 1 eV. The 5d electrons produce negative contributions to the hyperfine fields but positive or near zero contributions to the magnetic moments. It is postulated that the 5d electrons are transferred from the rare earth ions to those of the iron in the (Rare Earth) Fe₂ compounds. This leads to increases in the magnetic hyperfine fields because the negative 5d contributions are lost, but in detailed application of the model increases in the 6s contributions also play a large part. Published energy level and wave function analyses for atomic Gd, Tb, Dy, Er and Tm are used in order to apply the theory to these materials.     

Defect annealing in the hexagonal close packed heavy rare earth metals Gd,Tb, Dy,Ho, Er,Tm and Lu after electron irradiation at low temperatures

The recovery spectra of the heavy rare earths Gd, Tb, Dy, Ho, Er and Tm were determined after irradiation with electrons at low temperatures. In the cases of Er and Tm, strong thermal cycling phenomena were observed and attributed to hydrogen atoms changing configurations and interacting with the magnetic structure of the metal. After elimination of these effects in Er and Tm, all the spectra—together with those of the earlier treated Lu—had rather similar form: several close-pair peaks followed by a broad complex substage resembling the stage I _D in f.c.c metals and attributed to long-range interstitial migration. A correlation is found between the temperature of the maximum of this substage (normalized to the melting temperature) and the c/a-ratio of the metals.     

The influence off-d interactions on effective crystal fields seen by 4f electrons in the heavy rare earth metals

A method is given for determination of the influence off-d interactions on the effective crystal field seen by 4f shells in a previously published model for 5d electrons in the heavy rare earth metals. With the ratios of effectivei-th order fields felt by 4f shells to those acting on 5d electrons denoted by multiplication factorsM _i , it is shown thatM ₂ lies between ?2 and ?6 for the five metals Tb, Dy, Ho, Er and Tm. The factorM ₄ is also negative for all the heavy rare earths except Tm. It is larger in magnitude thanM ₂ , and varies quite sharply from one metal to another.     

Reinterpretation of narrow bandwidths in a model for 5d electrons in the heavy rare-earth metals

The narrow widthsW(1 eV) of 5d subbands occurring in a model for 5d electrons in the heavy rare-earth metals are reinterpreted as effective bandwidths for the purpose of determining the equilibrium occupation numbers of 5d subbands. It is shown that, if explicit consideration of interactions between 5d electrons is limited to intraatomic Coulomb interactions without exchange, then, to first order in the intraatomic repulsion energyU between 5d electrons, these effective widths satisfyW=W ₁-FU. HereW ₁ is a rectangular-subband equivalent of the overall subband width, andF is a numerical factor, the value of which depends on the local basis states and number of 5d electrons per atom. For the model being discussed it is estimated thatF=0.6±0.4. A valueW ₁ ^B 4 eV is inferred forW ₁ in Gd from results of band-structure calculations. The termFU cannot account for all of the difference betweenW ₁ ^B andW; the discrepancy is attributed to effects of correlation onW ₁.     

Magnetic and structural studies of the alloy system REIn0.5Ag0.5

The structural and magnetic properties of the alloy system REIn_0.5Ag_0.5 [RE = Gd, Tb, Dy, Ho, Er, Tm and Yb] are reported. All these alloys (except that of Yb) crystallize in a cubic CsCl type structure at room temperature. Low temperature X-ray diffraction data does not reveal any structural phase transformation down to 8 K. On the basis of magnetic susceptibility data at a different temperature (3–300 K) and applied magnetic field (2 × 10⁵ to 8 × 10⁶ A m^-1, it has been concluded that GdIn_0.5Ag_0.5 is ferromagnetic (T_c = 118 K), TbIn_0.5Ag_0.5 and DyIn_0.5Ag_0.5 are meta magnetic (T_N = 66 and 30 K, respectively) and alloys involving Ho, Er, Tm and Yb are ferrimagnetic with Néel temperatures (T_N) equal to 24, 22, 21 and 20 K, respectively. The evaluated effective magneton number (p) is found to be slightly larger compared to theoretical values for tripositive ions of Gd, Tb and Dy and a bit smaller for Ho, Er, Tm and Yb. The results have been qualitatively explained using appropriate theories.     

Hyperfine interactions in intermetallic compounds between Gd and 3d transition metals

Measurements of hyperfine interactions at ¹⁵⁵Gd nuclei in metallic compounds between Gd and 3d transition metals and at ⁶¹Ni in GdNi compounds by Mössbauer spectroscopy are reported. The results are discussed in terms of various models proposed for the electronic structure of these compounds. The Gd isomer shifts with respect to metallic Gd are at variance with the model of a strong d electron transfer from rare earths to transition metal ions. From the observation of a linear relation between magnetic hyperfine fields at Gd and at Dy nuclei in corresponding compounds it is inferred that crystal-field induced variations of Dy moments are neglible and that the conduction electron polarization induced by 4f moments is directly related to that caused by 3d moments.     

Electronic structure and magnetic susceptibility of nearly magnetic metals (palladium and platinum)

A self-consistent approach to calculations of the electronic structure and the magnetic susceptibility of nearly magnetic metals, such as palladium and platinum, has been developed in terms of the generalized s(p)d Hubbard model. The energy band structure has been calculated using the ab initio LDA + U + SO method with the additional inclusion of the interstitial s(p)d exchange interaction and spin-fluctuation renormalizations of the electronic spectra, which appear at finite temperatures. The developed approach makes it possible to quantitatively describe the density of states and unusual temperature dependences of the magnetic susceptibility of the nearly magnetic metals under consideration and to evaluate the basic parameters of the electron-electron interactions. The role of the spin-orbit interaction in the formation of the electronic and magnetic properties is enhanced when going from palladium (4d period) to platinum (5d period). The effects of the temperature redistribution of electrons between the s(p) and d states have been revealed.     

Separation of 5d and 4f contributions to the electric field gradient in heavy rare-earths

We calculate the conduction electron contribution to the electric field gradient in heavy rare earth metals assuming 5d virtual bound state character present. The results are in better agreement with experiment than found previously and predict antishielding parameters for Er and Tm in good agreement with those calculated directly.     

Magnetic properties of ternary gallides of type RNi4Ga (R=rare earths)

The magnetic properties of RNi₄Ga (R=La, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm and Lu) compounds have been investigated. These compounds form in a hexagonal CaCu₅ type structure with a space group P6/mmm. Compounds with the magnetic rare earths, R= Nd, Sm, Gd, Tb, Dy, Ho, Er and Tm, undergo a ferromagnetic transition at 5, 17, 20, 19, 12, 3.5, 8 and 6.5 K, respectively. The transition temperatures are smaller compared to their respective parent compounds RNi₅. PrNi₄Ga is paramagnetic down to 2 K. LaNi₄Ga and LuNi₄Ga are Pauli paramagnets. All the compounds show thermomagnetic irreversibility in the magnetically ordered state except GdNi₄Ga.     

Investigation of the magnetic hyperfine field at197Au impurities in the heavy Rare Earth metals

The Mössbauer effect has been used to measure the magnetic hyperfine field at the site of¹⁹⁷Au impurities in the heavy Rare Earth metals Gd, Tb, Dy, Ho and Er at 4.2 K. The magnetic hyperfine field decreases in a non linear way with decreasing spin of the Rare Earth host. For¹⁹⁷Au this decrease is stronger than for any other impurity investigated up to now. Possible reasons for this behaviour are discussed.     

Hydrostatic pressure derivatives of the single crystal elastic moduli of Gd,Dy and Er

The hydrostatic pressure derivatives of the single crystal elastic moduli of Gd, Dy and Er have been measured at 298°K, to pressures near 5 Kbar. The very small pressure derivatives of the adiabatic bulk moduli indicate that a small ion core model should be appropriate for interpreting the data. The long-range electrostatic contributions to the shear moduli have a dominant influence on the pressure derivatives of the shear moduli of Er, whereas the Gd and Dy data evidently reflect band structure contributions. The values of the longitudinal stiffnesses correspond remarkably well with the Bohm-Staver model for velocity of waves in an ion plasma dispersed in a sea of electrons, where the ionic interaction is perely Coulombic. This model is extended to provide an interpretation of the volume derivatives of the longitudinal moduli in terms of the volume derivative of the density of electron states at the Fermi energy.The Grüneisen parameters calculated from averages of the acoustic model gammas are in relatively poor agreement with those determined from thermal expansion data. An explanation based on the changes in c/a ratio with volume change is tested quantitatively and found to be reasonably successful. The values of dK_T/dP, where K_T is the isothermal bulk modulus, are applied to the Murnaghan equation of state and give excelent agreement with Bridgman's direct compression data for Dy and Er to 40 Kbar. For Gd, Bridgman's data indicate either that (dK_T/dP)_p=0 should be considerably larger than deduced from the adiabatic dK_s/dP measurements or that a phase change occurs near 20 Kbar. The occurrence of a phase change in Er at ～90 Kbar is definitely indicated when comparing the Murnaghan equation with X-ray diffraction data.     

Coexistence of superconductivity and magnetic ordering in the borocarbide superconducting system

The investigation of the intermetallic compounds Y_1−xRE_xNi₂B₂C and Y_1−xRE_xPd₅B₃C_0.4 (RE=Gd, Dy, Ho, Er; 0≤x≤1.0) demonstrates that the coexistence of the long-range-magnetic-ordering and the superconductivity is possible The normal-state magnetic susceptibility shows a Curie-Weiss-like temperature dependence and a small paramagnetic Curie temperature. The value of the effective moment is close to that of the free RE⁺³ ion. Both results suggest that the coupling between the conduction electron and magnetic ion is not strong, but the coupling is still strong enough to cause the depression of superconductivity. The Abrikosov-Gor‘kov magnetic pair-breaking theory can be applied to those systems. The corresponding magnetic-ordering temperature increases with increasing magnetic moment concentration while the superconducting transition temperature decreases. The long-range-magnetic-ordering is mainly caused by the Ruderman-Kittel-Kasuya-Yosida indirect exchange interaction via conduction electrons. This indirect exchange coupling strongly depends on the concentration of RE⁺³-moment and de Gennes factor.     

The Magnetic Hyperfine Field of 111Cd in the Rare Earth–Nickel Laves Phases RNi2

The magnetic hyperfine field of ¹¹¹Cd in the C15 Laves phases RNi₂ has been investigated by perturbed angular correlation (PAC) spectroscopy as a function of temperature for the rare earth constituents R = Nd, Sm, Gd, Tb, Dy, Ho, Er, and Tm.     

Magnetic studies of rare-earth tungstates

Measurement of magnetic susceptibility (χ_m) of rare-earth tungstates RE₂(WO₄)₃ with RE = Gd, Tb, Dy, Ho, Er, Tm and Yb have been reported in the temperature range 3–300 K at a magnetic field of 3.2 × 10⁵ Am^-1. All the tungstates are found to be ferrimagnetic with a ferrimagnetic Curie-temperature lying in the range 15–27 K. Evaluated magneton numbers agree fairly well with free tripositive rare-earth ions. The results are discussed on the basis of a new proposed model. Except for Tb all the evaluated parameters have systematic variation over the entire rare-earth series.     

Investigation of metamagnetic transitions in the itinerant d subsystem of the intermetallics RCo2 in superstrong magnetic fields up to 300 T

The time derivatives of the magnetization of intermetallic compounds RCo₂ (R = Y, Tm, Er, Ho, or Dy) with a metamagnetic subsystem of itinerant d electrons have been measured in pulsed magnetic fields up to 300 T generated by an explosive method. Peaks associated with abrupt demagnetization or magnetization of the d subsystem are found in the field dependences of dM/dH. The results obtained are compared with theoretical estimates and the values derived for the critical fields from measurements performed on substituted compounds. Pis’ma Zh. éksp. Teor. Fiz. 64, No. 3, 188–192 (10 August 1996)     

Intrinsic and extrinsic magnetic susceptibilities of some TCNQ complexes

The magnetic susceptibilities (χ) of quinolinium·(TCNQ)₂, N-methyl phenazinium·TCNQ and Li·TCNQ were measured from 2 to 300 K and are discussed in connection with the low-temperature specific heats (C) measured by other authors, χ is decomposed into three parts: χ_d the temperature-independent part, χ_c, Curie-Weiss type paramagnetism, and χ_p, the remainder. Correspondingly, C is composed of three terms, γT, H/T² and αT³. The electronic state of these substances is discussed in terms of each type of susceptibility.The model, on which the above separation of χ and C is based, defines two types of electrons: localized electrons associated with a magnetic moment and band electrons. Though this model is useful phenomenologically, it is shown that the analysis of χ on the basis of this model indicates less band electrons and more localized electrons or stronger magnetic interactions than does that of C.