Higher-order Anisotropy Research Articles

Superstrings in higher order extensions of Finsler superspaces

The work proposes a general background of the theory of field interactions and strings in spaces with higher order anisotropy. Our approach proceeds by developing the concept of higher order anisotropic superspace which unifies the logical and mathematical aspects of modern Kaluza-Klein theories and generalized Lagrange and Finsler geometry and leads to modelling of physical processes on higher order fiber bundles provided with non-linear and distinguished connections and metric structures. The view adopted here is that a general field theory should incorporate all possible anisotropic and stochastic manifestations of classical and quantum interactions and, in consequence, a corresponding modification of basic principles and mathematical methods in formulation of physical theories. The presentation is divided into two parts. The first five sections cover the higher order anisotropic superspaces. We focus on the geometry distinguished by non-linear connection vector superbundles, consider different supersymmetric extensions of Finsler and Lagrange spaces and analyze the structure of basic geometric objects on such superspaces. The remaining five sections are devoted to the theory of higher order anisotropic superstrings. In the framework of supersymmetric non-linear sigma models in Finsler extended backgrounds we prove that the low-energy dynamics of such strings contains equations of motion for locally anisotropic field interactions. Our work is to be compared with important previous variants of extensions of Finsler geometry and gravity. There are substantial differences, because we rely on modeling of higher order anisotropic interactions on superbundle spaces and do not propose some “exotic” Finsler models but a general approach which for trivial or corresponding parametrization of non-linear connection structures reduces to Kaluza-Klein and other variants of compactified higher-dimension spacetimes. The geometry of non-linear connections (not being confused with connections for non-linear realizations of gauge supergroups) is first considered for superspaces and possible consequences on non-linear connection fields for compatible propagations of strings in anisotropic backgrounds are analyzed. Finally, we note that the developed computation methods are general (in some aspects very similar to those for Einstein-Cartan-Weyl spaces which is a priority comparing with other cumbersome calculations in Finsler geometry) and admit extensions to various Clifford and spinor bundles.

Higher-order magnetic anisotropies and the nature of the spin-reorientation transition in face-centered-tetragonal Ni(001)/Cu(001)

Second- and fourth-order constants ${\mathrm{K}}_{2}$ , ${\mathrm{K}}_{4}$ of the magnetic anisotropy energy in tetragonally distorted Ni(001) ultrathin films on Cu(001) are determined as a function of temperature and film thickness. We measure a positive out-of-plane constant ${\mathrm{K}}_{4\mathrm{\ensuremath{\perp}}}$ \ensuremath{\approx}+0.24 \ensuremath{\mu}eV/atom and a negative in-plane constant ${\mathrm{K}}_{4\mathrm{\ensuremath{\parallel}}}$ near room temperature. This leads to a continuous (second or higher-order) spin-reorientation transition of the easy axis from an in-plane below \ensuremath{\approx}7 ML to a perpendicular orientation for thicker films. ${\mathrm{K}}_{4\mathrm{\ensuremath{\parallel}}}$ changes sign at ${\mathrm{d}}_{\mathrm{c}\mathrm{\ensuremath{\parallel}}}$ \ensuremath{\approx}6.7 ML which results in a change of the easy axis from the [100] direction below 6.7 ML to the [110] direction above. These results are quantitatively compared to Ni(111)/W(110) for which no out-of-plane reorientation is observed.

Analyses of sidereal and solar anisotropies in cosmic rays

The Sun's interplanetary magnetic field and the solar wind modulate the distribution of galactic cosmic-ray particles in the heliosphere. The particles diffuse inward, convert outward and have drifts in the motion of their gyro-centres. Irregularities in the IMF also scatter particles from their gyro-orbits. These processes are the components of solar modulation and produce streaming (and higher-order anisotropies) of particles in the heliosphere. The anisotropies can be investigated at the Earth by examining the count rates of cosmic-ray detectors. The anisotropic streams appear as diurnal variations in solar and sidereal time in the count rates. Higher-order anisotropies produce generally much smaller semi-diurnal and higher-order variations. Theoretical models of solar modulation predict effects that depend on the polarity of the Sun's magnetic dipole. The solar diurnal and north-south anisotropies can be used to test these predictions. This paper is a short review of analyses of 60 years of cosmic-ray data collected at the Earth for the solar and sidereal diurnal variations present. Past analyses have yielded interesting and controversial results regarding the rigidity spectra and components of these anisotropies. Some of the controversy remains today. Analyses of these anisotropies have also yielded quantitative information about parameters important to solar modulation, such as latitudinal and radial density gradients. The relatively new techniques used for these determinations are explained here. Calculations of these modulation parameters from Earth-based cosmic-ray detectors are reviewed and compared to spaceprobe measurements and theoretical predictions of their values. Recently, investigations of the sidereal and solar diurnal anisotropies have been combined to calculate mean-free-paths of cosmic rays in the heliosphere. The latest conclusions from these analyses are that the parallel mean-free-paths of cosmic rays may depend on the polarity of the Sun's magnetic field. The results of these investigations are included in this paper to indicate the present state of knowledge concerning this facet of cosmic-ray research.

Theory for the magnetic phase diagram of thin films: Role of domain formation.

The magnetic phase diagram of a thin film is determined at T=0. The exchange coupling, the magnetic dipole coupling, as well as the second- and fourth-order lattice anisotropies are taken into account. The long-range dipole coupling causes an instability of the uniform magnetization with perpendicular orientation and leads to a domain phase. Four different phases, in particular stripe domain structures with perpendicular and canted orientations of the domain magnetization, are found to be stable. The parameter region near the magnetic reorientation is studied in greater detail. It is shown that the higher-order lattice anisotropies may induce a continuous reorientation from a perpendicular to an in-plane magnetization, also for a domain phase. By carrying out lattice sums we conclude that a periodic array of straight domain walls should be stable against spatial dislocations. Our results suggest that the domain phases may alter the nature of the phase transitions at finite temperatures in thin films. Rather than phase transitions with critical phenomena a simple rotation of the magnetization might occur.

Spin configurations in the absence of a magnetic field in a two-sublattice system

The spin configurations in the absence of an external magnetic field have been systematically investigated for a two-sublattice system. Based on a two-sublattice model, the conditions for the existence of collinear and non-collinear spin structures were derived for either ferromagnetic or ferrimagnetic materials and the phase diagrams of the spin configurations at zero field were drawn, taking into account the second-order anisotropy parameters of both sublattices. The inclusion of higher-order anisotropy is discussed and it is shown that in some cases the same phase diagrams result.

Singularities in the magnetization processes of high anisotropy materials

Singularities of various kind are often observed In the magnetization curve of a ferromagnetic crystal. This is especially true for high anisotropy materials, such as the rare earth-intermetallic compounds. Examples are the First Order Magnetization Processes (FOMP), which are discontinuous rotations of the magnetization vector due to high order anisotropy terms. However, the hard direction anisotropy field of any ferromagnetic crystal is always a critical field at which it reaches saturation. The Singular Point Detection (SPD) technique allows detecting such singularities using polycrystalline samples by the observation of the successive derivatives d/sup 11//dH/sup 11/ The shape of the singularity and the order of differentiation n at which it becomes apparent depends on the symmetry of the hard axis. The SPD theory has been recently extended to multidomain crystallites on the basis of the Neel phase theory, and utilized for texture studies of permanent magnets. Moreover it has been proved to be valid also in the presence of canting between the sublattice moments caused by strong competition between anisotropy and exchange; in principle, in the case of ferrimagnetic compounds it is possible to detect spin flop transitions both of second and first order (high POMP). The application of the SPD approach to the general problem of the complex susceptibility tenser and its covariant derivatives can lead to further developments. As an example, the singularity observed in the transverse susceptibility of uniaxial materials is strongly sensitive to the presence of single domain crystallites. >

CONTINUOUS REORIENTATION TRANSITION OF THE MAGNETIZATION IN THIN FERROMAGNETIC FILMS

The change of the direction of the magnetization of very thin magnetic films from perpendicular at low temperatures to in- plane orientation at high temperatures is investigated. This transition is driven by the competition between uniaxial lattice anisotropy, favoring a perpendicular orientation, and the magnetic dipole interaction (shape anisotropy). The strong spin correlations in this two dimensional (2D) system have to be taken into account through a renormalization procedure. A continuous rotation of the magnetization and a vanishing long range ferromagnetic order around the reorientational temperature TR were observed experimentally. We propose that this may indicate a decomposition of the ordered 2D system into spin blocks. If the directional entropy of spin blocks near TR and higher order anisotropy terms are considered, a continuous perpendicular to inplane transition may result.

The two-dimensional Heisenberg ferromagnet with various types of interactions: temperature dependence of magnetic parameters

The temperature dependences of the anisotropy constant, spin-wave stiffness and spontaneous magnetizations for the two-dimensional (2D) Heisenberg model with various types of anisotropy and interactions have been calculated with the use of the low-temperature perturbation (spin-wave) theory. The authors argue that this theory is effective within a large portion (if not the main part) of the ferromagnetic phase. On the contrary, the Polyakov renormalization procedure is found to be inapplicable for the ferromagnetic phase at any anisotropy. They consider both the easy-axis and the easy-plane cases, taking into account one-site and exchange anisotropy as well as the higher-order magnetic anisotropy and the dipole-dipole interaction. For realistic models the latter is found not to change essentially the temperature dependence of the magnetic parameters. In particular, they did not find the reorientation (easy-axis to easy-plane) phase transition due to dipole-dipole interaction predicted by Pescia and Pokrovsky (1990). However, in general, the temperature dependence of the anisotropy constant proves to be much more sensitive to the type of the anisotropy and interactions than that of the spin stiffness and the spontaneous magnetization. In the easy-plane case the difference between the spin-wave stiffness for the in-plane and the normal polarizations is found to be extremely strongly temperature-dependent. At temperatures comparable with TC this difference may be of the order of magnitude of the stiffness itself, while at T=0 it is of relativistic origin. For the case of a very small anisotropy the authors also propose a renormalization procedure to perform a partial summation of the perturbation series.

Two-dimensional transport with highly anisotropic scattering

A finite element methodology is employed to discretize the direction coordinate for transport problems in integro-differential form. It is shown that the method proposed can satisfactorily handle particle streaming phenomena and, especially, high order anisotropy of emissions from collisions. As a consequence, the method is applicable to many problems of neutron and photon transport (shielding and blanket problems, multiplying systems). The method can be formulated in a way suitable to be handled by current discrete ordinate transport numerical algorithms.

Fourth-order elastic constants of nonideal hcp crystal Mg and Er

The expressions for the 19 fourth-order elastic constants of an hcp crystal with nonideal axial ratio have been derived with interactions extending up to second neighbors and using sublattice displacements to the second degree in strains. These formulas have been used to evaluate the fourth-order elastic constants of magnesium and erbium. The results reveal higher-order elastic anisotropy in erbium. The second pressure derivatives of the effective second-order elastic constants of magnesium and erbium have been obtained using their fourth-order elastic constants and are positive. The magnitudes of the second pressure derivatives of C112, C133, and C111 in magnesium and that of C112 in erbium are extremely large, lending support to the phase changes that occur in these metals when subjected to hydrostatic pressure.

Method of a singular point and determination of fields of anisotropy of polycrystalline hexagonal magnets

A numerical computation is performed on the magnetization curves and their derivatives for magnets of hexagonal syngony for isotropic polycrystalline specimens as well as materials with “sheet” texture in the basis plane. It is shown that investigation of the second derivatives ∂2M/∂H2 for a degree of texture fT > 0.4 permits obtaining information about the magnitude of the anisotropy field independently of its type. The singularities of M(H) and ∂2M/∂H2 are examined for magnetization processes of the first kind. In this case the singular point method permits measuring the magnitude of the critical fields, the critical magnetizations, and the anisotropy field on a textured specimen with fT>0.7, which yields information about the magnitudes of the high order anisotropy constants in the long run.

Interfacial anisotropy in magnetic superlattices.

We have observed large discrepancies between the dc magnetization and ferromagnetic resonance (FMR) magnetic anisotropy in Mo-Ni superlattices. We show that higher-order anisotropy is present in both measurements and develop an analysis by which the first-order and second-order energies can be treated independently. The low-temperature results show a systematic divergence of the first-order FMR and dc anisotropy as the Ni-layer thickness decreases. It is shown that the source of the divergence is an enhanced surface anisotropy measured by FMR. Possible sources of this enhancement are discussed.

Effect of heat treatment on the anisotropy of the compound Co2Z

The behavior of the anisotropy field of the hexaferrite Co2Z is studied as a function of the heat treatment regime. It is shown that the change in the magnitude of the anisotropy field and its temperature dependence observed during quenching are determined by the redistribution of Co2+ ions over octahedral positions with different orientations of the spin magnetic moments. The increase in the temperature of the spin-reorientation transition “cone of easy magnetization-plane of easy magnetization” indicates the increasing importance of the higher order anisotropy constants in the heat treatment.

Magnetization-vector measurements on a Ni-Mn spin-glass alloy

Measurements were made simultaneously of both the longitudinal and transverse magnetization components (ML,MT) of a field-cooled thin-disk sample of disordered Ni75Mn25 as it was rotated quasistatically in various fixed fields (Ha) up to 4 kOe. For Ha<1 kOe, the observed behavior at sample-rotation angles (θ) up to 90° is consistent with a purely unidirectional anisotropy field (HK=0.6 kOe), but at higher θ both ML and MT are anomalously low, signifying that the size of the rotating magnetization vector (MS) becomes substantially smaller than the thermoremanence. The normal demagnetizing fields are too weak to account for this effect. At higher Ha, MS rises slightly above the thermoremanence near θ=90°. This derives from the fact that while ML(θ) is normally behaved, the peak in MT(θ) is anomalously high, indicating that the effective unidirectional anisotropy grows with increasing Ha. Moreover, the shape of MT(θ) shows the additional presence of higher-order (e.g., uniaxial) anisotropy components.

Competing anisotropies and first-order magnetization processes in (Zn,Co)W-type hexaferrite

First-order magnetization processes (FOMP) and spin reorientation phase transitions are known to occur in hexaferrites as well as in rare-earth transition-metal compounds. This complex phenomenology has been described using high-order terms in the anisotropy energy expansion. In order to understand the microscopic origin of these high-order anisotropy constants in hexaferrites we have made a detailed analysis of the magnetic properties of BaZn1.1Co0.9Fe16O27 which shows, at the same temperature, easy cone anisotropy and a FOMP. We have found good agreement between the experimental data and the predictions of a simple two sublattice model which takes into account the effect of the competition between the exchange and the difference in the anisotropy of the sublattices. This strongly supports the hypothesis that such a competition and the subsequent noncollinear order are the origin of the observed large high-order anisotropy constants in hexaferrites as in intermetallic compounds.

IMF-sense-dependent cosmic ray anisotropy produced from diffusion-convection in heliosphere

An IMF-sense-dependent first order (or unidirectional) anisotropy of cosmic rays, which is produced perpendicularly to the ecliptic from the radial density gradient in solar system, has been confirmed by Swinson. In the present paper, we point out the existence of IMF-sense-dependent higher order anisotropies, based on the simulation of cosmic ray diffusion-convection in the heliomagnetosphere. In order to confirm their existence, we demonstrate some examples of the observed cosmic ray daily variation which is supposed to be due to these anisotropies.

Characterization of membrane lipids of a general fatty acid auxotrophic bacterium by electron spin resonance spectroscopy and differential scanning calorimetry.

Lipids in the plasma membrane of the general fatty acid auxotroph Butyrivibrio S2 pack as a bilayer that is characterized by a high order and high motional anisotropy and a low membrane fluidity compared to mammalian plasma membranes. Lipid packing as determined by the electron spin resonance (ESR) order parameter and membrane fluidity as measured by ESR correlation times are, however, comparable to those of other bacterial membranes. Membranes of the organism grown with saturated fatty acids of well-defined hydrocarbon chain length undergo a broad reversible endothermic phase transition, the peak temperature of which is well below the growth temperature; the end-point temperature of this thermal transition approximately coincides with the minimum temperature supporting significant growth of the organism. The lipid phase transition is also reflected in the temperature dependence of various ESR parameters, whereby the transition temperature thus derived is higher than the peak temperature of the endothermic transition but still lower than the growth temperature. ESR and calorimetry evidence taken together suggest that the endothermic transition is a gel to liquid-crystal transition and that, at the growth temperature, the plasma membrane of Butyrivibrio S2 is in the liquid-crystalline state. Similar values were measured for the order parameter of cell membranes of Butyrivibrio S2 regardless of whether the organism was grown on myristic, palmitic, or stearic acid. Butyrivibrio S2 has a mechanism enabling it to maintain membrane packing and fluidity at a fairly constant level.(ABSTRACT TRUNCATED AT 250 WORDS)

First-order transitions in anisotropic magnets with random fields and random uniaxial anisotropies.

Long-range magnetic order is known to be replaced by a spin-glass phase for rotationally invariant N-component spin systems with random fields or random uniaxial anisotropies. Using exact calculations in the limit N\ensuremath{\rightarrow}\ensuremath{\infty}, we show how long-range order is restored when a uniform anisotropy is added. For a uniaxial anisotropy g, long-range order is achieved above a critical value of g, at which a phase with an infinite susceptibility occurs. For higher-order (e.g., cubic or hexagonal) anisotropies long-range order is reached via first-order transitions. Scaling arguments and explicit calculations are used to obtain detailed predictions on the shape of the various phase boundaries.

Competing anisotropy in rare earth intermetallics

The manner in which crystalline electric fields compete in cubic ferromagnets is discussed. In the first section it is shown how competition between fourth and sixth-degree terms in the crystalline field energy lead to re-entrant behavior of the magnetization process in an applied magnetic field. For one special value of the ratio of sixth-to fourth-degree coupling strengths a new type of critical point is found to occur in a field applied in a [111] direction. The influence of temperature fluctuations is briefly discussed. The second section is devoted to a discussion of the effects of competition in alloys between ions with opposite signs of crystalline coupling. Using rare-earth Laves phase pseudobinary alloys as an example, the fluctuations in the sign of the crystalline field are shown to lead to unusual magnetic phases in concentrated alloys. The effects of mixing different angular momentum states is shown to lead, in special cases, to high-order anisotropy (eight and greater) and possibly, unusual magnetic phases in pure binary compounds. The influence of applied fields on canted-spin phases is also discussed and field-concentration phase diaplayed in a model case. In the last section, the important influence of randomness of the crystalline field is noted.

Effect of higher-order anisotropy constants on the ferromagnetic resonance in hexaferrites

In this paper the characteristics of the ferromagnetic resonance in hexagonal ferrites are investigated theoretically. It is shown that when the absolute magnitude of the anisotropy constants of second and higher order is comparable to or greater than that of the first-order constant, the angular dependences of the resonant field exhibit stationary directions which do not coincide with the symmetry axes of the crystal. As a result, additional maxima appear on the FMR curves of polycrystalline hexaferrites, which makes it more difficult to determine the anisotropy fields from experiments on polycrystals.