Brown's Equation Research Articles

NQR Studies of the chloro(amino)cyclotriphosphazenes, N 3P 3(NHCH 2·CH 2NH)Cl 4, N 3P 3(NHBu t) 4Cl 4 and N 3P 3(NHBu t) 4Cl 2

The frequencies and variable-temperature behaviour of 35Cl nuclear quadrupole resonance in three aminocyclophosphazene derivatives are reported. The observed frequencies and multiplicity are correlated with the disposition of the substituents and the crystal structure. The temperature-dependence data are discussed in the framework of Bayer-Kushida-Brown equations and low-lying torsional (librational) frequencies and their average temperature coefficients are estimated. Brown's parabolic equation provides a good fit to the experimental data. Variable-temperature proton FT-NMR measurements (at 270 MHz) have also been carried out. The results are consistent with the NQR data and indicate the presence of two-site chemical exchange of the -NH protons and hydrogen bonding.

Micromagnetism in Amorphous Alloys

Brown's micromagnetic equations are derived for the case of dipolar and magnetocrystalline fluctuations in amorphous ferromagnetic alloys. The influence of these fluctuations and the effects of internal stresses of so-called quasi-dislocation dipoles on the law of approach to ferromagnetic saturation is discussed in detail. It is shown that intrinsic fluctuations of magnetic properties lead to a 1/\sqrt{H} -term in the high-field range whereas internal stresses give rise to 1/H - and 1/H^{2} -power laws. A quantitative analysis of experimental results indicates that internal stresses dominate the effects of intrinsic fluctuations considerably.

PROCESSES OF DEBRIS TRANSPORT IN MOUNTAIN WATERSHEDS AND DEVELOPMENT OF MOUNTAIN SLOPES AND VALLEYS

A general form of the formulae for tractional load can be given in th formQB=cτα, where QB is the tractional load, the tractive force, a an exponent and c a constant. Most of the formulae for tractional load proposed thus far were obtained from the results of ex-periments carried in flumes with gentle gradients. The value of α which can describe the mechanism of debris transport in steep mountains was tried to obatin by comparison of measured amounts of erosion of actual mountains with calculated ones from Eq. (2). Eq. (2) which describes the changes of longitudinal profiles of average slope and valley bed was derived by using the above general form. High cor-relations are obtained when the value of α are 3 and 3.3 as shown in Table. 1. The value of α in Brown's equation is 2.5, and that in Sato, Kikukawa and Ashida's equation 1.5. As a result, it is confirmed that the basic processes of debris transport in the form of mass movement and by the action of running water in mountain watersheds with great gradients, can be approximated by the above general form. By using Eq. (4) of which α is 3.3 (m=2), topographic factors which reflect the rate of erosion of a mountain were studied, and the development of fault scarps and valley slopes were considered. Consequently, it is shown that the rate of erosion can be represented by a power function of mean slope of a mountain (Fig. 3). It is also shown that the average slopes of fault scarps which have been shaped by the interaction of uplift and erosion are straight, and the larger the rate of uplift is, the steeper the slope becomes (Fig. 5). From the laws of channel networks, it is derived that the longitudinal profile of the main valley of a basin which is in a steady state can be represented by a logarithmic curve. Profiles obtained from Eq. (4) and the longitudinal profiles of valleys which dissect fault scarps and volcanoes, can be represented by straight lines on semi-logarithmic paper when the locations of the divides are transferred horizontally.

Theoretical Evaluation of Limiting Storage Density by Thermomagnetic Recording

It is generally accepted that thermomagnetic writing allows in principle a far better density in digital storage than does conventional recording. The aim of this paper is to point out the physical limitations to storage density in thermomagnetic mass memories; some numerical evaluation will be performed in a highly idealized model, namely for a self-supporting homogeneous film with high coercivity, fully saturated in one easy direction, imbedded in a field opposite to the magnetization and heated in a spot by means of a high-energy well-focused beam. Nucleation of a reversed spot during cooling through Curie temperature is the recording phenomenon. For the density limitation the following reasons have been considered: (1) exchange interaction at the boundary between heated and cold region during cooling; (2) domain walls after cooling; (3) thermal conductivity. The approximate evaluation of the effect of exchange near Curie temperature has been possible by means of a new formulation of Brown's equation valid at high temperature. Considerations on thermal conductivity yield the result that the possible density is reduced if the recording speed increases.

Micromagnetics at High Temperature

The method of micromagnetics is extended to the cases in which the magnitude of the magnetization cannot be considered constant throughout a ferromagnetic body. The nonlinear equations, corresponding to Brown's equations in the standard treatment of micromagnetics, have been deduced with the proper boundary conditions, as well as their linearized form suitable for evaluation of the nucleation field and mode. Some relationships have been found between the nucleation field deduced in this way and the nucleation field for constant magnitude, emphasizing that a detailed knowledge of the magnetic equation of state of the material is needed for the determination of the true nucleation field. The case of a boundless plate with an applied field normal to its boundary planes has been considered in detail, and the existence of one nucleation mode, termed as "waving," peculiar to this treatment, has been recognized; the conditions for this mode of reversal have been deduced, with the result that it can be expected near the Curie temperature.

Approximate Solution for the Strain-Induced Deviation of the Magnetization from Saturation in Polycrystalline Cubic Ferrites

The strain-induced uniaxial magnetic anisotropy can cause a deviation in the magnetization from saturation along the applied magnetic field. An approximate variational solution is found for the necessary threshold strain or stress and for the amplitude of this deviation in thick, semi-infinite slabs of cubic polycrystalline ferro- or ferrimagnetic materials. The resulting solutions are compared to solutions obtained from the linearized Brown's equations for equilibrium. Specific calculations are made for nickel ferrite, YIG, and magnetite.

Micromagnetic Solutions for Ferromagnetic Spheres

Computer solutions are described for micromagnetic equilibrium conditions (Brown's equations). Computations relate to spherical particles and are made possible by assuming a constraint on the magnetization. Magnetic moment vectors are constrained to rotate horizontally with their angle of rotation varying in a vertical direction. This produces micromagnetic solutions of domains separated by Bloch walls. The study demonstrates that domains exist as equilibrium solutions and that the magnetization in each domain is not constant. Although these solutions do not produce uniformly magnetized regions, domains are identifiable because of their separation by sharp magnetization reversals. These transitions can be identified unambiguously as Block walls. Solutions exist for particles of different size, and the energies for double and triple domain states are calculated. The results of the computer study show the existence of critical radii. All computer solutions are checked for stability with a special stability matrix computed for this purpose.

Stray‐Field‐Free Magnetization Configurations

AbstractThe concept of stray‐field‐free micromagnetic configurations in applied to Bloch‐ and Néel‐walls, to stripe domains, and to the ripple structure in thin magnetic films. We propose a new structure of Bloch‐walls with a lower energy than the most general classical Bloch‐walls as calculated by LaBonte. In general a stray‐field‐free configuration has a lower symmetry than the simpler structures considered previously. It is concluded that the real structure, if not completely stray‐field‐free, should have at least the same symmetry as a possible stray‐field‐free solution. The significance of Brown's equations, which in general cannot be satisfied without admitting stray‐fields, is analysed with regard to the micromagnetic boundary condition.

Transformation of the Linear Equations of Magnetoelastic Interactions for an Infinitely Long Prism

Brown's linear equations of magnetostriction in a homogeneous cubic crystal and for an infinitely long prism along the applied external magnetic field are transformed into a new system of equations. In the transformed system, the two components of the magnetostatic scalar potential, the elasticity potential, and the three components of the elastic displacement vector, satisfy separate differential equations. The general solution is a linear combination of a finite number of functions, each one being a solution of the Helmholtz equation in two independent variables.

The Contribution of Dislocations to the Magnetocrystalline Energy and Their Effect on Rotational Hysteresis Processes

Internal stresses of dislocations interact in ferromagnetic crystals with the spontaneous magnetization through the magnetoelastic coupling energy. The magnetic free energy (MFE) of a straight dislocation line is calculated by means of Brown's micromagnetic equations. This energy corresponds to a magnetostrictive contribution to the magnetocrystalline energy which is superimposed on that of the ideal crystal. In a material with isotropic magnetostriction the MFE of a straight edge dislocation line possesses orthorhombic symmetry. The theoretical results may be compared with those of an investigation of magnetization processes of plastically deformed Ni single crystals by Köster. Analysis of measurements of the magnetostriction and of the increase of magnetization as a function of the applied field has shown that in deformed Ni crystals the MFE of the dislocations may become comparable to the magnetocrystalline energy constant K1 of the ideal crystal.

Nucleation Field of an Infinitely Long Elliptical Cylinder

The system of Brown's linear equations is rigorously solved for an infinitely long elliptical ferromagnetic cylinder magnetized to saturation. The problem is treated when the applied magnetic field and the easy direction of magnetization are assumed to be along the axis.

Transformation of Brown's Linear Equations for an Infinitely Long Prism

In the original system of Brown's linear equations the components of the magnetic vector normal to the applied magnetic field and the magnetostatic scalar potential are mixed. The present work shows how, for an infinitely long prism along the applied magnetic field, Brown's linear equations can be transformed into a new system of equations, where the components of the magnetic vector normal to the applied magnetic field and the magnetostatic scalar potential satisfy separate differential equations. The transformed system of equations can ease the calculation of the nucleation field for long ferromagnetic particles with different cross sections.

Possibility of Domain Wall Nucleation by Thermal Agitation

Very fine particles reverse their magnetization spontaneously, passing the energy barrier by means of thermal activation. In the same way one could argue that small portions of larger particles could be spontaneously reversed, thereby serving as nucleii for reversed domains. These can then grow, because it is known that the energy favors subdivision into domains. In treating such a possibility, however, one should take into account the additional barrier for forming a wall around the ``nucleus,'' which is usually left out in standard domain theory calculations when the wall energy is used. This energy is rather small but the barrier for formation of the wall is large. The model studied is a small sphere, centrally located inside a spherical particle, the magnetization in which starts to rotate gradually from the z direction in which the rest of the particle is magnetized. The angle between magnetization and the z axis is assumed to vary linearily in the radial coordinate, from zero on the surface of the inner sphere, to its maximum value in the center. This value eventually reaches π, thus completing a wall around a reversed domain of radius zero, which should then start to grow by further supply of energy. Studying the energy barrier for such a process, it is seen that the main contribution is from the exchange energy, which practically forbids the process. The probability for a domain wall to nucleate at room temperature is thus shown to be negligibly small, which justifies the neglection of temperature affects in Brown's equations.

Magnetization Curve of the Infinite Cylinder

The nucleation field for an infinite circular ferromagnetic cylinder, with the field applied along its axis, is rigorously calculated by exploring the whole eigenvalue spectrum of Brown's equations.