A general method for calculating the (nonuniform) demagnetizing field in ferromagnetic bodies of arbitrary shape is described. The theory is based upon the assumption that the magnitude of the magnetization vector is constant throughout the sample and that its direction coincides with the direction of the local magnetic field at any point within the sample. The total magnetic field is expressed as a series of ascending powers in M/H0, where M is the saturation magnetization and H0 the applied magnetic field. The first term of this series expansion (first-order theory) gives the demagnetizing field for very large applied fields, i.e., for a uniformly magnetized sample. The higher-order corrections (we consider in detail only the first correction term; second-order theory) take account of the fact that the sample is not in general uniformly magnetized. The general theory has been applied to rectangular slabs and circular cylinders. The first-order demagnetizing field has been calculated for rectangular slabs and circular cylinders of arbitrary dimensions. Our discussion of the second-order theory is restricted to the semi-infinite slab and the semi-infinite circular cylinder. For the semi-infinite slab the variation of the second-order demagnetizing field along the central symmetry axis and across the endface have been calculated. In cases of practical interest (spin-wave propagation experiments in YIG at 3 Gc/sec and applied magnetic field of about 1400 Oe), the second-order correction to the demagnetizing factor is approximately 20% of the first-order contribution.

The influence of inhomogeneities on the saturation of the ferromagnetic resonance is investigated. In the region of moderate power levels, the susceptibility at resonance χ′′ varies linearly with the square of the rf field h. The magnitude of the slope ∂χ′′/∂h2 depends on the nature of the dominant scattering mechanism. If the uniform mode scatters primarily to spin waves of very large wavelength, the slope should be negative. Scattering to spin waves of short wavelength gives a positive contribution to the slope and can lead to a reversal of the sign. The theoretical predictions agree with measurements at X band on various polycrystalline garnets and ferrites. At very high power levels the opening angle of the precessing magnetization vector approaches a limiting value, which is related to the “line width” ΔHk of z directed spin waves having the same frequency as the uniform mode. Experiments on single crystals and polycrystals of rare earth substituted garnets show that ΔHkincreases approximately linearly with the rare earth content. The materials investigated contain Gd, Yb, Er, Sm, Dy, Ho, or Tb and for a given ratio of substitution ΔHk increases in that order. The line width ΔHk of z directed spin waves is found to be approximately proportional to the line width ΔH of the uniform mode as measured in single crystals. Experimental results on cobalt and zinc substituted nickel ferrite are reported. ΔHk increases linearly with the cobalt content. For the nickel-zinc ferrites with a large magnetic moment the saturation curve (χ′′ vs h2) measured at X band shows a maximum well below the initial onset of nonlinearity. A theoretical explanation for this extraordinary behavior is given. A new nonlinear effect arising from spin wave instability in a microwave magnetic field applied parallel to the dc field has been observed. Spin waves which propagate in directions perpendicular to the dc field are most susceptible to this instability. The observed variation of the critical rf field strength agrees well with the theoretical predictions. It indicates that the spin-wave line width increases with increasing wave number and decreasing angle between propagation direction and dc magnetic field.

No abstract

The high frequency permeability of partially magnetized ferrites is calculated for some simple domain configurations, comprising only ``up'' and ``down'' domains. The method used is based upon the magnetostatic approximation and neglects exchange effects, but is otherwise substantially rigorous. The components of the effective permeability tensor (ratio of average induction to average magnetic field) in general depend upon details of the domain configuration in addition to the net dc magnetization. When the dc magnetization is cycled between the two states of complete magnetization the high frequency permeability, considered as a function of the dc magnetization, in general shows hysteresis. Detailed calculations of the high frequency permeability have been carried out for the case in which the domain configuration is cylindrically symmetric, i.e., invariant under rotation around the direction of magnetization. For any such domain configuration the two relevant components μeff and κeff of the effective permeability tensor obey the relation μeff2−κeff2=const, regardless of the number of domains and their relative size. This general theorem allows a simple derivation of the (isotropic) permeability in the completely demagnetized state, giving μeff=23{[(ω/γ)2−(Ha+4πM0)2]⧸[(ω/γ)2−Ha2]}1/2+13,where ω is the frequency, Ha the anisotropy field, and M0 the saturation magnetization.

The conversion of electromagnetic power into spin-wave power and vice versa is investigated from a theoretical point of view. The analysis applies to a rod of ferromagnetic material whose two ends are each in a resonant cavity that is connected to a waveguide. The dc magnetic field is assumed to be nonuniform in such a way that the effective wavelength of the spin waves becomes large in those regions of the sample that protrude into the cavities. The theoretical analysis of the excitation process leads to a differential equation which is of the same form as the well-known wave equation except that the wavenumber is a function of position. The conversion efficiency depends on the solution of this wave equation through a simple integral over the wavefunction which has the physical significance of a coupling length. A numerical estimate indicates that substantially complete conversion should be possible.