R. I. Joseph

Demagnetizing Field in Nonellipsoidal Bodies

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 slab...

Ballistic Demagnetizing Factor in Uniformly Magnetized Cylinders

A simple, closed‐form expression is obtained for the ballistic demagnetizing factor of a uniformly magnetized cylinder. Detailed comparison is made between the present results and both the previously derived series expansion (in which the radial variation of magnetization is neglected), valid for p(=length/diameter) ≥10, and the numerical extrapolation of the latter to the range 0≤p≤100. An alternate derivation of a result for the magnetometric demagnetizing factor of a uniformly magnetized cylinder is presented. Detailed comparison is made between the demagnetizing factors in the uniformly magnetized cylinder (ballistic, magnetometric) and the demagnetizing factor of an inscribed ellipsoid of revolution.

Theory of Magnetostatic Modes in Long, Axially Magnetized Cylinders

The characteristic equation determining the eigenfrequencies of the magnetostatic modes is derived from the equations of motion and the boundary conditions. The solutions may be classified as pertaining to surface and to volume modes. Surface‐mode solutions exist only for sufficiently small wave numbers, and their eigenfrequencies are larger than those of volume modes. The eigenfrequencies generally decrease with increasing wave number. Approximate, analytic expressions for the dependence of the eigenfrequencies on wave number have been obtained for the regions in which the wavelength is either much smaller or much larger than the cylinder radius. The approximate expressions are compared with numerical results obtained by means of an electronic computer.

Generation of Spin Waves in Nonuniform Magnetic Fields. III. Magnetoelastic Interaction

If a spin wave propagates through a region of nonuniform magnetic field in which the effective magnon wavenumber equals the phonon wavenumber, it is partially converted into an elastic wave. The conversion efficiency depends primarily on the field gradient (H′) at the crossover point. Theories have been developed both for the case of weak magnetoelastic coupling and strong magnetoelastic coupling. For weak coupling the magnon‐phonon conversion efficiency ηmp is ηmp=H′crit/|H′| (for |H′|>>H′crit), whereas for strong coupling ηmp = 1−π2exp(−H′crit/|H′|) (for |H′|<<H′crit). Here H′crit=πb22 ω/cMμ is a critical field gradient, b2 is one of the magnetoelastic constants, ω/2π the signal frequency, c the velocity of (transverse) sound, M the saturation magnetization, and μ the shear modulus. It has been assumed that the dc magnetic field is applied along a cube edge of a cubic crystal and that the material is elastically isotropic. For yttrium iron garnet (YIG) at room temperature and a signal frequency of 3×109...

Spin‐Wave Instability in Hexagonal Ferrites With a Preferential Plane

The large magnetic anisotropy field that is encountered in many planar ferrites markedly affects the rf magnetic field at which spin‐wave instability sets in. When the rf magnetic field is applied perpendicular to the dc field, spin‐wave instability can occur through the first‐or second‐order process. At ferromagnetic resonance, the first‐order process dominates if the frequency is smaller than a characteristic frequency ωc. This characteristic frequency has been calculated for ellipsoidal samples. For most shapes, the anisotropy field tends to increase ωc, thus favoring the occurrence of the first order process. In addition, the anisotropy tends to decrease the critical field at which spin‐wave instability sets in. Assuming the spin‐wave relaxation rates to be equal in the two situations compared, we find that for large anisotropy the second‐order threshold is reduced approximately in the ratio of (2/ζ)½ for relatively high frequencies; and in the ratio of (ω/ωMζ)½ for relatively low frequencies where ζ=...

Transient and Steady‐State Absorption of Microwave Power under Parallel Pumping. Theory

The transient growth of the spin‐wave population in a microwave magnetic field applied parallel to the dc field is investigated. It is assumed that the sample is in thermal equilibrium before the initiation of the microwave pulse, and that the rf field amplitude remains constant during the pulse. According to the theory the absorption coefficient approaches a steady‐state value below threshold, and grows indefinitely (as long as the field amplitude remains constant) above the threshold. The results of detailed calculations concerning the steady‐state and the transient absorption are described. Above threshold the time dependence of the absorption coefficient is characterized by a very fast initial rise to an intermediate plateau or hump, a subsequent more gradual rise, and an approximately exponential rise at later times. As the absorption coefficient increases, the field amplitude in the cavity decreases (assuming that the incident power remains constant). This reaction of the sample upon the field ampli...

Generation of Spin Waves in Nonuniform dc Magnetic Fields. II. Calculation of the Coupling Length

The coupling length is calculated by using two approaches: (a) For a simple model the wavefunction and the coupling length are determined rigorously; (b) More general, approximate results are obtained by means of the WKB method, which is valid if the dc magnetic field varies sufficiently slowly. It is shown that the coupling length is inversely proportional to the square root of the field gradient at the turning point.

Subthreshold Steady‐State Absorption under Parallel Pumping

If the rf and dc magnetic fields are parallel, the absorption coefficient μ″ at low power levels is theoretically expected (for YIG at room temperature, a frequency of 10 Gc/sec, and a dc field of 1 kOe) to be of the order of 2×10−4. We have calculated how the steady‐state absorption varies with dc and rf field strength below the instability threshold. Experiments on a high‐purity YIG single crystal are in reasonably good agreement with the theoretical expectations.

Microwave Magnetoelastic Resonances in a Nonuniform Magnetic Field

A new magnetoelastic resonance effect is described. A dc magnetic field was applied in the long direction of YIG slabs (1 cm long, 0.5 cm wide, and 0.1 or 0.07 cm high), each end of which was inserted in a stripline cavity. Under appropriate conditions a pulse of microwave power applied to one cavity resulted in a pulse in the second cavity delayed by several microseconds. Small variations were observed in the amplitude of the delayed pulse as the dc field was varied. Similar variations with the same periodicity (∼1 Oe) were also observed in the undelayed reflection from the input cavity. The amplitude variation is attributed to the excitation of magnetoelastic resonances in the nonuniform magnetic field near the endface of the sample. Because of the strong coupling between spin waves and elastic waves, the turning point (at which kspin waves=0) and the endface of the sample define a magnetoelastic quasicavity whose resonances give rise to the effect noted. According to this interpretation the separation ...

Transient Growth of Spin Waves under Parallel Pumping

We have experimentally observed the transient absorption during the application of a microwave pulse in a high‐purity single‐crystal YIG sphere at X‐band and dc fields on the order of 900 Oe when the rf power level exceeds the threshold only slightly (less than 1 dB). Detailed calculations of the transient absorption coefficient based upon a previously proposed theory of Schlomann and taking into account the reaction of the sample upon the field in the resonant cavity agree reasonably well with the experimental data.