268 The Electromagnetic Wave Equations Hence v' = ieE/mo). The.displacement r of the electron due to the field is given by ŕ = ťj and therefore r = - eE/mco2. The polarization P of the body is the dipóle moment per una volume. Summing over all electrons, we find P = Ier = -e2NE/m(o2, where N is the number of electrons in all the atoms in unit volume of the substance. By the definition of the electric induction, we have D = eE = E + 47tP. We thus have the formula e(oj) = 1 -4nNe2/mco2. (78.1. The range of frequencies over which this formula is applicable begins, in practice, at the far ultra-violet for light elements and at the X-ray region for heavier elements. If e M is to retain the significance which it has in Maxwell's equations, the frequency must also satisfy the condition wl/c)H, where / is the dimension rf the body. Putting e - 1 ~ 1, we have ÔP/Ôt ~ ojE ~ w2 IH/c. For the space derivatives rf §79 The dispersion of the magnetic permeability the magnetic moment M = XH we have c curl M ~ cXH/l. If \d P/5t| is small compared with \c curl M|, we must-have I2 < xc2/a2- (79.4) It is evident that the concept of magnetic susceptibility can be meaningful only if this inequahty allows dimensions of the body which are (at least) just macroscopic, i.e. if it is compatible with the inequality / > a, where a is the atomic dimension. This condition is certainly not fulfilled for the optical frequency range; for such frequencies, the magnetic susceptibility is always ~ v2/c2, where v is the electron velocity in the atom;t but the optical frequencies themselves are ~ v/a, and therefore the right-hand side of the inequality (79.4) is ~ a2 Thus there is no meaning in using the magnetic susceptibility from optical frequencies onward, and m discussing such phenomena we must put p. = 1. To distinguish between B and H in this frequency range would be an over-refinement. Actually, the same is true for many phenomena even at frequencies well below the optical range.* The presence of a considerable dispersion of the permeability makes possible the existence of quasi-steady oscillations of the magnetization in ferromagnetic bodies. In order to exclude the possible influence of the conductivity, we shall consider ferrites, which are non-metallic ferromagnets. The term "quasi-steady" means, as usual (§58), that the frequency is assumed to satisfy the condition co « c/l, where / is the characteristic dimension of the body (or the "wavelength" of the oscillation). In addition, we shall neglect the exchange energy related to the mhomogeneity of the magnetization resulting from the oscillations; that is, the spatial dispersion (§103) of the permeability is assumed to be unimportant. For this, the dimensions / must be much greater than the characteristic length for the inhomogeneity energy: / > ,/«, where a is of the order of the coefficients in (43 1) l ^ T-PUt H and B in the forms H = H0 + H', B = B0 + B', where H0 and B0 are the field and induction in the statically magnetized body, H' and B' the variable parts in the oscillations. When the displacement current is neglected, these variable parts satisfy the equations curlH' = 0, divB' = 0, (?95) which differ from the magnetostatic equations only in that the permeability is now (for a monochromatic field oc e to') a function of the frequency, not a constant. § A ferromagnetic medium is magnetically anisotropic, and its permeability is therefore a tensor fet (co), which determines the linear relation between the variable parts of the induction and the field. + This estimate relates to the diamagnetic susceptibility; the relaxation times of any paramagnetic or ferromagnetic processes are certainly long compared with the optical periods. It must be emphasized, however, that the estimates are made for an isotropic body, and must be used with caution when applied to ferromagnets. In particular the gyrotropic terms in the tensor pik which decrease only slowly (as I/o) with increasing frequency «see Problem 1) may be important even at fairly high frequencies. í This is discussed from a somewhat different standpoint in §103 below; see the second footnote to that section. .iPfeSeuOSCÍllatÍOnS are therefore called ma9netostatic oscillations. The theory has been given by C. Kittel 11!»/) tor homogeneous (see below) magnetostatic oscillations and by L. R. Walker (1957) for inhomogeneous 270 The Electromagnetic Wave Equations The first equation (79.5) shows that the magnetic field has a potential: H' = Substituting in the second equation B', = nikH'k = -fiikôil//dxk, we then obtain equation for the potential within the body: Hik(a))d2y/dxioxk = 0. ( Outside the body, the potential satisfies Laplace's equation A ý = 0; on the surface, and B'n must as usual be continuous. The first condition is equivalent to the continuity the potential ý itself; the second implies the continuity of /iik«,# /dxk, where n is a vector along the normal to the surface. Far from the body, we must have ij/ -»0. The problem thus formulated has non-trivial solutions only for certain values of the regarded as parameters. Equating the functions nik(co) to these, we find the nar oscillation frequencies of the magnetization of the body, called the inhomog ferromagnetic resonance frequencies. The simplest type of magnetostatic oscillation of a uniformly magnetized elli consists in oscillations which maintain the uniformity, the magnetization oscillating whole. To find their frequencies, it is not necessary to obtain a new solution of the equations; they can be derived directly from the relations (29.14): »! + "«"(**-»*)-&. (79,1| where nik is the demagnetizing factor tensor of the ellipsoid; H and B relate to the within it, and § is the external magnetic field. The latter is assumed to be uniform; in H i B, we again separate the oscillatory parts H' and B', which are now uniform throughout body. For these we have or H'i + nik(B'k-H'k) = 0 (Sik + 4nnaxlk)H'k = 0, ■ with the magnetic susceptibility tensor xik((o) defined by nik = ôik + 4nxik. Equating to i the determinant of this system of linear homogeneous equations, we find det\ôik + 4nnaxlk((o)\ = 0, the roots of which give the natural oscillation frequencies. These are called homogeneous ferromagnetic resonance frequencies. PROBLEMS Problem 1. Using the macroscopic equation of motion of the magnetic moment (the Landau-' equation; see SP 2, (69.9)), derive the magnetic permeability tensor for a uniformly magnetized use ferromagnet of the easy-axis type, in the absence of dissipation (L. D. Landau and E. M. Lifshitz, 19351 Solution. The equation of motion of the magnetization in a ferromagnet is M = y(H + ßM,v)xM, where y = g\e\/2mc (g being the gyromagnetic ratio), ß > 0 the anisotropy coefficient, and v a unit vector ata the axis of easy magnetization (the z-axis). We write H = H0 + H', where H' is a small variable field in i direction, and H0 a constant field which we take to be along the z-axis.t The transverse magnetization Mx,M,i + This field is used here with a view to applying the results in the subsequent Problems.