584 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1954 



the right of (2) is quite Avell su))stantiated by quantum mechanical con- 

 siderations. It is a vector normal to M and to the force Ht and is re- 

 sponsible for the precession. The second term is also a vector normal to 

 M, but is in the plane of M and ^ in a sense such as to reduce the angle of 

 the precession. It thus represents a damping. Not much is known about 

 the precise mechanism of the damping, so that its phenomenological 

 representation by the second term of (2) is still in doubt. 



Ht , the total field acting on the electron spins, is made up of terms 

 not all of which are of electromagnetic origin. It consists of the dc field 

 ^0 within the sample, the ac field H, the anisotropy field, and the field 

 ascribed to the quantum mechanical exchange forces between spins. 



Hq in the sample must be calculated from the applied dc field ^ext by 

 a purely magnetostatic calculation, which, in the case of sufficiently 

 simple shapes, can be carried out with the help of the appropriate de- 

 magnetizing factors. Throughout this paper it is assumed that this 

 problem has been solved, so that H^ is given. Furthermore it is assumed 

 that ^ext and ^o are uniform. Boundary effects due to non-uniformities 

 of ^0 are neglected. 



The microwave field H in the sample is one of the unknowns of the 

 problem of propagation, and will appear in the solution of Maxwell's 

 equations subject to the appropriate boundary conditions. 



The anisotropy field, a property of a single crystal of ferrite, arises 

 from the fact that through the medium of spin-orbit interaction, the 

 electron spins can "see" the orbital wave-functions. Since these have the 

 symmetry properties of the crystal, it is to be expected that the aniso- 

 tropy field will be a vector function of M, with the symmetry properties 

 of the crystal. The samples of ferrite used in practice contain a great 

 many small crystals randomly oriented, so that the net effect of the 

 anisotropy field on microwave propagation must be obtained by means 

 of an averaging procedure. The integrations involved are laborious and 

 have not been carried out so far. We shall therefore neglect anisotropy 

 altogether. Since anisotropy fields are usually of the order of a few 

 hundred gauss, this will put our results in error below frequencies of 

 about 3,000 mc/sec. (Corresponding to a precession frequency of yH^ = 

 3,000 mc/sec. Ho is about 1,100 gauss.) 



The field between two spins ascribable to exchange forces w^ill be zero 

 when the two are parallel, and thus arises out of differences of spin ori- 

 entation (that is, differences of M) from place to place. In fact, analysis 

 shows that this magnetic field is proportional to S7'M for cubic crystals. 

 Thus equation (2) really involves position coordinates as well as time. 

 Hence the ac part w of M at a point will depend not only on the ac field 



