1044 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



cylinder, Avhich is the integral of dw/dt over the cylinder, is 2MsHov 

 as in (11). The rate of dissipation of energy in this section of the wall 

 is somewhat more difficult to calculate. It is the sum of contributions 

 from all the small volumes along the cylinder. In order to calculate the 

 contribution from the small volume we are considering, we first note 

 that if the process is reversible, 



dg = '^ dd, (16) 



where 6 is the angle of rotation of M. Physically, the factor dg/dd repre- 

 sents a torque on the magnetization. We wull assume that (16) holds 

 even when the domain wall passes our system at a finite velocity and 

 we have departed slightly from isothermal equilibrium. The field He 

 defined in (6) transmits this torque to make the magnetization rotate 

 through the angle 6, but we will not go further into the details of this 

 process. 



As the domain wall goes by, dg/dd, which we Avill abbreviate as ^', 

 changes. If the process were reversible, g' would be zero at all times and 

 the torque on the magnetization would always have its e{iuilil)rium 

 value. Actually, as we deviate more and more from the reversible process 

 by moving the wall faster, the electrons are no longer able to rearrange 

 fast enough, and g' deviates from zero while continually relaxing toward 

 it. We must form our analysis in such a way that a maximum is estab- 

 lished for g', since even if the magnetization moves infinitely rapidly, 

 g' does not become infinite. Since the electrons minimize their free energy, 

 it must be positive, and we write g = gi^ — gi so that g' = gi^ — ^i, 

 where gi relaxes toward gi^ . Note that the torque relaxes downward 

 from a value, gi„ , associated with the adiabatic anisotropy energy (fast 

 motions of the magnetization) to a value zero, associated with the iso- 

 thermal anisotropy energy (slow motions of the magnetization). We 

 may now write: 



, / ' ' 



cf£i _ gix — g\ /,-x 



dt r ' ^ ^ 



Physically, this relation assumes that the torque on the magnetization 

 relaxes in the same way if ^i«, is a continuous function of time as it 

 would if gfioo and g^ differed but gi„ was constant. Here r is the relaxation 

 time associated with the rearrangement of divalent and trivalent iron 

 ions. If we write (17) in the form: 



^^ -f ^ = ^ , (18) 



dt T T 



