1040 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



magnetization in it is everywhere uniform even if we are inside the 

 domain wall, our equation of motion is: 



"^ = y[M XH] - \/MTM x(M X H)l (5) 



at 



Here 7 is the gyromagnetic ratio {ge/2 mc), and X is a parameter, as- 

 sumed to be characteristic of a given ferromagnetic material, which is 



determined by the magnitude of the damping effects in the motion of M. 

 The magnitude of the last term on the right in (5) is thus determined 

 by the amount of the damping losses. 



The rate of dissipation of energy in the small volume is H-(dM/dt), 



where AI is the magnetic moment of the volume, and H is the total 



magnetic field in the volume. The value of H requires some discussion. 



Outside the wall H = Ho where Ho is the applied field, which is parallel 

 to the wall. When the wall is moving, however, there is an additional 

 field He inside it. This field, which is normal to the wall, is a demag- 

 netizing field which arises from the tendency of M in the moving wall to 

 have a component normal to the wall. This field has just such a value 

 that the magnetization in the moving wall precesses about it with the 

 Larmor frequency. Its value is : 



He = -$/y){de/dz), (6) 



as Becker^^ has shown. Here v is the velocity of the w^all, 2 is a distance 

 coordinate normal to the wall, and 6 is the rotational angle of the mag- 

 netization as we pass through the wall along z. Inside the wall, He is 

 much larger than Ho , but in any case H = He -\- Ho . From (5) we find : 



H-dM/dt^\He, (7) 



as Kittel" first showed. In the theory of the domain wall it is shown 

 that 86/ dz in the wall is equal to the square root of the ratio of the in- 

 crease in anisotropy energy as the magnetization turns away from the 

 easy direction of magnetization, to the exchange energy constant. That 



• 14 



IS : 



^ = {W) - g{do)]/Ay^\ (8) 



oz 



The exchange energy constant A is defined by the following expression 

 for the exchange energy per unit volume due to gradients in the direction 



