MOTION OF IXniVIDTAL DOMAIN "WALLS 1041 



of magnetization : 



Exchange energy/unit vol = A[(VaiY + (Vao)' + (Vas)^, (9) 



where ai , ao , and 0:3 are the direction cosines of the magnetization. 

 g(d) is the anisotropy energy: 



g(d) = Ki (ai'a2 + a-ia^ + 0:3 «i ), (10) 



expressed in terms of 6, and g{9o) is the anisotropy energy along the 

 (Urection of easy magnetization. Note that [g(6) — g(6o)] is always posi- 

 tive. Ki is the first order anisotropy constant. 



If we use (6) and (8) in (7), and integrate over z along a cylinder of 

 unit cross-section normal to the wall to get the rate of energy dissipa- 

 tion for unit area of moving wall, we have: 



r H- ^ dz = iXv'/y'A"') r [g{d) - g(do)f' dd = 2HMsV, (11) 

 J-00 dt Jbi 



where di and 62 are the angular positions of M on the two sides of the 

 wall. 62 — di = T, of course, since we are considering a 180° domain 

 wall. In order to obtain (11), we have used (8) to transform from in- 

 tegration over z to integration over 6 as well as to evaluate (7). We set 

 our result equal to 2MsH (pressure on the wall) times v since this is the 

 rate at which the wall, considered phenomenologically, does work. We 

 may now write : 



2Msy'A'^' ^ 



X W) - g{e,)r dd ^'--^ 



This is the desired relation between wall velocity and applied field which 

 is to be compared with (4). In this way we find: 



^ = (\/y'A"') t [g{d) - g{9,)\"' dd. (13) 



We have thus shown the relation between the wall parameter jS and 

 the parameter X which measures in general the losses associated with 

 motions of the magnetization. 



The third part of our theoretical analysis is concerned with a cal- 

 culation of the damping parameter X, or rather the relation between 

 V and H itself, from an explicit physical mechanism. Such a calculation 

 has not been made in the past since the appropriate mechanism on 

 which to base it has remained obscure. Verwey and his co-workers 

 have explained the well known transition at about 115°K in Fe304 as 



