MO'l'IO.X OF l\l)l\ll)r \l, DOMAIN WALLS \(Y.V.) 



THEORY 



A theoretical aiialj'.sis of the experimental results given in the last 

 section dixicies itself rather naturally into three parts. First we chai- 

 acterize the data in terms of an equation of motion for unit, aica of 

 domain wall. This means we determine the constants of motion (viseous 

 resistance and coercive force in our case) of unit area of wall. Secondly, 

 we show the relation between the viscous resistance of unit area of 

 domain wall, and the constants which characterize the ferromagnetic 

 material in general (saturation magnetization, crystal anisotropy, etc.). 

 This is essentially an application of the recent work of Becker ' and 

 Kittel. Lastly, we calculate the magnitude of the damping from a 

 rela.xation mechanism which accounts for the low temperature effect 

 shown in Fig. 8 at least ciualitatively. 



Consider unit area of a 180° domain wall between two regions of 

 satiu'ated material. Such a system has an equation of motion for small 

 amplitudes of the applied magnetic field H which may be written: 



nu + f3z + az = 2-1/,//, (3) 



where z is the displacement of the domain wall along its normal, ni is 

 its mass per unit area, |3 is a parameter measuring viscous resistance, and 

 a is a stiffness parameter which has meaning only for small fields such 

 as those used in initial permeability measurements. When fields larger 

 than the coerci\e force are applied, as in our experiments, the term con- 

 taining a disappears and the field effectix'e in moving the wall is less 

 than the applied field by an amount e(iual to the coercive force; this is 

 shown b}^ the data given previously in the section on results. This re- 

 formulation of (3) is ({uite reasonable when one remembers the spikes 

 which pull back on the wall, in the experiments of Williams and Shockley, 

 for small wall motions and snap off entirely if the wall mo\'es a large 

 distance. Furthermore, since the velocity of the domain wall rises to 

 its stead^^ value in a negligible time in the.se experiments (Fig. 6) the 

 initial term in (3) is also negligible. As these remarks indicate, under the 

 conditions of the experiment in wall velocity, (2) takes the form: 



^k = 2il/.(//app - He). (4) 



This relation obviously fits the data given previously. 



The second step in the theoretical analysis starts from an equation of 



motion for the magnetization M in a small \'olume which was first used 

 by Landau and Lifshitz, and takes ad\-antage of more recent woik of 

 Becker'^ and Kittel." If we consider a volume small enough so that the 



