A critical assessment of Hoffmann´s linear theory of ripple

Hoffmann\´s theory of ripple in ferromagnetic films is based on an approximate treatment of magnetostatic interactions; it thereby achieves analytical simplicity but introduces errors. The present paper investigates these errors under the following conditions: the equations can be linearized, the magnetization is constrained to lie in the film plane and to be constant through the thickness, and the stray field varies negligibly through the thickness. Only the effect of a concentrated force (the Green\´s function of the problem) is examined. Hoffmann\´s formula is regarded as an empirical modification of the known formula for negligible magnetostatic interactions, and his Cyis regarded as a parameter to be adjusted for "best fit," according to some criterion, to the rigorous formula (a Fourier integral). Several criteria are examined; each is an integrated quantity that can be evaluated by Parseval\´s theorem. These lead to fairly consistent values of Cyand to a good fit as judged by the criterion. But as judged by the actual spatial variation of the magnetization, calculated to the first order in the magnetostatic interactions, the agreement is poor, especially at large distances, where a dipole-field behavior in the exact solution is replaced by an exponential decay in the Hoffman solution. Poor agreement is also found in the limiting case of very large magnetostatic interactions.