## Abstract

The Becker-Kersten treatment of domain boundary movements is widely applicable in the interpretation of magnetization curves, but it does not account satisfactorily for the higher coercivities obtained, for example, in permanent magnet alloys. It is suggested that in many ferromagnetic materials there may occur 'particles' (this term including atomic segregates or 'islands' in alloys), distinct in magnetic character from the general matrix, and below the critical size, depending on shape, for which domain boundary formation is energetically possible. For such single-domain particles, change of magnetization can take place only by rotation of the magnetization vector, I$_{0}$. As the field changes continuously, the resolved magnetization, I$_{H}$, may change discontinuously at critical values, H$_{0}$, of the field. The character of the magnetization curves depends on the degree of magnetic anisotropy of the particle, and on the orientation of 'easy axes' with respect to the field. The magnetic anisotropy may arise from the shape of the particle, from magneto-crystalline effects, and from strain. A detailed quantitative treatment is given of the effect of shape anisotropy when the particles have the form of ellipsoids of revolution (section section 2,3,4), and a less detailed treatment for the general ellipsoidal form (section 5). For the first it is convenient to use the non-dimensional parameter h, such that h = H/($|N_{a}-N_{b}|$) I$_{0}$, N$_{a}$ and N$_{b}$ being the demagnetization coefficients along the polar and equatorial axes. The results are presented in tables and diagrams giving the variation with h of I$_{H}$/I$_{0}$. For the special limiting form of the oblate spheroid there is no hysteresis. For the prolate spheroid, as the orientation angle, $ \theta $, varies from 0 to 90 degrees, the cyclic magnetization curves change from a rectangular form with $|h_{0}|$ = 1, to a linear non-hysteretic form, with an interesting sequence of intermediate forms. Exact expressions are obtained for the dependence of h$_{0}$ and $ \theta $, and curves for random distribution are computed. All the numerical results are applicable when the anisotropy is due to longitudinal stress, when h = HI$_{0}$/3$ \lambda \sigma $, where $ \lambda $ is the saturation magnetostriction coefficient, and $ \sigma $ the stress. The results also apply to magneto-crystalline anisotropy in the important and representative case in which there is a unique axis of easy magnetization as for hexagonal cobalt. Estimates are made of the magnitude of the effect of the various types of anisotropy. For iron the maximum coercivities, for the most favourable orientation, due to the magneto-crystalline and strain effects are about 400 and 600 respectively. These values are exceeded by those due to the shape effect in prolate spheroids if the dimensional ratio, m, is greater than 1 $ \cdot $1; for m = 10, the corresponding value would be about 10,000 (section 7). A fairly precise estimate is made of the lower limit for the equatorial diameter of a particle in the form of a prolate spheroid below which boundary formation cannot occur. As m varies from 1 (the sphere) to 10, this varies from 1 $ \cdot $5 to 6 $ \cdot $1 $ \times $ 10$^{-6}$ for iron, and from 6 $ \cdot $2 to 25 $ \times $ 10$^{-6}$ for nickel (section 6). A discussion is given (section 7) of the application of these results to (a) non-ferromagnetic metals and alloys containing ferromagnetic 'impurities', (b) powder magnets, (c) high coercivity alloys of the dispersion hardening type. In connexion with (c) the possible bearing on the effects of cooling in a magnetic field is indicated.