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MAGNETIC INTERACTIONS AND THE PREISACH MODEL
W. Brown, Jr
To cite this version:
W. Brown, Jr. MAGNETIC INTERACTIONS AND THE PREISACH MODEL. Journal de Physique Colloques, 1971, 32 (C1), pp.C1-1037-C1-1038. �10.1051/jphyscol:19711371�. �jpa-00214409�
JOURNAL DE PHYSIQUE Colloque C 1, supplkment au no 2-3, Tome 32, Fkvrier-Mars 1971, page C 1 - 1037
MAGNETIC INTERACTIONS AND THE PREISACH MODEL
W. F. BROWN, Jr
Department of Electrical Engineering, University of Minnesota, Minneapolis, Minnesota 55455, U. S. A.
RBsum6. - Sous certaines conditions, un groupe de particules monodomaines orient&, en interaction magnktique, produit un cycle d'hysterksis rectangulaire d6plack: comme dans le modkle de Preisach. Le groupe doit contenir au moins une particule exceptionnelle dont l'aimantation ne se renverse pas, dont le champ critique est supkrieur a ceux des autres d'une quantitk grande par rapport aux champs de couplage et dkpasse aussi le champ le plus grand appliquk a l'kchantillon ; les positions des particules doivent &re telles que les rotations rkversibles, produites par des composantes transversales des champs magnktiques mutuels, soient petites.
Abstract. - Under certain conditions, a cluster of oriented, magnetically interacting single-domain particles executes an approximately rectangular, displaced hysteresis loop, as in the Preisach model. The cluster must contain at least one exceptional nonreversing particle, whose critical field exceeds those of the others by an amount large in comparison with the mutual magnetic fields and also exceeds the maximum field applied to the specimen ; and the particle locations must be such that the reversible rotations, caused by transverse components of the mutual magnetic fields, are small.
The Preisach model [ l ] of a magnetic material is an assemblage of independent units, each of which has a rectangular hysteresis loop with jumps at H = b + a.
The values of a and b for the various units have a statistical distribution over the range 0 < a < coy
- co < b < co. Despite the model's empirical suc- cesses [2], attempts to justify it theoretically [3] have been unsatisfactory, especially for materials composed of single-domain particles.
In this case, it is usual t o identify the units with the particles. If the particles are uniaxial, are oriented with their axes of easy magnetization along the field direction, and have negligible magnetic interactions, their theoretical behavior corresponds to the Preisach model with every b = 0 [4]. The usual interpretation of a nonzero b is that - b is the cc interaction )> field due to particles other than the one under considera- tion. This interpretation encounters two difficulties.
First : when one of the other particles reverses its moment, its field reverses ; the interaction field - b is therefore not constant. This difficulty is commonly evaded by supposing that the distribution of b (and a) nevertheless remains approximately constant (cc sta- tistically stable )i) [5-71. It is not obvious why this should be so.
Second : when the field acting on a particle is partly due to other particles, whose moments also are free to rotate, its magnetization may reverse at a smaller net reversed field than if this field were directly control- led. This happens because of joint rotations of the moments of two or more particles. The importance of the effect has been demonstrated by analytical [8]
and numerical [9-101 calculations. In view of these results, the empirical success of the Preisach model remains mysterious.
In these calculations, however, the magnetically coupled particles were assumed to have identical critical field H,, for reversal in the absence of the coupling. When the H,,'s differ by amounts compara- ble with or larger than the interaction field, the situa- tion is different ; and under certain conditions, the behavior of a pair or cluster of particles can simulate that of a Preisach unit.
Consider first two particles (see Fig. 1) at (0, 0, 0) and (0, 0, r ) with volumes u, and v , ( N vl) respecti- vely, uniaxial positive anisotropy constants (crystal- line plus shape) K, and K2, and magnetization direc- tions (a,, PI, 7,) and (a,, B2, y,) ; the axes of easy
-Axes of easy magnetization
\ H
RG. 1. - Two magnetically interacting particles.
magnetization and the applied field H, are along Oz.
In the dipole approximation t o the magnetic interac- tion, the free energy is
(M, = spontaneous magnetization). When y, and y2 are close to + 1, G becomes, to the second order in the a's and Ks,
where
2 V l v, M : cii = 2 vi K i + vi M , H o +
r3 3 (3)
The behavior of the a's and of the p's may be treated independently. The state y, = y2 = 1 is an equilibrium state ; it is stable at large H,, and becomes unstable
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19711371
C 1 - 1038 W. F. BROWN, JR when Ho equals the larger root of the quadratic
equation obtained by setting the determinant 1 cij 1
equal to zero. When Kl = K2 and v1 = v2, this redu- ces to the result obtained before [8], which is incom- patible with the Preisach model ; a qualitatively simi- lar result is obtained when Kl # K2 and v1 # u2 but
I Kl - K2 I << vi M,2/r3. When, however,
I K1 - K2 I 9 vi M,2/r3 and K1 > K2, the critical field is
In the first case, the moments reverse simultaneously, by joint rotation ; in the second, moment 2 reverses as if in a fixed field due to the external sources and to moment 1.
By similar calculation when y1 is close to f 1 and y, close to - 1, we find that when
K1 + K2 9 vi M,2/r3
(which is certainly true when I K1 - K2 I 9 vi M?/r3) and Kl > K2, the critical field for return to the origi- nal state is
whereas the critical field for reversal of moment 1 (when moment 2 has already reversed) is
If Ho alternates between values + H,, with
H m < I H c 3 l 2
the system executes a hysteresis loop with its vertical parts at Hc, and Hc2, in accordance with the Preisach model as interpreted by NCel [11, 121. The above derivation provides sufficient conditions for the validity of that interpretation.
Now move particle 2 to (0, r sin O', r cos 8'). If Kl is large enough, particle 2 still executes hysteresis loops in the fixed field of particle 1 ; but unless O'=0 or 4 2 ,
[I] PREISACH (F.), Z. Physik, 1935, 94, 277.
[2] BIORCI (G.) and PESCETTI (D.), J. Phys., 1959, 20, 233.
[3] KNELLER (E.), Ferromagnetismus (Springer-Verlag, Berlin, 1962), 561.
[4] STONER (E. C.) and WOHLFARTH (E. P.), Phil. Trans.
Roy. Soc. London, 1948, A 240, 599-642.
r51 WOODWARD (J. G.) and DELLATORRE (E.). J. Auul.
- - , ,,
Phys., 1960, 31, 56-62.
[61 BATE (G.), J. Appl. Phys., 1962, 33, 2263-2269.
[7] PAUL (M.), 2. angew. Physik, 1970, 28, 321-325.
this field now has a y - as well as a z- component, and the irreversible jump is preceded by a reversible rotation. The loop is not only displaced but narrowed, and its top and bottom acquire curvatures of opposite signs. We may approximate this behavior, but only roughly, by superposing rectangular-loop and linear reversible )> magnetizations. Expansion of G to the second order fails (it prediyts an infinite transverse moment at the critical field) ; but we may find the critical fields from the familiar astroid (4 vs Hz) [13]
by treating Hz and H,, as the resultant of the applied field and the field of particle 1 :
2 K2 213 312
H~ = T - [I -
(3M
cos e' sin el)]
-M s 2 K2 r3
For a cluster of two or more ordinary particles and an exceptional nonreversing one, all coupled magne- tically, the behavior is again of Preisach type if all the particles are in a line along or perpendicular to the z axis. Otherwise, the loop is in general already roun- ded by reversible rotation even before the field of the exceptional particle is taken into account.
We may therefore justify the Preisach model under the following conditions. (1) The particles are oriented with their easy axes along the field direction. (2) The particles form clusters such that the interactions between members of different clusters are negligible. (3) If H,, (-- 2 KIMs) is a representative critical field of an isolated particle and if h ( ~ vMs/r3) is a repre- sentative mutual magnetic field within a cluster, the transverse component of h/Hc, is small enough so that a rough correction for reversible rotation is suffi- cient (4). Within a cluster, the spread AHco of Hco values for most of the particles 5 h ; but there is usually one exceptional particle (at least) whose Hco exceeds that of the others by an amount 9 h and also exceeds the maximum field applied to the specimen.
Under these conditions, the clusters behave approxi- mately as Preisach units.
[8] BROWN (W. F.), Magnetostatic Principles in Ferro- magnetism (North-Holland Publishing Company, Amsterdam, 1962), 1 12-1 16.
[9] BROWN (W. F.), J. Appl. Phys., 1962, 33, 1308.
[lo] BERTRAM (H. N.) and MALLINSON (J. C.), J. Appl.
Phys., 1969, 40, 1301.
[ll] N ~ E L (L.), Appl. Sci. Res., 1954, Sect. B, 4, 13.
[12] N ~ E L (L.), Phil. Mag. Suppl. 1955, 4, 191.
[13] OLSON (C. D.) and Porn (A. V.), J. AppI. Phys., 1958, 29, 274.
