Linear and cubic dynamic susceptibilities of superparamagnetic fine particles
Yuri L. Raikher and Victor I. Stepanov
Institute of Continuous Media Mechanics, Urals Branch of RAS, 614013, Perm, Russia
Received 21 February 1997
A consistent theory of linear and nonlinear cubic initial susceptibilities of an assembly of uniaxially
anisotropic noninteracting fine magnetic particles is presented. The expressions for the static equilibrium
susceptibilities are obtained directly from the pertinent statistical thermodynamics. The contributions of an-
isotropy emerge yet in the first order and are analyzed for random and axes-aligned distributions. The ac
susceptibilities are studied on the basis of the micromagnetic Fokker-Planck equation. Both a numerically
exact solution for arbitrary frequency and a reliable low-frequency approximation are given. The obtained
description proves to be more accurate as compared to the one based on the customary superparamagnetic
blocking model. The results are used for a quantitative interpretation of recently published set of data on Co-Cu
precipitating alloys. In this connection the choice of the particle size-distribution function is discussed.
S0163-18299704722-X
INTRODUCTION
Since the very first studies of fine-particle systems,
1
the
development of the micromagnetic science was inspired
mainly by the necessity to predict the magnetic properties
and response of a ferromagnetic particulate media. Beyond
argument, in this objective the fundamental and applicational
aspects are tied up very closely, if not inseparably.
The problem of prime interest while performing experi-
ments on or manufacturing fine-particle magnetic systems is
to characterize the magnetic content of the sample with as
few measurements as possible. Magnetic granulometry by
means of a quasistatic magnetization curve is very well
known and widely used.
1–3
The dynamic approach, where
simultaneously linear and nonlinear susceptibilities are taken
into account is more new, being most probably inspired by
its use in the spin-glass science. To justify the method, one
should process a good deal of experimental data with the aid
of an appropriate theory. Such a work has been attempted
recently in Refs. 4,5 with a precipitating Cu-Co alloy as a
test object. The authors had no difficulties in fitting the linear
susceptibility measurements with the aid of superparamag-
netic blocking model assuming that: i the particles are
single domain and their magnetization does not depend on
temperature, ii the magnetic anisotropy is uniaxial and has
one and the same value for all the particles, and iii the
magnetic dipole-dipole interaction is negligible. It was fitting
the nonlinear cubic susceptibility data where a problem
arose, since there was no theory for it insofar, consistent with
the aforementioned assumptions. To fill the place, in Refs.
4,5 were employed the formulas originally derived for an
isotropic superparamagnet. They were adjusted by replacing
the pertinent relaxation time with a one exponential in the
magnetic anisotropy constant K . However, the resulting
agreement turned out to be poor. From that the authors of
Refs. 4,5 concluded to that some of the basic assumptions
i – iii are wrong. From our viewpoint, in the first place this
reproach should be addressed not to the classic superpara-
magnetic theory as itself but to a rather ‘‘intuitive’’ manner
of its usage.
The incentive and the main goal of our paper is to con-
sistently extend the conventional theory on the case of a
nonlinear response and by that to confirm its validity. While
doing that we propose practical schemes both exact and ap-
proximate to handle linear and cubic dynamic responses in
the framework of classical superparamagnetism. Applying
our results to the reported data on the nonlinear susceptibility
of Cu-Co precipitates, we demonstrate that a fairly good
agreement may be achieved easily.
I. STATIC SUSCEPTIBILITIES
As a starting point we take an isolated single-domain par-
ticle of a ferro- or ferrimagnetic material rigidly trapped in
the bulk of a solid nonmagnetic matrix. Single domain has a
spatial uniformity of the spin alignment over the grain that
enables us to describe it by the net magnetic moment
= e, whose direction is given by a unit vector e. The
magnetic moment magnitude is =I v with I the saturation
magnetization of the material at given temperature and v
being the particle volume. Besides that, we assume the par-
ticle to possess a uniaxial magnetic anisotropy with an en-
ergy density K and a direction defined by a unit vector n.
If the external magnetic field H is not too high as to affect
the atomic magnetic structure, its only effect on a single-
domain grain is the magnetic moment rotation. Then the cor-
responding orientation-dependent part of the particle energy
U may be written as
U =-K v en
2
- eH . 1
The stationary distribution function of the particle magnetic
moment or if we neglect interactions of an assembly of
magnetic moments is determined by the Gibbs formula
W e =Z
-1
exp„ en
2
+ eh …,
Z
, =
exp„ en
2
+ eh …d e, 2
where
PHYSICAL REVIEW B 1 JUNE 1997-II VOLUME 55, NUMBER 22
55 0163-1829/97/5522/1500513/$10.00 15 005 © 1997 The American Physical Society