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Submitted on 1 Jan 1988
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DIMENSIONAL CROSSOVER IN AN ISING
CYLINDER WITH EXCHANGE AND DIPOLAR
INTERACTIONS
G. Gehring, M. Wragg
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplement au no 12, Tome 49, dCembre 1988
DIMENSIONAL CROSSOVER IN AN ISING CYLINDER WITH EXCHANGE AND
DIPOLAR INTERACTIONS
G. A. Gehring and M. J. Wragg
Department of Theoretical Physics, Oxford University, Oxford OX1 3NP, G.B.
Abstract.
-
In this paper we examine the crossover from three dimensional t o one dimensional behaviour for a finite cylindrical sample of ferromagnet in which dipolar interactions dominate. We consider the case in which the magnetic moments lie parallel to the cylinder.In the bulk three dimensional limit Aharony and
Fisher [I] showed that for t = T/T,-1
q;f
wheref
=(
-
)
$'
1
;;( S p ~ ) 2 / ~ a & J being the short range exchange cou-
t
;ling in three dimensions and ao the lattice spacing, adipolar regime exists in which the transverse correla- tion length
tL
scales at t-'I2 (as in short range mean field theory) while the longitudinal correlation lengthtII
scales asf
lI2t? 0 6511
7.
(1) 0 5For a finite system there is a crossover from be- haviour characteristic of an infinite system to a lower dimensional behaviour dominated by the shape and dimensions of the finite system. Privman and Fisher
[2] have discussed the behaviour of a short range Ising model in cylindrical geometry. In the problem we con- sider here we assume that the dimensional crossover occurs from the dipole dominated regime.
In three dimensions the dipolar interaction has two important features; first that it decays as T - ~ , and second that there is an angular dependence of
(3 cos2 0
-
1) where cos 0 =z/T, so that the interac- tions average to zero over a spherical shell. In the one dimensional cylinder the dipole interaction is restricted by geometry to be positive definite at long range. At low temperatures we consider a disc of spins (of area aL2 where L is the cylinder radius) as an effective spin and look at the exchange and dipole interactions be- tween effective spins lying on a chain. The exchange interaction between two such effective spins varies as the number of real spins, i.e. ( L / U ~ ) ~ J. The magnetic moment of each effective spin also varies as ( ~ / a o ) ~ so that the dipolar interaction between them varies like ( ~ / a o ) ~ .Hence we can define a modified ratio of dipolar to exchange energy
f'
= ( ~ / a o ) ~f
valid in the extreme low temperature one dimensional limit wheref'
>>
f
for a macrosco~ic
-
cvlinder. The interaction between " two effective spins is expected to vary as L ~ / z ~ only for z>>
L where cos 0=
1. In figure 1 is plotted the computed dipolar potential which shows the drop for z<
L as well as the long range behaviour.Fig. 1.
-
The dipolar potential between effective spins in the cylinder.A chain of spins interacting via a potential
U
(z)(l/zl+") orders at T
#
0 only for a<
1 [3]. Thusthere will be no long range order for the dipolar po- tential a = 2. Cardy [4] has shown that for a
>
lthe low temperature properties are similar to those for a short range model. There is an energy of a kink (domain wall) which is given by E u ZU (z) and
z
that at low temperatures the kinks are always unbound and the correlation length varies as exp (E/kT)
.
This contrasts with the high temperature behaviour where Griffiths [5] showed that the decay of the correlations cannot be faster than the range of interactions i.e. z - ~ in this case, rather than exponential. However since JII is already very long a t the point where crossover be- haviour begins we expect to be in the low temperature regime.Following Takahashi [6] we use the Bogoliubov in-
equality to develop an effective nearest neighbour model which enables us to calculate the low temper- ature properties. For the effective spins on a chain
C8 - 1402 JOURNAL DE PHYSIQUE
interacting with nearest neighbour and dipolar inter- actions
where J I m J ( ~ l a o ) ' and J2 ( z ) cc f ' ~ l z - ~ for z
>>
L. A Bogoliubov approximation is based on the nearestN
neighbour Hamiltonian Ho = -To a;ai+l where i
the free energy F is minimised with respect to Zo lead- ing to
where
and
to,
the correlation length, is given byFrom ( 3 ) we identify 2Zo as the kink energy for well separated kinks,
50
>>
1. The dipolar contribution to the domain wall energy completely dominates over the exchange and consequently ( 3 ) may be writtenwhere we have used an approxima1;e cutoff at z = 2L for the calculated interaction J2 ( z )
.
Therefore as50
-+1
co, To-;;
f ' ~
( L l a o ),
and so the correlation length is given byAexpressing a Boltzmann factor for clomain wall forma- tion when to
>>
l .We conclude that a crossover from three-dimensional t o one-dimensional behaviour near the bulk critical temperature is physically consistent, in terms of the domination of dipolar forces both above and below the crossover.
M. J. W. whishes to acknowledge the Science and Engineering Research council for the provision of a studentship.
111 Aharony, A., Fisher, M. E., Phys. Rev. B 8
(1973) 3323.
[2] Privman, V., Fisher, M. E., .I. Stat. Phys. 33 (1983) 385.
