An integral equation approach to orientational phase transitions in two and three dimensional disordered systems
The use of inhomogeneous Ornstein-Zernike equations to analyze phase transitions and ordered phases in magnetic systems is explored both in bulk three dimensional disordered Heisenberg systems and in a simple model for a two dimensional ferrofluid monolayer. In addition to closures like the Mean...
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Інститут фізики конденсованих систем НАН України
2001
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| Цитувати: | An integral equation approach to orientational phase transitions in two and three dimensional disordered systems / E. Lomba, F. Lado, J.J. Weis // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 45-66. — Бібліогр.: 34 назв. — англ. |
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irk-123456789-1197582017-06-09T03:04:48Z An integral equation approach to orientational phase transitions in two and three dimensional disordered systems Lomba, E. Lado, F. Weis, J.J. The use of inhomogeneous Ornstein-Zernike equations to analyze phase transitions and ordered phases in magnetic systems is explored both in bulk three dimensional disordered Heisenberg systems and in a simple model for a two dimensional ferrofluid monolayer. In addition to closures like the Mean Spherical Approximation, Hypernetted Chain and Zerah-Hansen approximation, the inhomogeneous Ornstein-Zernike equation must be complemented by a one-body closure, for which the Born-Green equation has been used in this paper. The results obtained prove that the proposed approach can furnish accurate estimates for the paramagneticferromagnetic transition in the three dimensional Heisenberg spin fluid, reproducing reliably the structure of the isotropic and ordered phases. In two dimensions, the results are fairly accurate as well, both for the dipolar film alone and in the presence of external perpendicular fields. At high densities/dipole moments the equation seems to predict a transition to a phase in which the dipoles lie mostly in the plane and are aligned into vortex-like structures. Evidence of this new phase is found in the simulation at somewhat higher couplings Використання неоднорідних рівнянь Орнштейна-Церніке для вивчення фазових переходів і впорядкованих фаз в магнітних системах досліджується як у невпорядкованих гайзенбергівських системах так і в простій моделі для двовимірного ферофлюїдного моношару. Неоднорідне рівняння Орнштейна-Церніке, крім таких замикань як середньосферичне, гіперланцюгове і наближення Зера-Гансена, мусить бути доповнене одно-частинковим замиканням, для якого було використано в цій статті рівняння Борна-Гріна. Отримані результати доводять, що запропонований підхід може давати точні оцінки для переходу парамагнетик-феромагнетик в тривимірному гайзенбергівському спіновому флюїді, надійно відтворюючи структуру ізотропної і впорядкованої фаз. У двох вимірах, результати є, безумовно, точними як для дипольної плівки без поля, так і в присутності зовнішніх перпендикулярно направлених полів. При високих густинах/дипольних моментах рівняння передбачають перехід до фази, в якій диполі лежать в основному в площині і утворюють вихороподібні структури. Наявність цієї нової фази є знайдена при дещо сильніших параметрах при моделюванні. 2001 Article An integral equation approach to orientational phase transitions in two and three dimensional disordered systems / E. Lomba, F. Lado, J.J. Weis // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 45-66. — Бібліогр.: 34 назв. — англ. 1607-324X PACS: 61.20.Gy, 68.15.+e, 75.10.-b, 75.30.-m DOI:10.5488/CMP.4.1.45 http://dspace.nbuv.gov.ua/handle/123456789/119758 en Condensed Matter Physics Інститут фізики конденсованих систем НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
The use of inhomogeneous Ornstein-Zernike equations to analyze phase
transitions and ordered phases in magnetic systems is explored both in
bulk three dimensional disordered Heisenberg systems and in a simple
model for a two dimensional ferrofluid monolayer. In addition to closures like
the Mean Spherical Approximation, Hypernetted Chain and Zerah-Hansen
approximation, the inhomogeneous Ornstein-Zernike equation must be
complemented by a one-body closure, for which the Born-Green equation
has been used in this paper. The results obtained prove that the
proposed approach can furnish accurate estimates for the paramagneticferromagnetic
transition in the three dimensional Heisenberg spin fluid, reproducing
reliably the structure of the isotropic and ordered phases. In two
dimensions, the results are fairly accurate as well, both for the dipolar film
alone and in the presence of external perpendicular fields. At high densities/dipole
moments the equation seems to predict a transition to a phase
in which the dipoles lie mostly in the plane and are aligned into vortex-like
structures. Evidence of this new phase is found in the simulation at somewhat
higher couplings |
| format |
Article |
| author |
Lomba, E. Lado, F. Weis, J.J. |
| spellingShingle |
Lomba, E. Lado, F. Weis, J.J. An integral equation approach to orientational phase transitions in two and three dimensional disordered systems Condensed Matter Physics |
| author_facet |
Lomba, E. Lado, F. Weis, J.J. |
| author_sort |
Lomba, E. |
| title |
An integral equation approach to orientational phase transitions in two and three dimensional disordered systems |
| title_short |
An integral equation approach to orientational phase transitions in two and three dimensional disordered systems |
| title_full |
An integral equation approach to orientational phase transitions in two and three dimensional disordered systems |
| title_fullStr |
An integral equation approach to orientational phase transitions in two and three dimensional disordered systems |
| title_full_unstemmed |
An integral equation approach to orientational phase transitions in two and three dimensional disordered systems |
| title_sort |
integral equation approach to orientational phase transitions in two and three dimensional disordered systems |
| publisher |
Інститут фізики конденсованих систем НАН України |
| publishDate |
2001 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/119758 |
| citation_txt |
An integral equation approach to orientational phase transitions in two and three dimensional disordered systems / E. Lomba, F. Lado, J.J. Weis // Condensed Matter Physics. — 2001. — Т. 4, № 1(25). — С. 45-66. — Бібліогр.: 34 назв. — англ. |
| series |
Condensed Matter Physics |
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2025-07-08T16:32:53Z |
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| fulltext |
Condensed Matter Physics, 2001, Vol. 4, No. 1(25), pp. 45–66
An integral equation approach to
orientational phase transitions in two
and three dimensional disordered
systems
E.Lomba 1 , F.Lado 2 , J.J.Weis 3
1 Instituto de Quı́mica Fı́sica Rocasolano, CSIC,
Serrano 119, E-28006 Madrid, Spain
2 Department of Physics, North Carolina State University,
Raleigh, North Carolina 27695-8202, USA
3 Laboratoire de Physique Théorique, Bâtiment 210, Université de
Paris-Sud, 91405 Orsay Cedex France
Received August 1, 2000
The use of inhomogeneous Ornstein-Zernike equations to analyze phase
transitions and ordered phases in magnetic systems is explored both in
bulk three dimensional disordered Heisenberg systems and in a simple
model for a two dimensional ferrofluid monolayer. In addition to closures like
the Mean Spherical Approximation, Hypernetted Chain and Zerah-Hansen
approximation, the inhomogeneous Ornstein-Zernike equation must be
complemented by a one-body closure, for which the Born-Green equa-
tion has been used in this paper. The results obtained prove that the
proposed approach can furnish accurate estimates for the paramagnetic-
ferromagnetic transition in the three dimensional Heisenberg spin fluid, re-
producing reliably the structure of the isotropic and ordered phases. In two
dimensions, the results are fairly accurate as well, both for the dipolar film
alone and in the presence of external perpendicular fields. At high densi-
ties/dipole moments the equation seems to predict a transition to a phase
in which the dipoles lie mostly in the plane and are aligned into vortex-like
structures. Evidence of this new phase is found in the simulation at some-
what higher couplings.
Key words: integral equations, ferromagnetic transitions,
antiferromagnetism, dipolar films
PACS: 61.20.Gy, 68.15.+e, 75.10.-b, 75.30.-m
1. Introduction
Orientational phase transitions in topologically disordered magnetic materials
have attracted a considerable interest over the years. Depending on the dominant
c© E.Lomba, F.Lado, J.J.Weis 45
E.Lomba, F.Lado, J.J.Weis
type of interaction, however, the experimental information is unevenly available.
Thus, in the case of leading magnetic dipole-dipole interactions, ferrofluids have
long been studied in detail [1]. On the contrary, the quest for liquid ferromagnetic
phases led by exchange interactions remained open until the advent of magnetic
levitation techniques [2]. Although evidence had already been found that Co/Au
melts exhibited a disordered paramagnetic phase [3], the systems involved lied more
properly in the glassy domain. Only recently [2], Albrecht and coworkers managed
to undercool a Co80Pd20 melt under its Curie line at zero field levitated in an elec-
tromagnetic container in order to avoid nucleation. This type of experiences have
renewed the interest to the Heisenberg spin fluid, which is the simplest model that
can give a reasonable account of these phase transitions.
But not only bulk systems have attracted the interest of experimentalists, a great
deal of work has been devoted to magnetically active films. This field has been partic-
ularly active given the progress in epitaxial growth techniques. In this way, a variety
of systems have been manufactured by deposition of thin films (a few atomic lay-
ers) of paramagnetic particles (metal oxides) on solid (usually metallic) surfaces [4].
The leading interactions here are the Heisenberg spin-spin exchange and spin-orbit
coupling with the substrate, which gives rise to a field-like term known as surface
anisotropy. Nonetheless, these materials, though disordered, should more properly
be considered as two-dimensional glasses. On the other hand, two-dimensional mag-
netic fluid systems have also recently been studied by Zahn, Mendez-Alcaraz and
Maret [5,6] who prepared a two-dimensional colloid by means of an arrangement of
superparamagnetic colloidal particles (doped with Fe2O3) confined to a water/air
interface. The behaviour of this system is controlled by the magnetic dipole-dipole
interaction and can thus be considered an 2D ferrofluid.
In this context, it is not surprising that theoretical investigations have also flour-
ished in the last years. Particularly, renewed effort has been devoted to devise so-
lution methods for the inhomogeneous Ornstein-Zernike equation [7], being evident
that the classical isotropic homogeneous integral equation approaches could never
throw light into the problem of the structured phases despite its ability to delimit
the boundaries of stability in various relevant systems, such as the dipolar hard
sphere fluid [8,9] and the ferromagnetic [10] and antiferromagnetic Heisenberg spin
fluid [11]. Computer simulation [12–15] and density functional theory [12,16] were for
some time the only available alternative to study the thermodynamics and structure
of the orientationally ordered phases, until Sokolovska [17] presented the solution of
the Mean Spherical Approximation (MSA) for the continuum ferromagnetic phase
of the Heisenberg spin fluid, using Lovett’s equation [18] to relate the one and two
body correlation functions. Independently, Lado and Lomba [19] solved a reference
Zerah-Hansen approximation for the same system, but using the Born-Green equa-
tion to connect one and two-body density correlations. In fact, Zhong and Petscheck
[20], had already solved the Percus-Yevick(PY) approximation for a nematic fluid,
but the assumption of rotational invariance on the Ward identity in the zero field
limit, introduced a completely uncontrolled approximation in their formalism [21].
A more thorough study of the structured phases of the Heisenberg fluid was later
46
Integral equations in magnetic systems
presented by Lado, Lomba and Weis [22] and Perera [23]. The main difference be-
tween these works lies in the fact that Lado and coworkers made use of an expansion
of the correlation function in terms of special orthogonal polynomials constructed
using the one body density correlation as weight function. This approach, inspired
on Lado’s treatment of internal degrees of freedom [24], renders a set of equations for
the anisotropic fluid which is formally identical to that of isotropic systems. Aside
from this computational aspect, Lado and coworkers [19,22] use the BG equation as
one-body closure, while Perera [23] uses Lovett’s equation. Whereas both relations
are exact, the final results are not necessarily equivalent since the approximations
introduced via the two body closure will certainly propagate differently through
each one particle equation. More recently, Holovko and Sokolovska [21] have also
presented the solution of the MSA for the nematic phase of a simple separable po-
tential model using Lovett’s equation as one particle closure. On the other hand,
dipolar fluids have proven more cumbersome to tackle, given the theoretical diffi-
culties associated with the long range of the dipolar interaction, and the additional
complication derived from dependence of the potential on the orientation of the in-
terparticle vector, but also quite recently Klapp and Patey [25] have presented the
solutions of the Reference Hypernetted Chain (RHNC) equation and the MSA for
the dipolar hard sphere ferroelectric phases, using both Lovett’s and the BG one
particle closures. Along the same lines, Lomba, Lado and Weis [26] have also consid-
ered a two-dimensional ferrofluid formed by dipolar hard spheres lying on a surface.
Here, anisotropy results from the constraint imposed on the dipolar spheres, which
although able to freely rotate in space, must have their centers situated in the plane,
and from the presence of external fields. Spontaneous symmetry ruptures, though
sought, do not seem to occur in this system under the conditions studied, even if
there seems to be evidence of the formation of vortex structures for high couplings.
It is our intention in this contribution to summarize the essential features of
the application of the inhomogeneous OZ approach to bulk Heisenberg magnetic
fluids, both with ferromagnetic [19,22] and antiferromagnetic [11,27] interactions.
The approach followed will be the one proposed by Lado and coworkers [19,22], al-
though we will attempt to make connections with other formalisms where needed.
The straightforward extension to deal with systems with three dimensional inter-
actions constrained in surfaces will also be sketched and its application to dipolar
ferrofluids illustrated. The next section will be devoted to a short presentation of
the method. Then section III will focus on the specifics of ferro and antiferromag-
netic spin fluids and finally in section IV we will briefly discuss the peculiarities of
ferrofluid monolayers.
2. The formalism
In order to make the derivations more explicit, we specialize here on the Heisen-
berg spin fluid. In the next section the specific features of the dipolar case in two
dimensions will be addressed.
The potential energy of a system composed of N hard spheres with embedded
47
E.Lomba, F.Lado, J.J.Weis
Heisenberg spins can be expressed
UN =
∑
i<j
[u0(rij) + uss(rij , ωi, ωj)] . (1)
Here u0(r) is the hard sphere potential for spheres of diameter σ and uss(r, ω1, ω2)
for r > σ is the Heisenberg spin-spin interaction,
uss(r, ω1, ω2) = −J(r) ŝ1 · ŝ2, (2)
βJ(r) = K
e−κ(r/σ−1)
r/σ
, (3)
where β = 1/kBT , with kB Boltzmann’s constant and T the Kelvin temperature. The
orientations ω = (θ, φ) of the unit spins ŝ are referred to a uniform field B0, which
defines the z direction. In the case of spontaneously broken symmetry in the zero field
case, following Bogolubov [28], one resorts to the infinitesimal field which stabilizes
the system [21]. In equation (3), the dimensionless coupling strength K may be read
as the inverse reduced temperature, K = 1/T ∗, while κ is a dimensionless range
parameter. A positive K will favour parallel alignments (ferromagnetic ordering)
and a negative value will favour antiparallel alignment (antiferromagnetic ordering).
The key quantities needed for a complete magnetic and thermodynamic descrip-
tion of this system are the one-body and two-body density functions,
ρ(1)(r, ω) =
ρ
4π
f(ω), (4)
ρ(2)(r, ω, r′, ω′) =
ρ2
(4π)2
f(ω)f(ω′)g(|r− r
′|, ω, ω′), (5)
where ρ = N/V is the density, f(ω) the one-body orientational distribution in the
interacting fluid, and g(r, ω, ω ′) the pair distribution function of the inhomogeneous
spin system in an external magnetic field.
The basic equations that determine the distribution functions f(ω) and g(r, ω, ω ′)
are well known. The one-body density can be differentiated with respect to x = cos θ
to give
d
dx
ln
[
f(ω)
f0(ω)
]
= − ρ
4π
∫
dr dω′ f(ω′)g(r, ω, ω′)
dβuss(r, ω, ω
′)
dx
, (6)
the first member of a Kirkwood-Born-Green-Yvon hierarchy. Here f0(ω) is the nor-
malized one-body orientational distribution of a noninteracting spin system,
f0(ω) =
eβµB0 cos θ
sinh(βµB0)/βµB0
, (7)
where µ is the spin dipole moment. Calculation of f(ω) from equation (6) requires
knowing g. Alternatively, one can use Lovett’s equation [18] which connects the
one-body orientational distribution and the direct correlation function c(r, ω, ω ′) via
∇ωf(ω) =
∫
c(r, ω, ω′)∇ω′f(ω′)drdω′. (8)
48
Integral equations in magnetic systems
Both relations are exact, but obviously will lead to different results as long as the
pair correlations functions are only approximated. In this regard, in classical liquid
state theory, the pair distribution and direct correlation functions are obtained from
the Ornstein-Zernike (OZ) equation and a closure relation respectively,
γ(r12, ω1, ω2) =
ρ
4π
∫
dr3dω3 f(ω3)[γ(r13, ω1, ω3) + c(r13, ω1, ω3)]c(r32, ω3, ω2), (9)
c(r, ω1, ω2) = exp[−βu0(r)− βuss(r, ω1, ω2) + γ(r, ω1, ω2) + b(r, ω1, ω2)]
− 1− γ(r, ω1, ω2). (10)
The first of these connects the indirect correlation function γ = g − 1 − c with the
direct correlation function c. The second, or closure, relation expresses c back in
terms of γ and the model’s pair interactions. This relation must be supplemented
with an approximation for b, the so-called bridge function. This can be approximated
in various ways giving rise to a variety of closures, like the reference-hypernetted
chain (RHNC) closure [29] or the reference version of the Zerah-Hansen (RZH)
closure [30]; for r > σ, these are
cRHNC(r, ω1, ω2) = exp[−βuss(r, ω1, ω2) + γ(r, ω1, ω2) + bHS(r; σHS)]
− 1− γ(r, ω1, ω2), (11)
cRZH(r, ω1, ω2) =
{
exp
(
m(r)
[
−βuss(r, ω1, ω2) + γ(r, ω1, ω2) + bHS(r; σ)
])
− 1
}
/m(r)− γ(r, ω1, ω2). (12)
In equation(11) bHS(r; σHS) represents the bridge function of a fluid of hard spheres
of diameter σHS. For the potential models considered in this contribution we have
simply used σHS = σ. The RZH closure features a mixing function m(r) = 1 −
exp(−αr) with a parameter α that is fixed by requiring consistency between the
virial and compressibility bulk moduli [30]. In the limiting case of α → 0, the RZH
closure simply reproduces the MSA closure.
The usual approach for orientation-dependent functions such as γ(r, ω1, ω2) is to
expand in spherical harmonics as both Sokolovska [17] and Perera [23] did, leading
to the appearance of matrix terms of the form
Rklµ =
∫
ρ(ω)Ykµ(ω)Ylµ̄(ω)d(ω) (13)
(or Aklµ in the notation of Klapp and Patey [25] and Y klµ in Sokolovska’s work [17])
once the OZ equation is decoupled into its components. These terms appear both
in the equation itself and in the expressions to evaluate thermodynamic properties.
Here we will instead expand in modified spherical harmonics,
γ(r, ω1, ω2) = 4π
∑
l1,l2,m
γl1l2m(r)Yl1m(ω1)Yl2m̄(ω2),
Ylm(ω) =
1√
4π
(−1)meimφPlm(cos θ). (14)
49
E.Lomba, F.Lado, J.J.Weis
The modified Legendre functions Plm(cos θ) are explicitly constructed using the
Gram-Schmidt method with the orthonormality condition
1
2
∫ 1
−1
dx f(x)Plm(x)Pl′m(x) = δll′, (15)
where f(cos θ) is the one-body distribution of the fluid. Obviously this construction
eliminates the Rklµ matrices from all subsequent expressions and as will be seen
yields equations for the coefficients completely analogous to those of the isotropic
integral equations.
Now, the OZ equation (9) deconvoluted by Fourier transformation and with the
pair functions expanded as in (14) becomes
γ̃l1l2m(k) = (−1)mρ
∑
l3
[γ̃l1l3m(k) + c̃l1l3m(k)]c̃l3l2m(k).
The significant feature here is that this OZ equation for an inhomogeneous fluid in
an external field is now identical to that of an ordinary homogeneous fluid and so,
along with a closure equation, can be solved for g(r, ω1, ω2) with the same familiar
algorithms already used for homogeneous systems [31].
This solution for g(r, ω1, ω2) is obtained using the polynomials P lm(x) generated
with the current f(x). We now return to equation (6) and update the one-body
distribution. In expanded form, this BG equation is
d
dx
ln
[
f(x)
f0(x)
]
=
∑
l1,l2,m
ξl1l2mPl1m(x)
dPl2m(x)
dx
, (16)
ξl1l2m = −ρ
∫
dr
∑
l3
gl1l3m(r)ul3l2m̄(r), (17)
where the ul1l2m(r) are the (known) coefficients of the spin-spin interaction
uss(r, ω1, ω2), so that finally
ln f(x) = ln f0(x) +
∞
∑
l=0
alPl0(x). (18)
Here al for l > 0 is determined by numerical (Gaussian) integration of equation (16)
and a0 by normalization.
The iterations for f(ω) and g(r, ω1, ω2) are continued until both functions are self-
consistently determined. We can now compute the complete magnetic and thermo-
dynamic properties of a Heisenberg spin fluid in a uniform magnetic field B 0 = B0k̂
or in the ferromagnetic phase. In particular, we find for the longitudinal and trans-
verse magnetic susceptibilities the following:
χzz/ρβµ
2 = σ2
x
[
1 + ρh̃110(0)
]
+ 〈x〉σx
[
ρh̃100(0) + ρh̃010(0)
]
+ 〈x2〉
[
1 + ρh̃000(0)
]
, (19)
χyy/ρβµ
2 =
1
2
(
1−
〈
x2
〉) [
1− ρh̃111(0)
]
, (20)
50
Integral equations in magnetic systems
where σ2
x ≡ 〈x2〉 − 〈x〉2 for x = cos θ. In connection with the susceptibilities there
is an important aspect that deserves to be born in mind. It is well known that the
spontaneous symmetry breaking associated with the ferromagnetic transition gives
rise to transverse fluctuations, known as Goldstone modes, which reflect in a di-
vergence of the transverse magnetic susceptibility. It has been proved that Lovett’s
equation automatically reproduces the presence of Goldstone modes in orientation-
ally ordered phases [17,23,21], but this is not evident when the BG equation is used.
In fact, we have found that the MSA transverse magnetic susceptibility remains
finite in the ferromagnetic phases at zero field, whereas the RHNC equation breaks
down at small but finite fields, signalling the presence of a divergence. Therefore,
we have taken advantage of these features to obtain zero field results from the RZH
approximation. Being a hybrid between MSA and RHNC one can always obtain a so-
lution for α = 0 and then increase its value until the divergence is reached. We have
found that the values in the immediate vicinity of the divergence agree extremely
well with zero-field computer simulation estimates. A more elegant approach would
certainly imply an analytic treatment of the divergence following Perera [23]. On the
other hand, one might perform the calculations at small field as Klapp and Patey
[25] suggested. Sokolovska [17] uses Baxter’s method to analytically solve the MSA
and consequently her treatment is not affected by the divergence.
3. Bulk Heisenberg fluids: ferromagnetic and antiferromagn et-
ic ordering
Here we present a set of calculations is for κ = 1, ρσ3 = 0.7, K = 1/T ∗ = 0 to
0.5, and two values of the external field, βµB0 = 1 and 0, using the RZH closure.
We find for these cases that the one-body orientational distribution function f(x)
continues to be well described by the functional form of f0(x), but with an effective
field B,
f(x) =
eβµBx
sinh(βµB)/βµB
. (21)
In the limit B0 → 0, the effective field B is zero for K < Kc and finite for K > Kc,
where Kc = 1/T ∗
c is then the computed Curie point.
For K > Kc, there is a singularity when B0 → 0 due to the presence of the
Goldstone modes. In fact, the B0 = 0 line corresponds in the B0–M plane (where
M is the magnetization per particle) to the spinodal line that indicates the equilib-
rium between phases with positive and negative magnetization. Consequently, the
transverse susceptibility χyy (but not the longitudinal component χzz) will diverge
as B0 → 0, reflecting the negligible cost of rotating an ordered sample in the absence
of an external field.
Using Monte Carlo simulation, Nijmeijer and Weis [13] found that the paramag-
netic-ferromagnetic transition for ρσ3 = 0.7 with the truncated potential occurs
at Kc = 0.264 ± 0.001. Our calculation in the paramagnetic phase and zero field
limit yields a divergence in χzz at Kc
l = 0.2645, in excellent agreement with the
simulation value.
51
E.Lomba, F.Lado, J.J.Weis
0.0
0.2
0.4
0.6
0.8
M
=
<
co
sθ
>
0.1 0.2 0.3 0.4 0.5
K
0.0
0.2
0.4
0.6
S
=
<
P 2(
co
sθ
)>
K
l
c
Figure 1. Magnetization per particle M and second order parameter S as func-
tions of inverse temperature K = 1/T ∗ obtained from MC simulation (squares
for βµB0 = 1, and white circles [2048 particle sample] and black diamonds [1372
particle sample] for βµB0 = 0) and the RZH integral equation (solid lines). The
dashed lines correspond to a power law fit to the RZH data just above Ku
c . The
dash-dot curves represent density functional theory estimates.
52
Integral equations in magnetic systems
1.0 1.5
−0.5
1.5
3.5
1.0 2.0 3.0 4.0 5.0
r/σ
−0.5
0.5
1.5
ρ11
0 (r
)−
3M
2
Figure 2. The ρ110(2) (r) angular correlation function from simulation (open circles)
and the RZH integral equation (solid lines) for K = 0.5 with and without an
external magnetic field. The long-range limiting value 3M 2 has been subtracted
from both theory and simulation results to ease the comparison between the two
temperatures. The discontinuity at r = 2.5σ is due to the potential truncation.
In figure 1 we show the values of the magnetization per particle M ≡ Mz =
〈cos θ〉 and the second order parameter S = 〈P2(cos θ)〉 obtained from the theory
and from a standard NVT Monte Carlo simulation using 864 (for B0 = 1), 1372 and
2048 particles (for B0 = 0), and averages over 4 × 104 configurations. Size effects
are noticeable for B0 = 0 close to the critical point. In the vicinity of the critical
temperature, one encounters convergence difficulties in solving the integral equation
as spin-spin correlations become long ranged. Both χzz and χyy in equations (19) and
(20) diverge, with χzz exhibiting the characteristic λ-type divergence of second-order
transitions.
In order to get an estimate of the critical inverse temperature from the ordered-
phase results, we fit the RZHmagnetization in the vicinity of the critical temperature
(K < 0.3) to a power law,
M = a(K −Ku
c )
β, (22)
which leads to a = 1.694, β = 0.397, andKu
c = 0.254. The fitted value of β is close to
the value β = 0.387 reported by Nijmeijer and Weis [13] and to the critical exponent
of the 3D lattice Heisenberg model, β = 0.362± 0.004 [32] and β = 0.3639± 0.0035
[33]. However, the available data from the integral equation magnetization are too
far away from Kc to draw here any conclusion with confidence and this agreement
53
E.Lomba, F.Lado, J.J.Weis
might simply be fortuitous. Nevertheless, a similar fit carried out on the second-
order parameter S produces the same estimate of Ku
c and so we might conclude that
this is the theoretical estimate of the critical inverse temperature derived from the
ordered-phase data. The agreement between the critical point estimatesKc
l = 0.2645
obtained in the isotropic phase and Ku
c = 0.254 obtained in the ordered phase can
be regarded as good.
In figure 2 we plot one of the averaged angular correlations most representative of
the orientational ordering and which in the isotropic case reduces to the standard h110
coefficient of the expansion of the pair distribution function in rotational invariants.
In the anisotropic case, the corresponding average can be related to the coefficients
of the expansion in special orthogonal polynomials by means of
ρ110(2) (r)/ρ
2 ≡ 3
〈
ρ(2)(12)/ρ2φ110(12)
〉
ω1ω2
= 3
[
〈x〉2 g000(r) + 2 〈x〉σxg
010(r) + σ2
xg110 − (1−
〈
x2
〉
)g111(r)
]
. (23)
We have found that both the structural and thermodynamics properties are
correctly predicted by the RZH equation in zero field. For finite fields the RHNC
equation is slightly better and has the advantage that properties like the free energy
can be directly evaluated from the correlation functions [22]. We are thus in a po-
sition to produce estimates for the phase diagram using the RZH equation at zero
field via thermodynamic integration, and applying a double tangent construction to
the RHNC free energies at finite fields. We also consider here a fully aligned spin
system, i.e., a spin fluid under the action of an infinitely strong external field. The
phase equilibria results are presented in figure 3, along with GEMC data. Comparing
the infinite field (i.e. pure Yukawa) and the zero field results in this figure, one notes
that the equilibrium densities are only slightly affected by the external field. We
have therefore chosen a relatively large field, βµB0 = 16, to perform an additional
set of calculations at nonzero but finite field. The results obtained from the RHNC
approximation, seen in figure 3, although relatively good, are somewhat worse than
those obtained for the pure Yukawa; this is a direct consequence of the neglect of the
optimization condition for finite field. The situation is slightly worse using the RZH
equation in the zero field case. Here the use of thermodynamic integration based
on the energy route, which yields extremely good thermodynamics, is too time con-
suming, since it has to be performed for every density needed to map the isotherms
required for the double tangent construction. Consequently, we have used the virial
route starting from low density results, which is somewhat poorer, since RZH viri-
al pressures are not as accurate as the corresponding internal energies. Therefore,
we have only calculated two equilibrium points (each implies one hundred integral
equation solutions) which are shown in the lower part of figure 3.
Finally, both theory and simulation show that the effect of an external field
on the spin system (and presumably also on dipolar fluids [34]) is a considerable
increase in the critical temperature, while equilibrium densities are not significantly
affected; i.e., external fields tend to stabilize the liquid phase.
Now, if the sign of the coupling constant K in equation(3) is changed the sys-
tem will now exhibit antiferromagnetic behaviour. Already in [11] it was found that
54
Integral equations in magnetic systems
1.0
1.2
1.4
1.6
T
*
0.0 0.2 0.4 0.6 0.8 1.0
ρσ3
1.0
1.2
1.4
T
*
βµB0=∞βµB0=16
βµB0=0
Figure 3. Phase diagram of the Heisenberg spin fluid in the presence of an ex-
ternal field (upper figure) and at zero field (lower figure). Simulation data are
shown as black circles and RHNC results as solid lines. In the lower figure, the
dash-dot line indicates density functional theory results taken from [12] while
white diamonds represent two equilibrium points obtained from thermodynamic
integration of the RZH results.
55
E.Lomba, F.Lado, J.J.Weis
Table 1. Computed order parameter and thermodynamic properties of the anti-
ferromagnetic Heisenberg spin fluid for ρσ3 = 0.8 and κ = 6.
K 〈P2(cos θ12〉 a βP/ρ βU/N
RZH MC RZH RZH RZH MC
–4.0 0 – 0 3.612 –2.834 –
–5.0 0 – 0 2.034 –4.144 –
–5.5 0 – 0 0.625 –4.650 –
–5.7 0 – 0 0.135 –4.910 –
–5.8 0.502 – –3.50 –0.303 –5.894 –
–5.9 0.553 – –3.96 –0.685 –6.181 –
–6.0 0.581 0.65 –4.24 –0.835 –6.470 –7.8
–7.0 0.745 – –6.66 –3.170 –8.942 –
–8.0 0.811 0.81 –8.65 –5.589 –11.232 –12.7
the paramagnetic-antiferromagnetic transition occurs at much lower temperatures
(higher couplings) than the the ferromagnetic transition. This was particularly evi-
dent given the difficulties encountered by the simulation to attain antiferromagnet-
ic ordering, which apparently takes much longer to build. This is connected with
the fact that, whereas the propagation of ferromagnetic ordering is energetically
favourable (all layers of parallel particles tend to lower the energy of the system)
in the case of antiferromagnetic interactions successive layers of ordered spins have
opposing contributions, i.e., first neighbours are antiparallel (lowering the energy)
whereas second neighbours being parallel tend to increase the energy, and so on.
The conditions for the formation of antiferromagnetic phases are then too severe to
be easily handled by the simulation procedures and hinder the convergence of inte-
gral equations. However, we have found that decreasing the range of the interaction
translates into a considerable lowering of the critical Kc. Thus, whereas isotropic
RHNC calculations indicate that, at ρσ3 = 0.8, the transition occurs for Kc ≈ 10
for a screening factor κ = 1, changing this to κ = 6 (which implies that practically
only nearest neighbours are involved in the interaction) reduces the critical coupling
to Kc ≈ 4.3. Nonetheless, these conditions also pose important problems to the sim-
ulation procedures, in particular since the low temperature combined with the short
range translates into a certain ‘sticky’ character that induces a considerable amount
of clustering. Thermodynamics and order parameters obtained for this model using
RZH and simulation for various states are collected in table I. These results are still
very preliminary but indicate that despite the difficulties the method can still be
applied to antiferromagnetic systems. It turns out that the one particle distribution
function in these systems follows a functional form
f(x) =
eax
2
2eD(1)
, (24)
where D(t) is a Dawson’s integral and a > 0. In the relative large region around
56
Integral equations in magnetic systems
the antiferromagnetic transition the integral equation experiments considerable con-
vergence difficulties, and we have found that a simple density functional theory of
the type developed in [12] predicts this transition to be slightly first order. All this
clearly indicates that the antiferromagnetic system deserves further study.
4. Ferrofluid monolayers
The interaction energy for a configuration of N sphere centers rj with embedded
three-dimensional magnetic moments µj in a plane area A is
U =
∑
i<j
u0(rij) +
∑
i<j
udd(rij, ωi, ωj)−
∑
j
µj ·B0, (25)
where
udd(r12, ω1, ω2) = − µ2
r312
[3(r̂12 · µ̂1)(r̂12 · µ̂2)− µ̂1 · µ̂2] (26)
is the dipole-dipole potential, and B0 a uniform magnetic field perpendicular to the
plane. Even in the absence of the magnetic field, the planar arrangement of the point
dipoles produces anisotropy.
With the particles laying on a plane and placing the x-axis along the intermolec-
ular axis, (26) reduces to
udd(r12, θ1, θ2, φ) = −µ2
r3
[
3
2
sin θ1 sin θ2 cos(φ1 + φ2)
+
1
2
sin θ1 sin θ2 cos(φ1 − φ2)− cos θ1θ2
]
, (27)
where θi denotes the angle formed by dipole i with the plane normal and φ i is the
azimuthal angle. Clearly one can consider this as purely one dimensional problem
with particles carrying an additional internal degree of freedom quantified by θ i. We
can then simply make use of the expressions introduced in the previous section with
minor changes to reflect the different symmetry of the problem and the reduction in
dimensionality. Thus, first we will have to expand the correlation functions in our
new set or orthogonal polynomials as,
γ(r, ω1, ω2) = 4π
∑
l1,l2,m1,m2
γm1m2
l1l2
(r)Yl1m1
(ω1)Yl2m̄2
(ω2). (28)
Similarly, Fourier transforms, which will be used to deconvolute the OZ equation,
are also conveniently expanded with the x axis along the planar vector k. A two-
dimensional Fourier transform of the correlation functions is then evaluated as fol-
lows. Choose the x axis along k. Then using (28) we have
γ̃(12) =
∫
dr γ(12)eikr cosϕr
= 4π
∑
l1,l2,m1,m2
∫ ∞
0
dr rγm1m2
l1l2
(r)
57
E.Lomba, F.Lado, J.J.Weis
×
∫ 2π
0
dϕre
ikr cosϕre−i(m1−m2)ϕrYl1m1
(ω1)Yl2m̄2
(ω2)
= 4π
∑
l1,l2,m1,m2
γ̃m1m2
l1l2
(k)Yl1m1
(ω1)Yl2m̄2
(ω2), (29)
where
γ̃m1m2
l1l2
(k) = 2πim1−m2
∫ ∞
0
dr rγm1m2
l1l2
(r)J|m1−m2|(kr). (30)
Here, Jm(x) is the Bessel function of order m generated by the integral over ϕr.
Similarly, an inverse transform is
γm1m2
l1l2
(r) =
1
2πim1−m2
∫ ∞
0
dk kγ̃m1m2
l1l2
(k)J|m1−m2|(kr). (31)
It follows from the circular symmetry of the system that |m1−m2| must be an even
integer.
We find by calculation that in this case the angular distribution function can be
cast into the form
f(x) =
1
C
exp
(
βµ2Kx2 + βµBx
)
, (32)
In equation (32), K and B are effective values of the magnetic surface anisotropy
and external magnetic field, respectively, while the normalization constant C is given
by
C =
1
4
√
π
a
eb
2/4a
[
erf
(
2a− b
2
√
a
)
+ erf
(
2a + b
2
√
a
)]
. (33)
where a = βµ2K and b = βµB. In contrast with the antiferromagnetic fluid, here
a < 0 reflecting the symmetry of the angular distribution at zero field.
A comparison between our Monte Carlo data for f(x) and the results from the
RHNC, integral equation is displayed in figure 4 for various external fields. Given
that the magnitude of common magnetic dipoles is appreciably smaller than that of
their electric counterparts, we have only considered a relatively moderate reduced
dipole, µ∗ = (βµ2/σ3)1/2 = 1. The agreement seen in the figure between the inte-
gral equation and the simulation results is excellent. The most noticeable feature
observed is the breaking of the symmetry of the distribution with respect to the
plane (x = cos θ = 0) due to the external field. We have found that obviously the
response of the dipoles to the field is weaker at higher densities. This effect of density
is readily understandable, since the dipole-dipole interaction favouring head-to-tail
alignments increases the tendency of the dipoles to remain in-plane. The same effect
can also be seen in the zero-field truncated Gaussian distribution of figure 4, which
becomes progressively more sharply peaked about x = 0 as the density increases,
approaching a fully coplanar distribution. A similar effect can be induced by an
increase of the dipole moment.
As to the pair structure of the system, in dipolar fluids, in addition to the center-
to-center pair distribution function g000(r) ≡ g0000(r), the most significant angular
correlations are represented by the ensemble shell averages – projections of the
normalized two particle density ρ(2)(12)/ρ2– of the standard rotational invariants
58
Integral equations in magnetic systems
0.2
0.3
0.4
0.5
0.6
0.7
f(
co
sθ
)
MC
MC (LSF) y=0.61exp(−0.71x
2
+0.34x)
RHNC
DFT
0.3
0.4
0.5
0.6
0.7
MC
MC (LSF) y=0.61 exp(−0.67x
2
)
RHNC
DFT
−1 −0.5 0 0.5 1
cosθ
0
0.5
1
1.5
2
2.5 MC
MC (LSF) y=0.47exp(−0.70x
2
+1.45x)
RHNC
DFT
ρσ2=0.8, βµ2/σ3=1.0
βµΒ0=0
βµΒ0=1
βµΒ0=4
Figure 4. Angular distribution function f(cos θ) in a ferrofluid monolayer of dipo-
lar hard spheres at ρσ2 = 0.8 and various external fields. Open circles denote MC
data and solid line RHNC results. The dotted line tracking the MC points is a
least squares fit (LSF) whose coefficients are shown in the figure.
59
E.Lomba, F.Lado, J.J.Weis
φ110 = µ̂1 ·µ̂2 and φ112 = 3(µ̂1 · r̂)(µ̂2 · r̂)−µ̂1 ·µ̂2, which in the isotropic case reduce
to the coefficients h110(r) and h112(r). These quantities are connected respectively
to the relative orientation of a pair of dipoles at a given separation and to the
contribution of a given spherical shell to the dipolar excess energy. In terms of the
calculated coefficients gm1m2
l1l2
(r) of this work, the relevant averaged quantities are
given by
ρ110(2) (r)/ρ
2 ≡ 3
〈
ρ(2)(12)/ρ2φ110(12)
〉
ω1ω2
= 3
[
〈x〉2 g0000(r) + 2 〈x〉σxg
00
01(r) + σ2
xg
00
11 − (1− 〈x2〉)g1111(r)
]
(34)
and
ρ112(2) (r)/ρ
2 ≡ 3
2
〈
ρ(2)(12)/ρ2φ112(12)
〉
ω1ω2
= −3
2
[
〈x〉2 g0000(r) + 2〈x〉σxg
00
01(r) + σ2
xg
00
11(r) +
1
2
(1− 〈x2〉)g1111(r)
− 3
2
(1− 〈x2〉)g1−1
11 (r)
]
. (35)
The correlation functions obtained from the RHNC integral equation are com-
pared with MC simulation data in figures 5–6 for various densities and external fields.
It can be appreciated that the optimized RHNC integral equation provides an excel-
lent description of the microscopic structure of the dipolar fluid, with and without
external field, except in the case of ρσ2 = 0.8. Here the integral equation renders a
much more pronounced orientational structure while the spatial ordering is exactly
reproduced. This discrepancy will be further discussed below when analyzing the
thermodynamic properties. The behaviour at large separations is well reproduced,
including the crossover of ρ110(2) (r) and ρ1122 (r). We have also found that the contact
values of the correlation functions decrease slightly as the field is augmented; this
is due to the fact that the out-of-plane alignment induced by the field introduces
repulsive dipole-dipole interactions. This effect is obviously less significant at low
densities.
It is worth noting that we have encountered convergence problems in the RHNC
solutions for reduced densities above 0.8. As can be seen in figure 7, this corre-
sponds to a region where the transverse (i.e., in-plane) susceptibility starts to rise
appreciably and where the orientational order (see figure 6) predicted by the integral
equation is more pronounced than what is actually found in the simulation. This
orientational structure disagreement is in consonance with the discrepancies in the
behaviour of the transverse susceptibility depicted in figure 7. Thus, whereas the
increase in the theoretical susceptibility and the difficulties in convergence seem to
indicate that the system might be close to some sort of in-plane order-disorder tran-
sition, the simulation predicts no anomalous behaviour for these high density states.
Nonetheless, at somewhat higher dipole moment the simulation results start to show
a clear head-to-tail in-plane alignment of the dipoles with the formation of vortex
structures, as can be appreciated in figure 8. It thus might happen that the integral
equation underestimates the value of the transition dipole moment. Alternatively,
60
Integral equations in magnetic systems
1 2 3 4 5 6
r/σ
0
0.5
1
ρkl
m
(r
)
1 1.25
0
1
2
3
1
2
3
4
g00
0 (r
)
ρσ2=0.7, βµΒ0=4, βµ2/σ3=1
Figure 5. Radial projection g000(r) and angular projections ρklm(2) (r) = ρ110(2) (r)
& ρ112(2) (r)/ρ
2 of the normalized pair density function ρ(2)(12)/ρ
2 in a ferrofluid
monolayer of dipolar hard spheres at ρσ2 = 0.7 and βµB0 = 4. Here and in
figure 6 the angular projection curve with the largest contact value corresponds
to ρ112(2) (r) for βµB0 = 0 and ρ110(2) (r) for βµB0 = 4. Open circles denote MC data
and solid lines represent RHNC results.
61
E.Lomba, F.Lado, J.J.Weis
1 2 3 4 5 6
r/σ
0
0.2
0.4
hkl
m
(r
)
1 1.25
0
1
2
3
4
5
1
2
3
4
5
g00
0 (r
)
ρσ2=0.8, βµΒ0=0, βµ2/σ3=1
Figure 6. Same as figure 5 for ρσ2 = 0.8 and βµB0 = 0.
62
Integral equations in magnetic systems
0 0.2 0.4 0.6 0.8
ρσ2
0
0.5
1
1.5
χ T
/ρ
βµ
2
0
0.2
0.4
0.6
χ L
/ρ
βµ
2
Figure 7. Longitudinal and transverse magnetic susceptibilities of a ferrofluid
monolayer in the RHNC approximation (lines) and from MC simulation (sym-
bols). Solid lines (circles) correspond to βµB0 = 0, dotted lines (squares) to
βµB0 = 1, and dash-dotted lines (diamonds) to βµB0 = 4. The error on the MC
results for the susceptibilities is about 5%.
63
E.Lomba, F.Lado, J.J.Weis
10 0 10
10
0
10
X
Y
10 0 10
10
0
10
X
Y
ρσ2 = 0.8, βµ2/σ3 = 1 ρσ2 = 0.8, βµ2/σ3 = 4
Figure 8. Planar projection of simulation snapshots of a ferrofluid monolayer
for two different dipole moments without external field. The arrows indicate the
projection of the three dimensional magnetic dipole.
it could be the case that the ordering process for a low dipole moment in a finite
system is an extremely slow one that is only captured by the simulation at lower
temperatures, although calculations for various systems sizes and lengths of the run
do not seem to support this possibility. The situation thus remains inconclusive,
though we are of the opinion that the lack of convergence in the integral equation
procedure is indicating the onset of some in-plane ordering.
In summary, in this contribution, it has been shown that the inhomogeneous
Ornstein-Zernike relation, coupled with a one body relation, either BG or Lovett’s
equation, is a powerful tool to study orientational transitions and orientationally
ordered phases in a wide range of systems and physical conditions. Still, a more
thorough comparison between different one body closures remains to be done and
our information on the antiferromagnetic system is far from complete.
5. Acknowledgments
E.L. thanks the Dirección General de Enseñanza Superior e Investigación Cien-
t́ıfica (Spain) for support of this work under Grant No. PB97-0258-C02-02.
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E.Lomba, F.Lado, J.J.Weis
Підхід інтегральних рівнянь до орієнтаційних
фазових переходів у дво- і тривимірних
невпорядкованих системах
Е.Ломба 1 , Ф.Ладо 2 , Дж.Дж.Вейс 3
1 Інститут фізичної хімії
Серрано 119, Е-28006 Мадрід, Іспанія
2 Державний університет північної Кароліни,
27695-8202, США
3 Лабораторія теоретичної фізики,
Паризький університет, 91405 Орсей, Франція
Отримано 1 серпня 2000 р.
Використання неоднорідних рівнянь Орнштейна-Церніке для вив-
чення фазових переходів і впорядкованих фаз в магнітних системах
досліджується як у невпорядкованих гайзенбергівських системах так
і в простій моделі для двовимірного ферофлюїдного моношару. Не-
однорідне рівняння Орнштейна-Церніке, крім таких замикань як се-
редньосферичне, гіперланцюгове і наближення Зера-Гансена, му-
сить бути доповнене одно-частинковим замиканням, для якого бу-
ло використано в цій статті рівняння Борна-Гріна. Отримані резуль-
тати доводять, що запропонований підхід може давати точні оцін-
ки для переходу парамагнетик-феромагнетик в тривимірному гай-
зенбергівському спіновому флюїді, надійно відтворюючи структуру
ізотропної і впорядкованої фаз. У двох вимірах, результати є, без-
умовно, точними як для дипольної плівки без поля, так і в присутнос-
ті зовнішніх перпендикулярно направлених полів. При високих густи-
нах/дипольних моментах рівняння передбачають перехід до фази, в
якій диполі лежать в основному в площині і утворюють вихороподібні
структури. Наявність цієї нової фази є знайдена при дещо сильніших
параметрах при моделюванні.
Ключові слова: інтегральні рівняння, феромагнітні переходи,
антиферомагнетизм, дипольні плівки
PACS: 61.20.Gy, 68.15.+e, 75.10.-b, 75.30.-m
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