Phase Transitions in Systems with Finite Number of Atoms

1.5 MB·English

by  Yury Kirienko

Phase Transitions in Systems with Finite Number of Atoms
Yury Kirienko1,∗ and Leonid Afremov2
Far-Eastern Federal University, Vladivostok, Russia
[email protected], [email protected], ∗corresponding author
Keywords: ordering, Curie temperature, magnetic phase transition, concentration phase transition,
particles with a finite number of atoms.
2
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Abstract.PropertiesofnanoparticleshavebeenstudiedwithintheframeworkofIsingmodelandthe
2
method of random-field interactions: the average magnetic moment and position of critical points of
n
a themagneticandtheconcentrationphasetransitionsdependingontheirsize.ItisshownthattheCurie
J
temperature is inversely proportional to the size of the particle. Critical concentration of the ordered
7
state decreases with increasing of size of nanoparticles until the percolation threshold of massive

] particles.
l
l
a
h
- Introduction
s
e
m Influenceofsurfaceonthemagneticpropertiesofnanoparticlesbecomessubstantialwithdecreasing
. of their size, if the number of atoms on the surface is comparable (or exceeds) the number of atoms
t
a
insidetheparticle [1].Thusthelow-temperature(97K)measurements [2,3]revealedthatthespecific
m
magnetic moment (cid:104)µ(cid:105) decreases with increasing of number of atoms in nanoparticles of iron, cobalt
-
d andnickeluptothevaluesofthemagneticmomentofmassivesample.Incontrasttonanoparticlesof
n
3d-metals mentioned above, for rare-earth elements the magnetic moment increases with the growth
o
c ofnumberofatoms [2,4].Itisshown [5],thatthesaturationmagnetizationofnanoparticlesofα-Fe
[
(with sizes up to 7 nm) is much smaller than in massive iron. We should note the fact, that the Curie

1
temperaturedecreaseswithdecreasingoftheparticlesize [6].
v
2 These features of the magnetic properties of nanoparticles may be associated with a reduction
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(compared to the massive particles) in the number of nearest neighbors and, consequently, the weak-
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1 eningoftheinteractionbetweentheatoms.
.
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0
Model
2
1
: Magnetic atoms are distributed over N3 sites of the cubic lattice with a probability p. Thus p is the
v
i concentration ofmagnetic atomsandN is thetotal numberof atomson theedge of thecubic-shaped
X
nanoparticle.Accordingto [7],thedistributionfunctionforrandominteractionfieldsH onaparticle
r
a
locatedattheorigincanbedefinedas:
(cid:32) (cid:33)
(cid:90)
(cid:88) (cid:89)
W (H) = δ H − h (r ,m ) F (m )δ(r −r ) dr dm , (1)
k k k k k k0 k k
k k
where δ(x−x )— Dirac delta function, h = h (m ,r )— field created by atoms with magnetic
0 k k k k
momentsm locatedatcoordinatesr ,r —thecoordinatesoflatticesites,F(m )—thedistribution
k k k0 k
function for the magnetic moments, which in the approximation of Ising model for a ferromagnets
canberepresentedasfollows:
F (m ) = (α δ(θ )+β δ(θ −π))((1−p)δ(m )+pδ(m −m )). (2)
k k k k k k k 0
Here θ — the angle between m and OZ-axis, α and β — relative probabilities of the spin ori-
k k k k
entation along and against OZ-axis (θ = 0 and θ = π, respectively); m — the magnetic moment
k k 0
of the magnetic atom. In the approximation of nearest neighbors and the direct exchange interaction
betweenmagneticatoms,theequation(1)canberepresentedas:
(cid:88)z C(cid:88)zz−n (cid:88)2n
W (H) = pz−n(1−p)n ω δ(h−M J), (3)
j lν lν
n=0 ν lν∈L(Czn(kj))
where k is a set of nearest neighbors of the magnetic atom numbered as j, z = dimk — its co-
j j
ordination number; Cn(k ) — sample of n atoms of the total number of z nearest neighbors of jth
z j
atom; L(Ω) is a binomial set of permutations of an arbitrary set Ω with the amount of elements
equal to 2dimΩ. Introducing the notation α ≡ 1 − α we have got ω = (cid:81)z α and
(cid:80) (cid:80) −n n lj (ν=1;lν∈kj) ±lv
M = ±m = m ±|2α −1|.Finally,J istheconstantofexchangeinteraction.
lj n∈kj n 0 n∈kj n
Using the expression for the distribution function of the interaction fields (3), one can obtain
equationsthatdetermineaveragerelativemagneticmomentsateachlatticesite:
(cid:90) (cid:18)m H(cid:19) (cid:88)z C(cid:88)zz−n (cid:88)2n (cid:18)M J(cid:19)
µ = tanh j W (H)dH = pz−n(1−p)n ω tanh lν . (4)
j k T j lν k T
B B
n=0 ν lν∈L(Czn(kj))
Expression(4) allows to investigate the dependence of the magnetic moment M = (cid:80)N3 µ on the
j=1 j
temperature T and concentration p, as well as to determine the dependence of the phase transition
temperatureandthepercolationthresholdonthenumberN ofatoms.Systemofequations(4)canbe
solvednumericallywithNewton’smethod.
Results
Fig.1 represents the temperature dependence of the specific (per atom) magnetic moment (cid:104)µ(cid:105) =
M/N3 oftheparticleswithdifferentnumberofmagneticandnon-magneticatoms.Calculationswere
performed for a simple cubic lattice. As expected, reducing the number of magnetic atoms leads to a
reductioninthenumberofnearestneighborsand,consequently,toloweringtheCurietemperature.
NotethattheincreaseinCurietemperaturewithincreasingnumberofmagneticatomsisdescribed
fairlywellbyalinearfunctionof1/N (seeFig.2),whichisconsistentwithexperimentalresults [8].
DependenceoftheCurietemperatureonthenumberofatomsontheedgeofcubic-shapedparticle
is shown on Fig.3. Decrease in the Curie temperature is associated both with the decrease in the
concentration of magnetic atoms and the weakening of the interaction (decrease in the number of
nearestmagneticneighbors).
Reducingtheconcentrationofmagneticatomspand,consequently,theinteractionbetweenthem
leads to loss of ordering at a certain critical concentration p — point of the concentration phase
c
transition (see Fig.3, on the right). One can see that the curve p = p (N) asymptotically tends to
c c
p (N → ∞) = 0.31—thewell-knownpercolationthresholdforasimplecubiclattice [10].
c
References
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Fig. 1: The dependence of the specific magnetic moment of nanoparticles of different sizes on the
reducedtemperaturet = k T/m J (fordifferentconcentrationsp)
B 0
Fig. 2: Dependence of relative change in the Curie temperature on the inverse number of atoms on
the edge 1/N. Here T = T (N → ∞) is the Curie temperature of a massive particle, and ∆T =
C∞ C C
T − T (N). The dots mark the experimental results [8]. We should notice that the experimental
C∞ C
points correspond the lattice,that differs from simple cubic. But according to our previous work [9],
behaviour of the curve on this figure in such the coordinates conforms to the critical exponent λ and
isindependentonthetypeofthelattice
Fig. 3: DependenceoftheCurietemperature(ontheleft)andthecriticalconcentration(ontheright)
onthenumberN ofatomsontheedgeofcubic-shapedparticleatdifferentconcentrationsofmagnetic
atoms
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