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Disordered lattice gas
Volume 84A, number 4
PHYSICS LETTERS
27 July 1981
DISORDERED LATTICE GAS S. INAWASHIRO~and N.E. FRANKEL School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia
and C.J. THOMPSON Department ofMathematics 2 University ofMelbourhe, Parkvile, Victoria 3052, Australia and The Institute for Advanced Study, Princeton, NJ 08540, USA Received 18 May 1981
A statistical mechanicalmodel, the disordered magnetic/nonmagneticlattice gas, for studying amorphous systems is reported. The results for the disordered, nonmagnetic, lattice gas are presented in detail.
One of the most exciting and important problems lii modern statistical mechanics is the study of disordered systems. Disordered magnetic systems of current physical interest are amorphous ferromagnets and spin glasses. When the disordered system is nonmagnetic, the systems of physical interest are amorphous solids, metals and glasses. In many respects, as we will discuss herein, the physical properties of amorphous systems resemble those of real fluids. In this paper, we will present a brief synopsis of a disordered lattice gas model for describing such systems. We will then present the statistical mechanical results for a disordered, nonmagnetic, lattice gas and discuss in detail its properties.
The most widely followed approach for treating disordered magnetic systems is to consider an ordered lattice in which each site is occupied by a magnetic atom. The disorder in the system is then modelled by choosing an appropriate probability distribution for the magnetic coupling constants. Special techniques must be introduced to study the statistical mechanics of these disordered magnetic systems. Edwards and Anderson [1] (EA) introduced the “n-replica” method
2
which has since been employed by numerous workers [2] in the field. On the other hand, in treating a disordered magnetic system, Matsubara [3] introduced the method of the distribution function and derived a nonlinear integral equation which it must satisfy. This technique has also been developed and refined by numerous workers, see e.g. ref. [4] , in the field. It is this technique which we have utilized in the results reported in this letter. In real amorphous substances, such as glasses and amorphous ferromagnets, however, their structure is irregular with no underlying structural long-range order corresponding to a precise, regularly ordered crystalline lattice - These systems resemble more closely a network of atoms interlaced with a network of vacancies. If the diffusion coefficient for these systems is small enough (relaxation time large enough) we can think of applying equilibrium statistical mechanics to studies of these systems even though these networks are continuously changing their irregular topology in time. In many respects these systems resemble a real liquid wherein the molecules array themselves in something like irregular networks (which vary in time), as seen most vividly in computer simulations using molec-
On leave from Department of Applied Physics, Tohoku University, Aza Aoba, Sendai 980, Japan.
ular dynamics [5]. Several years ago, two of us (FT) [6] tried to simu-
Permanent address.
late the above picture of disordered magnetic systems
0031—9163/81/0000—0000/s 02.50 © North-Holland Publishing Company
205
Volume 84A, number 4
PHYSICS LETTERS
27 July 1981
by treating a continuum model of magnetic atoms. Here disordered systems were thought of as dense gases/liquids where the spin and spatial configurations were treated on an equal footing. Since then, we have mapped the model of (FT) onto an appropriate spin-I lattice gas with magnetic and nonmagnetic interactions [7] We then studied the model for infmitely longranged interactions (mean-field-approximation) with random probability functions [8] for the magnetic and nonmagnetic coupling constants, respectively. This disordered magnetic lattice gas can, therefore, be viewed as our attempt to incorporate the abovementioned irregularities in real substances into the original lattice gas [9] with its underlying structurally ordered lattice. Similarly, when the magnetic interaction is turned off, the disordered nonmagnetic lattice
thermodynamic quantities of the disordered magnetic lattice gas are given elsewhere [8]. In this letter we will exhibit the formal expressions for the disordered, nonmagnetic, lattice gas [eq. (1) with = 0 and B = 0] , and study them in much detail. The equation of state for the disordered lattice gas is
gas may also be viewed as our attempt to incorporate structural irregularities, inherent in nonmagnetic amorphous substances and correspondingly real fluids, within the original lattice gas model [9].In many ways the disordered lattice gas displays a close analogy to the cellular theories of liquids [10] We consider the disordered magnetic lattice gas where particles with spins interact with each other through magnetic and nonmagnetic interactions which are randomly distributed over the lattice with a certain specified probability. Spin-l operators are used to represent a particle with up or down spin at a lattice site by S = 1 or —1, respectively, and a vacant site by S = 0. Then the disordered magnetic system is described by
X)13(i~U) and p’ = j.L + 13~ln 2 with (3 = ters p and X are given by
.
j.s.s— E ~ (i
•)
~
~
1
~ ~
f
(3 .J
Ll
2/2d
1
lnl
—
1 e_~ P01)]
—
2117 + 0
!
2
—
2~
~2
2
2\,~j7\2 11~ )1” 1
where =
1
—
{l + exp 13[p + pU 0 2+ r~-s,/~ ~17U~1,
+.}(p —
C (ii) ~ V’~~ ~
=~
~
x
=
1/kT. The order parame-
d
(3)
—i—— C I j
~ 2 e~2h’2d iP~71jj 7).
(4
We note that in the limit of no disorder, t~U-* 0,
the equation of state reduces to the standard van der Waals lattice gas result, p”1F1 ln(l —p)1 —~-p2U 0,
(j ~) if I I
(5)
with critical values given by Pc = ~Qn 2 ~) U0, kTc = ~ U0 and p~ = ~.. To study the effect of disorder we introduce the parametery i~U/U0.It is straightforward to develop a small-y asymptotic expansion for p~Tc and p~,and to find that they all decrease from their2). y = values with the correction term valTo0 get a feeling for leading what happens for large 0(y of y. we have studied eq. (2) with U ues 0 = 0. The isotherms critical isotherm exhibited in fig. 1. Itaround is first the interesting to observeare that even with —
—
B
where
I~Si
—
p ~ S~,Si = 1,0, —1,
(1)
denotes the magnetic interaction between
lattice sites i and /, U~1the nonmagnetic (potential) interaction field, between sites i andpotential. 1, B an external magnetic andlattice p the chemical The interactions J~1and U~1are distributed 1around their mean with the variances values J0 = .T0z—1 and ~ U0z /~J~t~Jz1/2 and U ~ z~12in gaussian probabiity distributions. Here the coordination number, z, goes to infmity in the limit of infinitely long-ranged interactions. The method of the distribution function is generalized and used in carrying out the statistical mechanical calculations. The general results for the 206
U 0 = 0, that a gas—liquid phase transition occurs; equal competition between repulsive and attractive values of the interaction still allows for a condensation phenomenon. Furthermore, we can see qualitatively that Tc will vary linearly withy asy -+ To see what effect intermediate values of disorder
Volume 84A, number 4
PHYSICS LETI’ERS
27 July 1981
0-10
______ 0.21001 ~~Tr~.22
0.5
‘V
0.19
0.005
0
____________________
0
5
10
____________
15
Fig. 1. Isotherms around the critical isotherm for Uo
20 =
0 and
~~U=i.o.
0
0:5
y
1.0
Fig. 2. The criticaldensity p~as a function of the parameter
y’~U/U0.
IC
k
0.30
0-~0
0.25 J
0.38 0.36
o’r______ 0 0.5 1.0
0
0
y
1.0
~,,
Fig. 3. The critical temperature Tc given in units of U0/k as a function of the parameter y = ~ U/U0.
Fig. 4. The critical parameter pcJ(kTcpc) as a function of the parametery = ~ U/Ut).
have on the critical parameters, we have exhibited the numerical results for their dependence on y in figs. 2— 4. The critical density, p~,varies monotonically from
References
= 0) = to ~ = 0.1336. The critical temperature first decreases slightly and then increases rapidly with increasingy. The critical parameter Pc/ (kTc p~)has an interesting but relatively insensitive dependence ony for intermediate values. In summary, we have reported on a new model for studying disordered magnetic/nonmagnetic systems. We have displayed detailed results for the disordered, nonmagnetic, lattice gas and shown that the presence
Pc(Y
00)
of disorder produces interesting and novel effects. We are grateful to the Australian Research Grants Committee for their support. One of the authors (S.I.) thanks the University of Melbourne for its hospitality during his stay.
[1] S.F. Edwards and P.W. Anderson, J. Phys. P5 (1975) 965.
[21R. Balian, R. Maynard and G. Toulouse, eds., Ill-condensed Matter, Les Houches Lectures (North-Holland, Amsterdam, 1979).
[3] F. Matsubara, Prog. Theor. Phys. 51(1974)1694; 52 (1974) 1124. [4] 5. Inawashiro and S. Katsura, Physica 100A (1980) 24. [5] T.W. Wainwright and B.J. Alder, Nuovo Cimento 9 Suppl. 1(1958)116. [6] N.E. Frankel and C.J. Thompson, I. Phys. C 8 (1975) 3194. [7] S. Inawashiro, N.E. Frankel and C.J. Thompson, a set of papers submitted to Phys. Rev. [8] S. Inawashiro, N.E. Frankel and C.J. Thompson, subto and Phys. Rev. [9] mitted T.D. Lee C.N. Yang, Phys. Rev. 87 (1952) 410. [10] J.A. Barker, Lattice theories of the liquid state (Pergamon, Oxford, 1963).
207
PHYSICS LETTERS
27 July 1981
DISORDERED LATTICE GAS S. INAWASHIRO~and N.E. FRANKEL School of Physics, University of Melbourne, Parkville, Victoria 3052, Australia
and C.J. THOMPSON Department ofMathematics 2 University ofMelbourhe, Parkvile, Victoria 3052, Australia and The Institute for Advanced Study, Princeton, NJ 08540, USA Received 18 May 1981
A statistical mechanicalmodel, the disordered magnetic/nonmagneticlattice gas, for studying amorphous systems is reported. The results for the disordered, nonmagnetic, lattice gas are presented in detail.
One of the most exciting and important problems lii modern statistical mechanics is the study of disordered systems. Disordered magnetic systems of current physical interest are amorphous ferromagnets and spin glasses. When the disordered system is nonmagnetic, the systems of physical interest are amorphous solids, metals and glasses. In many respects, as we will discuss herein, the physical properties of amorphous systems resemble those of real fluids. In this paper, we will present a brief synopsis of a disordered lattice gas model for describing such systems. We will then present the statistical mechanical results for a disordered, nonmagnetic, lattice gas and discuss in detail its properties.
The most widely followed approach for treating disordered magnetic systems is to consider an ordered lattice in which each site is occupied by a magnetic atom. The disorder in the system is then modelled by choosing an appropriate probability distribution for the magnetic coupling constants. Special techniques must be introduced to study the statistical mechanics of these disordered magnetic systems. Edwards and Anderson [1] (EA) introduced the “n-replica” method
2
which has since been employed by numerous workers [2] in the field. On the other hand, in treating a disordered magnetic system, Matsubara [3] introduced the method of the distribution function and derived a nonlinear integral equation which it must satisfy. This technique has also been developed and refined by numerous workers, see e.g. ref. [4] , in the field. It is this technique which we have utilized in the results reported in this letter. In real amorphous substances, such as glasses and amorphous ferromagnets, however, their structure is irregular with no underlying structural long-range order corresponding to a precise, regularly ordered crystalline lattice - These systems resemble more closely a network of atoms interlaced with a network of vacancies. If the diffusion coefficient for these systems is small enough (relaxation time large enough) we can think of applying equilibrium statistical mechanics to studies of these systems even though these networks are continuously changing their irregular topology in time. In many respects these systems resemble a real liquid wherein the molecules array themselves in something like irregular networks (which vary in time), as seen most vividly in computer simulations using molec-
On leave from Department of Applied Physics, Tohoku University, Aza Aoba, Sendai 980, Japan.
ular dynamics [5]. Several years ago, two of us (FT) [6] tried to simu-
Permanent address.
late the above picture of disordered magnetic systems
0031—9163/81/0000—0000/s 02.50 © North-Holland Publishing Company
205
Volume 84A, number 4
PHYSICS LETTERS
27 July 1981
by treating a continuum model of magnetic atoms. Here disordered systems were thought of as dense gases/liquids where the spin and spatial configurations were treated on an equal footing. Since then, we have mapped the model of (FT) onto an appropriate spin-I lattice gas with magnetic and nonmagnetic interactions [7] We then studied the model for infmitely longranged interactions (mean-field-approximation) with random probability functions [8] for the magnetic and nonmagnetic coupling constants, respectively. This disordered magnetic lattice gas can, therefore, be viewed as our attempt to incorporate the abovementioned irregularities in real substances into the original lattice gas [9] with its underlying structurally ordered lattice. Similarly, when the magnetic interaction is turned off, the disordered nonmagnetic lattice
thermodynamic quantities of the disordered magnetic lattice gas are given elsewhere [8]. In this letter we will exhibit the formal expressions for the disordered, nonmagnetic, lattice gas [eq. (1) with = 0 and B = 0] , and study them in much detail. The equation of state for the disordered lattice gas is
gas may also be viewed as our attempt to incorporate structural irregularities, inherent in nonmagnetic amorphous substances and correspondingly real fluids, within the original lattice gas model [9].In many ways the disordered lattice gas displays a close analogy to the cellular theories of liquids [10] We consider the disordered magnetic lattice gas where particles with spins interact with each other through magnetic and nonmagnetic interactions which are randomly distributed over the lattice with a certain specified probability. Spin-l operators are used to represent a particle with up or down spin at a lattice site by S = 1 or —1, respectively, and a vacant site by S = 0. Then the disordered magnetic system is described by
X)13(i~U) and p’ = j.L + 13~ln 2 with (3 = ters p and X are given by
.
j.s.s— E ~ (i
•)
~
~
1
~ ~
f
(3 .J
Ll
2/2d
1
lnl
—
1 e_~ P01)]
—
2117 + 0
!
2
—
2~
~2
2
2\,~j7\2 11~ )1” 1
where =
1
—
{l + exp 13[p + pU 0 2+ r~-s,/~ ~17U~1,
+.}(p —
C (ii) ~ V’~~ ~
=~
~
x
=
1/kT. The order parame-
d
(3)
—i—— C I j
~ 2 e~2h’2d iP~71jj 7).
(4
We note that in the limit of no disorder, t~U-* 0,
the equation of state reduces to the standard van der Waals lattice gas result, p”1F1 ln(l —p)1 —~-p2U 0,
(j ~) if I I
(5)
with critical values given by Pc = ~Qn 2 ~) U0, kTc = ~ U0 and p~ = ~.. To study the effect of disorder we introduce the parametery i~U/U0.It is straightforward to develop a small-y asymptotic expansion for p~Tc and p~,and to find that they all decrease from their2). y = values with the correction term valTo0 get a feeling for leading what happens for large 0(y of y. we have studied eq. (2) with U ues 0 = 0. The isotherms critical isotherm exhibited in fig. 1. Itaround is first the interesting to observeare that even with —
—
B
where
I~Si
—
p ~ S~,Si = 1,0, —1,
(1)
denotes the magnetic interaction between
lattice sites i and /, U~1the nonmagnetic (potential) interaction field, between sites i andpotential. 1, B an external magnetic andlattice p the chemical The interactions J~1and U~1are distributed 1around their mean with the variances values J0 = .T0z—1 and ~ U0z /~J~t~Jz1/2 and U ~ z~12in gaussian probabiity distributions. Here the coordination number, z, goes to infmity in the limit of infinitely long-ranged interactions. The method of the distribution function is generalized and used in carrying out the statistical mechanical calculations. The general results for the 206
U 0 = 0, that a gas—liquid phase transition occurs; equal competition between repulsive and attractive values of the interaction still allows for a condensation phenomenon. Furthermore, we can see qualitatively that Tc will vary linearly withy asy -+ To see what effect intermediate values of disorder
Volume 84A, number 4
PHYSICS LETI’ERS
27 July 1981
0-10
______ 0.21001 ~~Tr~.22
0.5
‘V
0.19
0.005
0
____________________
0
5
10
____________
15
Fig. 1. Isotherms around the critical isotherm for Uo
20 =
0 and
~~U=i.o.
0
0:5
y
1.0
Fig. 2. The criticaldensity p~as a function of the parameter
y’~U/U0.
IC
k
0.30
0-~0
0.25 J
0.38 0.36
o’r______ 0 0.5 1.0
0
0
y
1.0
~,,
Fig. 3. The critical temperature Tc given in units of U0/k as a function of the parameter y = ~ U/U0.
Fig. 4. The critical parameter pcJ(kTcpc) as a function of the parametery = ~ U/Ut).
have on the critical parameters, we have exhibited the numerical results for their dependence on y in figs. 2— 4. The critical density, p~,varies monotonically from
References
= 0) = to ~ = 0.1336. The critical temperature first decreases slightly and then increases rapidly with increasingy. The critical parameter Pc/ (kTc p~)has an interesting but relatively insensitive dependence ony for intermediate values. In summary, we have reported on a new model for studying disordered magnetic/nonmagnetic systems. We have displayed detailed results for the disordered, nonmagnetic, lattice gas and shown that the presence
Pc(Y
00)
of disorder produces interesting and novel effects. We are grateful to the Australian Research Grants Committee for their support. One of the authors (S.I.) thanks the University of Melbourne for its hospitality during his stay.
[1] S.F. Edwards and P.W. Anderson, J. Phys. P5 (1975) 965.
[21R. Balian, R. Maynard and G. Toulouse, eds., Ill-condensed Matter, Les Houches Lectures (North-Holland, Amsterdam, 1979).
[3] F. Matsubara, Prog. Theor. Phys. 51(1974)1694; 52 (1974) 1124. [4] 5. Inawashiro and S. Katsura, Physica 100A (1980) 24. [5] T.W. Wainwright and B.J. Alder, Nuovo Cimento 9 Suppl. 1(1958)116. [6] N.E. Frankel and C.J. Thompson, I. Phys. C 8 (1975) 3194. [7] S. Inawashiro, N.E. Frankel and C.J. Thompson, a set of papers submitted to Phys. Rev. [8] S. Inawashiro, N.E. Frankel and C.J. Thompson, subto and Phys. Rev. [9] mitted T.D. Lee C.N. Yang, Phys. Rev. 87 (1952) 410. [10] J.A. Barker, Lattice theories of the liquid state (Pergamon, Oxford, 1963).
207