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The phase transition in the simple cubic ising lattice
Physica 62 (1972) 109-118 0 North-Holland Publishing Co.
THE PHASE TRANSITION IN THE SIMPLE CUBIC ISING LATTICE M. HELM * Physikalisches Institut der Universitiit, Freiburg, Deutschland
Received 28 January 1972
Synopsis A general method for the calculation of correlation functions in the king model near the critical temperature will be specialized to the simple cubic lattice in order to obtain numerical results.
1. Introduction. In previous
work1*2) a method for the calculation of the thermodynamic properties of a general Ising lattice near the critical temperature T, was developed. In this second paper the general equations of paper I will be specialized to the simple cubic lattice in order to compare the numerical results with the experimental values of the critical exponents. In section 2 we obtain the algebraic equation for the long-range order parameter s. Then, in section 3, we solve the difference equation for the pair correlation function and calculate the decoupling factors 2qiM and 8c in the geometry of the cubic lattice. We deduce the equation for the consistency parameter u(T) in section 4. The numerical values for various thermodynamic quantities are represented in section 5. Finally a critical discussion of the theory and its results will be given in section 6. 2. Long-range order parameter. Although the quadratic lattice is the simplest case showing a phase transition (N = 4), we have chosen the simple cubic lattice (N = 6) in order to test the method of calculation developed in paper I’) against experimental values. [In the one-dimensional case (N = 2) our method is exact.] We obtain from eq. (1.5.10)’
(c&Y:+
Qoc:&)
s* + (&Y; +
+ (a? + Q&q%)
Q&3,$ s4 +
+ (a: +
(cx:d:+
Q&4,)
s6
+ G
m = 0, 1,2,3.
= 0, (2.1)
+ Present address: 815 Holzkirchen, Alpenblickstr. 9, Germany. 9
Q&)
Equation numbers from paper I will be preceded by I. 109
110
M. HELM
The coefficients LY:and S,” are easily calculated from the formulas (1.35) and (1.48). The numerical values for N = 6 are tabulated in the appendix. Using the four equations of system (2.1) we eliminate in a straightforward calculation the higher moments (C,-(4)), (G(6)) and (S(8)), obtaining an equation for the long-range order parameter s2 : s2 (As4 - 2Bs2 + C) = 0,
(2.2)
where : A = (1 - U)(l5t,
+ 3t, -
B = (1 - a) (15t, -
12t, + 9Ut2t4 + 5Ut4t6 -
16ut2ts),
5t, - 3ut2t4 - 8ut,t, + 5ut4t6),
(2.3) (2.4)
c = (a + 15) t2 + (4u + 12) 14 + (13u + 3) I6 - a (a + 15) t2t4 - u(5u + 11) t4tb -
16ut2t4 + 16u2tzt4t6 -
16.
(2.5)
We see from eq. (2.2) that .s2 = 0 is a solution and we suppose that this solution is valid above T,. In order to obtain the solution below T, we use the condition 0
(2.6)
If C = 0, we have from eq. (2.6) s2 = 0. This is a condition for the critical temperature T,. Since s2 contains still the consistency parameter u(T) we have to solve the equation for this quantity in order to obtain the numerical values for s2. 3. Pair correlutionfinction.
We shall calculate now the pair correlation function and the reciprocal correlation length x = (n’ - 6)‘. Using eq. (1.5.5) we obtain for N = 6:
+ [&&‘&
+ &&?, (a$‘& + Qcd’d:)] s’} (rr,M
+ [d?% + uA,Qo~?b:l
(4)) + a? (S(8)),
(2)) = 0, (3.1)
If we insert the explicit values of 01: and SZ:and use the four equations (3.1) to eliminate the higher moments (rr& (2)),.
(3.2)
PHASE TRANSITION
111
IN THE S.C. ISING LATTICE
Above T,n(T)is given by: n=
6 [16 - a (tz + 4t, + 13t,) + a2 (f&, + 5t,t,)] (15t2 i- 12t, -I- 3t, - 15&2t, - 16~tzfe - llt,t,
. (3.3)
+ l6~‘tzt,+te)
Below T, n (a, T) is a rather complex expression. The reciprocal correlation lenght x “) is given by x = (n - 6)*. Since x must vanish for T = T, we have another condition for T, which is equivalent to the first condition C = 0, as it is easily proved. Solving the difference equation (3.2)j) we obtain an expression fur the shortrange correlation (rooOrl & :
<~ooor1oo) = (1 - s2)z100/z000,
(3.4)
where :
I’.o=4*jK(._:sosy)dy-2~‘,
(3.5)
0
Z000
= 4x
4 s n - 2cosy
K(n
-
24cosy)dy’
(3.6)
0
K(x) is the complete elliptical integral of the first kind. The last integration must be performed numerically, except for n = 6 4*5). In (1.5.4) we have defined the
decoupling factors & and tic in the following manner:
6, (r, k A =
6~ Q-3k A =
(s2’rrRM (2j - 2))
.P (rr& (sZkrM(2j
sZk (rM(2j
(2j - 2)) ’ - 1)) - 1)) .
(3.7)
(3.8)
These factors form part of the reciprocal correlation length and will be calculated’ now using several approximations. It is clear that for independent short- and longrange order the factors (3.7) and (3.8) must be equal to one. In the region near T, we expect, however, a coupling between s2 and rr, so that 6, and 9, will be + 1. First we neglect the dependence of (3.7) and (3.8) on the order of the moments (k and j) assuming $j&)
=
s2
s2
(3.9)
M. HELM
112
,il
=
6
(3.10)
,
<~2r0,0,0rm+l,n,l) + (s2ro,o,orm-l,,,J+
etc.
s2
In order to clear up the matter we use in (3.9) and (3.10) the explicit coordinates (m, II, 1) for the point R and (0, 0,O) for the origin. Now we neglect the point dependence 8(r) moving the lattice point (m, n, 1) in the neighbourhood of the origin i.e. (m, n, I) -+ (1, 0,O).Thus we define:
(s2r0.0.0r1.0.0) , s2(rO.O,orl .o,o>
@, =
,&
=
(3.11)
G-2.0.0 + rO.l.O + (s2ro.o.o s2(r0.0.0 (r2,0,0 + ro,l.o +
+ ro.o,l
+
rO,-l.O + ro.o,l
+
rO,-l.o
ro.o.-l)) + ro,o,-l)) + s2(ro2,o.o)' (3.12)
In (3.11) and (3.12) we use the special symmetry of the simple cubic lattice and obtain : (s2r000r100)
6, =
s2
(3.13) ’
(S’rO00r200)
99)M=
s2
((~ooOr200)
4
+ +
4
>
+ +
S2
(3.14)
b2i30>’
In order to evaluate the expressions (3.13) and (3.14) we need an equation for the unknown quantities (s2rtrk). However, it is not difficult to see that the expressions (C(4)), and (C(4)), contained in the equation for the long-range order (1. l), are just the desired quantities. Let us remember the definitions (1.5.8): (C,(4))
= & (s2M(2)) = 1 (s2rtrk)
(C,(4))
+ S: (s2r,M(1)) +
5C
= S: (s2M (2)) = 4 C (s2rirk).
(3.15) (3.16)
The summation is again carried out over the basic cell. Using the symmetry of the cubic lattice we obtain : C
(3.17)
c
(3.18)
= 6 <~2~ooor100).
PHASE TRANSITION
IN THE S.C. BING
LATTICE
113
Comparing (3.15), (3.16), (3.17) and (3.18) we obtain: 4
+ (s2~100~--100) (3.19)
= (l/12) = (a/9) ?
(l/30) (G(4))> - (l/120) (G(4))
=
(3.20)
= (l/30) (1 - a/3)
In the last step we used the approximation (1.5.9). From the long-range order system (2.1) (C,(4)) is easily calculated. We find: (C,(4)) 8=
= s* (1 - S2) 8, 15
2 (1 - at,) (1 - at,J x [2 - (1 + s*) t4 + 2at2t4 - 2at,s* - (2t, + at*) (1 - s*)] .
(3.21)
We obtain from (3.13), (3.20) and (3.21): 1 -s*
$+$1-a
( > 3
(3.22)
(r000r100)’
To evaluate the moments (rtoo) and (s*r*ooo) contained in (3.14) we use the identity (Safe) = 1. We have (r~oo) = 1 - s* and assuming further (s2r,‘oo) = s* (1 - s*) we find from eqs. (3.19) and (3.14): 1 - s*
6M =
1+$e.
n (~ooorloo) (
(3.23) >
In (3.23) we have used the correlation eq. (3.2) for the point (m, 12,I) = (1, 0, 0), i.e., n
=
1 -
s*
+
4 ~~ooorloo) + (rooor200).
(3.24)
4. Consistency parameter. In the same way as in the preceding sections we now evaluate eq. (1.5.13) for N = 6 to obtain an equation for the consistency parameter a(T). We have:
+ [&%c + [$6:
+ uB4GM(oLTB:+ Qo~zSz)]s’} (r&f + a/?6QcG$] (r,M
(5)) = 0,
(3))
m =0,1,2,3.
(4.1)
114
M. HELM
Similarly to (3.17) and (3.18) we write:
(4.2)
(M(2))
(4.3)
= 1 QFk) = 3n <~000~100)- 3 (1 - s2).
In (4.3) we have used (3.24). We see from the coefficients given in the appendix that the system (4.1) contains only three equations. Thus eliminating the moments
F(T, a,,44, s’(4) = 0.
(4.4)
This last equation of our theory will be used to calculate the consistency parameter a(T). 5. Thermodynamic quantities. Eqs. (2.6) (3.2) and (4.4) are a closed system, which is solved in the following iterative way. At first we assume a value for a(T) in the range 0 < a < 1. Introducing this value in eqs. (2.6) and (3.3) and (3.4), we calculate the quantities sz, n and (roOOrlOO).Then we insert all these values together with the chosen a(T) in eq. (4.4) and obtain for various values u(T) a plot of the function F[a(T)]. The zeros of this function F(u) = 0 determine the true values of the consistency parameter a(T) and consequently the values of all the other thermodynamic quantities. 1 F (01
Fig. 1. The function F(a) in the vicinity of T, (kT/J = 4.540; kT,/J = 4.442).
Let us consider now some typical plot of F[a(T)] near T, (see fig. 1). Well above or below T, we have a unique zero F(u) = 0, which gives the value u(T). In the immediate vicinity of T, we see, however, that the abscissa is crossed by the
PHASE TRANSITION
IN THE S.C. ISING LATTICE
115
pronounced peak of F(u), so that three zeros 0 < a < 1 appear. This difficulty is caused by the overestimation of the short-range order (rO,,,,rlOo). That is why we suppose that the equation for the correlation function (3.2) has too simple a form in the vicinity of T,. This seems to be caused by the neglect of the point dependence in a (r, T) [see eq. (1.5.2.)].
10 8
a
-
6 ‘ 2
-
0 0
1
2
3
L
5
kT/J Fig. 2. The consistency
parameter a(T) over the temperature for the two- (curve II) and the three-dimensional (curve III) lattice.
The plot of u(T) over the reduced temperature kT/J is presented in fig. 2. The mentioned difficulty produces a tilt-over of the curve if we approach T, from below. Thus we determine T, by means of the conditions n = 6 and s2 + 0 approaching the critical point from above. We have found kT,lJ = 4.442.
001
1 IO-‘
10-3
10-z
10-I
I
Tc -T
Ts-----Fig. 3. The logarithmic plot of the long-range order parameter s over the reduced temperature, compared with experimental values. a: present theory; b: Ni; c: FeF,; d: COz (see ref. 6); and e: classical theories.
In order to obtain unique results in the region near and below T,, we raise the values of u(T) in this region, so that the two branches of u(T) above and below T, are connected at T = T,. Thus we multiply u(T) below T, by the factor 1.710,
116
M. HELM
0'
OS
10
1.5
TIT,
Fig. 4. The specific heat C, against temperature.
whereas the values a(T) correction falsifies the consider the logarithmic (Tc - T)ITc = 10m3 the
above T, remain unchanged. It is clear, that this rough thermodynamic quantities near and below T,. Let us plot of the long-range order parameter s (fig. 3). Up to critical index /? is given by the value 0.33 f 0.01. Below
ooll 10.'
lb"
10-2
10-1
IT-T,)IT, -
Fig. 5. Logarithmic
plot of C, against the reduced temperature.
however, ,I!?assumes again the classical value of 0.5. Clearly this behaviour is caused by our correction of a(T). Another interesting quantity is the specific heat C, presented in fig. 4. The logarithmic plot ?f C, above T, (fig. 5) yields the value of the critical exponent
this value,
PHASE TRANSITION
IN THE S.C. ISING LATTICE
117
& = 0.10 + 0.01. Below T, we have found 0 < 0~’< 0.2. In fig. 6 we show the plots of the reciprocal correlation length x = (n - 6)‘, which give the exponents Y = 0.78 +_ 0.01 and Y’ = 0.50 + 0.01 (see ref. 6).
5.0
I
4.0
-
3.0
-
20
-
x
I.0 -
0
1.0
0.5
TIT,
Fig. 6. The reciprocal correlation
1.5
-
lenght x = (n - 6)* against T/T,.
6. Discussion and conclusions. We have shown that the numerical results of our theory are in encouraging agreement with the experimental values. On the other hand near and below .T, serious difficulties remain which are caused by the decoupling of the short- and long-range order and especially by the neglect of the point dependence of a (r, T). Let us remember, however, that the principal equation of the theory in (1.2.1) together with its moment representation in (1.3.6) and (1.3.7) is of a completely general character. It seems possible that in a larger basic cell (next-nearest neighbours, etc.) the strongly coupled inner moments may be eliminated exactly, whereas the smaller outer moments may be more insensitive against the approximation (1.5.2). Perhaps our calculation, when using the smallest lattice cell, may be interpreted as a first approximation but clearly the complication of the calculations in a higher approximation (i.e., using a larger lattice cell) will grow rapidly.
118
M. HELM APPENDIX
I
The numerical values of the coefficients c$’ and 6: for the cubic lattice, calculated from eqs. (1.3.5) and (1.4.8) are the following: Cx;= 1
a; = 1
a; = 1
‘%p= 1
as = 1
a; = 1
a; = 6
a; = 4
a; = 2
a: = 0
a:=
-2
a:=
a; = 15
a: = 5
a?,=
-1
a:=
fx:=
-1
a: = 5
3; = 20
,0.; = 0
(*:=
-4
fx: = 0
s; = 1
8; = 6
s;=
s; = 1
-3
-4
Lx: = 0
a: = 4
s; = 20
&=
15
6; = 6
ST = 5
s; = 10
s;l = 10
s; = 5
s; = 1 s; = 1
s; = 1
s: = 4
S; = 6
s; = 4
s; = 1
s: = 1
s; = 3
s: = 3
83”= 1
8;: = 1
s: = 2
s: = 1
15
s: = 1
sg = 1 a:=
REFERENCES 1) 2) 3) 4) 5) 6)
Helm, M., Phys. Letters 33A (1970) 513. Helm, M., Physica 57 (1972) 46. Zernike, F., Physica VII (1940) 565. Watson, G., Quart. J. Math. 10 (1939) 266. Byrd, P. and Friedmann, H., Handbook of Elliptic Integrals (Berlin, 1954). Heller, P., Reports on Progress in Physics XXXII (1967) 731.
1
THE PHASE TRANSITION IN THE SIMPLE CUBIC ISING LATTICE M. HELM * Physikalisches Institut der Universitiit, Freiburg, Deutschland
Received 28 January 1972
Synopsis A general method for the calculation of correlation functions in the king model near the critical temperature will be specialized to the simple cubic lattice in order to obtain numerical results.
1. Introduction. In previous
work1*2) a method for the calculation of the thermodynamic properties of a general Ising lattice near the critical temperature T, was developed. In this second paper the general equations of paper I will be specialized to the simple cubic lattice in order to compare the numerical results with the experimental values of the critical exponents. In section 2 we obtain the algebraic equation for the long-range order parameter s. Then, in section 3, we solve the difference equation for the pair correlation function and calculate the decoupling factors 2qiM and 8c in the geometry of the cubic lattice. We deduce the equation for the consistency parameter u(T) in section 4. The numerical values for various thermodynamic quantities are represented in section 5. Finally a critical discussion of the theory and its results will be given in section 6. 2. Long-range order parameter. Although the quadratic lattice is the simplest case showing a phase transition (N = 4), we have chosen the simple cubic lattice (N = 6) in order to test the method of calculation developed in paper I’) against experimental values. [In the one-dimensional case (N = 2) our method is exact.] We obtain from eq. (1.5.10)’
(c&Y:+
Qoc:&)
s* + (&Y; +
+ (a? + Q&q%)
Q&3,$ s4 +
(cx:d:+
Q&4,)
s6
m = 0, 1,2,3.
= 0, (2.1)
+ Present address: 815 Holzkirchen, Alpenblickstr. 9, Germany. 9
Q&)
Equation numbers from paper I will be preceded by I. 109
110
M. HELM
The coefficients LY:and S,” are easily calculated from the formulas (1.35) and (1.48). The numerical values for N = 6 are tabulated in the appendix. Using the four equations of system (2.1) we eliminate in a straightforward calculation the higher moments (C,-(4)), (G(6)) and (S(8)), obtaining an equation for the long-range order parameter s2 : s2 (As4 - 2Bs2 + C) = 0,
(2.2)
where : A = (1 - U)(l5t,
+ 3t, -
B = (1 - a) (15t, -
12t, + 9Ut2t4 + 5Ut4t6 -
16ut2ts),
5t, - 3ut2t4 - 8ut,t, + 5ut4t6),
(2.3) (2.4)
c = (a + 15) t2 + (4u + 12) 14 + (13u + 3) I6 - a (a + 15) t2t4 - u(5u + 11) t4tb -
16ut2t4 + 16u2tzt4t6 -
16.
(2.5)
We see from eq. (2.2) that .s2 = 0 is a solution and we suppose that this solution is valid above T,. In order to obtain the solution below T, we use the condition 0
(2.6)
If C = 0, we have from eq. (2.6) s2 = 0. This is a condition for the critical temperature T,. Since s2 contains still the consistency parameter u(T) we have to solve the equation for this quantity in order to obtain the numerical values for s2. 3. Pair correlutionfinction.
We shall calculate now the pair correlation function and the reciprocal correlation length x = (n’ - 6)‘. Using eq. (1.5.5) we obtain for N = 6:
+ [&&‘&
+ &&?, (a$‘& + Qcd’d:)] s’} (rr,M
+ [d?% + uA,Qo~?b:l
(4)) + a? (S(8)),
(2)) = 0, (3.1)
If we insert the explicit values of 01: and SZ:and use the four equations (3.1) to eliminate the higher moments (rr& (2)),
(3.2)
PHASE TRANSITION
111
IN THE S.C. ISING LATTICE
Above T,n(T)is given by: n=
6 [16 - a (tz + 4t, + 13t,) + a2 (f&, + 5t,t,)] (15t2 i- 12t, -I- 3t, - 15&2t, - 16~tzfe - llt,t,
. (3.3)
+ l6~‘tzt,+te)
Below T, n (a, T) is a rather complex expression. The reciprocal correlation lenght x “) is given by x = (n - 6)*. Since x must vanish for T = T, we have another condition for T, which is equivalent to the first condition C = 0, as it is easily proved. Solving the difference equation (3.2)j) we obtain an expression fur the shortrange correlation (rooOrl & :
<~ooor1oo) = (1 - s2)z100/z000,
(3.4)
where :
I’.o=4*jK(._:sosy)dy-2~‘,
(3.5)
0
Z000
= 4x
4 s n - 2cosy
K(n
-
24cosy)dy’
(3.6)
0
K(x) is the complete elliptical integral of the first kind. The last integration must be performed numerically, except for n = 6 4*5). In (1.5.4) we have defined the
decoupling factors & and tic in the following manner:
6, (r, k A =
6~ Q-3k A =
(s2’rrRM (2j - 2))
.P (rr& (sZkrM(2j
sZk (rM(2j
(2j - 2)) ’ - 1)) - 1)) .
(3.7)
(3.8)
These factors form part of the reciprocal correlation length and will be calculated’ now using several approximations. It is clear that for independent short- and longrange order the factors (3.7) and (3.8) must be equal to one. In the region near T, we expect, however, a coupling between s2 and rr, so that 6, and 9, will be + 1. First we neglect the dependence of (3.7) and (3.8) on the order of the moments (k and j) assuming $j&)
=
s2
s2
(3.9)
M. HELM
112
,il
=
6
(3.10)
,
<~2r0,0,0rm+l,n,l) + (s2ro,o,orm-l,,,J+
etc.
s2
In order to clear up the matter we use in (3.9) and (3.10) the explicit coordinates (m, II, 1) for the point R and (0, 0,O) for the origin. Now we neglect the point dependence 8(r) moving the lattice point (m, n, 1) in the neighbourhood of the origin i.e. (m, n, I) -+ (1, 0,O).Thus we define:
(s2r0.0.0r1.0.0) , s2(rO.O,orl .o,o>
@, =
,&
=
(3.11)
G-2.0.0 + rO.l.O + (s2ro.o.o s2(r0.0.0 (r2,0,0 + ro,l.o +
+ ro.o,l
+
rO,-l.O + ro.o,l
+
rO,-l.o
ro.o.-l)) +
In (3.11) and (3.12) we use the special symmetry of the simple cubic lattice and obtain : (s2r000r100)
6, =
s2
(3.13) ’
(S’rO00r200)
99)M=
s2
((~ooOr200)
4
+ +
4
+ +
(3.14)
b2i30>’
In order to evaluate the expressions (3.13) and (3.14) we need an equation for the unknown quantities (s2rtrk). However, it is not difficult to see that the expressions (C(4)), and (C(4)), contained in the equation for the long-range order (1. l), are just the desired quantities. Let us remember the definitions (1.5.8): (C,(4))
= & (s2M(2)) = 1 (s2rtrk)
(C,(4))
+ S: (s2r,M(1)) +
5C
= S: (s2M (2)) = 4 C (s2rirk).
(3.15) (3.16)
The summation is again carried out over the basic cell. Using the symmetry of the cubic lattice we obtain : C
(3.17)
c
(3.18)
= 6 <~2~ooor100).
PHASE TRANSITION
IN THE S.C. BING
LATTICE
113
Comparing (3.15), (3.16), (3.17) and (3.18) we obtain: 4
+ (s2~100~--100) (3.19)
= (l/12)
(l/30) (G(4))> - (l/120) (G(4))
=
(3.20)
= (l/30) (1 - a/3)
In the last step we used the approximation (1.5.9). From the long-range order system (2.1) (C,(4)) is easily calculated. We find: (C,(4)) 8=
= s* (1 - S2) 8, 15
2 (1 - at,) (1 - at,J x [2 - (1 + s*) t4 + 2at2t4 - 2at,s* - (2t, + at*) (1 - s*)] .
(3.21)
We obtain from (3.13), (3.20) and (3.21): 1 -s*
$+$1-a
( > 3
(3.22)
(r000r100)’
To evaluate the moments (rtoo) and (s*r*ooo) contained in (3.14) we use the identity (Safe) = 1. We have (r~oo) = 1 - s* and assuming further (s2r,‘oo) = s* (1 - s*) we find from eqs. (3.19) and (3.14): 1 - s*
6M =
1+$e.
n (~ooorloo) (
(3.23) >
In (3.23) we have used the correlation eq. (3.2) for the point (m, 12,I) = (1, 0, 0), i.e., n
=
1 -
s*
+
4 ~~ooorloo) + (rooor200).
(3.24)
4. Consistency parameter. In the same way as in the preceding sections we now evaluate eq. (1.5.13) for N = 6 to obtain an equation for the consistency parameter a(T). We have:
+ [&%c + [$6:
+ uB4GM(oLTB:+ Qo~zSz)]s’} (r&f + a/?6QcG$] (r,M
(5)) = 0,
(3))
m =0,1,2,3.
(4.1)
114
M. HELM
Similarly to (3.17) and (3.18) we write:
(4.2)
(M(2))
(4.3)
= 1 QFk) = 3n <~000~100)- 3 (1 - s2).
In (4.3) we have used (3.24). We see from the coefficients given in the appendix that the system (4.1) contains only three equations. Thus eliminating the moments
F(T, a,
(4.4)
This last equation of our theory will be used to calculate the consistency parameter a(T). 5. Thermodynamic quantities. Eqs. (2.6) (3.2) and (4.4) are a closed system, which is solved in the following iterative way. At first we assume a value for a(T) in the range 0 < a < 1. Introducing this value in eqs. (2.6) and (3.3) and (3.4), we calculate the quantities sz, n and (roOOrlOO).Then we insert all these values together with the chosen a(T) in eq. (4.4) and obtain for various values u(T) a plot of the function F[a(T)]. The zeros of this function F(u) = 0 determine the true values of the consistency parameter a(T) and consequently the values of all the other thermodynamic quantities. 1 F (01
Fig. 1. The function F(a) in the vicinity of T, (kT/J = 4.540; kT,/J = 4.442).
Let us consider now some typical plot of F[a(T)] near T, (see fig. 1). Well above or below T, we have a unique zero F(u) = 0, which gives the value u(T). In the immediate vicinity of T, we see, however, that the abscissa is crossed by the
PHASE TRANSITION
IN THE S.C. ISING LATTICE
115
pronounced peak of F(u), so that three zeros 0 < a < 1 appear. This difficulty is caused by the overestimation of the short-range order (rO,,,,rlOo). That is why we suppose that the equation for the correlation function (3.2) has too simple a form in the vicinity of T,. This seems to be caused by the neglect of the point dependence in a (r, T) [see eq. (1.5.2.)].
10 8
a
-
6 ‘ 2
-
0 0
1
2
3
L
5
kT/J Fig. 2. The consistency
parameter a(T) over the temperature for the two- (curve II) and the three-dimensional (curve III) lattice.
The plot of u(T) over the reduced temperature kT/J is presented in fig. 2. The mentioned difficulty produces a tilt-over of the curve if we approach T, from below. Thus we determine T, by means of the conditions n = 6 and s2 + 0 approaching the critical point from above. We have found kT,lJ = 4.442.
001
1 IO-‘
10-3
10-z
10-I
I
Tc -T
Ts-----Fig. 3. The logarithmic plot of the long-range order parameter s over the reduced temperature, compared with experimental values. a: present theory; b: Ni; c: FeF,; d: COz (see ref. 6); and e: classical theories.
In order to obtain unique results in the region near and below T,, we raise the values of u(T) in this region, so that the two branches of u(T) above and below T, are connected at T = T,. Thus we multiply u(T) below T, by the factor 1.710,
116
M. HELM
0'
OS
10
1.5
TIT,
Fig. 4. The specific heat C, against temperature.
whereas the values a(T) correction falsifies the consider the logarithmic (Tc - T)ITc = 10m3 the
above T, remain unchanged. It is clear, that this rough thermodynamic quantities near and below T,. Let us plot of the long-range order parameter s (fig. 3). Up to critical index /? is given by the value 0.33 f 0.01. Below
ooll 10.'
lb"
10-2
10-1
IT-T,)IT, -
Fig. 5. Logarithmic
plot of C, against the reduced temperature.
however, ,I!?assumes again the classical value of 0.5. Clearly this behaviour is caused by our correction of a(T). Another interesting quantity is the specific heat C, presented in fig. 4. The logarithmic plot ?f C, above T, (fig. 5) yields the value of the critical exponent
this value,
PHASE TRANSITION
IN THE S.C. ISING LATTICE
117
& = 0.10 + 0.01. Below T, we have found 0 < 0~’< 0.2. In fig. 6 we show the plots of the reciprocal correlation length x = (n - 6)‘, which give the exponents Y = 0.78 +_ 0.01 and Y’ = 0.50 + 0.01 (see ref. 6).
5.0
I
4.0
-
3.0
-
20
-
x
I.0 -
0
1.0
0.5
TIT,
Fig. 6. The reciprocal correlation
1.5
-
lenght x = (n - 6)* against T/T,.
6. Discussion and conclusions. We have shown that the numerical results of our theory are in encouraging agreement with the experimental values. On the other hand near and below .T, serious difficulties remain which are caused by the decoupling of the short- and long-range order and especially by the neglect of the point dependence of a (r, T). Let us remember, however, that the principal equation of the theory in (1.2.1) together with its moment representation in (1.3.6) and (1.3.7) is of a completely general character. It seems possible that in a larger basic cell (next-nearest neighbours, etc.) the strongly coupled inner moments may be eliminated exactly, whereas the smaller outer moments may be more insensitive against the approximation (1.5.2). Perhaps our calculation, when using the smallest lattice cell, may be interpreted as a first approximation but clearly the complication of the calculations in a higher approximation (i.e., using a larger lattice cell) will grow rapidly.
118
M. HELM APPENDIX
I
The numerical values of the coefficients c$’ and 6: for the cubic lattice, calculated from eqs. (1.3.5) and (1.4.8) are the following: Cx;= 1
a; = 1
a; = 1
‘%p= 1
as = 1
a; = 1
a; = 6
a; = 4
a; = 2
a: = 0
a:=
-2
a:=
a; = 15
a: = 5
a?,=
-1
a:=
fx:=
-1
a: = 5
3; = 20
,0.; = 0
(*:=
-4
fx: = 0
s; = 1
8; = 6
s;=
s; = 1
-3
-4
Lx: = 0
a: = 4
s; = 20
&=
15
6; = 6
ST = 5
s; = 10
s;l = 10
s; = 5
s; = 1 s; = 1
s; = 1
s: = 4
S; = 6
s; = 4
s; = 1
s: = 1
s; = 3
s: = 3
83”= 1
8;: = 1
s: = 2
s: = 1
15
s: = 1
sg = 1 a:=
REFERENCES 1) 2) 3) 4) 5) 6)
Helm, M., Phys. Letters 33A (1970) 513. Helm, M., Physica 57 (1972) 46. Zernike, F., Physica VII (1940) 565. Watson, G., Quart. J. Math. 10 (1939) 266. Byrd, P. and Friedmann, H., Handbook of Elliptic Integrals (Berlin, 1954). Heller, P., Reports on Progress in Physics XXXII (1967) 731.
1