- Email: [email protected]
An ε-convergence method for the evaluation of critical parameters
Volume 101, number 1
CHEMKAL PHYSICS LETTERS
AN E-CONVERGENCE METHOD FOR THE EVALUATION
M.V. SANGARANARAYANp;N
and SK
7 October 1983
OF CRITICAL PARAMETERS
RANGARAJAN
Deparment of Inorganic and Physical Chemistry. Indian Institute of Science, Barrgaiore .&Xl 012, Itrdfa Received 16 May 1983; in final form 8 July 1983
A simple cconversence technique is applied to obtain accurate estimates of critical temperatures and critical it\ponmts of a few two- and threpdiniensional king models. When applied to the virial series for hard spheres and hard discs, this method predicts a divergence of the equation-of-state at the density of closest packing.
1. Introduction
In this note, we report how a simple algorithm, termed the ‘&convergence method“ [ 1.2]_ can be used to obtain values of critical parameters. We illustrate tltis claim by applying the method to determine critical tentperatures and critical exponents associated with some well-studied Ising models, starting front the respective series expansions. We also demonstrate the use of the method in obtaining information about the divergence and the asymptotic limit of the virial series for hard spheres and hard discs. The primary question addressed in the E algorithm is the following- Given the partial sequence aI, a?, .._, a,l. what is its likely limit, Le. what is limn_,ma,z ? What we do in this paper is to map several questions into problems of limits: (A) The problem of evaluating the radius of convergence xc of a class of power series. 22~~~~~~ can be converted into one of evaluating the limit as 12+ 00 of an/all+1 _ (B) In the problem of asymptotic matching, for example. we verify if the power series I&.x’ diverges as (1 x/x,)@ and estimate fl_ (C)Yet another interesting problem is to extrapolate from partial sums information regarding the infinite sum. Examples of pltysical counterparts of (A)-(C) given above are: (A) the prediction of critical temperatures or densities, as the case may be, (B) the prediction of critical exponents, and (C) estimating from the first few virial coefficients the values of pressure at high densities. We discuss here (A) and (B) only. The E algorithm 131 constructs a two-dimensional array ef) front a given partial sequence ~8~. i = 0, 1. ___)N- 1 entered as a column. The superscript j refers to an eiement in column X-_The various elements of the table are cd-
culated by using the algorithm:
(1) We define the column tii , 3 0, and set the given sequence to be eo. i The resulting table is not unlike a difference table but with a non-linear transformation law [eq. (l)] (also called the “rhombus rule”).
0 OOQ-~614/83/000~-OOOOIS 0300 0 1983 Norm-Hoard
49
Volume lOl_ number 2. Evaluation
CHEMICAL PHYSICS LETTERS
1
7 October 1983
of critical temperatures
3. I. i?vo dimensions (i) Let us write
the magnetisation M for a two-dimensional Ising model [4] as a power series in x, where x = The radius of convergence is detemlined using the E-convergence table, givingx, = exp(-lJ/kT,) T, is the critical temperature,
esp (-4J/kT)_ where
- &XT3- 34x4 - 152x5 - 714x6 - 3472x7
Al= 1 - 2s
--- -
(3
By forming the successive ratios mr/mr,l and applying the E table. it is observed that the value ofx, Onsager’s exact values [S] by --5%. As a strategy to improve the accuracy. we resort to transforming In flf = -2~~
- 8x3 - 36~~ - 16&x5 - 816.67x6
-4080x7
differs from M to In Mr
.
(3)
Differentiating the above series with respect to x and then using the successive ratios of the coefficients together with the E table yields a radius of convergence of 0.1715781 (table l), in excellent agreement with Onsager’s exact value ]5] of (3 - Z3/‘) = (0.1715728). (ii) It can be demonstrated that the above result is not fortuitous by applying the method to another well-known series for wlucl~ the esact solution is available_ Consider the truncated, high-temperature susceptibility series x for a triangular lattice ]6] : ~=1~6w+3Ow2+I3Sw3~6O6w4+2586~5~1O818w6+44574o7+181542~8~..., where w = tan11 (.J/kT)_ As before. d(ln x)/da = 6 + 24~
we convert the above series into logarithmic
+ 90w2 + 336~~
+ 12660~
By making use of the successive ratios of the coefficients exact value is 0.26795 ]6] _
In the case of three-dimensional
(4)
+ 4752~5
form and differentiate:
+ 17645.999&
+ 657600~
+ --- _
in the E table, we is found to be 0.26736,
models, for which esact solutions
(5)
whereas the
are not available, a comparison
can be made
with l’ade estimates [7] _As a case study. the high-temperature susceptibility series for the simple-cubic (SC) and body-centred-cubic (bee) lattices have been considered and the critical temperatures evaluated as shown below_ (_i) SC lartice: Starting with the susceptibility series. it is easy to obtain
Table 1
CvJuarion of the critical temperature for the two-dimensional Isin_emodel
0 0.1666666666 0 0.1666666666 0
0.1666666666 109.9999895
0.171428571 -1428571.429
0 0.171527871 0.171569321
50
0.1715781 7075.6889
0_171428567 7069.635914
0 0
420.0308535 0.171427871
CHEMICAL PHYSICSLETTERS
Volume 101, number 1
d(ln x)]dw = 6 + 24~ + 126~~ -t 528~~
i- 11160~~
f 2646~~
Use of the e-convergence table yields a value of 0.21869
7 October 1983
+ 54942&
+-236447.8640~
+ . .. _
for o; while the Pad5 estimate is 0.218156
(6)
171. Again,
there is good agreement_ (ii) &cc Zattice: The bee lattice can be treated in a manner similar to that above by using the series [7] :
[email protected] x)/do = 8 -I-48~ + 34402
+ 2016~~
+ 13928w4
+ 833760s
+ 5675120~
+3443136w7
+ _.. _
(7)
wc is determined to be 0.15647 (compared with the Fade prediction of 0.15617 [7]). For brevity, we have not shown the tables corresponding to triangular, SCand bee lattices. 2.3. Virial s&es (i) Hard discs: The virial series for hard discs may be written as 181 (the 8th and 9th virial coefficients are from ref_ [9]): p/pkT=
1 + bp + 0.782b2p’
+ 0_0647b7p7
t- 0.0359b8p8
-+ 0.53223b3p3
f- 0.33355b4p4
= 1+ %3rpr-1
-I-0.1989bsp5
+ 0.1 148b6p6
,
@I
were used in the second cdwhere B2 = b = $m2 and a is the hard-disc diameter. The successive ratiosB#$_L urnn of the E table, and the radius of convergence is found to be 1/055402543b, corresponding to p = 0995pe (table 2). The divergence at p/p= = 1 as inferred from the above method supports the earlier conjecture of Baram and Luban [S] . (ii)Hard spheres: In a similar manner, the virial series for hard spheres may be written as [LO] (the value of B, is from ref. [9]) p/pkT
= 1 + bp + 0.625b’p’
+ 0.2869b3p3
+ 0.1 103b4p4 + 0.0386b5p5
i- 0.01 27b6p6 + 0.00445b’p’
) (9)
where b = 1 n$ , u being the hard-sphere diameter. Using the ratios of successive coefficients 13,/B,_ I as in the case of hard discs, the radius of convergence is found to be l/0321 171 b, corresponding to p/p, = I_ _
Table 2 Estimation of the radius of convergence of the vixialseries for hard discs 0
b
0 0
0.782b
0.680601B
0
-4.58715593 OS92422Ob -9.8620302b -18.5573515b
0.6267140b 0
0.565596628
-47.1401600b -133.5617443
0.55402543b
0.5569013b -32.8814004b
OS9630167b 0
51
Volume 101. number 1
CHEMICAL
PHYSICS
7 October
LETTERS
1983
3. Estimation of critical exponents Let us consider a typical asymptotic matching of ln M with j3 ln (1 -x/x,)_ If we write In &f = 2$= I fr (x/x,.)‘, our aim is to verify that the sequence f--r&.] converges as c + a; the convergence limit can then be identified as +3 in the series as M = (1 -s/xx,)~. lx1 an analogous manner, the esponent 7 can be obtained in the asymptotic Emit viz. x: * (I - w&+_)~ _
(i> Ising model: Tfle ma&netisation in nr = -0.OSSS745
(.r/y,)2
series is transformed
- 0.0-1040507 (s&)3
- O.O2081;235(_Y~~,)6 - 0.0178566
(x/x,)’
using Onsager’s esact vaIue of s, into - 0.03 I 1958 (x/s,)4
= x
I,.(s/xJ
- 0.0249777(x/x,)5
_
(10)
Using the appropriate values of [-rl,] to form the initial sequence of the E table. the critical exponent p is found to be 0.125 (table 3). in agreement with Onsager*s esact value of l/S [5] _ (ii) Trimgzrkw larricc: The sme procedure is employed to obtain the exponent y from the hi&temperature susceptibility series [eq. (J)] _y is found to be 1.74796. in good agreement with the estimated value of 1.749 +o.oo_?, Ill]_
fi) bee lattice: From cq. (7). it is easy to obtain In s = 1.3493752(0~~,) + 0_25s781?9(w~w,)~
+ 2.331075S6(w/wJ2
i- 0.4367726(w/~,)~
+ 0.~016077(~~~~~6
+ 0_~~0~~7G~~~~=)7
+ ~.~99S07~~7(~/~~)4 =iz
f,(W/W,)’
_
0 1)
As explained cxlier. tire sequence [--r&j teuds to a limit of - IA%95 in the E table (not given here). This may be compared with ,m earlier theoretical estimate of -1.3 1 +- 0.05 [I I] and a more recently reported value of --I 2402 + 0.0009 1 I 21 (see also ref. [ 131 for more details). (ii) SCZarricc: IJI the c’tw of the simple-cubic lattice. use of the same procedure with eq. (6) yields a value for y of -1 ._W99S .
-0.lI7749 -1?SSAP8710 -0.12121517
+0.0003
19795
-2SO.270629 -0.1’37832
-368.298121 -0.11489170
-9196.67616 -0.123sss5
-0.OS7SZ1SQ -9169_69697
-0.1’49911
-0.125000 -9469.6993
CHEMICAL PHYSICS LETTERS
Volume101, number1
7 October 1983
In the case of hard spheres and hard discs, the value of the critical exponent, 7’;: for the vilria1series is somewhat uncertain. Baram and Luban are of the opinion that 7” = -1 in both cases in the asymptotic limit, vi& p/pk? = (1 - p/r;&-1 f8), whereas, according to the scaled particle theory, r’ = -3 f?4fI To estimate +y’the method mentioned earlier (section 3.1) is applied to the virial series. (i) Hard discs: The virial series is rewritten as
In@JpkT) = 1.813799@1~,) + 0_21~86473~/~~)5
+ 0.92774281
(j&-Q2 + 0.4986338(&J3
+ 0_~6414OS~/~=)6
+ 0.~23307584~/~~)7
+ 029833157@i~e)~ + .__= c
$XPf&)’
-
(12)
The sequence f-&r], where I- varies from 3 to 7, was used as the entry in the f table, and the convergence Emit of ’ the sequence yields the value of the exponent 7’ in p/pkT x (1 - p/p,)‘)“. The reason for forming the specific sequence is to achieve smooth convergence to the limit. The value of 7’ is -1.05, viz_ the asymptotic form of p/pkT is *(l - p/~,)-~. (ii) Hard spheres: The value of the critical exponent for hard spheres is calculated in an analogous manner from the series:
h(@plpkT)= 2_9619219(p/p,)+
1.09662266
C--P/P,)”
* W2386127@/~,)~
+
023763006~~~~)~
The sequence [--r&f using all of the values of jr is employed, and the critical exponent is found to be -0,92_ Because of the uncertainties in the estimated value of the higher-order virial coefficients (B7 and IQ), there is some inaccuracy in the fmal value of y’; nevertheless, it is reasonable to believe that r’ = -1 in the case of hard spheres also. The above analysis demonstrates the relevance of E convergence to the questions raised here, and shows that the E algorithm succeeds well, when it works! The e-convergence technique is no doubt related to Pad& approximants [3, IS], but the inherent simplicity and elegance is a point in favour of the former. (It may be mentioned that ah calculations reported here were performed using a non-programmable pocket calculator!) Definitely, there is no “overkill” in this method! The potentials of this method are enormous, and the utility of this algorithm for the evaluation of pressure from partial virial series has been demonstrated by us elsewhere [ 161. We have also shown (~pub~~ed~ how critical parameters for many other systems, e-g. ferrimagnets 1171, can be obtained by this route. The input coefficients in the examples considered here have all been from series approximations (e.g. virial, high-temperature expansion, etc_) obtained theoretically. There is no reason why reliable polynomial fits to experimental observations (in appropriate variables) cannot be used to predict the onset of critica@inguXar features (eg activity coefficients of single- or multi-components, specific heats, etc.), Finally, a word about rigour. Many of these methods have not had any theoretical footing frmer than those of the Pad& approximations 131. The intuitive elements in the analysis (e.g. use of cumulants for better convergence properties) are valuable but need a more satisfactory theoretical support. We are investigating these aspects.
Refe*nces [I] z). Shanks, J. Math. Plxys.34 (1955) f. [ 21 P. Wynn, Math. Tables and Other Aids to Computation 10 (I 956) 91_ f3] C-A- Baker Jr. and P. Gxaves-Morris.in: Encyclopedia of maihematics,Vol. 13, ed. G.-C. Rota (Ad~on-W~s~e~~ Reading, 1981).
53
Volume [4] [5] [6] [7] [S] [9j [ lo]
101. number
1
CHEMICAL
PHYSICS
T.L. Hill, Statistical mechanics (McGraw-Hill, New York, 1956). L. Onsager. Nuovo Cimento (Suppl.) 6 (1949) 261. C-J. Thompson. Mathematical statistical mechanics (MacMillan, New York. 1972)). G.A. Baker Jr.. Phys. Rev. 124 (1961) 768. A. Baram and hi. Luban. J. Phys. Cl Z (1979) L659. KU’. Kmtky. J. Cbem. Phqs. 69 (1978) 2751. C.A. Crouton, Introduction to liquid-state physics (Wiley, New York, 1975).
[ 111 G.A. Bskrr Jr. and D.S. Gaunt. Phys. Rev. 155 (1967) 545. [ 171 J.C. Le Guillou and J. Zinn-Justin. Phys. Rev_ Letters 39 (1977) [ 131 [ 141 [ 151 [ 161 [ 171
54
LETTERS
95. A. Kumar, H.R. Krishnamurrhy and E.S.R. Gopal, Phys. Rep. (1983). to be published. H. Reiss, 1-1. Frisch and J.L. Lebowitz, J. Chem. Phys. 31 (1959) 369. J.R. hlacDonald_ J. Appl. Phys. 35 (1964) 3034. hl_V. Sxn_eunnarayan.m and SK. Ran_garajan. Phys. Lrtrers 96A (1983) 339. R-G. Bo\\ crs and B.Y. Y ousif. Phys. Letters 96A (1983) 49.
7 October
1983
CHEMKAL PHYSICS LETTERS
AN E-CONVERGENCE METHOD FOR THE EVALUATION
M.V. SANGARANARAYANp;N
and SK
7 October 1983
OF CRITICAL PARAMETERS
RANGARAJAN
Deparment of Inorganic and Physical Chemistry. Indian Institute of Science, Barrgaiore .&Xl 012, Itrdfa Received 16 May 1983; in final form 8 July 1983
A simple cconversence technique is applied to obtain accurate estimates of critical temperatures and critical it\ponmts of a few two- and threpdiniensional king models. When applied to the virial series for hard spheres and hard discs, this method predicts a divergence of the equation-of-state at the density of closest packing.
1. Introduction
In this note, we report how a simple algorithm, termed the ‘&convergence method“ [ 1.2]_ can be used to obtain values of critical parameters. We illustrate tltis claim by applying the method to determine critical tentperatures and critical exponents associated with some well-studied Ising models, starting front the respective series expansions. We also demonstrate the use of the method in obtaining information about the divergence and the asymptotic limit of the virial series for hard spheres and hard discs. The primary question addressed in the E algorithm is the following- Given the partial sequence aI, a?, .._, a,l. what is its likely limit, Le. what is limn_,ma,z ? What we do in this paper is to map several questions into problems of limits: (A) The problem of evaluating the radius of convergence xc of a class of power series. 22~~~~~~ can be converted into one of evaluating the limit as 12+ 00 of an/all+1 _ (B) In the problem of asymptotic matching, for example. we verify if the power series I&.x’ diverges as (1 x/x,)@ and estimate fl_ (C)Yet another interesting problem is to extrapolate from partial sums information regarding the infinite sum. Examples of pltysical counterparts of (A)-(C) given above are: (A) the prediction of critical temperatures or densities, as the case may be, (B) the prediction of critical exponents, and (C) estimating from the first few virial coefficients the values of pressure at high densities. We discuss here (A) and (B) only. The E algorithm 131 constructs a two-dimensional array ef) front a given partial sequence ~8~. i = 0, 1. ___)N- 1 entered as a column. The superscript j refers to an eiement in column X-_The various elements of the table are cd-
culated by using the algorithm:
(1) We define the column tii , 3 0, and set the given sequence to be eo. i The resulting table is not unlike a difference table but with a non-linear transformation law [eq. (l)] (also called the “rhombus rule”).
0 OOQ-~614/83/000~-OOOOIS 0300 0 1983 Norm-Hoard
49
Volume lOl_ number 2. Evaluation
CHEMICAL PHYSICS LETTERS
1
7 October 1983
of critical temperatures
3. I. i?vo dimensions (i) Let us write
the magnetisation M for a two-dimensional Ising model [4] as a power series in x, where x = The radius of convergence is detemlined using the E-convergence table, givingx, = exp(-lJ/kT,) T, is the critical temperature,
esp (-4J/kT)_ where
- &XT3- 34x4 - 152x5 - 714x6 - 3472x7
Al= 1 - 2s
--- -
(3
By forming the successive ratios mr/mr,l and applying the E table. it is observed that the value ofx, Onsager’s exact values [S] by --5%. As a strategy to improve the accuracy. we resort to transforming In flf = -2~~
- 8x3 - 36~~ - 16&x5 - 816.67x6
-4080x7
differs from M to In Mr
.
(3)
Differentiating the above series with respect to x and then using the successive ratios of the coefficients together with the E table yields a radius of convergence of 0.1715781 (table l), in excellent agreement with Onsager’s exact value ]5] of (3 - Z3/‘) = (0.1715728). (ii) It can be demonstrated that the above result is not fortuitous by applying the method to another well-known series for wlucl~ the esact solution is available_ Consider the truncated, high-temperature susceptibility series x for a triangular lattice ]6] : ~=1~6w+3Ow2+I3Sw3~6O6w4+2586~5~1O818w6+44574o7+181542~8~..., where w = tan11 (.J/kT)_ As before. d(ln x)/da = 6 + 24~
we convert the above series into logarithmic
+ 90w2 + 336~~
+ 12660~
By making use of the successive ratios of the coefficients exact value is 0.26795 ]6] _
In the case of three-dimensional
(4)
+ 4752~5
form and differentiate:
+ 17645.999&
+ 657600~
+ --- _
in the E table, we is found to be 0.26736,
models, for which esact solutions
(5)
whereas the
are not available, a comparison
can be made
with l’ade estimates [7] _As a case study. the high-temperature susceptibility series for the simple-cubic (SC) and body-centred-cubic (bee) lattices have been considered and the critical temperatures evaluated as shown below_ (_i) SC lartice: Starting with the susceptibility series. it is easy to obtain
Table 1
CvJuarion of the critical temperature for the two-dimensional Isin_emodel
0 0.1666666666 0 0.1666666666 0
0.1666666666 109.9999895
0.171428571 -1428571.429
0 0.171527871 0.171569321
50
0.1715781 7075.6889
0_171428567 7069.635914
0 0
420.0308535 0.171427871
CHEMICAL PHYSICSLETTERS
Volume 101, number 1
d(ln x)]dw = 6 + 24~ + 126~~ -t 528~~
i- 11160~~
f 2646~~
Use of the e-convergence table yields a value of 0.21869
7 October 1983
+ 54942&
+-236447.8640~
+ . .. _
for o; while the Pad5 estimate is 0.218156
(6)
171. Again,
there is good agreement_ (ii) &cc Zattice: The bee lattice can be treated in a manner similar to that above by using the series [7] :
[email protected] x)/do = 8 -I-48~ + 34402
+ 2016~~
+ 13928w4
+ 833760s
+ 5675120~
+3443136w7
+ _.. _
(7)
wc is determined to be 0.15647 (compared with the Fade prediction of 0.15617 [7]). For brevity, we have not shown the tables corresponding to triangular, SCand bee lattices. 2.3. Virial s&es (i) Hard discs: The virial series for hard discs may be written as 181 (the 8th and 9th virial coefficients are from ref_ [9]): p/pkT=
1 + bp + 0.782b2p’
+ 0_0647b7p7
t- 0.0359b8p8
-+ 0.53223b3p3
f- 0.33355b4p4
= 1+ %3rpr-1
-I-0.1989bsp5
+ 0.1 148b6p6
,
@I
were used in the second cdwhere B2 = b = $m2 and a is the hard-disc diameter. The successive ratiosB#$_L urnn of the E table, and the radius of convergence is found to be 1/055402543b, corresponding to p = 0995pe (table 2). The divergence at p/p= = 1 as inferred from the above method supports the earlier conjecture of Baram and Luban [S] . (ii)Hard spheres: In a similar manner, the virial series for hard spheres may be written as [LO] (the value of B, is from ref. [9]) p/pkT
= 1 + bp + 0.625b’p’
+ 0.2869b3p3
+ 0.1 103b4p4 + 0.0386b5p5
i- 0.01 27b6p6 + 0.00445b’p’
) (9)
where b = 1 n$ , u being the hard-sphere diameter. Using the ratios of successive coefficients 13,/B,_ I as in the case of hard discs, the radius of convergence is found to be l/0321 171 b, corresponding to p/p, = I_ _
Table 2 Estimation of the radius of convergence of the vixialseries for hard discs 0
b
0 0
0.782b
0.680601B
0
-4.58715593 OS92422Ob -9.8620302b -18.5573515b
0.6267140b 0
0.565596628
-47.1401600b -133.5617443
0.55402543b
0.5569013b -32.8814004b
OS9630167b 0
51
Volume 101. number 1
CHEMICAL
PHYSICS
7 October
LETTERS
1983
3. Estimation of critical exponents Let us consider a typical asymptotic matching of ln M with j3 ln (1 -x/x,)_ If we write In &f = 2$= I fr (x/x,.)‘, our aim is to verify that the sequence f--r&.] converges as c + a; the convergence limit can then be identified as +3 in the series as M = (1 -s/xx,)~. lx1 an analogous manner, the esponent 7 can be obtained in the asymptotic Emit viz. x: * (I - w&+_)~ _
(i> Ising model: Tfle ma&netisation in nr = -0.OSSS745
(.r/y,)2
series is transformed
- 0.0-1040507 (s&)3
- O.O2081;235(_Y~~,)6 - 0.0178566
(x/x,)’
using Onsager’s esact vaIue of s, into - 0.03 I 1958 (x/s,)4
= x
I,.(s/xJ
- 0.0249777(x/x,)5
_
(10)
Using the appropriate values of [-rl,] to form the initial sequence of the E table. the critical exponent p is found to be 0.125 (table 3). in agreement with Onsager*s esact value of l/S [5] _ (ii) Trimgzrkw larricc: The sme procedure is employed to obtain the exponent y from the hi&temperature susceptibility series [eq. (J)] _y is found to be 1.74796. in good agreement with the estimated value of 1.749 +o.oo_?, Ill]_
fi) bee lattice: From cq. (7). it is easy to obtain In s = 1.3493752(0~~,) + 0_25s781?9(w~w,)~
+ 2.331075S6(w/wJ2
i- 0.4367726(w/~,)~
+ 0.~016077(~~~~~6
+ 0_~~0~~7G~~~~=)7
+ ~.~99S07~~7(~/~~)4 =iz
f,(W/W,)’
_
0 1)
As explained cxlier. tire sequence [--r&j teuds to a limit of - IA%95 in the E table (not given here). This may be compared with ,m earlier theoretical estimate of -1.3 1 +- 0.05 [I I] and a more recently reported value of --I 2402 + 0.0009 1 I 21 (see also ref. [ 131 for more details). (ii) SCZarricc: IJI the c’tw of the simple-cubic lattice. use of the same procedure with eq. (6) yields a value for y of -1 ._W99S .
-0.lI7749 -1?SSAP8710 -0.12121517
+0.0003
19795
-2SO.270629 -0.1’37832
-368.298121 -0.11489170
-9196.67616 -0.123sss5
-0.OS7SZ1SQ -9169_69697
-0.1’49911
-0.125000 -9469.6993
CHEMICAL PHYSICS LETTERS
Volume101, number1
7 October 1983
In the case of hard spheres and hard discs, the value of the critical exponent, 7’;: for the vilria1series is somewhat uncertain. Baram and Luban are of the opinion that 7” = -1 in both cases in the asymptotic limit, vi& p/pk? = (1 - p/r;&-1 f8), whereas, according to the scaled particle theory, r’ = -3 f?4fI To estimate +y’the method mentioned earlier (section 3.1) is applied to the virial series. (i) Hard discs: The virial series is rewritten as
In@JpkT) = 1.813799@1~,) + 0_21~86473~/~~)5
+ 0.92774281
(j&-Q2 + 0.4986338(&J3
+ 0_~6414OS~/~=)6
+ 0.~23307584~/~~)7
+ 029833157@i~e)~ + .__= c
$XPf&)’
-
(12)
The sequence f-&r], where I- varies from 3 to 7, was used as the entry in the f table, and the convergence Emit of ’ the sequence yields the value of the exponent 7’ in p/pkT x (1 - p/p,)‘)“. The reason for forming the specific sequence is to achieve smooth convergence to the limit. The value of 7’ is -1.05, viz_ the asymptotic form of p/pkT is *(l - p/~,)-~. (ii) Hard spheres: The value of the critical exponent for hard spheres is calculated in an analogous manner from the series:
h(@plpkT)= 2_9619219(p/p,)+
1.09662266
C--P/P,)”
* W2386127@/~,)~
+
023763006~~~~)~
The sequence [--r&f using all of the values of jr is employed, and the critical exponent is found to be -0,92_ Because of the uncertainties in the estimated value of the higher-order virial coefficients (B7 and IQ), there is some inaccuracy in the fmal value of y’; nevertheless, it is reasonable to believe that r’ = -1 in the case of hard spheres also. The above analysis demonstrates the relevance of E convergence to the questions raised here, and shows that the E algorithm succeeds well, when it works! The e-convergence technique is no doubt related to Pad& approximants [3, IS], but the inherent simplicity and elegance is a point in favour of the former. (It may be mentioned that ah calculations reported here were performed using a non-programmable pocket calculator!) Definitely, there is no “overkill” in this method! The potentials of this method are enormous, and the utility of this algorithm for the evaluation of pressure from partial virial series has been demonstrated by us elsewhere [ 161. We have also shown (~pub~~ed~ how critical parameters for many other systems, e-g. ferrimagnets 1171, can be obtained by this route. The input coefficients in the examples considered here have all been from series approximations (e.g. virial, high-temperature expansion, etc_) obtained theoretically. There is no reason why reliable polynomial fits to experimental observations (in appropriate variables) cannot be used to predict the onset of critica@inguXar features (eg activity coefficients of single- or multi-components, specific heats, etc.), Finally, a word about rigour. Many of these methods have not had any theoretical footing frmer than those of the Pad& approximations 131. The intuitive elements in the analysis (e.g. use of cumulants for better convergence properties) are valuable but need a more satisfactory theoretical support. We are investigating these aspects.
Refe*nces [I] z). Shanks, J. Math. Plxys.34 (1955) f. [ 21 P. Wynn, Math. Tables and Other Aids to Computation 10 (I 956) 91_ f3] C-A- Baker Jr. and P. Gxaves-Morris.in: Encyclopedia of maihematics,Vol. 13, ed. G.-C. Rota (Ad~on-W~s~e~~ Reading, 1981).
53
Volume [4] [5] [6] [7] [S] [9j [ lo]
101. number
1
CHEMICAL
PHYSICS
T.L. Hill, Statistical mechanics (McGraw-Hill, New York, 1956). L. Onsager. Nuovo Cimento (Suppl.) 6 (1949) 261. C-J. Thompson. Mathematical statistical mechanics (MacMillan, New York. 1972)). G.A. Baker Jr.. Phys. Rev. 124 (1961) 768. A. Baram and hi. Luban. J. Phys. Cl Z (1979) L659. KU’. Kmtky. J. Cbem. Phqs. 69 (1978) 2751. C.A. Crouton, Introduction to liquid-state physics (Wiley, New York, 1975).
[ 111 G.A. Bskrr Jr. and D.S. Gaunt. Phys. Rev. 155 (1967) 545. [ 171 J.C. Le Guillou and J. Zinn-Justin. Phys. Rev_ Letters 39 (1977) [ 131 [ 141 [ 151 [ 161 [ 171
54
LETTERS
95. A. Kumar, H.R. Krishnamurrhy and E.S.R. Gopal, Phys. Rep. (1983). to be published. H. Reiss, 1-1. Frisch and J.L. Lebowitz, J. Chem. Phys. 31 (1959) 369. J.R. hlacDonald_ J. Appl. Phys. 35 (1964) 3034. hl_V. Sxn_eunnarayan.m and SK. Ran_garajan. Phys. Lrtrers 96A (1983) 339. R-G. Bo\\ crs and B.Y. Y ousif. Phys. Letters 96A (1983) 49.
7 October
1983