Download A Lattice Model of Liquid Helium» II I. Introduction

19
Progress of Theoretical Physics, Vol. 17, No. 1, January 1957
A Lattice Model of Liquid Helium» II
Hirotsugu MATSUDA
Quantum Chemistry Laboratory, Chemistry Department, Kyoto University, Kyoto
and
Takeo MATSUBARA
Research Institute for Fundamental Physics, Kyoto University, Kyoto
(Received August 28, 1956)
The partition function of the lattice model of liquid helium introduced in the preceding paper is
calculated by making use of the Kikuchi's approximation. Setting two parameters, the effective mass
m=1/7m 0 (mo is the mass of He atom), and the lattice spacing d=3.1A, an excellent agreement of
the density dependence of A-temperature with experiment is obtained. The general trend of other
thermodynamical quantities calculated in this paper is in good accordance with the observation.
I.
Introduction
In the preceding paper/> hereafter referred to as I, we have introduced Hamiltonian
of the lattice model of liquid helium and proved its equivalence to that of a spin system
with anisotropic exchange coupling. The A-transition was shown there to correspond to
the spin system, and making use of the molecular field approximation, we could obtain
right dependence of the A-temperature on density. Phonons in liquid helium were proved
to correspond to spin waves in the spin system.
All the properties of our lattice model, however, were derived by translating the
results obtained for the spin system, under certain assumptions which were not necessarily
without question open to criticism. For example, in the molecular field approximation we
assumed the existence of the long range order of magnetization in the 'ICY plane, but there
is some difficulty in defining the long range order because of the fact that the magnetization in the x y plane is not constant of motion owing to the anisotropic exchange
coupling. Furthermore, the crude approximation adopted in I fails to afford not only
quantitative conclusions on specific heat and pressure, etc., but also qualitative explanation
for the last of the three questions proposed at the begining of I. Namely as to the
question why liquid helium has negative thermal expansion coefficient just below the J.-point,
the previous treatment seems unsatisfactory for obtaining a correct solution.
To remedy these points, we shall employ in this paper a rather direct method of
approximation without referring to the spin system for calculating the partition function
of the lattice model, and try to understand qualitatively the thermodynamical behavior of
H. Matsuda and T. Matsubara
20
the liquid helium near the A-point.
§ 2.
Reduction of the partition function
The Hamiltonian of the lattice model has been taken in I as
(1)
where m is the effective mass of an atom, d is the lattice constant and ( ij) means to
take the summation over all the nearest neighbor pair points. The operators a; and a;*
satisfy the commutation relations :
[a;,
aj]- =[a;*, a,*]_ =[a;, aj]_ =0 for i ~ j,
[a;, a;]+=[a;*, a;*]+=O,
[a;, a-:]+=1 for
i=j.
(2)
(3)
Here, for the sake of simplicity, we put v 0 = 0, because the weak attractive potential will
not play any essential role in the peculiar behaviors of thi~ substance. In evaluating the
partition function, we regard the total number of atoms N = :8 a,* ai as a given constant.
i
Then, the Hamiltonian may be written apart from the additive constant as
H= -B:8 a;* aj,
(4)
<ij>
where
B=flj2md2 •
The problem is to calculate the partition function
Z=Trace [exp( -fJH)].
In the representation where a;* a;'s are diagonal, a[ a,~ is such an operator that transfer
an atom from the j-th to the i-th lattice point if the j-th pG:ition is occupied by an
atom and the i-th by a hole. Otherwise it gives a zero. Therefore, when we expand
Zas
Z=Trace [:8(-fJ H)njn!]
(5)
n
and substitute ( 4) into ( 5) , we see that there appear in the right-hand side of ( 5)
infinitely many terms, to each of which certain transfers of atoms correspond. If we
represent each transfer of an atom by a line connecting the initial and final position of
the atom, each term of (5) can be described by a set of closed polygons which we shall
call a graph (Fig. 1). That is, Z can be written as
Z= 2j G2n (fJ B) 2n/ (2n)!
(5')
n
and the problem is reduced to counting the number of graphs G2n produced by 2n
transfers. In a graph there may be double or triple, etc., line3 corresponding to twice
or three times, etc., repeated transfers of an atom along the same path and this sitnation
introduces a formidable complication which prevents us from counting G2,. in a simple
A Lattice Model of Liquid Helium, II
21
manner. In what follow~, we shall give up the rigorous calculation of G2n and make a
simplification so a- to attain an approximate expression of Z on the basis of the theory of
prob2.bility.
First, we shall define for
each graph its skeleton which is
a figure obtained from a graph
~
by replacing all the multiple
lines with single lines (Fig. 2).
When we transfer atoms only
along the lines of skeleton, a
set of closed polygons is again
(c)
(a)
(b)
(d)
formed corresponding to a graph
Fig. 1. Examples of allowed graphs
with the smallest number of
transfers in a given skeleton
(Fig. 3) . We call the graph
thus obtained a fundamental
figure (abbreviated as F. G.)
corresponding to the original
graph. We can always reduce
each graph to its F. G. by the
above mentioned procedures, and
(a)
(b)
(c)
(d)
therefore we may classify the
Fig. 2. The skeleton corresponding to Fig. 1.
graphs by the corresponding
D c;]
--
·D
CJ
F. G.'s.
Now let us fix the reference
configuration of atoms and holes
at any one of [M!j (M- N) !N !]
configurations where M denotes
the, total number of lattice points.
If we are concerned with the
(a)
(b)
(c)
(d)
term of order 2n in (5'), the
Fig. 3. The fundamental figures corresponding to Fig. 1.
centra~ problem we have to solve
is : In how many ways can we transfer atoms on a F. G. and return to the initial
configuration after 2n transfers ?
Before we enter into this problem, we shall define several notations. By the term
"unit line" we mean a line connecting nearest neighbor lattice points, and let the total
number of unit lines in a skeleton be m (the number of bonded pairs), and the total
number of lattice points forming the skeleton be s (the number of bonded apices) .
Further, let the number of unit lines included in the F. G. be 2m which we call the
side length of the F. G.* p0 = (M-N)jM represents the probability of finding a hole at
• This is necessarily an even number.
H. Matsuda and T. Matsubara
22
a lattice point and p 1 =MIN the probability of finding an atom at a lattice point.
In order to settle the above mentioned problem, we shall base our calculation on
the following assumptions :
Assumption I
When we want to transfer an atom along a unit line, this unit line should have
one end occupied by an atom and the other end by a hole. The probability that a
unit line is " transferable" (i. e. along which one can transfer an atom) is, therefore,
equal to 2p0 p1 • Among m unit lines in a skeleton, there are 2mpoP1 transferable unit
lines. If we regard for a moment 2n transfers of atoms on a graph as independent
events to each other, there are [2mp 0 p1] 2" ways of arranging these 2n transfers.
Assumption II
We have to correct the error produced by assuming 2n independent transfers. As
a nature of trace, atoms should return to their initial configuration after 2n transfers. If
we assume that each atom almost forgets its initial configuration during each travel, the
probability for returning to the initial configuration will be approximated by the reciprocal
of the number of ways that s p 1 atoms and sPo holes can be distributed among s lattice
points of the skeleton, or by [s!j (sp0)! (sp1) !]-1.
Assumption III
In order that a graph drawn by arranging 2n unit lines according to assumptions I
and II belongs to the original F. G. with side length 2m, it is necessary that each of
2m unit lines of F. G. is used up, at least once, as the path of transter. This restriction
will be approximately taken into account by a factor f(m, n), which is the probability
that 2n indistinguishable particles can be taken out of a set composed of equal number
of a great many particlea of 2m species, with a restriction that each species should appear
at least once in 2n particles. Evidently for n < m
f(m, n) =0,
and for
(6)
m<n~2m
f(m, n),....., [ (2n) !l22(n-mlJ(2 (n2:m)) (2m) -2n,
Multiplying these three factors together with the temperature factor ({?B) 2" I (2n)! and
the total number of configurations (M!j (M- N)! N!), and summing up with respect to
both n and all possible F. G., we have
z
Mf
(M-;)! N!
"'
s..~f(m,n)[2mpop1]2n
({?B) ?n
(2n)!
[
J-l,
s!
-(s-Po-)-!(-sp-1)_!_
(7)
where S means the summation over all possible F. G.'s.
Each F. G. will be classified by a set of parameters {~.} and {y,} which are to be
and s then become functions of {x.}
introduced in the next section. The values of m,
and {y.}. Denoting by g ( {x.}, {y,}) the number of F. G.'s belonging to the set of
parameters {xi} {y,}, (7) can be rewritten as
m
A Lattice Model of Liquid Helium, II
23
\XTe further approximate Z by the maxtmum term of the right-hand side of (7 1) , the
log<:.rithm of which is denoted by P( {xi}, {yi}). Then from oP( {xi}, {yi} )jon=O we
hav~
--=--]
1
n=m[1 + 1+4m/r
(8)
for m < n ~2m, where r =2m p0 p1 fiB. We shall see in the next section that at the
i.-point r<m/2, so that n<m (1 1/9). So far as we are concerned with the temperature
range near the A-point, we may therefore put n:::::::..m.
+
§a. Introduction of Kikuchi's approximation
In evaluating g ({xi}, {yi} ) , we shall use so-called Kikuchi's approximation. In his
paper2> on the A-transition of liquid helium, Kikuchi undertook a calculation of the total
number of ways g(L) that polygons can be drawn by connecting nearest neighbor points
of a simple cubic lattice so that the total number of sides of all the polygons altogether
is L Since he made use of the Feynman's formulation, 3> the rule of drawing polygons
in his case is somewhat different from that in our F. G. For instance, such a polygon
as (d) in Fig. 3 is not allowed in Kikuchi's case while it cannot be excluded according
to our definition of F. G. But in the temperature region near the A-point where, as we
shall see presently, the polygons of one-sided type like (b) and (c) in Fig. 3 have just
become important, it will not lead to a serious error even if we neglect such complicated
polygons as (d) in Fig. 3. Then we may put the analogous restrictioru, as Kikuchi's
case concerning polygons. They are :
(i) between two adjacent lattice points only one side is drawn except for two sided
polygons, and
(ii) polygons must not come into contact.
We need not distinguish two directions of cyclic transfer because its effect is already
taken into account by a factor 2 in the probability factor [2mp 0 p1] 2".
Under these restrictions concerning polygons, we have now table I for the possible
configurations of a lattice point and table II for that of a bond.
The x's and y's are the probabilities of finding one of the respective configurations and
the values of a's and {i's are th~ numbers of different configurations having the same
probability on the basis of symmetry requirement. With the aid of tables I and II, we
can express m, in and s in terin.s of x' s and y' s. When m < n ~ 2m
2n:::::::..2m =3M(2y4 + 25y8) ,
m=3M(y4 +25y8 ) ,
s=M(6x2 + 15x3) .
(9)
24
H. Matsuda and T. Matsubara
Table I
Configuration
Configuration of a point
Probability
•
1
11:2
6
xa
15
Table II Configuration of a bond
•
•
::::.
.
c:
u
-===• L
Qt
..J
{1,
Probability
Configuration
r
r
~
11
1
12
10
1s
25
14
1
15
20
16
100
'Y'1
100
25
Ys
According to Kikuchi's approximation for the case of simple cubic lattice, g ( {x;}, {y,}) is
approximated by
g ( {x,}, {y,}) =X6 y- 3 (M!) 2,
(10)
where
8
3
X=ll(x, M)!'\
i=l
Y=ll(y.
M)!~•.
i=l
Putting (10) into (7'), we finally obtain for the logarithm of the maximum term of
the partition function per lattice point
(11)
where we put
F= ln ro =ln CPoP1 jj B),
(12)
A=Po 1np0 +p1 1np1"
In deriving (11) we have made a simplification that in::::::. m which will be justified later
on.
A Lattice Medel of Liquid Helium, II
§ 4.
25
Region just below the A-point
From the tables I and II, the following geometrical relations among x's and y's are
obtained:
x1 =y1 +5 y2 + 10yr;,
(13)
x2=y2+ 5 Ys+ 10 'Yo='Yo
Xs=y, +5 Ya+ 10 'Y7=5 'Ys·
If we choose y3, y4 , y6 , y1 and y8 as independent variables, the other parameters can be
expressed as linear combinations of them, as shown in table III.
Table III. Relations between the dependent and independent variables.
The meaning of this table is, for example, Xl = 1 - 6y4 - 75ys.
1
Y7
)'3
-6
1
Ys
-75
1
5
1
25
-11
-5
1
100
-10
100
-125
-5
-10
5
Maximizing P( {xi}, {yi}) given by (11) with respect to these independent variables,
one gets
(14a)
'Y1 Yo= 'Y2Yo•
(14b)
'Yl'Y1='Jr;2,
(14c)
exp [- 2 (A+ F)] (yi 0yiy/1 )
exp [-(A+ I')] (y54 y8/y15)
= (x2jx1) 10,
= (xix
1)
5•
(14d)
(14e)
These equations together with the: relations in t2.ble III can be easily solved by substitutions
Yi'Y1 =v,
(15)
yielding
v=1/5 exp [A+P]=1/5 exp(A)('0 /'J~B,
(16)
w=1/10(4 exp [A+l']-1) =2v-l/10.
(17)
The other variables can be written as. functions of v and w.
Since both y5 and h are positive, w should be also positive, and accordingly eq. (17)
26
H. Matsuda and T. Matsubara
shows that the solution obtained above is valid only when the temperature is lower than
the critical temperature TJ. defined by
4 exp [4 + r] -1 = 0
or
(18)
At T=T}., it holds that
r=2mp0 p1 (3 B~l_m exp (-/f)< mj2.
(19)
This justifies the approximation n ~ m in a self-consistent way.
Just below the critical temperature where our approximation is valid, the logarithm
of the partition function per lattice point is given by
P=3ln(1+5v+10:w) -2ln(1+6v+15:w) -4.
(20)
The energy E, heat capacity at constant volume Cu, and pressure p of the system can
be readily calculated as
E=_!_5 MeAPoPtB[---24_ __
1+6v+15:w
J
25
-,
1+5v+10:w
C -3M(I
B )2 1 [
864
nsPoPt eA kT2 (1+6v+15:w)2
625
(21)
J
(22)
kT- -oP
3 A PoP1 B
p=
- - =1- [kr(P-+ Pt 1nP1)
- +-e
d 3 oM
d3
Po
5
75
x{ 1+5v+10:w
72
}{ Pt -1-p In Pt }]
1
1+6v+15:w
Po
Po .
(23)
According to (22), the magnitude of specific heat at the critical point amounts to
Cu/Mk=33j43.
(24)
§ 5. The region above the A-point
When w is zero, x3 and y5 to 'Ys also vanish as is seen from (13) and (14).
Returning to the meanings of these parameters, we see that this corresponds to a state
in which only two-sided polygons exist. Then, of the five equations (14a) to (14e),
all but (14a) and (14d) become trivial identities. We can still use v as defined in
(16) for this case, but the results are different, namely
v= (r-1)/10
(25)
with
In this region, the partition function per lattice point, the total energy, heat capacity at
A Lattice Model of Liquid Helium, II
27
constant volume, and pressure of the system are respectively given by
P=31n(1+5v) -2ln(1+6v) -A,
E=2M(
12
1+6v
15
1 +5v
-
) (
A
e
PoPt
(26)
B)2 1
rkT
2( A
(.IB)2{
e PoPw
P=MkT [P+pl In A+(-1_5da
Po
1+5v
25
( 1 + 5v) 2
(27)
(28)
12 )
1+6v
(29)
At the critical temperature, (28) gives
Cv/Mk=ll/169.
(30)
Comparing this with Cn derived in § 4 and noting that E and p are continuous at T "'
we can conclude that this temperature T .. corresponds to the critical point of second-kind
transformation and may be identified with the A temperature of liquid helium.
§ 6.
Comparison with experiments and discussions
In order to compare our results with experiments, we must first fix two parameters~
the effective mass m and the lattice spacing d. If we choose
d=3.1A
(31)
where m0 is the mass of He4 atom. Eq. (18) gives an excellent agreement with experiment of the density dependence of the A temperature as is shown in Fig. 4. Along the
.?.-line, the value of p1 changes from 0.65 under the pressure of 1 atm. to 0.78 under
the pressure of 25 atm.
We have obtained rather small value for m. One of the reasons for this will be
due to too severe restriction of F. G. or neglect of the complicated polygons such as (d)
in. Fig. 3. The restriction, indeed, will impede possible particle transfers when the temperature is lowered. In other words, it will make one underestimate the quantum effect
and consequently will tend to lower the A-temperature, resulting in smaller value of m in
order to have the agreement with experiments.
H. Matsuda and T. Matsubara
28
Figs. 5 and 6 show respectively
the curve3 of heat capacity and
isopycnals calculated basing on the
\
present approximation. The general
trend of these curves agrees fairly well
with observations in a qualitative
01?
sense. Although Fig. 6 does not
show the existence of negative thermal
expansion coefficient in the density
range 0.6 < p1 < 0.8, to which the
0.16
"
density of actual liquid helium is
supposed to correspond, the gradient
of each curve has at the A-line a
X
discontinuity which becomes larger
0./5
with increasing density as is observed
in liquid helium.
The negative thermal expansion
coefficient
will be expected to come
I
out from the fact that the lower the
0.14 --=-------''--------'-------'
1.0
1.5
2.0
2.5"K
temperature is, the more important
T
becomes the contribution to the parFig. 4. The ,!-line calculated from eq. (18).
tition
function from a graph with
X correspond to. the experimental values due
large polygons which is sensitive to
to Keesom & Miss Keesom. 4>
the density variation.
For p1 = 0.9
the pressure increases considerably with decreasing temperature just below the A-point,
showing the possibility of the existence of negative thermal expansion coefficient.
As we have omitted the attractive potential from the beginn;ng, numerical values
of pressure can not be compared with experiments. Neglect of certain types of F. G.
makes one underestimate the value of P less and less as the temperature is lowered below
the A-point. This will be one of the main reasons why the values of specific heats in Fig. 5
are smaller than those of experiments.
Finally we note that the formulation in this paper has a certain similarity with FeynmanKikuchi's paper.. 2>3> The interpretation of the transition is just the same,. and actually
we owe our calculation in § 3 to § 5 much to Kikuchi's paper. The main difference lies
in the reduction of the partition function. Feynman introduced the effective mass m'
which is a complicated function of density and temperature, and he separated the part
of the partition function that is not essential for the transition. Our calculation will
correspond to the estimation of density and temperature dependence of the part left out
of consideration in his theory, thus enabling us to see qualitatively the phenomena
depending on density.
density sr jr.r.
0.18
\
\~
\
\
\,..
/1 Lattice Model of Liquid Helium, II
0.8
29
Cv caljgr deg
0. 7
0.65
4
0.6
0.5
3
0.4
0.3
0.2
2
0.1
0
1. 0
Fig. 5.
1. 5
2.0
2.5'K
T
Specific heats at constant volume derived
from (22) and (28).
I
L---------------------~-------------1.0
1.5
2.0
U"K
T
Fig. 6.
Isopycnals of liquid helium calculated from
(23) and (29).
References
1)
p)
3)
4)
T. Matsubara and H. Matsuda, Prog. Theor. Phys. 16 (1956), 569.
R. Kikuchi, Phys. Rev. 81 (1951), 9ss. Phys. Rev. 96 (1954), 563.
P. R. Feyman, Phys. Rev. 90 (1953), 1116. Phys. Rev. 91 (1953), 1291.
W. H. Keesom "Helium" Elsevier Publishing Company, Amsterdam (1942)