CiteSeerX — Calculation Of The Density Matrix Of Liquid

CALCULATION OF THE DENSITY MATRIX

OF LIQUID

4

He

I.Vakarchuk, K.Vasylyna,

Ivan Franko Lviv State University Department for Theoretical Physics

12 Dragomanov St., UA{290005, Lviv{05, Ukraine Received January 27, 1997

A system of ^N atoms of liquid ⁴He is under consideration. The calculation of the one-particle density matrix based on the pair-corre- lation function method is presented. Number of terms is factorized and summized with extracting the small terms. The results obtained have the form of the two terms sum where the rst term have the structure similar to the density matrix of ideal bose-gas. The equations for calculating the temperature of bose-condensation are presented.

1. Introduction

In this paper we present the result of a calculation of the single-particle reduced density matrix of super uid⁴He. Our method of calculation is to apply pair-correlation function method, developed in [1,2].

In section III and IV the dierent ways to obtain the equation for the temperature of phase transition in⁴He based on density matrix are outlined.

We try to obtain the results rested on the measured values only. It can be fullled by representing the results into the forms, where the liquid- structure function

S

(^k) is the single parameter. Then we can take

S

(^k) for example from the experimental studies of liquid helium.

2. Basic assumption

We consider a system of

N

atoms of ⁴He, enclosed in the volume

V

at the temperature

T

. By denition the

s

-particle reduced density matrix of such system is:

F

^s(^r1

;:::;

^rsjr01

;:::;

^r0s) =

V

^s

Z

^N

Z d^rs+1

:::

^Z d^rN

R

^N(^r1

;:::;

^rNjr01

;:::;

^r0N)

;

where the general density matrix:

R

^N(^r1

;:::;

^rNjr01

;:::;

^r0N) =^X

n

^n(^r01

;:::;

^r0N)

e

^{?H^N n}(^r1

;:::;

^rN);(2.1)

Z

^N =^Z d^r1

:::

^Z d^rN

R

^N(^r1

;:::;

^rNjr1

;:::;

^rN)

c

I.Vakarchuk, K.Vasylyna, 1997

ISSN0452{9910. Condensed MatterPhysics 1997No 9(117{126) 117

| partition function of the system described by the Hamiltonian operator

H

^^N. For the

s

-particle reduced density matrix

s

=1,2,3,

:::

, and in the bulk limit

s=N

^!0; with the condition

N

^!1 ,

V

^!1 ,

N=V

=



^!const.

The momentum distribution of atoms is given by the single-particle re- duced density matrix

F

¹(^rjr0) :

N

^q=

N V

Z

e

^?iqR

F

¹(

R

)d^R

;

^R=^r1?r01

;

where

F

¹(^rjr0) =

F

¹(^jr?r0j)

:

We start from the expression for the general density matrix obtained in [3]:

R

^N(

x

^j

x

⁰) =

P

^N(

x

^j

x

⁰)

R

^0N(

x

^j

x

⁰)

;

R

^0N(

x

^j

x

⁰) = 1

N



m

^ 2



^~2

 3

2 N

X

Q

exp

8

<

:

?

m

^ 2



^~2

N

X

j=1

(^rj?r0Qj)²

9

=

;

:

(2.2) where the sum goes over all permutations of the coordinates^r1

;:::

^rN

:

P

^N(

x

^j

x

⁰) =

e

^{?E0 0}(

x

⁰) ⁰(

x

)

V

^N

;

where

⁰(

x

)^{ 0}(^r1

;:::;

^rN) = 1^p

V

^{N exp}

8

<

:

12

X

q6=0

a

²(

q

)



^q



^?q

9

=

;

| the wave function of ground state of the system,

E

⁰ =

N

(

N

^?1)

2

V 

^0?1₄^X

q6=0

(



^q?1)^2~2

q

² 2

m ; 

^q =

s

1 + 2

N

V 

^q

=

^~2

q

²

2

m ; a

²(

q

) =^?



^q?1 2

:

The density matrix

R

^0N corresponds to the formation of the bose-con- densate fraction. It also should be noted, that the mass of particles was transformed (

m

^!

m

^), according to the equation:

m m

^{ = 1?}3¹

N

X

q6=0

(

S

^q?1)²

S

^q+ 1

;

it is clear that

m

^

> m

, that means the decrease of bose-condensation temperature. This equation must be solved numerically. Calculation of this type have been performed in [4-6]. For the eective mass of liquid ⁴He at the density



= 0

:

02185

A

^?3 yet

m

^0= 1

:

7

m

[17,18].

The factor

P

^N(

x

^j

x

⁰) in (1.2) takes into account the hard-core repulsion of atoms due to the decrease of the free volume per particle that results in the increase of the phase transition temperature.

3. One-particle redused density matrix

Single-particle density matrix can be obtained by integration of general density matrix:

V F

1 ^{1(r1jr01) = 1}

Z

^N

Z

d^r2

:::

d^rN

R

^0N(^r1

;:::;

^rNjr01

;

^r2

:::;

^rN)^ exp

8

<

:

?

E

⁰_{+ 12}^X

k6=0

a

²(

k

)^



^k



^?k+



^0k



^0?k

9

=

;

:

(3.1)

Note that the structure of density matrix

R

^0N, mentioned in (2.2) is similar to the structure of

N

range determinant, in which all terms will be taken with "+":

R

^0N(^r1

;:::;

^rNjr01

;

^r2

:::;

^rN) = 1

N





















K

¹⁰¹⁰

K

¹⁰²

::: ::: K

^10N

K

²⁰¹⁰ 1

::: ::: K

^20N

K ::: :::: ::: ::: :::

³⁰¹⁰

K

³⁰² 1

::: K

^30N

K

^{N0 10}

K

^{N0 2}

::: :::

1





















+

:

(3.2)

Here a matrix element is

K

^i0j =^

m

^

2



^~2

3

2

e

^{? m}



2^{~2 P}N j^=1(ri^?rj⁾²

:

(3.3)

Let us expand the determinant (3.2) and extract the terms with indexes 1⁰ and 1 in each term. Next we shall construct the number of elements containing all permutations of rst

s

elements. Extracting the rst row of the determinant, next, the rst column, we can separate the multiplier depending on ^r1 and ^r01 in each term. Afterwards we shall continue this process with the minors of rank

N

^?2,

N

^?3, etc.

The set of terms received has the structure of product of rst

s

elements of the determinant into the new determinant rank

N

^?

s

. All new deter- minants are composed according to the following rules: the determinant rank

N

^?2 does not contain rst and

j

-th rows, and rst (to index 1⁰) and

j

-th columns, etc. Arranging the sum over the indexes of the elements with simultaneous addition of all permutations of elements of mute indexes we can symmetrize all terms of the sum, beginning from the third one.

The density matrix (3.2) can be rewriten in the form of number of terms containing the density matrices of the system of

N

^?

s

particles, where

s

= 1

;

2

;::: ;N

^?1. Let us note these matrices are

R

^0N?s. It should be also noted that all obtained density matrices do not depend on^r1 and ^r01.

Thus we can write:

R

^0N(^r1

;:::;

^rNjr01

;

^r2

:::;

^r0N) = 1

N K

¹¹⁰⁰

R

^0N?1+ 1

N

(

N

^?1) ^

X

26j6N

K

^1j0

K

^j10

q

⁰

R

^0N?2+ 1

N

(

N

^?1)(

N

^?2)

X

26j6=i6N

h

K

^1j0

K

^ij0

K

^j100 + +

K

^1j0

K

^ji0

K

ⁱ¹⁰⁰ⁱ

R

^0N?3+ 1

N

(

N

^?1)(

N

^?2)(

N

^?3) ^

X

26i<j<l6N

h

K

^1j0

K

^ij0

K

^jl0

K

^l100 +

K

^1j0

K

^ji0

K

^il0

K

^l100 +

K

¹ⁱ⁰

K

^il0

K

^lj0

K

^j100 +

K

^1j0

K

^jl0

K

^li0

K

ⁱ¹⁰⁰ + +

K

^1l0

K

^lj0

K

^ji0

K

ⁱ¹⁰⁰+

K

^1l0

K

^li0

K

^ij0

K

^j100i

R

^0N?4+

:::

(3.4) It is helpful now to introduce the collective variables



^kand



^kaccording to the relations (see [1,2]):



^N?sk = 1

p

N

^?

s

N

X

j=s+1

e

^?ikrj;



^k = 1^p

N

s

X

j=1

e

^?ikrj

;

(3.5)

that is



=

s

N

^?

s

N 

^{N?sk +}



^k

:

Substituting (3.5) into (3.1) we are able to reduce the density matrix to the sum of the terms including the density matrices

R

^0N?s, according to (3.4).

Thus we can separate the integration over (

N

^?

s

) variables (on which

R

^0N?s depend) and next step is an integration over ^r1

;:::;

^rs.

Introduction of the collective variables permit us to split the density matrix into parts, each one being dependent on the^rs+1

;:::;

^rN or on the rst

s

coordinates. Only after that we have a possibility to separate these parts completely.

Let us introduce the notation follows. We can dene the function de- pendent only on the rst

s

coordinates by:

g

^s(1

;

2

;::: ;s

^j1⁰

;

2

;:::;s

) = exp

8

<

:

12

X

k6=0

a

²(

k

)^?



^k



^?k+



^0k



^0?k

9

=

;





*

exp

8

<

: X

k6=0

a

²(

k

)

s

N

^?

s N 

^N?sk

?



^?k+



^0?k

9

=

; +

N?s

where^h

:::

^iN?s denote:

h

:::

^iN?s = 1

Q

^N?s

Z d

r

^s+1

:::

^Z d

r

^N(

:::

)

R

^N?s0 (

s

+ 1

;:::;N

)^

exp

(

X

k

a

²(

k

)

N

^?

s

N 

^N?sk



^N?s?k

)

;

Q

^N?sis the part of density matrix depending on the rest (

N

^?

s

) coordinates, which are introduced by the relations

Q

^N?s = ^Z d^rs+1

:::

^Z d^rN

R

^0N?s(

s

+ 1

;:::;N

)^ exp

8

<

:

2

a

⁰+^X

k6=0

a

²(

k

)

N

^?

s

N 

^N?sk



^N?s?k

9

=

;

:

(3.6)

It should be also noted that

Q

^N?sis quite similar to the partition func- tion of the system of (

N

^?

s

) particles, but diers in by some constants. It follows immediately from (3.6) .

Q

^N?s goes to the partition function only in the bulk limit, thus we will keep our denition.

After all transformations we have to rewrite the one-particle density matrix in the form:

F

¹(^r1jr01) =

V N Q

^N?1

Q

^N

K

¹¹⁰⁰

g

¹(1^j1⁰) +

V N Q

^N?2

Q

^N

Z d^r2

K

¹²⁰

K

^{2100 }

g

²(1

;

2^j1⁰

;

2) +

V

N Q

^N?3

Q

^N

Z d^r2Z d^r3

K

¹²⁰

K

²³⁰

K

³¹⁰⁰

g

³(1

;

2

;

3^j1⁰

;

2

;

3) +

N V Q

^N?4

Q

^N

Z d^r2Z d^r3Z d^r4

K

¹²⁰

K

²³⁰

K

³⁴⁰

K

^{4100 }

g

⁴(1

;

2

;

3

;

4^j1⁰

;

2

;

3

;

4) +

::: :

(3.7) Next let us evaluate

g

^sand

Q

^N?s. For this goal we are to nd

g

^srst. We suppress the dependence on coordinates and use the abbreviated notations:

g

^s=

g

^s(1

;

2

;::: ;s

^j1⁰

;

2

;::: ;s

)

:

g

^s may be considered as exponent of irreducible "averages". Due to the translational invariancy of



^N?sk the rst irreduceble average is equal to zero. First term tends to zero because



^N?sk= 0.

It follows also that



^N?sk1



^{N?sk2 N?s6}= 0 only for

k

¹=^?

k

². Finally one can write

g

^s by:

g

^s = exp

8

<

:

12

X

k6=0



a

²(

k

)^?



^k



^?k+



^0k



^?k0
+

a

²²(

k

)

N

^?

s

N S

^N?s(k)j



^k+



^k0j2

;

(3.8) where we consider only two rst terms. The liquid-structure function

S

^N?s(^k) for

N

^?

s

particles is

S

^N?s(^k) =



^N?sk



^{N?s?k N?s}

:

(3.9) In the same approximation for

Q

^N?s we have:

Q

^N?s =

Z

^N?s0 exp

8

<

:

?

12

X

k6=0

ln^1^?2

a

²(

k

)

N

^?

s

N S

^0N?s(k)

 9

=

;

(3.10)

Z

^N?s0 = ^Z d^rs+1

:::

^Z d^rN

R

^N?s0 (

s

+ 1

;:::;N

)

;

where

S

^0N?s(^k)|the liquid-structure function of ideal Bose-gas.

As was mentioned above

Q

^N?s is not exactly the partition function of the system. It has an additional exponent as it follows immediately from the right-hand side of the equation (3.10).

Next it is possible to expand

Q

^N?s

=Q

^N in powers of the

s=N

lim

s=N!0

Q

^N?s

=Q

^N =

0

@

z

⁰exp

8

<

:

?

21

N

X

k6=0

2

a

²(

k

)



^dS0(k)d

1^?2

a

²(

k

)

S

⁰(^k)

9

=

; 1

A S



exp

8

<

:

?

21

N S

X

k6=0

2

a

²(

k

)

S

⁰(^k) 1^?2

a

²(

k

)

S

⁰(

k

)

9

=

;

;

(3.11) where

z

⁰ =

Z

^N?10

=Z

^N0 and it can be dened by

z

⁰ =

e

^0,



⁰|the chemical potential.

Let us turn back to (3.9). The liquid-structure function

S

^N?s(^k) may be obtained in the form of derivative of

Q

^N?s so that the nal result is:

S

^N?s(^k) =

S

^0N?s(^k)

1^?2

a

²(

k

)^N?sN

S

^0N?s(^k)

:

We are still faced with the task of collecting

F

¹(^rjr0). For this goal we have to decompose

g

^s.

For the decomposition of the

g

^s we shall factorize it using the collective variables



^k for each

s

. So equation(3.6) can be modied to read

g

^s=

F

^s0exp

8

<

:

?

21

N s

X

k6=0

2

a

²(

k

)(

S

⁰(^k)^?1) 1^?2

a

²(

k

)

S

⁰(^k)

9

=

;

;

(3.12)

where

F

^s0 =

F

¹⁰(1^j1⁰)^Ys

j=2

q

F

²⁰(1

;j

)

F

²⁰(1⁰

;j

) ^Y

26i<j6s

F

²⁰(

i;j

)

; F

¹⁰(1^j1⁰) = exp

8

<

:

N

1

X

k6=0

a

²²(

k

)

S

⁰(^k) 1^?2

a

²(

k

)

S

⁰(^k)



e

^ik(r1?r01)?1^

9

=

;

; F

²⁰(

i;j

^j

i;j

) ^

F

²⁰(

i;j

) =

e

^{? (jr~ i?rjj)}

:

Exponent in (3.12) may be added to

z

⁰ in each term of density matrix.

Let us label this part

z

.

Thus, adopting the mentioned factorization, we arrive at the expression:

F

1(^r1jr01) = ^V

N F

0

1(1^j1⁰)



zK 0

11 0+^z2

Z

d^{r2K120 qF20}(1^;2)^K2100qF20(1^0;2) +

z 3

Z

d^r2

Z

d^{r3K120 qF20}(1^;2)^K230F20(2^;3)^K3100qF20(1^0;3)^

q

F 0

2(1^;3)^F20(1^0;2) +^z4

Z

d^r2

Z

d^r3

Z

d^r4K120qF20(1^;2)^

K 0

23 F

0

2(2^;3)^K340F20(3^;4)^K4100qF20(1^0;4)^qF20(1^0;4)^F20(1^;3)^

q

F 0

2(1^;4)^F20(1^0;2)^F20(1^0;3)^F20(2^;4) +^:::



:(3.13)

4. Summation of the density matrix

Let us transform (3.13) in the form which allows to collect

F

¹(^r1jr01). In this case, we shall separate the leading terms in the one-particle density matrix.

Let us introduce

h

³ = ^q

F

²⁰(1

;

3)

F

²⁰(1⁰

;

2)^?1

;

h

⁴ = ^q

F

²⁰(1

;

3)

F

²⁰(1

;

4)

F

²⁰(1⁰

;

2)

F

²⁰(1⁰

;

3)

F

²⁰(2

;

4)^?1

;

...

Thus

F

¹(

r

^1j

r

¹⁰) =

F

¹⁰(1^j1⁰) 1

N

X

q

e

^iqR

(

z

²(

K

^q)²

1^?

zK

^{q +}

zK

^q0

)

+

F

¹⁰(1^j1⁰)

V

N

X

n>3

z

ⁿ

D

ⁿ

;

(4.1)

D

ⁿ =^Z d^r2

:::

^Z d^rN

K

^{120 q}

F

²⁰(1

;

2)

K

^100nq

F

²⁰(1⁰

;n

) ^Y

26i<j6n

h

K

^ij0

F

²⁰(

i;j

)

h

ⁿⁱ

: K

^q = ^Z

K

^(^R)

e

^?iqRd^R

; K

^(^R) =

K

⁰(^R)^q

F

²⁰(

R

);

K

^q = ^Z

K

(^R)

e

^?iqRd^R

; K

(^R) =

K

⁰(^R)

F

²⁰(

R

);

K

^q0 = ^Z

K

⁰(^R)

e

^?iqRd^R

;

R = ^r?r0;

Or, under the assumption

D

ⁿ = 0:

F

¹(

r

^1j

r

⁰¹) =

F

¹⁰(1^j1⁰) 1

N

X

q

e

^iqR

(

z

²(

K

^q)²

1^?

ZK

^{q +}

ZK

^q0

)

:

(4.2)

Note, that equation (4.2) for ideal Bose-gas passes into

F

^1(

r

^1j

r

⁰¹) = 1

N

X

q

e

^iqR

zK

^q0 1^?

zK

^q0

:

From the normalization condition the equation (4.2) we obtain for

z

ⁿ 1 = 1

N

X

q

(

z

²(

K

^q)²

1^?

zK

^{q +}

zK

^q0

)

;

that yields 1^?

z

^c

K

⁰= 0 in the



-point. Therefore for

z

^cwe have

z

^c = 1

=K

⁰, where

K

⁰=

K

^q(

q

= 0).

The equation for the critical temperature after substituting

z

^c yet:

1 = 1

N

X

q

((

K

^q

=K

⁰)² 1^?

K

^q

=K

^{0 +}

K

^q0

K

⁰

)

:

(4.3)

Now we turn to our assumption for

D

ⁿ. Let us consider a few rst terms

D

³ = ^Z d^r2Z d^r3

K

^{120 q}

F

²⁰(1

;

2)

K

²³⁰

F

²⁰(2

;

3)

K

^3100q

F

²⁰(1⁰

;

3)

h

³

D

⁴ = ^Z d^r2Z d^r3Z d^r3

K

^{120 q}

F

²⁰(1

;

2)

K

²³⁰

F

²⁰(2

;

3)

K

³⁴⁰

F

²⁰(3

;

4)

K

^{4100 }

q

F

²⁰(1⁰

;

4)

h

⁴ ...

Unfortunately, these terms are dicult to be estimated. The terms involving

h

ⁱ are expected to be small in interesting temperature range. We remain the numerical calculations of this problem for the future.

5. Dierent form of equation for density matrix

An alternative form of equation (4.1) can be obtained by rewriting equation (3.13) in the dierent form. We shall write the coecients

h

^2 = ^q

F

²⁰(1

;

2)

F

²⁰(1⁰

;

2)^?

F

²⁰(1

;

2)

F

²⁰(1⁰

;

2);

h

^3 = ^q

F

²⁰(1

;

2)

F

²⁰(1

;

3)

F

²⁰(1⁰

;

2)

F

²⁰(1⁰

;

3)

F

²⁰(2

;

3)^?

F

²⁰(1

;

2)

F

²⁰(2

;

3)

F

²⁰(1⁰

;

2);

h

^4 = ^q

F

²⁰(1

;

2)

F

²⁰(1

;

3)

F

²⁰(1

;

4)

F

²⁰(1⁰

;

2)

F

²⁰(1⁰

;

3)

F

²⁰(1⁰

;

4)^

F

²⁰(2

;

3)

F

²⁰(3

;

4)

F

²⁰(2

;

4)^?

F

²⁰(1

;

2)

F

²⁰(2

;

3)

F

²⁰(3

;

4)

F

²⁰(1⁰

;

4) ...

In this case for

F

¹ we obtain:

F

¹(

r

^1j

r

¹⁰) =

F

¹⁰(1^j1⁰) 1

N

X

q

e

^iqR

z

²(

K

^q)²

1^?

zK

^{q +}

zK

^q0+

F

¹⁰(1^j1⁰)

V

N

X

n>3

z

ⁿ

D

^n

;

(5.1)

where

D

^2 =^Z d^r2

K

¹²⁰

K

¹⁰⁰²

h

^2

; D

^3=^Z d^r2Z d^r3

K

¹²⁰

K

¹⁰⁰²

h

^2

;

and

D

^n=^Z d^r2

:::

^Z d^rN

K

¹²⁰

K

^{100n Y}

26i<j6n

K

^ij0

h

^n

:

K

^q and

K

^q0 are dened by relations in (4.1).

The equation for the



-point temperature yet 1 = 1

N

X

q

( (

K

^q

=K

⁰)² 1^?

K

^q

=K

^{0 +}

K

^q0

K

⁰

)

:

(5.2)

where

D

^n are neglected.

Note, that dierent ways to collect the equation for density matrix exist, as we have shown above. Thus we obtain some similar equations for nding

T

^. It is clear that it requires more detailed numerical calculations of

D

ⁿ and

D

^n which were neglected without sucient proves. A nal remark must be made concerning the use of formal parameters. We need only the liquid- structure factor of system to solve equation (4.3) and equation (5.2). So it is possible to make use of data from the experimental works of a number of papers [7-11] or from the theoretical calculations.

References

[1] Yukhnovskii I.R Statistical operator and collective variables. // Ukr. Fiz.

Zhurn., 1964, vol. 9, No 7, p. 827-838 (in Russian).

[2] Yukhnovskii I.R Displacement and collective variables method. // Ukr. Fiz.

Zhurn., 1967, vol. 12, No 10, p. 1677-1694 (in Russian).

[3] Vakarchuk I.O. Statistical operator of a system of identical interacting parti- cles in coordinate representation. // Journal of Physical Studies, 1996, vol. 1, No 1, p. 25-38 (in Ukrainian).

[4] Vakarchuk I.O. On the microscopic theory of ^-transition in super uid ⁴He.

// Teor. Mat. Fiz., 1978, vol. 36, No 1, p. 122-135.

[5] Vakarchuk I.O. Bose-condensate in liquid ⁴He // Teor. Mat. Fiz., 1985, vol.

65, No 2, p. 285-295 (in Russian).

[6] Vakarchuk I.O. Density matrix of super uid helium-4. // Teor. Mat. Fiz., 1989, vol. 80, No 3, p. 439-451 (in Russian); Teor. Mat. Fiz., 1990, vol. 82, No 3, p. 438-439 (in Russian).

[7] Achter E.K., Meyer L. X-Ray scattering from liquid helium. // Phys. Rev.,1969, vol. 188, No 1, p. 291-300 [8] Robko H.N., Hallock R. Structure-factor measurements in⁴He at saturated vapor pressure for 1^:38^{6T 6}4^:24^K. // Phys. Rev. B, 1981, vol. 24, No 1, p. 159-182.; Robko H.N., Hallock R. Structure-factor measurements in ⁴He as a function of density. // Phys. Rev. B, 1982, vol. 25, No 3, p. 1572-1588.

[9] Sears V.F., Svensson E.C., Woods A.D.B., Martel P. Neutron-diffraction study of the static structure factor and pair-correlation functions in liquid

4He. // Phys. Rev. B, 1980, vol. 21, No 8, p. 3638-3651.

[10] Wirth F.H., Hallock R.B. X-Ray determination of the liquid structure factor and pair-correlation function of⁴He. // Phys. Rev. B, 1987, vol. 35, No 1, p.

[11] Chang C.C., Campbell C.E. Energy and structure of the ground state of liquid89.

4He. // Phys. Rev. B, 1977, vol. 15, No 9, p. 4238-4255.

ROZRAHUNOK MATRIC GUSTINI RDKOGO 4

He

.^Vakarquk,^K.^Vasilina

V statt rozglnuta sistema^N atomv rdkogo^4He. ^{Z povnoÝ}

matric gustini sistemi, pri dopomoz metodu parnih kore-

lcnih funkc, otrimana odnoqastinkova matric gustini u vigld sumi elementv, k pdlgat~ faktorizacÝ. ^Psl

vidlenn malih qlenv ta zgortann rdu buv otrimani vi-

raz dl odnoqastinkovoÝ matric gustini u vigld sumi dvoh dodankv,perxi z kih po struktur podbni do matric gu-

stini deal~nogo boze-^azu. Naveden formuli dl rozrahunku temperaturi boze-kondensacÝ.