ROTON LIFETIME EFFECTS ON THE THERMODYNAMIC PROPERTIES OF He II

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ROTON LIFETIME EFFECTS ON THE THERMODYNAMIC PROPERTIES OF He II

G. Carneiro

To cite this version:

G. Carneiro. ROTON LIFETIME EFFECTS ON THE THERMODYNAMIC PROPERTIES OF He II. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-144-C6-145. �10.1051/jphyscol:1978664�.

�jpa-00218111�

JOURNAL DE PHYSIQUE Colloque C6, suppl&ment au no 8, Tome 39, aolit 1978, page C6-144

ROTON

LIFETIME

EFFECTS ON

THE

THERMODYNAMIC

PROPERTIES

^{OF He} I 1 G.M. ~arneiro'

Departmento de ~ i s i c a , Pontifzkia Universidade Cato'lica Cx. P. 38071, Rio de Janeiro, RJ, BrasiZ

R6sumb.- Ce travail traite d'effets introduits par le temps de vie fini des rotons sur les proprib- t6r: thermodynamiques de l'hblium 11. On utilise une approximation conservative et self-consistante pour l'entropie, laquelle tient compte de la diffusion multiple entre les rotons d'une paire.

Abstract.- This paper investigates the effects of finite roton lifetime on the thermodynamic proper- ties of the He I1 using a self-consistent and conserving approximation for the entropy that takes in- to account repeated roton-roton scattering.

Recent work on the thermodynamic properties of related to G by Qyson equation He I1 /1,2,3/ show that the entropy expression pro-

posed by Bendt, Cowan and Yarnell (BCY)/4/, fails to reproduce the experimental data for temperatures above 1 K. These authors attribute this result to the fact that the BCY assumption does not take into account effects of finite elementary excitation li- newidths. ,

In this paper we calculate the entropy of He I1 taking into account repeated roton-roton scatte- ring. We find that the BCY expression fails even when the linewidth of rotons can be neglected and

that a finite rdton linewidth increases the entropy and makes the agreement with experiment worse. We also find that the entropy has an additional contri-

Go being the "bare" roton propagator,

G,'(~,z) = z

-

^{E' E' =} A(o)+(~-~~)~/~~.I is thewbare"

P' P

roton energy, and A(p,w) = -2ImG(p,w+in), W . The functional

@[GI

has the property

GQ[G]/GG(~,z)

-

⁼

^-

^{C(P,Z) . .. (7)}

To generate a self-consistent and conserving appro ximation/7/ for S one starts from an approximate

@[GI and obtain G from equations(6) and(7). We ap- proximate Q by the diagrams of figure 1 (A).

bution from bound roton pairs, not considered pre- ( A )

viously.

The starting point of our calculation is the

expression for the roton entropy, S, in terms of the ( 0 )

fully renormalized roton propagator F(p,w)/5/ : S = S

+ s t

DQ where

m

( ' I _{Fig. 1 :} Diagramatic representation of (A) Q and

(B) C.

sDQ

=

6 [

^{$(w) B(p7w), (2)} The lines in these diagrams represent G and the

" J-a,

a

circles represent the bare roton-roton vertex yo

B(p,w) = {21rnlnE~-l (p,w+ina

-

^{+ ZReG(p,w)}

-

which we assume constant yo= 2g.

~mZ(p, io)

1

m

( 3 ) In this approximation S' is given by

and

aa[c]

Ifan(u) a

S t = -(-I*

+ ; -

2r

-

aT *(~,Y)R~C(~.'Y) (4) St=-2;

1:

^{dw 2a} +1~0[-l+g~ex(~~w)] an(w) (sign g)

-

^arctan

- a~ -. -

^{P, P}

Here k(P,w)1mx(P7w+i~)1

- -

⁺

pp'

^C

]*

²⁸

^{) $}

A(p.u)A(pl ,<I

O(W) = -E(w)ln n(u)

-

^(I+n(w)) ln(l+n(w)fl (5)

--

^-a

n(w) = {exp(w/T)-1 ,I is the self-energy part,

-

E(~)-n(E+wfl Ret(p+p'.<w) (8)

+ Wor& su ported in part b the Br+zilian National where Councll &r Research and &chnologlcal Development

(CNPq). k(p7')-l-gR~x(~,Cd)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978664

where

Now we assume that G(p,w) can be written as

where E~=A(T)/(~-~~)~/~U D and 'I are related respec- tively "t the real and imaginary parts of C in the usual way. In this case B, equation(3) is given by 121

I 121'

For I'<<T, B(p,w) in equation(2) may be replaced by

D

-

^D

~T~(w-E ) and,consequently SDQ = SBCy E

$

^o(E~)

P

which is the BCY expression. For finite T, SD;), equation(2), is greater than SBCy (as can be seen

-

by expanding o(w) around sU) by a factor which is P

at least of order (I'/TI2. "

Now we consider S', equation(8). First we se- parate from it the contribution from bound roton pairs, Sb, which comes from the singular part of k(P,Q) at = Eb(P) (Eb<2A(T)),

a(;)

k(P,Q)

-

=

- -

^S2-E,,

^'

^{W E b ) (14)}

We find

o(Q) C(P,Q)

2w

-

near E b where

is the spectral function associated with bound pairs. We see from equation(l6) that 2a(P)Imx(P,E )

-.

-

^b

is the width of the bound pair. For

r<< 1

~ b - 2 ~ 1 (and for P

2

d v ) we find that the width of C equals 2I'p2/p%2I'. Thus for I'<<~E -2~1 and r<<T

b

we may replace C in equation(l4) by 2s6(Q-E ) and b Sb = Ca(Eb).

P

The remaining contribution to S', which we de- note by S1(S'=S'-S ), can be written as

pC

F

^b

{arctan k(P,w) Irn~(P,w+iq)-

f(_P'"

>=

-

^I~X(P,Q)

-

-a0 El+gRex(~,Qu w (sign g )

3

⁽¹⁸⁾

For r<<T we may replace the A(p,w)

-

in equation(l7) by 2s6(w-~ D ) and SL becomes

P

According to equations(l7) and(lO), S' is in general different from zero. Thus the BCY expression is not correct even in the case were the linewidth can be neglected and there are no bound pairs.

In conclusion then we showed that S has three contribution, S = S +S +S' The first S

DQ b c' DQ

reduces to the BCY expression for I'<<T. Experiments show that the BCY expression overestimates the en- tropy. We found that finite I' increases S thus

DQ'

worsening agreement with experiment. The second term S is positive and therefore increases S further.

b

The third term S' which we find to be non-zero even for r<<T may be positive or negative depending on wether D - ~ e g is negative or positive. In this sim- ple model equations (1 7) 'and(l0) show that IZf-Reg is negative, thus making's' positive.and increasing S even further. It is possible that more complex forms of the roton-roton interaction can make IIf-~ea positive. An investigation along these line is now under way and the results will be reported elsewhere.

References

/ 1 / ~ietrich,~'.~.

,

Graf ,E.H., Huang,C.H. and Passel, L., Phys. Rev.

A5

(1972) 1377

/ 2 / Maynard,J., Phys. Rev. E(1976) 3868

/3/ D0nnel1y~R.J. and Roberts,P.H., J.Low.Temp.Phys.

27(1977) 687

-

/4/ Bendt,P.J., Cowan,R.D. and Yarnell,J.L., Phys.

Rev.

113

(1959) 1386

151 Carneir0,G.M. and Pethick,C.J., Phys.Rev.

(1975) 1106

/6/ The system of units used in this paper is such that Boltzmann's constant k and the volume of the system are equal to unit

/7/ Baym,G., Phys.Rev.

127

(1962) 1391