QUASIPARTICLE INTERACTIONS AND SUPERFLUID TRANSITION TEMPERATURES IN LIQUID 3He

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QUASIPARTICLE INTERACTIONS AND

SUPERFLUID TRANSITION TEMPERATURES IN

LIQUID 3He

K. Levin, O. Valls

To cite this version:

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JOURNAL D E PHYSIQUE Colloque C6, suppl6ment au no 8, Tome 39, aozit 1978, page C6-33

3 t

QUASIPARTICLE INTERACTIONS AND SUPERFLUID TRANSITION TEMPERATURES

IN

LIQUID

He K. Levin and 0. ~alls'

The James Franck I n s t i t u t e and The Department of Physics, The Univepsity of Chicago Chicago, IZZinois, 60637

Resum&.- Nous examinons les cons6quences d'un modsle ph6nomenologique decrivant l'interaction des quasiparticules dans l1h6lium-3. Le modsle ne d6pend que de l'amplitude des transferts d'impulsion et d'bnergie. Avec cette hypothsse nous pouvons expliquer qualitativement (i) la grandeur des para- mgtres de Landau mesures expgrimentalement (ii) la validite de l'approximation s-p pour les calculs de proprietss de transport (iii) la stabilisation de la phase A et (iv), calculer la temperature de la transition superfluide et sa variation avec la pression.

Abstract.- We examine the consequences of a phenomenological model for the quasiparticle interaction in 3 ~ e which depends only on the magnitudes of the momentum and energy transfer. Within this ansatz it is possible to understand qualitatively (i) the magnitude of the measured Landau parameters, (ii) the validity of the s-p approximation for transport calculations, (iii) the A state stabilization and (iv) quantitatively the magnitude and pressure dependence of the superfluid transition temperature.

In order to explain a number of properties ring sum rule is used. The transport properties in- of liquid 3 ~ e which are based on the Landau scat- volve integrals of A(8,@) which weight the region tering amplitudes, we investigate a phenomenologi- around 8 = ~ very heavily. Consequently, in the pre-

cal model for the low (but finite) frequency inter- sent model they will differ from their respective action (vertex function) rk, between two incoming values in the s-p approximation by at most 30 to spin 112 particles on the Fermi surface of four 40 %.

momenta k and k', of the form To obtain the we expand A(8,O) in terms of

R

Legendre polynomials and define JR, VR as the Rth I r iplet =

-

~ ( q )

+

J(q+k-k')

+

V(q)

-

V(q+k-k') ₍₁₎ _{Legendre polynomial projections of J and V (divided} and

':inslet = 3J(q) + 3J(q+k-kt) + V(q) + V(q+k-kt) (2) Here J and V are functions to be determined from experiment ; they are assumed to depend only on the magnitude of the momentum and energy transfer

q =

( 1

q

1

,w )

.

In terms of the usual angles

(e,@)

we can compute the scattering amplitudes

where Z 5 (l-a~/awl ) and C is the self energy ;

kF

by 2R+1)

z ' A ~ ~ = 2N(O)V(O)

-

N(0)Vo + 3N(0)Jo ₍₄₎ zlAoa=

-

2N(O)J(O)

-

N(0)Vo

-

N(0) Jo (5)

z'AS =

r

N(0)VR + 3N(O)J (2R+l),R>O

!&

4

(6)

Z'A~ =

r

N(O)V~

-

N(O)J~ (~R+I),

a

>o _{( 7 )} The paramagnon model is based on a particu- lar form for the functions J(q) and

1

V(q) : J(q) = T~/(I-I~o(q)) and 1

V(q)= TI/(I+I~o(q)), where I 5 Y/N(O)~S the Hub- bard interaction parameter and xo(q) the Lindhard

N(0) is bare single particle density of states and susceptibility. An important check on the consis- tency of our identification of the paramagnon Lan- mx/m is the effective mass ratio. Comparison with

the s-p approximation may be made by expanding dau parameters is the observation (based on eq.(5) a

-

A(r,$) in Legendre polynomials PR($). It follows that X/X, % mK/m( 1 + Fo ) = mW/m(l-Aoa)

-(l-Y)-l

after some algebra from eqs. (1). and (2) that the This result underlines the importance of including present result truncated at

R

= 1 is equivalent to the quasiparticle renormalization factor

z

= mx/m the result obtained for A(=,@) using the s-p appro- in the

f

A ~ I

.

ximation, providing also that the forward scatte- Under the most plausible circumstances of V

t

+ Supported by NSF and NSF-MRL

a positive constant and J(K) peaked around K=O, we Present address : Phys. Department, University find by comparing eqs.(6)-(9) wikh experiment that of California, Berkely, CAL.

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(i) V/J(O)

2

e(0. I) and that (ii) %

o(0.

I)

,

where J(K) + 0 when ~ ~ / >> 4

<.

k ~Both these num- ~ bers are characteristic of spin fluctuation theo- ries.

-

A more quantitative statement can be made about the 9. = 1 partial waves of V using measurements of A:. From eqs.(6) and (7) it follows that V1 a (A '13

+

Ala). All the present, admittedly un-

I

certain, measurements /I/ are consistent with the

values V1 = 0 so that Ala =

-

0.68 at p=O and

-

0.84 at p = 34 bars.

To determine the superfluid transition temperature we note using eqs.(l), (6) and (7) that the pairing coupling constant

A

is

A

A r

A

/Z = (Ata + AlS)/6

2

AIS/9 for V,

2

0 (8)

The s-p approximation and forward scattering sum rule also yields /2/ eq.(8) :

AS-'

= -(A: +A:)/6 =

a s

(A1 +A1 ) / 6 . Comparison of the experimental /I/ values of -(A>:) 16 with AlS/9 shows, remarkably, that they differ by less than 10 % at all pressures.

To obtain Tc for general J(q), V(q) (follo- wing reference 131) we numerically solved the Elia- shberg equations. As might be expected, in a varie- ty of models we investigated, T has approximately the form

where

w

characterizes the width of the peak in J(K)

Choosing J(K) to be a Lorentzian or one of several Gallilean invariant [mx/m = (1 -A1 '/3)-'] paramagnon models we found that the best fit to the )Ak? was for = 0.1. For the Lorentzian case the numerically obtained transition temperatures, which fit eq.(9) to within a few percent, are plotted in fi- gure 1. They are in very good agreement with experiment. Because there are no adjustable parameters in obtaining Tc, this represents a strong confirma- tion that the pairing interaction in 3 ~ e is media- ted by spin fluctuation exchange.

Fig. 1 : Pressure dependence of the theoretical and experimental (reference /I / ) superf luid transition temperatures.

References

/I/ Wheatley,J.C., Rev. Mod. Phys.

5

(1975) 415 /2/ Patton,B. and Zaringhalam,A.,Phys. Lett.

A55

(1975) 95