THE SCATTERING FUNCTION S33 (q, ω) OF DILUTE HeII-He3 MIXTURES

HAL Id: jpa-00218367

https://hal.archives-ouvertes.fr/jpa-00218367

Submitted on 1 Jan 1978

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

THE SCATTERING FUNCTION S33 (q, ω) OF DILUTE HeII-He3 MIXTURES

W. Götze, M. Lücke, A. Szprynger

To cite this version:

W. Götze, M. Lücke, A. Szprynger. THE SCATTERING FUNCTION S33 (q, ω) OF DI- LUTE HeII-He3 MIXTURES. Journal de Physique Colloques, 1978, 39 (C6), pp.C6-196-C6-197.

�10.1051/jphyscol:1978687�. �jpa-00218367�

JOURNAL DE PHYSIQUE

Colloque C6, supplbment au no 8, Tome 39, aotit 1978, page C6-196

THE SCATTERING FUNCTION

Sg3

(q,w) OF DILUTE H ~ I I - H e 3 MIXTURES W. G6tze, M. Liicke and A. Szprynger

Max-Planck-Institut

filr

Physik and Physik Department der Teehnischen Universit?it, Milnehe, Germany I n s t i t u t e of

Low

Temperature and Structure Research, Polish Academy

o f

Sciences, Wroclaw, Poland

Resum6.- La fonction de diff~action S33 (q,w) des melanges de H~II-He3

1 tempgrature zdro est cal-

culde dans la limite de concentration zdro de ~e~ dans le cadre d'une thdorie des modes couplds. La masse effective de He3 dependant du vecteur d'onde et la structure du spectre des excitations conti- nues sont d6terminges.

Abstract

,-

The scattering function S33 (q,w) of H~II-He3 mixtures at zero temperature is calculated in the zero He3 concentration limit wlthin a mode coupling theory. The momentum dependence of the effective He3 mass and the structure of the continuous excitation spectrum is predicted.

A He3 atom is He11 experiences a considera- ble mass enhancement by inducing motion in the HeII part of which consists of backflow

/ I / .

One common- ly views this motion in terms of density excita- tions /2/ created by and coupled to the He3 movement.

These concepts have been extended /3/ to a nonli- near, first principles theory for the He3 motion coupled selfconsistently to He11 density excita- tions at T

= 0

K by adopting mode coupling methods which proved to be successfui for pure He11 f4/.

The motion of a single He3 particle is des- cribed by the dynamical susceptibility x(q,z) for the He3 density fluctuation. Its spectrum is given by the zero concentration contribution of the dyna- mica1 He3 structure function to the scattering law of the mixture

X" (q,w)

=

lim Sg3 (q,w

_;

x) .

^{( 1}

X

+o

The susceptibility

tion of a He3 density fluctuation and a He11 densi- ty fluctuation

W

W

N ~ I (q,~) J ^% J ^d

^E

^{\ v}

~ & % ) , 2

x;=(~,E)

0

xtt (Z-Z,

^WE)

⁽⁴⁾

This golden rule type formula contains the emission probability for HeII modes of momentum k -+

and energy

E

by the He3 atom. The decay vertex can be expressed /3,4/ approximately in terms of liquid structure factors which were taken from the lite- rature /5/. In the limit x

+ 0

the He11 spectrum X;I(q,w) is unchanged. For it we use the results of our previous theory 141.

Equations (2-4) are, together with the Kramers-Kronig formula for M, a system of nonlinear integral equations for X (q,z). The solution descri- bing the motion of a ~e~ atom coupled selfconsis- tently via M to a hybridization of a "dressed" He3 particle and He11 density excitations has been ob- tained by iteration.

can be expressed by a polarization kernel M(q,z)

As expected there are, for wavenumbers below and a characteristic frequency Oo(q) which is, for

a threshold qc, undamped elementary excitations given M(q,z), most conveniently determined by the

defined bv voles of - .

(2)

.

.

on the real axis spectrums normalization

'

⁰

_{MIt ( q , ~ (q))}

= 0 .

(5b)

The absorptive part of the relaxation Kernel

Mn(q,u) is approximated within mode coupling theory The excitation energy

E

(q) of these He3 by the decay of the IIe3 motion into a pair excita- quasiparticles defines their effective mass m*(q)

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978687 Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1978687

according t o E (q) = q2/ (2m2(q)). I n a d d i t i o n t o t h e

&-functions from t h e s e He3 s i n g l e modes o f energy

~ ( q ) , t h e r e a r e broad c o n t r i b u t i o n s t o t h e spectrum

2" (q,w) from m u l t i p l e mode e x c i t a t i o n s . T h e i r in-

t e n s i t y i n c r e a s e s with i n c r e a s i n g wavenumber on c o s t of t h e s i n g l e mode i n t e n s i t y u n t i l f o r q

qc s i n g l e mode He3 e x c i t a t i o n s a r e no longer p o s s i b l e .

can a l s o be t r a c e d back t o s p e c i f i c combinations of He3 and He11 e x c i t a t i o n s .

F i g u r e l a shows a l s o t h e c u r v e 92 (q)/w - w whose i n t e r s e c t i o n w i t h M' ( q , w ) y i e l d s according t o

(5a) t h e z e r o s i n t h e r e a l p a r t of t h e denominator o r ( 2 ) . F i . r s t t h e r e i s a t w = 6.7 K t h e undamped He3 e x c i t a t i o n (arrow a ) which c a r r i e s i n our example 59 % o f t h e t o t a l s p e c t r a l weight of X" (q,w).There i s a second i n t e r s e c t i o n (arrow b) where Mv'(q,w) i s f i n i t e . T h i s y i e l d s a resonance i n X" (q,w) (Fig. Ib) near 25 K. A t a t h i r d frequency (arrow c ) e q u a t i o n

(5a) i s n e a r l y f u l f i l l e d w i t h M" (q,u) simultaneous- l y being small. The r e s u l t i s a n o t h e r resonance i n

X"

(q,w) l o c a t e d n e a r 35 K. Note t h a t t h e i n t e r s e c -

The s t r u c t u r e i n MU(q,w), shown f o r t h e re- p r e s e n t a t i v e example q = 1.5 !-I i n f i g u r e 1 , i s due t o phase space kinematics of opening and c l o - s i n g decay channels and t o t h e momentum dependence of t h e v e r t e x . Since energy and moment9 conserva- t i o n does n o t a l l o w two-mode s t a t e s with frequencies

t i o n a t 16 K does not l e a d t o a resonance s i n c e ^MI' i s l a r g e t h e r e and M' shows anomalous d i s p e r s i o n . The s t r u c t u r e i n X" (q,w), consequently, i s due t o t h e kinematics of He11 d e n s i t y f l u c t u a t i o n s b e i n g generated by t h e movement of t h e "dressed" He3 atom.

Fig. 1 : M" (q,u) ( f u l l curve) and M' (q,w) (dasked curve) a s a f u n c t i o n of frequency f o r q = 1.5 A-l.

The dash-dotted curve i n d i c a t e s t h e f u n c t i o n

a:

^(Q) /w-w

Fig. Ib : Normalized spectrum ^{X I '} (q,w) ^/.T f o r q = 1.5 A - ~

l e s s than t h e s i n g l e mode energy

-

i n our example

E = 6.7 K

-

one f i n d s M " ( q , w l ~ ( q ) ) = 0. The smooth i n c r e a s e of M"(q,u) j u s t above E ( q ) i s cau- sed by s o f t phonon emission by t h e moving He3 atom.

The s t r o n g peak a t w = 15 K i s due t o h y b r i d i z a t i o n with He11 maxons 9 f o r which t h e phase space t u r n s o u t t o be l a r g e s t . The o r i g i n o f t h e o t h e r l e s s conspi- cuous f e a t u r e s of t h e coupled mode spectrum M1'(q,w)

The elementary e x c i t a t i o n d i s p e r s i o n E (q) t u r n s o u t t o b e a monotonously i n c r e a s i n g f u n c t i o n of momentum w i t h an approximate p a r a m e t r i z a t i o n f o r t h e e f f e c t i v e mass

& ( q ) = m S ( I

+

a q ) ; a = 0.1 (6) The v a l u e of m* = 2.35 m a g r e e s w e l l w i t h t h e one known from thermodynamical d a t a f o r d i l u t e H~II-He3 s o l u t i o n s . It h a s been o b t a i n e d e a r l i e r by q u i t e a d i f f e r e n t t h e o r e t i c a l approach / 6 / . The undamped ~e~

q u a s i p a r t i c l e e x c i t a t i o n s c e a s e t o e x i s t f o r

q + q c % 2

1-I

w i t h t h e d i s p e r s i o n having a P i t a e v s k i endpoint /7/ a t qc. For q > qc, however, t h e spec- trum c o n t i n u e s a s a resonance o f X" (q,w)

References

/ I / Feynman, R.P., S t a t i s t i c a l Mechanics (W.A. Benjamin I n c . , Reading) 1972

/ 2 / M i l l e r , A . , P i n e s , D., Wozieres, P., Phys. Rev.

127 (1962) 1452

-

131 GEtze, W., ~ G c k e , M., Szprynger, A , , p r e p r i n t / 4 / GGtze, W., ~ G c k e , M., Phys. Rev.

B13

(1976) 3825 151 Achter, E.K., Meyer, L., Phys. Rev.

,188

(1969) 291

Hallock, R.B., Phys. Rev.

A5

(1972) 320 ; Massey, W.E., Woo, C.W., Tan, H . T . , Pnys. Rev.

A1 (1970) 519

-

/6/ Feenberg, E . , Theory of Quantum F l u i d s (Academic P r e s s , Nev7 York)

,

1969

171 P i t a e v s k i , L.P., Sov. Phys. JETP,

2

(1959) 8 3 0 ; 12 (1961) 155

-