HAL Id: jpa-00218377
https://hal.archives-ouvertes.fr/jpa-00218377
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.
ROTON SECOND SOUND AND THE
ROTON-ROTON INTERACTION POTENTIAL
B. Castaing, A. Libchaber
To cite this version:
B. Castaing, A. Libchaber.
ROTON SECOND SOUND AND THE ROTON-ROTON
JOURNAL DE PHYSIQUE Colloque C6, supplement au n" 8, Tome 39, aout 1978, page C6-219
ROTON SECOND SOUND AND THE R0TON-R0TON INTERACTION POTENTIAL
B. Castaing and A. Libchaber
Groupe de Physique des Sol-ides de I'Ecole Normale SupSrieure, 24 rue Lhomond, 7S2S1 Paris OS,France
+ Address for correspondence : Institut Laue-Langevin, 1S6X, 38042 Grenoble Cedex, France.
Résumé.-Nous avons mesuré la vitesse et 1' absorption du second son de rotons dans sa région d'existence à haute pression (25/bars) et basse température (autour de 0, 5 K).Les mesures
d'absorption nous donnent la fréquence de collision roton-roton. Cette fréquence de collision est en bon accord avec celle obtenue par d'autres mesures à plus haute température, compte tenu de la modification du nombre de rotons. Afin d'expliquer la grande valeur obtenue, qui n'était pas bien
comprise, nous proposons un nouveau potentiel d'interaction entre rotons, qui prend en compte l'échange de rotons virtuels entre eux. Nous obtenons alors un très bon accord avec l'expérience sans l'aide d'aucun paramètre ajustable.
A b s t r a c t . - We have measured the roton second sound v e l o c i t y and absorption a t high pressure and low
temperature. The absorption measurements give a roton-roton c o l l i s i o n frequency i n good agreement
with the previous measurements a t higher temperature. In order to explain i t s v a l u e , we have b u i l t
a new i n t e r a c t i o n p o t e n t i a l between r o t o n s , taking into account the exchange of v i r t u a l rotons
between them. We then obtain a very good agreement with the experiment without any adjustable
parameter.
1. INTRODUCTION.- The experimental discovery / l / of
the roton second sound predicted by Khalatnikov and
Chernikova / 2 / has raised many i n t e r e s t i n g problems
/ 3 - 6 / .
In a recent paper 111, we r e p o r t roton second
sound v e l o c i t y and roton-roton c o l l i s i o n time
mea-surements by heat pulses experiment. We will
con-c e n t r a t e here on the roton-roton con-c o l l i s i o n time
measurements. Indeed the shape of the i n t e r a c t i o n
p o t e n t i a l between rotons i s , in our view an
import a n import quesimportion. Laudau and Khalaimportnikov / 8 / had p r o
-posed a 5 - l i k e i n t e r a c t i o n , in order to explain the
v i s c o s i t y measurements, but J . Yan and M.J. Stephen
191 have shown t h a t t h i s type of p o t e n t i a l gives a
maximum effective cross section too small to i n t e r
-pret the experimental r e s u l t s . In f a c t , the long
range i n t e r a c t i o n must be dipole-dipole l i k e 19/,
but a t small d i s t a n c e s (< 3 A) t h i s p o t e n t i a l i s
probably inadequate. With a reasonable cut-off (5A)
19/, the cross section becomes too small. Yan and
Stephen / 8 / then introduce an a r b i t r a r y short
range potential
-.
2. EXPERIMENTAL RESULTS.- The experimental set-up
has been described elsewhere / 3 / . We thus w i l l only
describe the measurements p r i n c i p l e s . On figure 1,
we see the typical shape of the roton second
sound s i g n a l .
1 t(ms)
Fig. 1 : Typical line shapes of roton second sound pulse, for a propagation length of 6 mm :
a) P = 2 5 bars, T = 0.51 K. b) P = 15 bars, T=0.57 K The points correspond to the arrival time.
In order to have a good precision for the velocity v of this mode, following /3/, we assimilate this
shape to the sum of a gaussian and its derivative. The c'entre of this shape gives the propagation ti-me t = — where L is the propagation length. If the maximum and the minimum are well marked, the time interval between them is, with a good approximation
?u2L TJ-T-V/Z
equal to 2 3— where T is the roton-rotoi
*•
vJ k B
T rrcollision frequency, u = and y the roton effec-tive mass. It is our way of measuring Tr r. t Laboratoire associe au C.N.R.S.
As t h e most probable c o l l i s i o n p r o c e s s e s a r e of t h e type : 2R + 2R where R s t a n d s f o r a r o t o n , t h e c o l l i s i o n frequency i s expected t o be l i k e :
-
1 = B nr where nr i s t h e r o t o n number d e n s i t y rrand B slowly dependson t h e temperarure. We have measured B a t two p r e s s u r e s : P = 25 b a r s and P = 20 b a r s . The r e s u l t s a t T = 0.5K a r e : -16 3 P = 25 b a r s B = ( 1 - 7 ' 0 . 2 ) 10 m / s P = 20 b a r s B =
+
0 . 2 ) 10- 16 m 3 / S These r e s u l t s a r e i n f a i r agreement w i t h t h e v i s c o s i t y /10/ and r o t o n l i f e t i m e /11/ measurements a t h i g h e r temperature which g i v e a v a l u e o f B c l o s e t o 2x10 -16 m 3 / s3. ROTON-ROTON INTERACTION POTENTIAL.- I n t h i s s e c t i o n we propose a new form f o r t h e p o t e n t i a l between r o t o n s , s t a r t i n g from t h e i d e a t h a t a r o t o n
i s s t r i c t l y e q u i v a l e n t t o a d i p o l e , t h a t i s t o a c l o s e s o u r c e and w e l l o f %e. We t h u s s e a r c h f o r t h e i n t e r a c t i o n p o t e n t i a l between such d i p o l e s .
Let u s assume f i r s t t h a t t h e o n l y elementary e x c i t a t i o n s i n l i q u i d % e ( t h a t i s t h e only f r e e o s c i l l a t i o n s of t h e v e l o c i t y f i e l d ) a r e t h e phonons, t h e d i s p e r s i o n r e l a t i o n of which i s q2
-
w2 = 0.7
C
An e x a c t p a r a l l e l can b e made t h e n between t h e e l e c - t r i c f i e l d i n t h e vacuum and t h e v e l o c i t y f i e l d i n %e. T o r example, i n t h e presence of a s o u r c e of %e, t h e F o u r i e r t r a n s f o r m V (q,u) of t h e v e l o c i t y p o t e n t i a l ' w o u l d be g i v e n by :
( $ - a ~ ( q , w ) = c o n s t .
2
I n a s t a t i c problem (without tempdral dependence) : V(q) =
m.
I n t h e presence of a d i p o l eP:
2
a
=
a
61;;)
and t h e i n t e r a c t i o n p o t e n t i a l b e t - q2ween two d i p o l e s P and P i s :
+ +
+ - t
2 V(q) = pl ( p l q ) (XJ2q)q2
The r o t o n s , because of t h e i r d i p o l a r backflow
+
and t h e form of t h e i n t e r a c t i o n energy : $.vS,
T h i s p o t e n t i a l corresponds t o t h e exchange of " v i r t u a l phonons" between two r o t o n s / g / . But r o t o n s a r e a l s o elementary e x c i t a t i o n s of 4 ~ e . The phonon-roton d i s p e r s i o n r e l a t i o n can b e approximated by t h e formula :
(k i s t h e wave v e c t o r a t t h e r o t o n minimum, p t h e e f f e c t i v e mass and Athe gap of t h e r o t o n s ,
K =-- ('U)
'l2
.
We t h u s assume t h a t t h e v e l o c i t yM
p o t e n t i a l i n t h e presence of a source of 4 ~ e i s g i v e n by : 2 2 ( K ~+
(q-ko) )-
u2 V(q,w) = c o n s t .l
and t h e i n t e r a c t i o n p o t e n t i a l between two r o t o n s
becomes :
+ +
3
+
fi2k: (klq) (k2q) V(k k ,q) =-
1 2 P q2 ( ~ ; + ( q - k ~ )
n e g l e c t i n g t h e temporal dependence which y i e l d s r e - t a r d e d p o t e n t i a l s . For small q ( o r t h e l o n g r a n g e c a s e i n t h e d i r e c t space) we r e c o v e r t h e preceding formula. The d i f f e r e n c e between t h i s p o t e n t i a l and t h e preceding one i s a k i n g of Yukawa p o t e n t i a l , of
1 r a n g e
-
o s c i l l a t i n g w i t h t h e s p a t i a l frequency k o .KO'
It corresponds t o t h e exchange of " v i r t u a l r o t o n s " between t h e two r o t o n s . I n o r d e r t o e s t i m a t e t h e t - m a t r i x f o r roton-roton d i f f u s i o n and t h u s t h e roton-roton c o l l i s i o n frequency, we have used t h e Lippmann-Schwinger 1121 v a r i a t i o n a l method. The f o r - mula o b t a i n e d f o r ;:'c i s :I1
VC' E, i s a n energy cut-off of t h e o r d e r of
-
2.
The v a l u e s o b t a i n e d f o r B a t T = 0 . 5 K a r e :P = 25 b a r s : B = 1 . 6 ~ 10-16m3/s P = 20 b a r s :
