THE PHONON ANOMALY IN b.c.c. He4 : A PHONON SELF-TRAPPING ?

HAL Id: jpa-00221355

https://hal.archives-ouvertes.fr/jpa-00221355

Submitted on 1 Jan 1981

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 PHONON ANOMALY IN b.c.c. He4 : A

PHONON SELF-TRAPPING ?

M. Héritier

To cite this version:

JOURNAL DE PHYSIQUE

CoZihque C6, suppZdment

ax

n012, Tome 42, ddcembre 1981 page c6-890

THE PHONON ANOMALY I N

b.c.c.

~e~

:

A PHONON SELF-TRAPPING

?

Labomtoire de Physique des SoZides, Universitd de Paris-Sud, 91405 Orsay, France

Abstract.- It is proposed that a longitudinal phonon in b.c.c. He4,at wavevec- W z 2 . 3

k 1

can be self-trapped in a liquid drop and form a roton-like

excitation. This can account for anomalies observed in the phonon spectrum,in a sphere of momentum space.

The concept of self-trapping,well-known in the case of a single particle,can

be extended to collective excitations (1): consider a system where one can define

two phases which have almost equal thermodynamical potential,GS in the stable phase

and G in the excited one. Suppose than an elementary excitation as a lower energy

E

w in the excited phase than in the stable phase wS,so that Aw = w - w >>AG= G - G

E S E E S'

The excitation energy lowering Aw tends to induce the formation of a volume of the

excited phase, limited to a finite size by A G . Taking into account the excitation

localization energy Eloc,the excitation can be self-trapped if the thermodynamical potential balance is favourable.

We consider here longitudinal phonons in He4,either in the b.c.c. solid or

in the superfluid (the phonon-roton spectrum). This system seems a good candidate

for collective excitation self-trapping:first,longitudinal phonons exist with a long lifetime in both phases [as "rotons" at large wavevectors in the superfluid). Secondly, in many respects,solid and liquid Helium behave very similary: Delrieu has

noted striking analogies between the two phases (2) (orders of magnitude,types of in-

teractions,t;rpes of defects). Castaing and Nozieres have pointed out the "nearly

4

solid" character of liquid He3 (3) (the same idea applies to liquid He ) . A roton in

superfluid He4 has been described as a longitudinal phonon self-trapped in a solid- like region of the liquid (4). Here, the stable phase is the b.c.c. solid at pressu- res only slightly higher than the melting pressure, so that the melting free en-

thalpy is much smaller than a typical atomic kinetic energy in the liquid

'kin = fi2k2/?m, where m is the atomic mass and k% n/a (a is the interatomic spacing).

The b.c.c. structure corresponds to the.lowest solid-liquid surface tension.

The excitation spectrum of the liquid exhibits a minimum (thc "roton mini- +

mum") ,at a wavevector

I q l

= q = 2 . 1 k 1 , w = w R = 7.3K. In the larger part of the

momentum space, the phonon energies in the solid w (q) are much larger than w

R that:

Aw(q) = wS

-

w >) f i 2 k 2 > > A G ( p , T )

R 2m

Therefore, conditions for self-trapping in a superfluid drop of radius R are favou- rable at wavevectors around the roton minimum. The free enthalpy balance condition can be written:

-

Aw (qo)

+

Eloc (R)

+

(4n/3) 3 3

+

~4aaR2 ~ < 0

The second term is the energy necessary to localize the roton within the drop.The last term is the solid-liquid interface energy,trhich remains finite on the melting curve,and, therefore,predominates on the volume term. As estimated from ion mobility

measurements (5), o = 0.04 erg/cm2. A roton can be pictured as a vortex ring of dia-

meter 2R ,with a dipolar velocity field, except at small distances-In the Feynman

model, 2Ro 2. a. However, in this work,it is more consistent to use the picture of a

phonon self-trapped in a solid-like region of diameter 23

.

The estimate of R is

about 4-5 interatomic distances on the melting curve. To localize the roton within the drop,it is necessary to perturb the dipolar backflow,which increases the fluid

kinetic energy. Imposing that the velocity vanishes for r > R,and considering the

fluid as continuous,perfect and incompressible,we obtain,as leading term:

Minimization with respect to R,gives the free enthalpy of the self-trapped state:

GsT = - AuYqo)

+

11/21 (~~~;/2ml?E) I" x (810) 2/ 3

+he optimum drop radius is

R a

(9

/ ~nok:]"~

We expect about 120 atoms in the drop and a self-trapped phonon energy about 10K

above the roton energy. Given the rough approximations involved these figures should be considered as orders of magnitude, rather than precise determinations.

Have these self-trapped phonons been observed expermentally? Indeed,anomalies have been seen in the phonon spectrum of b.c.c. He4 in neutron scattering experiments

(6):at equivalent points of the momentum space,non identical neutron profiles were

+

observed. At

I q

1

2. 2.3k1,0n a sphere of the momentum space,the profiles were dissy-

metric, with a higher intensity,and occured at lower energy.While dissymetries and

changes in the peak intensities may be due to interference effect in the one-phonon scattering function (seeGlyde(7)) the peak energy lowering cannot be interpreted so easily. the fact that the anomaly occurs at a constant value of the modulus of the

wavevector,inde?endently of its direction,agrees with our interpretation,since the

roton spectrum of the liquid is isotronic. The value of the anomalous wavevector

0- 1 - 1

=

2.3A is also in agreement with the roton wavevector,located at 2.12 on

the melting curve and,therefore,at a higher value in the metastable liquid at pres-

sures above the melting pressure. The value of the minimum e n e r g y , ~

-

15K is in

remarkable agreement with our rough estimate

,

10K above the roton energy,i.e.

about 17K. Werthamer (8) has shown that the Debye-Waller factor in the solid closely

4

resembles the structure of superfluid He

,

exhibiting characteristic oscillations

in the longitudinal modes (but much smaller ones in the transverse modes).

C6-892 JOURNAL DE PHYSIQUE

we want tc point out in this model are consequences of high non linearities. In this quantum crysta1,the equations of motion should be non linear. Indeed,the physical

condition Aw >> AG implies that the degrees of freedom involved in the excitation

are strongly coupled to the order parameter of the phase transition (here the fouriff components of the density at wavevectors belonging to the reciprocal lattice). The coupling is so large that a linearization is not possible. It is well known that,in such a case, a local solution is possible: the case of an electron in an ionic crys- tal embedded in a lattice polaron is a well known example. An electron in a magnetic insulator with a large s-d exchange, forming a magnetic polaron,is another one. In our case the collective excitation becomes local both in real space and in momentum space. In this local solution,the anhamonicity is so large that surprising simila- rities with the liquid phase are observed. This is what we want to modelize with our liquid drop picture.

REFERENCES

1) M. Heritier,P. Lederer and G. Montambaux, J. of Phvsics c , G , L 703,(1980)

2) J . M . Delrieu,private ccmmunication

3) B. Castaing and P. Nozieres, J. de Physique (Paris),s,257 ,(1979)

4 ) M.HBritier,G. Montambaux and P. Lederer J. de Physique (Paris) %,L493,(1979)

5) R.M. Ostermeier and K.W. Schwartz,Phys. Rev. g,2510, (1972)

6) E.B. Osgood, V.J. Minkiewicz, T.A. Kitchens and G. Shirane,Phys. Rev.*,1537,

(1972)

V.J. Minkiewic2,T.A. Kitchens and E.B. Osgood,Phys. Rev.

g,

1513,(1973)

7 ) H.R. Glyde, Can. J:Phys,E,761, (1971)

H. Horner, Phys. Rev. Letters,

2,

556,(1972)