DYNAMICAL INTERACTION OF NEGATIVE IONS WITH SUPERFLUID 4He

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DYNAMICAL INTERACTION OF NEGATIVE IONS

WITH SUPERFLUID 4He

F. Sheard, R. Bowley

To cite this version:

JOURNAL DE PHYSIQUE CoZloque C6, supplgment au no 8, Tome 39, aofit 1978, page

C6-81

DYNAMICAL INTERACTION OF NEGATIVE IONS

WITH

SUPERFLUID

4 ~ e F.W. Sheard and R.M. Bowley

Deparhent of Physics, U n i v e ~ s i t y of Nottingham, University Park, Nottingham NG7 2RD, England

RGsum6.- L'interaction entre les ions et les excitations de l'h6lium liquide est calcul6e par llap- proximation Born-Oppenheimer.

Abstract.- The coupling between ions and the excitations of liquid helium is calculated using the Born-Oppenheimer approximation.

It was first suggested by Reif and Meyer / I / that, at sufficiently low temperatures or high fields, the velocity of ions in superfluid k e would be li- mited by the Cerenkov emission of rotons from ions whose drift velocities

7

were greater than the Landau critical velocity vL. This idea has been con- firmed in recent experiments using pressurized li- quid helium at 0.35 K in which negative ions were accelerated to supercritical velocities /2,3/.

However a quite unexpected result emerged from a comparison of the observed behaviour of the excess drift velocity

-

vL as a function of elec- tric field E, with the theory of roton emission pro- cess /4,5/. If the dominant process by which an ion loses energyandmomentum is creation of single ro- tons, then the dependence

7

-

vL

'

E~~~ is predic- ted, whereas if it is creation of pairs of rotons then

7

-

vL %

~ ~ 1 ~ .

For electric fields between 3 and

lo3

kV m-' the experimental data /2,3/ fit the Ell3 law and cannot possibly be reconciled with the E2I3 law. Thus negative ions moving through superfluid 4 ~ e create rotons only, or at least predominantly, in pairs whereas in neutron scattering from 4 ~ e the creation of single rotons seems to present no dif- ficulty.

The theoretical results quoted above were based on a perturbation treatment of roton'emission

locity for an ion of effective mass m* emitting a roton of momentum p at the minimum of the '~e ex- citation spectrum.

The apparent complete absence of single-roton processes has not been accounted for theoretically since the magnitudes of the matrix elements havenot yet been calculated. Analysis of existing data on the basis of the two-roton theory /5/ gives

K, = 0.43 x 10" s-' and shows alsa that K1/K2 must be <

lo-'

161. An earlier theoretical estimate by Takken 171, using a quantum hydrodynamical approach, suggested that K1 'L 5 x 1017 s-', which is so large

that one-roton processes should completely dominate. Since this is manifestly not the case a theory of the ion-roton interaction is urgently needed.

In this paper we propose a model for the dy- namic interaction of negative ions with the complete excitation spectrum (phonons and rotons) of liquid 4 ~ e . The interaction with phonons gives rise to the backflow around the moving ion and accounts for the hydrodynamic effective mass of the ion. The inter- action with rotons is more sensitive to the model chosen to describe the excitation spectrum of the superfluid. We also report the results of prelimi- nary calculations of the ion-roton coupling.

The Hamiltonian for the ion-liquid He system is

with the assumption of constant interaction matrix

H =

%

+ (p2/2m) + HiL elements. The transition rates for one- and two-

roton processes were found to be /4,5/ where HL is the Hamiltonian for the unperturbed (v > v') liquid He, P , .. = i M V R is the momentum operator of the

bare ion of mass m-(essentiallv the electron mass). R,(v) = ,[(vK

-

v;)/v~' (V > v;) _{The in-teraction is given by}

where v: = vL + (~~/2m*), vi = vL + (Po/m*) are the

corresponding threshold velocities and K1, K, are HiL =

I

~ ( $ 1

U(r

-

!Id3?

the rate constants which are related to the respec-

where U(r

-

R)

is the interaction potential between tive matrix elements. Here po/mf is the recoil ve-

a helium atom at

r

and the ion localised about a

point

5

and p(r) is the number density operator of .-,

the helium. We now define adiabatic states of the system

where I$@)> is a state vector of the helium cor- responding to an ion fixed at R and the ion state is a plane wave @ % exp(iK.R).

-

-

Operation of the Hamiltonian (I) on the state (3) gives rise to a term

which induces transitions between these states.This term may be manipulated into the form

where E is the adiabatic energy of the state I$(R)> and the momentum MK is taken to be along the z axis.

A negative ion is an electron inside a bubble of radius s (% 1 1 at 25 bar) so for simplicity we take the interaction to be a simple step potential

p

(S < so)

v(s)

=

(s > so)

where

2

=

:

-

R.

This enables the term p(:)au/azd3r,

I

which is the force exerted on the liquid helium due to the ion, to be replaced by a term involving the stress-tensor operator a in the helium, integrated

Z Z

over the surface of the ion. A similar procedure has been used to derive the interaction between a vi- brating solid surface and liquid helium 181 whichis relevant to the Kapitza resistance problem. The ef- fective ion-helium interaction may then be written in the form

I

Here is the energy difference between the final and initial states of the helium (which is the ener- gy of the emitted excitations) and dSZ is the pro- jection of an element of surface area of the ion on a plane perpendicular to the z axis. For emission of excitations with total momentum

Uj?,

izz(z) con- tains a term with phase factor exp(-i9.51, which is averaged over the surface of the ion. This gives

We have studied the dynamics of a negative ion in liquid 'He with the above interaction using the intermediate coupling theory of the polaron 191.

We shall not discuss the details here but describe briefly the procedure and main conclusions. To begin with we consider the effect of the single-phonon emission and absorption terms in the interaction (5).

The theory proceeds via two unitary trans- formations of the Hamiltonian. The first of these removes the dynamical coordinate

fS

of the ion from the Hamiltonian and the second is a displaced oscil- lator transformation which is designed to removethe one-phonon terms from the Hamiltonian. This has the effect of introducing the dipolar backflow in the helium around the moving ion. Furhtermore the mo- mentum associated with the backflow is equivalent to modifying the mass of the ion. The interaction we have derived gives precisely the same backflow and effective mass m* is found in classical hydro- dynamics which confirms in large measure our method of derivation of the ion-helium interaction Hamiltonian.

These transformations also modify the roton terms in the Hamiltonian so that calculation of the emission rate for one-roton processes is quite in- volved. We calculate that KI/K2 % 0.3 using the va-

lue of K2 obtained from a comparison with experiment. This is too large by an order of magnitude at least. However this reSult is derived using the theory of Sunakawa et a1 /lo/. For the excitation spectrum of 4 ~ e and we have not taken into account the renorma- lization of the excitations due to their mutual in- 'teractions which is important in the roton region. Nevertheless our theory provides a firm basis for understanding the dynamical behaviour of negative ions in superfluid '~e and further work is in pro- gress.

References

/I/ Reif, F. and Meyer, L., Phys. Rev.

2

(1960) 1164 /2/ Allum, D.R., Bowley, R.M., McClintock, P.V.E. and

Phillips, A., Phil. Trans. Roy. Soc. Lond. A

284

(1977) 179

/3/ Allum, D.R., Bowley, R.M. and McClintock, P.V.E., Phys. Rev. Lett.

2

(1976) 1313

/ 4 / Bowley, R.M. and Sheard, F.W., Proceedings of the Fourteenth International Conference on Low Tempe- rature Physics, edited by M. Krusius and M. Vuorio

(North Holland, Amsterdam and Oxford 1975) Vol. I, p. 165

/5/ Bowley, R.M. and Sheard, F.W., Phys. Rev. B

16

(1977) 244

/6/ Sheard, F.W. and Bowley, R.M., Phys. Rev. B, to be

published

171 Takken, E.H., Phys. Rev. A 1 (1970) 1220

181 Sheard, F.W., Bowley, R.M. and Toombs, G.A., Phys. R e v . A Z ( 1 9 7 3 ) 3135

-

191 Lee, T.D., Low, F. and Pines, D., Phys. Rev.

2

(1953) 297

/lo/ Sunakawa, S., Yamasaki, S. and Kebukawa, T.,