ON PAIR BREAKING BY ELECTRON BUBBLES IN SUPERFLUID 3He

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ON PAIR BREAKING BY ELECTRON BUBBLES IN

SUPERFLUID 3He

M. Salomaa

To cite this version:

JOURNAL DE PHYSIQUE Colloque C6, supplément au n° 8, Tome 39, août 1978, page C6-24

ON PAIR BREAKING BY ELECTRON BUBBLES IN SUPERFLUID

He

M.M. Salomaa

Research Institute for Theoretical Physios Siltavuorenpenger 20 C, 00170 Helsinki 17, Finland

Résumé.- Nous étudions la vitesse des ions négatifs dans l'hélium superfluide (3He) en fonction du champ électrique appliqué. Pour les vitesses supérieures à la vitesse de Landau le mouvement des ions est limité par les processus qui ne conservent pas les quasiparticules.

Abstract.- The drift velocity of negative ions in superfluid 3He is considered as function of elec-tric field. For speeds in excess of the Landau critical velocity, quasiparticle nonconserving pro-cesses limit the ion's motion.

According to experiments /1,2/, the mobility of negative ions in liquid 3He stays strikingly

constant down to T . The temperature-independent normal state mobility is due to nearly recoilless

3He-ion scattering. This is a consequence of the

effects, first envisaged by Josephson and Lekner/3/, of the quasiparticle scattering process on the ionic recoil spectrum, which is essentially diffusive / 4 / .

In the superfluid, the ion mobility rises re-markably quickly /1,2/ and a novel resonant beha-viour in quasiparticle scattering, which does not allow for usual Born approximation methods, under-lies the data /4/. The theory was also extended for the A phase / 4 / ion mobility tensor 151.

This is a preliminary report on the calcula-tion of the ionic velocity in the complete nonli-near regime observed in the experiments. The addi-tion of the common drift velocity q/2 (see below) to the paired 3IIe atoms with momenta p and -p gives

these states different kinetic energies, thus lift-ing the exact degeneracy, due to time-reversal sym-metry, of £•+ and £ ->-. The total rate of momentum

P -P

transfer from the ion to excitations equals

ff - - 2,_y?.-Pf

\

>

-

f

i-

f)

- 2lr I (-P_i-Pf)W+ (-ijfXl-n^Ml-n )6(-E_.j-Ef)

i,f D

-+ + ( 1>

- 2ir I (Pi+P_f)Wj (i;-f)nin_f6(Ei+E )

i,f D

- 2ir I (-p_£+p_f)Wj (-i;-f)(l-n_i)n_f6(-E_i+E_f)

i,f D -»-

->-where p. and p , are the initial and final state qua-siparticle momenta and p_. = -p. but with spin un-changed. Since q « k „ , the quasiparticle excitation

o -*• -*•

spectrum is given by E. = E.-p..v , where

E.=/5Z +AZ,A is the energy gap v the ionic drift

velocity and n. denotes the Fermi distribution

1 i

evaluated with E.. We use eq. (1) to compute the

rate of momentum transfer for a given vn and to then determine the field required to balance the

losses /6/.

Calculating the transition probabilities W*

corresponding to the collision channels

illustra-ted in figure I is most conveniently formulaillustra-ted in the rest frame of the moving ion, where a constant

superflow of 3He excitations passes by the ion, resembling a state of persistent current flow in a

superconductor.

?

~*\

\ J

P+ -Rl -Pi

Fig. 1 : The quasiparticle scattering, pair crea-tion, pair breaking and quasihole scattering pro-cesses which limit the ionic motion in the super-fluid

Space only permits mentioning here some

fea-tures of the quasiparticle scattering phenomena.

The solution to the 4 x 4 equation for 3 ~ e colli- sions off an ion

+

IT(

;,.

)

.

1;

Iv~&

4'

41

I;>

+

hole space as an axially symnetric (about vD = fiz/m) eigenfunction expansion. The coefficient, which are 2 x 2 matrices in spin space (due to the nontrivial structure of the B phase order parameter), can be expressed as continued matrix fractions 161.

In eq.(2) V is the ion's bare potential and the quasiparticle Green's function matrix is

+ + +

where p+ = p fq/2 and each entry is a 2 x 2 spin

-

matrix. Using Galilean relativity it is convenient to write the diagonal and offdiagonal parts of the propagator in terms of the relative and center of mass coordinates, respectively. To obtain the rates of the physical processess of figure 1 the appro- priate Bogoliubov scattering amplitudes are used.

Figure 2 illustrates the phase space for pair breaking and the occurence of gapless excitations.

Fig. 2 : Fermi surface for 3 H e - ~ (above) seen in the rest frame of the ion with q < q c (left) and with q > qC (right) where qc = AlpF gives the+criti- cal veloc~ty. .Same for A phase (below) with

q l

1

rapid rate of growth in the drag force above the critical velocity. For increasing drift velocity the initial bound levels below the superfluid ener- gy gap gradually turn into quasibound virtual sta- tes 161.

The nonlinear scattering equations for the ABM state are for the general situation more com- plicated than in the B phase. Evidently, in the A phase pair breaking by the moving ion is important at lower drift velocity due to the vanishing of the energy gap seen by excitations in the direction of the orbital anisotropy axis. In the particular case

+

with qllR (see figure 2) the critical velocity for the onset of pair breaking is zero. There exists an interesting analogy of the anisotropic motion of ions in 3 ~ e - ~ with the Hall effect in metals, or with the propagation of light in birefringent crys- tals. A measurements of the associated Hall voltage, if experimentally feasible, would provide yet ano- ther verification of the anisotropy of this remar- kable superfluid phase. Further, the ionic motion, being sensitive to the local orientation of the anisotropy axis, could at least in principle be ex- ploited to provide a microscopic probe of the spa-

+

tial L-vector textures.

ACKNOWLEDGEMENTS.- I am indebted to Professors G. Baym and C.J. Pethick for suggesting this problem and for instruction.

References

/I/ Ahonen,A., Kokko,J., Lounasmaa,O., Paalanen,M. Richardson,R., Schoepe,W. and Takano,Y., Phys. Rev.Lett.

37

/ 2 / Ahonen,A., Kokko,J., Paalanen,M.,Richardson,R., Schoepe,W. and Takano,Y. J.Low Temp.Phys.

30

/ 3 / Josephson,B. and Lekner,J.,Phys.Rev.Lett.z (1969)111

/ 4 / Baym,G., Pethick,C.J. and Salomaa,M.,Phys.Rev. Lett.g(1977)845 and to be published

/5/ Roach,P., Ketterson,J. and Roach,P.,Phys.Rev. Lett.

39

161 For full details see Salomaa,M. to be published