Positive Ion Mobility in Normal and Superfluid 3He* J. Kokko, M. A. Paalanen,? W. Schoepe,$ and Y. Takano

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Journal of L o w Temperature Physics, Vol. 33, Nos. 1/2, 1978

Positive Ion Mobility in Normal and Superfluid 3He*

J. Kokko, M. A. Paalanen,? W. Schoepe,$ and Y. Takano Low Temperature Laboratory, Helsinki University of Technology, Espoo, Finland

(Received March 21, 1978)

The mobility of positive ions has been measured in the normal and superfluid

3 , ,

phases of He at several pressures. Below 100 m K the normal phase mobthty increases logarithmically with decreasing temperature down to the superfluid transition temperature T~; it shows an anomalous fump near 100 inK. A t low temperatures the drift velocity is nonlinear for electric fields exceeding 30 V / c m . In the superfluid the mobility, normalized to its value at To, is much less than for negative ions. We have also observed the anisotropic mobility in the A phase and the L a n d a u critical velocity for pair-breaking in both superfluid phases.

In o u r previous m e a s u r e m e n t s ~2 of negative ion mobility in liquid 3He, no trace of ionic recoil effects, which should reduce the scattering of quasiparticles at low t e m p e r a t u r e s , could be distinguished in the normal phase. Positive ions are e x p e c t e d to be lighter than negative ions, offering further insight into the effects of energy exchange b e t w e e n 3He quasiparticles and the ion. For this reason we have extended the m e a s u r e m e n t s to positive ions b o t h in the normal and the superfluid phases. I n d e e d we have found that the positive ion mobility increases with decreasing t e m p e r a t u r e in the n o r m a l phase. In the superfluid phases we o b s e r v e d qualitatively similar b e h a v i o r as in the case of negative ions. T h e e n h a n c e m e n t of the mobility, normalized to the value at the superfluid transition t e m p e r a t u r e , however, was clearly less p r o n o u n c e d for positive ions. During the course of o u r work, R o a c h et al. 3"4 r e p o r t e d similar e x p e r i m e n t s with negative and positive ions. We will c o m p a r e their results with ours. O u r n o r m a l phase data at low t e m p e r a t u r e , in particular, disagree with theirs and we will discuss the reasons for this discrepancy. Most recently A l e x a n d e r et al. 5 studied positive ions in the normal phase and o b s e r v e d novel features

*Work supported by the Academy of Finland.

tPresent address: Bell Laboratories, Murray Hill, New Jersey.

:~On leave of absence from Regensburg University, Regensburg, West Germany, supported by Deutsche Forschungsgemeinschaft.

69

0022-2291/78/1000-0069505.00/0 ~ 1978 Plenum Pubhshing Corporation

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70 J. Kokko, M. A. Paalanen, W. Schoepe, and Y. Takano

related to the creation of several ion species. Their mobility data cannot be directly c o m p a r e d with ours, because their electric fields were higher than ours by an order of magnitude.

T h e t e m p e r a t u r e independence of negative ion mobility was qualitatively understood in the framework of the theory by Josephson and Lekner. 6 The positive ion results show an essential effect of t e m p e r a t u r e and should shed more light on the kinetics of ions in liquid 3He. The situation is somewhat similar in the case of the superfluid phases; the mobility calculation in the B phase, assuming elastic quasiparticle-ion collisions, by Baym et al., 7 was in good agreement with experiment for negative ions, whereas our positive ion results seem clearly out of the range of validity of the theory.

The 3He sample was cooled by nuclear demagnetization of copper.

T h e t e m p e r a t u r e was d e t e r m i n e d by measuring the nuclear magnetic sus- ceptibility and the spin-lattice relaxation time of platinum powder. Ions were produced by applying voltages between 800 and 2000 V to three tungsten field-ionization tips. Details of the refrigeration and ther- mometry, as well as the time-of-flight m e t h o d of measuring the mobility, have been previously reported. 2"8 An advantage of the current apparatus is that the external magnetic field (28 roT) could now be oriented to form any angle with the ion velocity. We were thus able to study the anisotropic mobility in the A phase. The drift space was 2.4 mm long. In o r d e r to estimate the absolute accuracy of our mobility data, we measured the negative ion mobility in the normal phase at 28 bar and obtained excellent

w

agreement (within 4 % ) with our previous value.

Figure l a shows the t e m p e r a t u r e d e p e n d e n c e of the positive ion mobility/.~ in the normal phase at 6 and 28 bar. Important features of the data are an anomalous jump in the mobility at about 100 m K and the logarithmic increase toward decreasing temperatures, the increase being steeper at the higher pressure. We also found at the lowest t e m p e r a t u r e that the drift velocity t, was not proportional to the applied field E at E as small as 30 V / c m (see Fig. 2). Care was taken therefore to determine the mobility from the ratio v i e in the linear velocity regime. T h e data points at the lowest temperatures were obtained from the v - E plots shown in Fig. 2.

For each pressure we then went on using a safe field at higher temperatures, occasionally checking, by constructing the whole v - E curve again, that increasing space charge effects had not warped the results. This led to higher allowed fields at higher temperatures. At 6 bar, for instance, we used E = 19 V / c m for temperatures below 4.2 inK; the largest field was 51 V / c m . Roughly 20% of the points shown in Fig. l a were taken from the v - E plots.

Regarding the anomalous jump in the mobility, Roach et,al. 4 have recently demonstrated that the anomaly is caused by tiny traces of 4He

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Positive Ion Mobility in Normal and Superfluid 3He 71

t )

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Fig. 1. Mobility of positive ions in the normal phase. (a) Temperature dependence of the mobility for 0.5 (+), 6 (O), and 28 bar 4(D). The solid line shows the smoothed data of Roach et al. (b) Pressure d e p e n d e n c e at 24 mK.

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20 i I f I i [ i

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Fig. 2. Dependence of the drift velocity on the applied electric field in the normal phase. The inelastic threshold velocity kBT/pF is indicated with arrows. Broken lines are extrapola- tions from the l~w-velocity region. +, 1.5 inK, 0.5 bar. At 6bar: O, 2.66 mK; O, 5.14 mK; 73, 8.42 inK. At 28bar: A, 3.11 mK; A, 5.71 mK.

i m p u r i t i e s in t h e s a m p l e , * o n l y at t e m p e r a t u r e s b e l o w t h e a n o m a l y w a s t h e m o b i l i t y u n a f f e c t e d b y t h e i m p u r i t y c o n c e n t r a t i o n . T h e r e f o r e w e will n o t d w e l l u p o n t h e t e m p e r a t u r e r e g i o n a b o v e t h e a n o m a l y .

T h e s o l i d l i n e in F i g . l a s h o w s t h e s m o o t h e d d a t a o f R o a c h et al. f o r t h e i r p u r e s t 3 H e c l o s e to t h e v a p o r p r e s s u r e . W e h a v e t w o d a t a p o i n t s at 0 . 5 b a r m a r k e d b y c r o s s e s in t h e f i g u r e . T h e d a t a o f R o a c h et al. a g r e e w i t h o u r p o i n t at 2 4 m K b u t lie s i g n i f i c a n t l y l o w e r t h a n o u r s at 1.5 m K . Q u i t e p o s s i b l y t h e y h a v e u n d e r e s t i m a t e d t h e m o b i l i t y b y d e t e r m i n i n g t h e r a t i o v i e at t o o h i g h a n e l e c t r i c field. T h e i r e a r l i e r o b s e r v a t i o n 1~ t h a t t h e

*The 4He concentration in our sample was 30 ppm. The volume of the sample cell was 13 cm 3 and the surface area of thecopp4er s~onge in the cell was 10-15 m 2. By using a typical surface density of 10 atoms/cm for He, we estimate that the sponge surface absorbs as much as 400 ppm of 4He impurities in the sample until the monolayer coverage is completed. The mobility anomaly seems to be extremely sensitive to the minute concentration of 4He left in the liquid due to thermal fluctuations.

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Positive Ion Mobility in Normal and Superfluid 3He 73

mobility was i n d e p e n d e n t of electric field below 200 V / c m is in dis- a g r e e m e n t with our results in Fig. 2. T h e mobilities o b t a i n e d by A l e x a n d e r et al. 5 f r o m their fastest ion signals are qualitatively similar to ours, although their non-gating m e t h o d introduces uncertainty in interpreting the signals.

T h e pressure d e p e n d e n c e of the mobility at 24 m K is shown in Fig. lb.

T h e mobility drops monotonically as the pressure increases and the change is m o r e rapid at higher pressures. T h e same m e a s u r e m e n t for negative ions shows an opposite pressure d e p e n d e n c e . 2 T h e quite different behaviors of the two ion species are due to the r e m a r k a b l y different ion structures.* T h e negative ion is a " b u b b l e " of v a c u u m containing an electron; the bubble radius shrinks with increasing pressure, which enhances the mobility. T h e negative ion radius o b t a i n e d f r o m our mobility m e a s u r e m e n t s is in very good a g r e e m e n t with the values predicted by the bubble model. 2 T h e positive ion, on the other hand, is a " s n o w b a l l " of 3He a t o m s solidified by the attractive electrostatic force of the positive charge. 12 Since a higher pressure enhances the solidification, the ion radius increases at higher pressures; the decreasing mobility is due to this as well as to the fact that the Fermi m o m e n t u m and the density of the quasiparticles increase toward higher pressures. T h e radius of positive ions is not quantitatively known.

O n e m a y try to estimate the positive ion radius with the aid of a result by Bowley, 13 who fits the slope of the mobility as a function of In T using the ion radius R as a p a r a m e t e r . Taking the slope f r o m the data below 60 m K for 6 bar and below 20 m K for 28 bar, we obtain R = 7.05 and 6 . 1 8 . ~ . C o n t r a r y to the prediction of the snowball model, the radius is smaller for the higher pressure. F u r t h e r m o r e , taking all.the p a r a m e t e r s 14 n e e d e d to fit the absolute value of our 6 - b a r data (R = 7.05 ~k, M / m 3 = 45 + 1, the ratio of the ion mass to the 3He atomic mass), we can use the theory to predict the mobility at 28 bar. T h e result at 3.1 m K is tx = 0.17~ c m 2 / V sec, 27% lower than our e x p e r i m e n t a l result. Increasing R and M would lower the prediction further. Obviously, then, the pressure d e p e n d e n c e of the theory by Bowley is in d i s a g r e e m e n t with the snowball model. T h e larger slope of the higher pressure data can, however, be due to the t e m p e r a t u r e d e p e n d e n c e of the 3He melting pressure, which leads to t e m p e r a t u r e - d e p e n d e n t radii according to the snowball model. Neglecting the surface tension of the snowball, the model predicts R ( P ) o c [nm/(Pm _p)]1/4, where Pm is the melting pressure, P is the pressure of the liquid, and nm is the liquid density at the melting pressure. 11'~2 With the values f r o m H a l p e r i n ' s data on the melting curve ~5 we estimate that the ion radius decreases by 13% when the liquid cools f r o m 70 to 3 m K at 28 bar;

at 6 b a r the decrease is only 2% in the s a m e t e m p e r a t u r e range. A

*For a review on ion structures and mobilities see Fetter. 11

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74 J. Kokko, M. A. Paalanen, W. Schoepe, and Y. Takano

decreasing ion radius not only reduces the n u m b e r of colliding quasiparticles, but also makes the ion mass smaller, which enhances the ion's recoil.

Both effects would tend to boost the growth of the mobility with decreasing t e m p e r a t u r e b e y o n d what would happen with a constant radius. Thus, inferring the ion radius from the mobility data is not straightforward, particularly at high pressures.

Fetter and Kurkij~irvi ~6 have pointed out that the drift velocity becomes nonlinear above the threshold velocity k B T / p F for inelastic i o n - quasiparticle scattering, where PF is the Fermi m o m e n t u m . This critical velocity, as indicated by arrows in Fig. 2, seems to agree with the onset of the nonlinearity. We can express their prediction by the formula

/~E = v[1 + a ( p F V / k B T ) 2 + 9 9 9 (1) where the coefficient a depends on/x. Since their explicit expression for the mobility p, is not self-consistent and does not agree with our data, we l e t / z and a be fitting parameters and compare only the functional form with the experiment. Bowley has also used their theory for his model 13 and has numerically found the relation

1 / v ~ 1 / t z ' E + 1 / v ~ (2)

for velocities exceeding k B T / p F , where the constants tt' and voo depend on the temperature. T h e numerical values of the fitting parameters are listed in Table I, which should serve for checking the models. The fit with Eq. (2) was p o o r at low electric fields, where the proportionality between the drift velocity and the field holds. This is reflected in the fact that the coefficient /z' in the table overestimates the mobility tz. Note that in regimes where the drift velocity is not proportional to the field, Alexander e t al. extracted the mobility by a fit equivalent to Eq. (2). We avoided such an uncertainty by determining the mobility strictly in the linear regime.

Figures 3a and 3b show the mobility in the superfluid phases divided by the value at Tc. All data were taken in the linear velocity region; we used no extrapolation from the nonlinear regime as was done by Roach e t al. 3 T h e data at 28 bar were taken with the external magnetic field B

T A B L E I Low-velocity Formula (1)

value

P, bar T, mK /x, em2/V sec /z, cm2/V sec a

Formula (2) tz', cm2/V sec voo, cm/sec

0.5 1.5 0.309 0.311 1.31 x 10 -2 0.327 74.3

6 2.66 0.276 0.276 2.24 x 10 -2 0.286 151

28 3.11 0.243 0.244 5 . 2 4 x 10 -2 0.253 110

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Positive Ion Mobility in Normal and S u p e r f l u i d 3 H e 7 5

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Fig. 3. (a) M o b i l i t y in t h e s u p c r f l u i d p h a s e s at 28 b a r n o r m a l i z e d to t h e v a l u e a t Tc for t w o o r i e n t a t i o n s of t h e m a g n e t i c f e l d . (b) M o b i l i t y in the B p h a s e a t 18 a n d 28 bar. T h e s o l i d line is t h e c a l c u l a t i o n in Ref.

18.

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oriented both parallel and perpendicular to the electric field E. We note the following features: Going down in t e m p e r a t u r e starting at To, the mobility starts rising rapidly. In addition, the mobility with BIlE soon grows about 13% higher than with B • E. At the AB transition the mobility with B][E goes smoothly over to the B-phase mobility within the experi- mental scatter. T h e r e appears to be a pressure d e p e n d e n c e in the normalized mobility in the B phase. In both phases the increase of the normalized mobility is considerably less than for negative ions; in the B phase, in particular, it is only one-half of the negative ion increase.

The anisotropy of the A-phase mobility agrees with the observation by Roach e t al., 3 except that the reduction of the mobility with B • E is larger in our case. We believe that the discrepancy is due to a texture effect. Of the two configurations, only the one with BNE orients the anisotropic energy gap uniquely. The other leaves room for texture effects.

Normalizing the mobility to the value at Tc is a reasonable procedure because the t e m p e r a t u r e d e p e n d e n c e of the mobility in the normal phase is negligible c o m p a r e d to the superfluid phases. For instance at 28 bar, the normal phase mobility extrapolated into the superfluid would change only by 4% in the t e m p e r a t u r e range of our superftuid data. We measured the normal phase mobility at 18 bar only close to To. Effects such as an artificial pressure d e p e n d e n c e in the normalized mobility could arise from the extrapolated growth being different from 4% at 18 bar. Such an uncertainty, however,,is not sufficient to account for the observed pressure d e p e n d e n c e of the B-phase mobility. Note that no pressure d e p e n d e n c e was observed for negative ions. 2

Current theoretical treatments of ion mobility in the superfluid phases have mostly assumed an elastic ion-quasiparticle scattering model. For negative ions in the B phase the theory of Baym e t a l . 7 agreed well with our previous experiments and demonstrated that the earlier calculations by Soda 17 and by Bowley 18 underestimated the mobility as a result of neglecting the superfluid effects on the scattering cross section. Our data for positive ions, on the other hand, lie lower than any of the theoretical predictions. T h e discrepancy is less for the theories assuming a constant scattering cross section; the result first derived by Bowlcy is shown in Fig.

3b for comparison. This fact may suggest that the effect of the singularity in the density of quasiparticle states on the scattering cross section, which was envisaged by Baym e t al., is overwhelmed by the energy exchange in the scattering process, leaving the n u m b e r of thermally excited quasiparticles as the major difference between the normal and superfluid phases.

Although a calculation for freely recoiling ions was recently done by Soda 19 with a constant scattering cross section, the result showed no reduction from the elastic case.

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Positive Ion Mobility in Normal and S u p e r l l u i d a H e 7 7

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The nonlinearity of the drift velocity in the superfluid is illustrated in Fig. 4, where data taken at 28 bar with BIlE are shown. The features are similar to what one finds with negative ions: The onset of the nonlinearity is consistent with the L a n d a u critical velocity for pair-breaking

A/pF,

where A is the superfluid energy gap. A b o v e this threshold the slope becomes parallel to the normal phase data.

In summary, the mobility of positive ions in liquid 3He at low temperatures shows remarkable dissimilarity to that of negative ions. The mobility is strongly temperature d e p e n d e n t in the normal phase and decreases with increasing_pressure. A t low temperatures the drift velocity becomes nonlinear as a function of the applied electric field for unexpec- tedly small fields. In the superfluid the normalized mobility increases much less rapidly than for negative ions. A theoretical calculation qualitatively agrees with the temperature d e p e n d e n c e of our normal phase mobility.

Quantitative understanding of our superfluid results must await a theory that properly treats inelastic scattering. It is h o p e d that our data will stimulate further theoretical efforts in this direction.

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A C K N O W L E D G M E N T S

W e a r e g r a t e f u l t o L i F u - z h e n g a n d L e i l a R e h n f o r v a l u a b l e a s s i s t a n c e d u r i n g t h e m e a s u r e m e n t s . W e a l s o a p p r e c i a t e s t i m u l a t i n g d i s c u s s i o n s w i t h M a r t t i S a l o m a a , H e n r i k S m i t h , a n d e s p e c i a l l y J u h a n i K u r k i j i i r v i , a n d h e l p f u l c o m m u n i c a t i o n s a n d s u g g e s t i o n s f r o m R o g e r B o w l e y , M i k e C r o s s , a n d D o u g O s h e r o f f .

R E F E R E N C E S

1. A. I. Ahonen, J. Kokko, O. V. Lounasmaa, M. A. Paalanen, R. C. Richardson, W.

Schoepe, and Y. Takano, Phys. Rev. Lett. 37, 511 (1976).

2. A. I. Ahonen, J. Kokko, M. A. Paalanen, R. C. Richardson, W. Schoepe, and Y. Takano, J. Low Temp. Phys. 30, 205 (1978).

3. P. D. Roach, J. B. Ketterson, and P. R. Roach, Phys. Rev. Lett. 39, 626 (1977).

4. P. D. Roach, J. B. Kenerson, and P. R. Roach, Phys. Lett. 63A, 273 (1977).

5. P. A. Alexander, C. N. Barber, P. V. E. McClintock, and G. R. Pickett, Phys. Rev. Lett.

39, 1544 (1977).

6. B. D. Josephson and J. Lekner, Phys. Rev. Len. 23, 111 (1969).

7. G. Baym, C. J. Pethick, and M. Salomaa, Phys. Rev. Lett. 38, 845 (1977).

8. A. I. Ahonen, P. M. Berglund, M. T. Haikala, M. Krusius, O. V. Lounasmaa. and M. A.

Paalanen, Cryogenics 16, 531 (1975).

9. J. G. Daunt, S. G. Hedge, and E. Lerner, in Monolayer and Submonolayer Helium Films, J. G. Daunt, and E. Lerner, eds. (Plenum Press, New York, 1973), p. 19.

10. P. D. Roach, J. B. Ketterson, and P. R. Roach, in Quantum Fluids and Solids, S. B.

Trickey, E. D. Adams, and J. W. Dufty, eds. (Plenum Press, New York, 1977), p. 259.

11. A. L. Fetter, in The Physics of Liquid and Solid Helium, K. H. Bennemann and J. B.

Ketterson, eds. (Wiley, New York, 1976), pp. 242-305.

12. K. R. Atkins, Phys. Reu. 116, 1339 (1959).

13. R. M. Bowley, J. Phys. C, to be published; and preprint.

14. R. M. Bowley, private communication.

15. W. P. Halperin, thesis, Cornell University 1975.

16. A. L. Fetter and J. Kurkij/irvi, Phys. Rev. B 15, 4272 (1977).

17. T. Soda, Prog. Theor. Phys. 53, 903 (1975).

18. R. M. Bowley, J. Phys. C 2, L151 (1976).

19. T. Soda, Prog. Theor. Phys. 58, 1096 (1977).