ULTRASONIC STUDIES OF 4He SOLID-LIQUID INTERFACE MOBILITY

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ULTRASONIC STUDIES OF 4He SOLID-LIQUID INTERFACE MOBILITY

M. Manning, M. Moelter, C. Elbaum

To cite this version:

M. Manning, M. Moelter, C. Elbaum. ULTRASONIC STUDIES OF 4He SOLID-LIQUID IN- TERFACE MOBILITY. Journal de Physique Colloques, 1985, 46 (C10), pp.C10-801-C10-804.

�10.1051/jphyscol:198510175�. �jpa-00225384�

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JOURNAL DE PHYSIQUE

Colloque C10, suppl6ment au n o 12, Tome 46, d6cembre 1985 page _C10-801

ULTRASONIC STUDIES OF 4 ~SOLID-LIQUID INTERFACE MOBILITY e

M.B. MANNING, M.J. MOELTER AND C. ELBAUM

Physics Department and Metals Research Laboratory, Brown University, Providence, RI 02912, U.S.A.

R6sum6

-

Nous avons effectue les premiares mesures de la' reflexion acoustique, ainsi que de la transmission acoustique, 2 l'interface solide- liquide de 4 ~ e . Nous utilisons ces mesures pour de'terminer l'absorption de l'e'nergie acoustique totale, relative, 2 l'interface, en fonction de la temperature. En nous basans sur ces re'sultats, nous proposons un nouveau mgchanisme, comprenant la ge'ne'ration et les arrangements de lacunes, pour la cine'tique de la croissance et de la mobilitg de l'interface en fonction de la tempgrature.

Abstract - The first measurements of acoustic reflection have been carried out, together with acoustic transmission at the solid-liquid interface of 4 ~ e and are used to determine the total relative acoustic energy absorption at the interface as a function of temperature. Based on these results a new mechanism involving vacancy generation and rearrangements is proposed for the temperature dependence of the kinetics of growth and interface mobility.

Crystal growth from its liquid can be described phenomenologically by a relation v = KAp, where v is the growth rate of the solid,

Av

is the difference in chemical potential between the liquid and the solid, and K is a kinetic coefficient. For a classical system in which the growth process is diffusion and/or nucleation depen- dent, K increases with temperature, usually according to an Arrhenius law. For quantum solids (such as helium) Andreev and parshinl proposed that crystal growth proceeds in an entirely different way, that the solid-liquid interface has very high mobility and that at T = OK the process is continuous and reversible, i.e., without dissipation. Thus at T = O.the coefficient K is infinite; as the temperature increases, thermal excitations in the liquid and in the solid interact with the solid-liquid interface and cause dissipation. Consequently, unlike in classical systems, the coefficient K decreases with increasing temperature. This propos- al led to a number of experimental and theoretical investigations on helium crystal growth.*-l1 It was r e a l i ~ e d , ~ , ~ in particular, that pressure changes associated with a sound wave propagating from liquid to solid would be taken up by the advanc- ing or receding interface, where the pressure would be near or at the equilibrium melting value. Thus sound transmission between the two media would be substantial-

ly reduced or even suppressed. This reduction in transmission provides a method of studying growth kinetics. Here we report results of the first measurement of the reflection of sound, as well as an extension to higher temperatures and at a higher frequency (10 MHz) of earlier results4 on transmission of sound across the He-4 liquid-solid interface. Helium crystals were grown under constant pressure condi- tions. Vertical sound transmission, liquid to solid, was monitored by measuring the received signal heights of the first transmitted signal and the two reflected signals from the fnterface. (See fig. 1). The amplitudes of the first transmitted signals (TI, in fig. 1) with solid filling 10%-50% of the cell were normalized to the liquid calibration signal by division. Corrections for signal attenuation in the solid and liquid He-4 and for any focussing effects in the solid were made by determining the effective attenuation from two different heights of solid present

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:198510175

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(210-802 JOURNAL DE PHYSIQUE

in the cell. The reflection and transmission coefficients (obtained from the relative amplitudes of reflected and transmitted signals) are shown in figs. 1 and 2 for h.c.p. and b.c.c. crystals grown from the superfluid. First we note the strik- ing result that in the temperature interval 1.25 K '2 T

2

1.38 K the amplitude of

the signal reflected from the interface on the 'liquid side, labeled R in fig. 1, is

zero

within the resolution of our observations. At the same time &he signal R (reflection on the solid side) is large and generally in excess of the value ex- 2 pected from acoustic impedance mismatch.12 Since R is not zero (and is, in fact, larger than the value expected from acoustic impedance mismatch) in the temperature 2 interval where R is zero, the vanishing of the latter cannot be attributed to such artifacts as a solid-liquid interface inclined to the direction of sound propaga- 1 tion. The observed zero value of R is evidently not related to the roton attenuation peak13 either, since for the ultrasonic frequency of 10 MHz used here this 1 peak occurs at a lower temperature (see fig. 1). Moreover, these observations are independent of which (top or botton) transducer is used as a transmitter. The transmission coefficient for h.c.p. crystals grown from the superfluid (fig. 2), is found to vary monotonically with temperature and to reach very small (s 0.1) values below 0.9 K. At any given temperature the relation12

-

1 is generally not satisfied. The following exceptions are noted: 1) A !gs&f grown at 1.07 K dis- played substantial agreement, both in reflection and transmission, with acoustic impedance mismatch expectation. We do not have an explanation for this result. we can only conjecture that in this case the solid-liquid interface was "faceted.;14 2) The only two crystals of b.c.c. 4 ~ e , grown from the superfluid, had transmission coefficients substantially smaller than expected from acoustic impedance mismatch and from extrapolation of values for h.c.p. crystals (see fig. 2). This case will be treated separately. Here we confine the discussion to h.c.p. crystals grown

from superfluid, excluding the crystal grown at 1.07 K (see above). First we note that our transmission coefficients (fig. 2) have values similar to those of

Castaing et a ~ .(CBL) obtained at 1 MHz in the temperature range, 0.8 ~ K

2

T 2 1 K, in which the two experiments overlap. We analyzed our transmission results through the expressions developed by Castaing and ~oziSres3 and used by (CBL). In particular, we extracted from our transmission data values of the qyantity 5-I

( 5

^{is the}

interface mobility), according to Eq. (7) of ref. (3), i.e.

TLS = 2z2/(z1

+

Z2

+

Z1Z2S) (1) GIe note, however, that in deriving3 Eq. 1, R = T

-

1, was implied and that our data do not satisfy this condition. The behavior of 5-1 obtained here should be considered, therefore, as an indication of a trend, rather than a precise result. With this reservation,

5-1

was analyzed by fitting temperature dependences of the form 5-I =

AT^ +

^~;$,~xp(-^TIT) in an attempt to identify separately phonons and roton contributions

,

as wgll as

5-I

=

' , 5

exp(-TOIT), in keeping with the analysis of ref. (4). The first form yielded a negative fonstant A, while the second one pro- vided a good fit with To = 7.5 f 0.3 K and

: 5

= (1.6

+

^{0.5) x} ^{in c.g.s.}

units. Since our results generally do not satisfy the relation R = TLS

-

^{1, we}

calculated from our data the relative acoustic energy dissipation E at the interface, as a function of temperature, using the energy conservation expression12*15

E represents all of the acoustic energy dissipation, including that3''' consistent with the condition R = T

-

1, as well as the additional dissipation found when R = T

-

1 is not satisEHed (i.e., the case of our results). Fig. 3 shows a plot of E ta~elative Acoustic Energy Loss") as a function of temperature. The shape of this plot is strongly suggestive of a relaxation mechanism. We fitted, therefore, the data to a relation of the form

where C is an "energy relaxation strength", T a T o exp(U/T) and T is taken as the reciprocal of the Debye frequency, i.e., T, = 1.6 x 10-l2 sec. ?he resulting acti-

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vation energy is U = 11.52

+

0.04 K and the prefactor C = 1.520 10.036. The curve in Fig. 3 represents this fit. Previous deduction of interface mobility, 5, from the damping of melting-freezing waves2 and from acoustic transmission4 measurements show that up to % 1 K, 5-I is thermally activated. We obtain similar behavior of 5-I from our acoustic transmission measurements alone. From our acoustic transmission together with the acoustic reflection (which has not been previously mea- sured), we calculate the total relative energy dissipation, E, at the interface, as a function of temperature, and we find that E displays relaxational behavior (fig.

3). The raw data for this energy dissipation are obtained directly from the experimental results using only an energy balance equation (Eq. 2), without reference to any specific model. This is in contrast to the deduction of the interface mobility, 5, discussed above. In view of these observations, we offer the following thoughts. The observed energy dissipation is related to a diffusion process at the interface. The activation energy of % 11.5 K can be interpreted as the sum of a vacancy formation energy16 of % 8 K and a barrier VM to vacancy (or atom) motion of

% 3.5 K. This barrier would result in a vacancy tunneling rate of wT =

%

e-A, X = (d/X) (2m vCi)lI2 (where d is the barrier width, w is the Debye frequency) and an activated gump rate of w =

%

exp(-V IT). The ?emperature, TCR, at which the two rates become equal is % $.3 K, when t#e mass of a helium atom IS used for m, i.e.;

on the assumption that a vacancy "jump" involves the displacement of a single atom.

(While this may be an oversimplification for delocalized vacancies in a quantum solid, it should give a reasonable approximation to TCR). The melting-freezing process is thus considered to be limited by the accommodation of thermally gener- ated vacancies. At low temperatures (below T ) vacancy migration and rearrangements are governed by (temperature independen@ tunneling. The decrease in interface mobility with increasing temperature is thus determined by the increasing vacancy concentration alone, i.e., exp(-V IT), where VF is the formation energy of

% 8 K. Above T vacancy (or atom) migraFion is thermally activated with an energy barrier V % 3.5%. Thus, in this case the interface mobility is limited by concentration and migration of vacancies, with an effective activation energy V M

F ⁺V~

% 11.5 K. The observed relaxational behavior of the relative acoustic energy loss (fig. 3) is consistent with .this interpretation. For consistency with our as well as

ear lie^?,^

results, we only have to assume that TCR % 1 K, rather than 'l. 1.3 K estimated above. In view of the simplification used m our estimate this should certainly not be considered a serious discrepancy. We also note that the location and magnitude of the peak (fig. 3) are determined primarily by values of E above

Q, 1 K and are insensitive to the values of E below % 1 K, where a change to tunneling is expected. This research was supported by the National Science Foundation through Grant No. DMR 8304224.

A.F. Andreev and A.Y. Parshin, Zh. Eksp. Teor. Fix.

75,

1151 (1978) [SOV.

Phys. JETP

3,

763 (1978)l.

K.O. Keshishev, A.Y. Parshin and A.V. Babkin, Pism'a Zh. Eksp. Teor. Fiz.

30,

63 (1979) [SOV. Phys. JETP Lett.

30,

56 (1979)

1.

B. Castaing and P. ~ozisres, J. Physique

41,

701 (1980).

B. Castaing, S. Balibar and C. Laroche, J. Physique

3,

697 (1980).

T.E. Huber and H.J. Maris, Phys. Rev. Lett.

47,

1907 (1981).

D. Savignac and P. Leiderer, Phys. Rev. Lett.

49,

1869 (1982); J. Bodensohn, P. Leiderer and D. Savignac, in "Phonon Scattering in Condensed Matter", Ed.

W. Eisenmenger, K. Lassman and S. ~gttinger (Springer-Verlag, 1984).

A.F. Andreev and V. G. Knizhnik, Zh. Eksp. Teor. Fiz.

3,

416 (1982) [SOV.

Phys. JETP

56,

(I), 226 (1982)

1.

R.M. Bowley and D.O. Edwards, J. Physique

44,

723 (1983).

P.E. Wolf, D.O. Edwards and S. Balibar, J. Low Temp. Phys.

51,

(5161, 489 (1983).

.-

^-., -

10. L. Puech and B. Castaing, J. Physique-Letters

43,

L-601 (1982).

11. V.N. Grigor'ev, N.E. Dyumin and S.V. Svatko, Fiz. Nizk. Temp.

9,

649 (1983) [SOV. J. Tamp. Phys.

2,

(6), 331 (1983)l.

12. In acoustic mismathh theory the transmission and reflection coefficients for a pressure wave, originating in medium a and entering medium b at normal inci- dence are respectively:

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C10-804

JOURNAL DE

PHYSIQUE

Tab = 2zb/(za + Zb) and Rab = (Zb

-

^Za)/(( ^Zb

+

Za)) where Zi = pici, are mass density and sound velocitv.

13. P.R. Roach, J.B. Ketterson and M. Kuchnir, Phys. Rev.

e,

(51, 2205 (1972).

14. A faceted interface is expected to have low mobility and thus transmit sound according to acoustic impedance mismatch. See, for example, K.O. Keshishev, A.Y. Parshin and A.I. Shal'nikov, Sov. Scient. Rev., A&, 155 (1982).

15. L. Kinsler and A. Frey, Fundamentals of Acoustics (Wiley, New York, 1962), ch.

5, 6.

16. H.A. Goldberg, Physics Letters

2,

^A(1), 45 (1976).

Fig. 1. Sound reflection coefficient for waves propagating from liquid to solid (R1) as a function of temperature; X- for h.c.p., 0- for b.c.c. helium. Inset shows schematically the interior of the cell and labels the wave prop- agation directions. The horizontal lines marked "A-M" indicate the range of reflection coefficients expected from acoustic impedance mismatch between the liquid and all crystal

".

? 2

⁰

}

"O'O" PWk

a I, #

" ;

^sCC

I

orientations of h.c.p. 4 ~ e .

. 8 1 1 . 2 1.4 1.6 1.8

TEMPERATURE (K)

nj.1

Fig. 2. Sound transmission coefficient for waves propagating from liquid to solid (TI), as a function of temperature; X- for h.c.p., 0- for b.c.c. helium. The horizontal lines marked "A-M" indicate the range of transmission coefficients expected from acoustic impedance mismatch between liquid and all crystal orientations of h.c.p. 4 ~ e .

Fig. 3. Relative acoustic energy loss E (see Eq. 2) at the solid-liquid interface, as a function of temperature. The curve represents a fit of the relaxation function, Eq. 3.

; y . 4 See text.

5" . 3

W U

a -

,-

_m. 2 _{. I}

2 ⁰

U

. 8 . 9 1 1.1 1.2 1 . 3 1 . 4 1.5 TEMPERATURE (K)