Abstract We have studied the elasticity of solid 4 He in relation with its possible supersolidity. For this we have measured acoustic resonance frequencies in a 1 cm 3 cell filled either with polycrystals or with single crystals of 4 He. We have observed a large stiffening at low temperature as first observed by Day and Beamish in polycrystals. The 3 He impurity content has been varied from 300 ppb to 0.4 ppb. When kept in equilibrium with liquid helium, single crystals should be impurity free. In these crystals, a large stiffening is observed, which should not be the result of the pinning of dislocations by impurities

X Rojas

· C Pantalei

· H J Maris

· S Balibar

CiteSeerX

J Low Temp Phys (2010) 158: 478–484
DOI 10.1007/s10909-009-9966-z
Acoustic Properties of Solid 4He in the Limit of Zero
Impurity
X. Rojas · C. Pantalei · H.J. Maris · S. Balibar
Received: 30 June 2009 / Accepted: 7 September 2009 / Published online: 23 September 2009
© Springer Science+Business Media, LLC 2009
Abstract We have studied the elasticity of solid 4He in relation with its possible su-
persolidity. For this we have measured acoustic resonance frequencies in a 1 cm3 cell
filled either with polycrystals or with single crystals of 4He. We have observed a large
stiffening at low temperature as first observed by Day and Beamish in polycrystals.
The 3He impurity content has been varied from 300 ppb to 0.4 ppb. When kept in
equilibrium with liquid helium, single crystals should be impurity free. In these crys-
tals, a large stiffening is observed, which should not be the result of the pinning of
dislocations by impurities.
Keywords Solid helium 4 · Elasticity · Supersolidity
PACS 67.80.-s
1 Introduction
The existence of supersolidity is well established now. In three different types of
experiments, several groups have found anomalies in the rotation properties [1–14],
in the specific heat [15] and in the elastic properties [13, 16, 17] of 4He crystals.
All these anomalies are very sensitive to the disorder of the samples and to the con-
centration of 3He impurities. Surprisingly, Day et al. [13, 16] have discovered that
This work was supported in part by the French ANR grant BLAN07-1-215296 and by the US
National Science Foundation grant No. DMR-0605355.
X. Rojas () · C. Pantalei · S. Balibar
Laboratoire de Physique Statistique de l’Ecole Normale Supérieure, associé au CNRS, à l’UPMC
Paris 06 et à l’Université Paris Diderot, 24 rue Lhomond, 75231 Paris Cedex 05, France
e-mail: rojas@lps.ens.fr
H.J. Maris
Department of Physics, Brown University, Providence, RI 02912, USA
J Low Temp Phys (2010) 158: 478–484 479
Fig. 1 Two photographs of the single crystal X6 in the 1 cm3 cell. Left: during growth. Right: after growth
and in equilibrium with a few liquid droplets. The two black rectangular objects are the piezo-electric
transducers with their electrical connexions on top
when supersolidity appears, the shear modulus μ of solid samples increases. Day et
al. [16] have attributed this increase to 3He impurities which bind to dislocations and
pin them below a temperature in the range 50 to 200 mK which depends on their
concentration.
It seems somewhat surprising that an interpretation in terms of dislocations is
relevant for samples which are mostly polycrystalline with a large density of grain
boundaries [4, 18]. We have made acoustic measurements similar to those previously
done by Day et al. in order to compare polycrystals to single crystals. By doing so,
we have found evidence that the interpretation in terms of pinning of dislocation is
not sufficient because we have observed stiffening in single crystals which are in
equilibrium with liquid helium so that their 3He impurity concentration should be
negligible.
2 Experimental Setup
As shown by the photographs in Fig. 1, we have placed two piezo-electric transducers
in a nearly cubic cell made by cutting a square hole (11 × 11 mm2) in a 10 mm
thick copper plate which is closed by two glass windows (6 mm thick). Between
the windows and the copper plate, indium rings allow the cell to be leak-tight up to
65 bar at least. Thanks to large windows on the cryostat itself, helium crystals can be
observed from the outside with an ordinary camera. The transducers are transverse
PZT ceramics1 providing a displacement in the vertical direction (about 0.8 Å/V at
low temperature). They resonate around 500 kHz but they are used near 30 kHz to
excite and detect acoustic resonances inside the cell. We have used very low levels of
excitation, typically 1 mV, in order to stay below the threshold for non-linear elastic
behavior even at resonances whose quality factor is of order 1,000. We have found
1PZT “shear bars” made of material C5800 by Channel Industries Inc.
480 J Low Temp Phys (2010) 158: 478–484
Fig. 2 (Color online) The shift
in resonance frequency of solid
samples prepared with 4He
containing 300 ppb 3He
impurities. For polycrystals
(diamonds and triangles) the
frequency increases as the
temperature is reduced below an
onset temperature of about
200 mK. For single crystals, the
onset temperature is much
lower, about 50 mK
that this threshold corresponds to a strain of order 10−8, in agreement with Day et
al. In classical crystals, this threshold is of order 10−2, comparable to the plasticity
threshold which is itself close to the fracture limit. Finding a threshold lower by 6
orders of magnitude is a strong indication that the mobility of defects in quantum
crystals plays an important role in their elastic properties.
3 Poly- and Single Crystals Grown from Natural 4He
We first used natural 4He which is assumed to contain 300 ppb 3He impurities. We
prepared polycrystals by the “blocked capillary method” as done by Day et al., that
is by crystallizing a liquid sample at 63 bar and constant volume. As shown previ-
ously [18], this method ends up with a polycrystalline sample at 33 bar with grain
sizes in the micrometer range. We have also prepared polycrystals by fast injection of
mass at 100 mK through the fill line of the cell, in order to compare with a polycrystal
at 25 bar. This type of polycrystal was shown by Sasaki et al. [18] to have grain sizes
of order 50 µm. Both polycrystals (X3 and X4 on Fig. 2) show a stiffening at low
temperature which is very similar to what has been found before by Day et al. The
amplitude of the stiffening varies from one sample to the other but, once normalized
by the maximum variation, the data fall on a universal curve. The effect is apparently
independent of pressure and frequency (data are taken at 25 and 33 bar, and we work
at 30 kHz while Day et al. work from 20 Hz to 8 kHz).
We then prepared single crystals which had to be obtained by slowly injecting
mass into the cell at constant temperature and pressure. One difficulty was that the
walls are preferentially wet by the liquid phase (the contact angle of the liquid-solid
interface is about 45° [18, 19]) so that the fill line blocks before the liquid crystallizes
in several corners. A future cell will have a fully convex geometry and the orifice of
the fill line at the highest point of the cell [20] but, with the present cell, we had to fill
J Low Temp Phys (2010) 158: 478–484 481
the cell as much as possible at 1.45 K before cooling down so that the rest of the liquid
regions crystallizes. When cooling from 1.45 K to 1 K, the liquid-solid equilibrium
pressure decreases from 26.3 to 25.3 bar so that stresses damage the crystal quality. If
done too quickly, the crystal breaks. In most cases, the final sample is a single crystal
in equilibrium with a few liquid droplets which have a typical volume of 1 mm3 (see
the picture on the right of Fig. 1).
We have studied the acoustic resonances in these low quality single crystals. Fig-
ure 2 shows the frequency shift normalized by its low temperature value. The real
shift varies from sample to sample. No systematic difference in shift amplitude was
observed between poly- and single crystals but it occurs at much lower temperature
in single crystals: the onset is around 50 mK instead of 200 and the transition was
not yet completed at the lowest temperature we could achieve (36 mK). There are
at least 3 possible interpretations for this observation: either the binding energy of
3He impurities is much smaller on dislocations—which may explain the anomalies
in single crystals—than on grain boundaries which may be relevant for polycrystals,
or the purity of single crystals is different from that of polycrystals, or the stiffening
is not, or not only, due to the pinning of dislocations proposed by Day et al. In order
to progress, we have repeated the experiment with ultrapure 4He containing 0.4 ppb
3He only.
4 Ultrapure Samples
When preparing ultrapure polycrystals, we found a stiffening once more in agreement
with the results previously obtained by Day et al. (this time with 1 ppb 3He): the
onset of stiffening is around 70 mK. Note that, for comparison, in Fig. 2 we have
multiplied Day’s normalized variation by 1,000 because, in our cell for the mode
studied, we have calculated numerically that the frequency shift is one third of the
modulus shift (f/f = 0.33μ/μ) so that Day’s change by 10% in μ should induce
a change by 1,000 Hz in our experiment where f is around 30 kHz. The ultrapure
single crystals gave the same sharp stiffening below 40 mK which we had already
observed with normal purity single crystals. At this stage, we realized that the energy
of 3He atoms is lower by more than 1 K in the liquid than in the solid. We have
calculated this effect [20] by using the theory by Edwards and Balibar [21]. The ratio
of the concentration in the hcp solid to the concentration in the liquid is
Xh3
XL3
= 4.42
T 3/2
exp
[
−1.359
T
]
(1)
which is 10−12 at 40 mK. Furthermore, the diffusion coefficient D of 3He atoms in
solid 4He is large, especially for low Xh3 , as shown by Schratter and Allen [22] who
found DXh3 = 2.4 × 10−11 cm2/s. It means that even if during growth 3He atoms
enter the solid, the impurities diffuse fast enough on a macroscopic distance (1 cm)
for the equilibrium concentrations to be reached in a small time. As a consequence, in
crystals in equilibrium with liquid helium, the 3He concentration is extremely small:
starting with 300 ppb (respectively 0.4 ppb), Xh3 should be below 10
−15 (respectively
482 J Low Temp Phys (2010) 158: 478–484
Fig. 3 (Color online)
Frequency shift for ultrapure
crystals prepared with 0.4 ppb
3He content except for Day’s
polycrystal (1 ppb) and for
single crystals Xa to Xf
(300 ppb)
4×10−19) as soon as a 1 mm3 liquid drop is in contact with our 1 cm3 single crystals,
even less if the liquid volume is larger. This is probably why our single crystals show
the same stiffening temperature for 300 ppb as for 0.4 ppb He gas: the final crystals
are free of impurities. Our results open the way to studies with no impurities at all
(one can reach 10−21 if the liquid volume is about the same as the solid volume in
contact with it). Note that, even if 3He atoms bind to dislocations with an energy
in the range 0.3 to 0.7 K [23, 24], the dislocations should also be free of impurities
below 50 mK as soon as some liquid 4He is in contact with the solid.
We then succeeded in recrystallizing the single crystals by annealing them at
1.45 K with the fill line open. This method allowed us to study “recrystallized” sam-
ples at the melting pressure but with no apparent liquid region in equilibrium with
them. Figure 3 shows that such recrystallized samples stiffen at a temperature similar
to that of polycrystals with the same purity (0.4 ppb). Since crystals show a large
stiffening even in the absence of impurities, the pinning of dislocations by impurities
cannot be the only mechanism in all crystals.
Unfortunately, we consider the above results as preliminary because our ther-
mometer apparently disconnects from the cell temperature below 40 mK. This is
probably because either light or RF radiation are not sufficiently well filtered and
warm up the thermometer (a Speer carbon resistance) when its thermal contact with
the cell becomes too weak (the Kapitza resistance diverges as T tends to zero). If this
is true, our observation of a stiffening below 40 mK is qualitatively correct but the
transition may take place in a broader temperature domain, between 40 and 10 mK
(the base temperature of our refrigerator).
In conclusion, what could happen in the absence of impurities? A first possibility
is that there is a binding energy for dislocations among themselves: the dislocation
network would be connected or the nodes of their network would stop moving below
a certain temperature only. However, it is usually assumed that dislocations form a
J Low Temp Phys (2010) 158: 478–484 483
connected network with fixed nodes [25–27]. It could be that there is a roughening
transition temperature of order 40 mK below which dislocations stop moving [28].
A third possibility is that stiffening is due to vortex proliferation at the supersolid
transition. The latter hypothesis assumes that supersolidity is an intrinsic property
of ideal 4He crystals which is only enhanced and displaced to higher temperature
by the presence of disorder. This would agree with the predictions by P.W. Ander-
son [29] but contradict many others [30–32]. For further progress, it looks interesting
to run experiments in ideal crystals containing no impurities and no structural defects,
something which now appears possible [20]. One would see if ideal crystals show a
stiffening at low temperature and a rotation anomaly at the same temperature. Note
that, in Lin’s experiment [15], a small peak in the specific heat was measured with
crystals in equilibrium with the liquid. They were single crystals probably not free
of dislocations but free of impurities. One also has to repeat the dc-flow experiments
by Sasaki et al. [33, 34] which were done down to 50 mK only in crystals in equilib-
rium with liquid helium while the present results show that there is a transition below
40 mK for impurity free crystals.
Acknowledgements We are grateful to F. Caupin, E. Rolley, C. Guthmann, Yu. Mukharsky, E. Varo-
quaux, A. Braslau, J. Bossy, J. Dupont-Roc and P. Jacquier for helpful discussions.
References
1. E. Kim, M.H.W. Chan, Nature (London) 427, 225 (2004)
2. E. Kim, M.H.W. Chan, Science 305, 1941 (2004)
3. E. Kim, M.H.W. Chan, Phys. Rev. Lett. 97, 115302 (2006)
4. S. Balibar, F. Caupin, J. Phys.: Condens. Matter 20, 173201 (2008)
5. A.S.C. Rittner, J.D. Reppy, Phys. Rev. Lett. 97, 165301 (2006)
6. A. Penzev, Y. Yasuta, M. Kubota, J. Low Temp. Phys. 148, 677 (2007)
7. M. Kondo, S. Takada, Y. Shibayama, K. Shirahama, J. Low Temp. Phys. 148, 695 (2007)
8. Y. Aoki, J.C. Graves, H. Kojima, Phys. Rev. Lett. 99, 015301 (2007)
9. A.S.C. Rittner, J.D. Reppy, Phys. Rev. Lett. 98, 175302 (2007)
10. A.C. Clark, J.T. West, M.H.W. Chan, Phys. Rev. Lett. 99, 135302 (2007)
11. E. Kim, J.S. Xia, J.T. West, X. Lin, M.H.W. Chan, Bull. Am. Phys. Soc. 52, 610 (2007)
12. E. Kim, J.S. Xia, J.T. West, X. Lin, A.C. Clark, M.H.W. Chan, Phys. Rev. Lett. 100, 065301 (2008)
13. J.T. West, M.H.W. Chan, J. Beamish, O. Syshchenko, Nat. Phys. 5, 598 (2009)
14. J.T. West, X. Lin, Z.G. Cheng, M.H.W. Chan, Phys. Rev. Lett. 102, 185302 (2009)
15. X. Lin, A.C. Clark, Z.G. Cheng, M.H.W. Chan, Phys. Rev. Lett. 102, 125302 (2009)
16. J. Day, J. Beamish, Nature 450, 853 (2007)
17. Y. Mukharsky, O. Avenel, E. Varoquaux, J. Low Temp. Phys. 148, 689 (2007)
18. S. Sasaki, F. Caupin, S. Balibar, J. Low Temp. Phys. 153, 43 (2008)
19. S. Balibar, H. Alles, A.Ya. Parshin, Rev. Mod. Phys. 77, 317 (2005)
20. X. Rojas, C. Pantalei, H.J. Maris, S. Balibar, J. Low Temp. Phys. (2009), doi:10.1007/s10909-
009-9966-z
21. D.O. Edwards, S. Balibar, Phys. Rev. B 39, 4083 (1989)
22. J. Schratter, A.R. Allen, J. Low Temp. Phys. 57, 179 (1984)
23. I. Iwasa, H. Suzuki, J. Phys. Soc. Japn. 46, 1722 (1980)
24. M.A. Paalanen, D.J. Bishop, H.W. Dail, Phys. Rev. Lett. 46, 664 (1981)
25. N.F. Mott, Philos. Mag. 43, 1151 (1952)
26. J. Friedel, Philos. Mag. 44, 444 (1953)
27. A. Granato, K. Lücke, J. Appl. Phys. 27, 583 (1956)
28. D. Aleinikava, E. Dedits, A.B. Kuklov, D. Schmeltzer, Phys. Rev. B (2009), submitted.
29. P.W. Anderson, Science 324, 631 (2009)
484 J Low Temp Phys (2010) 158: 478–484
30. D.M. Ceperley, B. Bernu, Phys. Rev. Lett. 93, 155303 (2004)
31. N. Prokof’ev, B. Svistunov, Phys. Rev. Lett. 94, 155302 (2005)
32. N. Prokof’ev, Adv. Phys. 56, 381 (2007)
33. S. Sasaki, R. Ishiguro, F. Caupin, H.J. Maris, S. Balibar, Science 313, 1098 (2006)
34. S. Sasaki, F. Caupin, S. Balibar, Phys. Rev. Lett. 99, 205302 (2007)


Acoustic Properties of Solid 4 He in the Limit of Zero Impurity

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Acoustic Properties of Solid 4 He in the Limit of Zero Impurity

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