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H↓ on He : sticking and 2d-superfluidity
B. Collaudin, B. Hebral, M. Papoular
To cite this version:
B. Collaudin, B. Hebral, M. Papoular. H↓on He : sticking and 2d-superfluidity. Journal de Physique, 1986, 47 (9), pp.1503-1506. �10.1051/jphys:019860047090150300�. �jpa-00210346�
H~ on He : sticking and 2d-superfluidity
B. Collaudin, B. Hebral and M. Papoular
Centre de Recherches sur les Très Basses Températures, C.N.R.S., BP 166 X, 38042 Grenoble Cedex, France
(Reçu le 6 mars 1986, accepti le 29 avril 1986)
Résumé. 2014 Le coefficient de collage, qui détermine le temps de séjour en surface 03C4s, est évalué à taux de couverture élevé. Pour que le système présente les propriétés de superfluidité bidimensionnelle, il faut que 03C4s reste grand devant
un temps caractéristique de diffusion de vortex.
Abstract. 2014 The sticking coefficient, which governs the sticking time 03C4s, is discussed for high surface-coverage conditions. We point out that 03C4s must remain large compared to a characteristic vortex diffusion time, if the system is to display 2d-superfluidity.
Classification Physics Abstracts 67.40 - 67.70 - 05.40
1. Time scales.
In contrast to metals and superfluid Fermi liquids,
there have been few studies of kinetic properties in
Bose systems [1]. Moreover the systems considered
were, naturally, particle-conserving. This is not the
case with polarized atomic hydrogen, one of the prototypes of a dilute Bose gas [2]. Even without
considering depletion due to recombination, there is permanent particle exchange between bulk gas and the helium-adsorbed Hi layer. This exchange should
be rapid on the scale of processes of interest such as
relaxation or recombination, in order for these processes to be described in terms of equilibrium adsorption. The adsorption isotherm, far from satura- tion, i.e. for n. nsat N 1014 at.cm-2 (see Sect. 3),
has the simple form :
Here n. stands for the two-dimensional H I layer
atomic density for a bulk gas density n,; sa = 1 K is the adsorption energy of H upon liquid 4He ; and AB = h(2 nmkb T)- 1/2 is the thermal de Broglie wave- length of an H atom at temperature T. Under the present practical conditions, the total number of atoms in the bulk will be much greater than the total number of adsorbed atoms :
where V is the volume, and A the area of the H! cell.
In thermal equilibrium, the average times an atom
spends in the bulk, r, and on the surface, Tg, are related by
This inequality ensures that, whatever 1:V’ the Bose condensate - when any can be reached - will be able to get to equilibrium with the excitations. Eckem
[1] has derived an expression for ’to, this condensate
equilibration time, valid for a dilute Bose gas at a
fi2 n2/3 temperature close to the critical T, = 3.3 v
(but outside the critical region AT/Tr -- ao n’/3
where ao = 0.7 A is the H(-H( scattering length and
sets the scale for the weak interatomic interactions.
For n, = 10’9 at. CM- 3, T,,; = 75 mK and io is actual- ly rather short : 10- 7 s.
The situation is not so simple if one considers the Kosterlitz-Thouless transition to 2d superfluidity.
For one thing there is no Bose condensate. On the other hand, for the 2d hydrogen layer to actually display superfluid behaviour, the average lifetime
on the surface should be much larger than a characte-
ristic 2d-superfluidity correlation time ioa :
Otherwise, phase coherence in the hydrogen layer
could not be established
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019860047090150300
1504
We shall see (Sect. 4) that this inequality is not systematically fulfilled in practice. Let us start with a
discussion of the physics of -rÕd.
2. 2d-superfluidity cut-off frequency.
Dissipation effects near the Kosterlitz-Thouless (KT)
transition are known to be governed - just as in 3d-superfluidity - by the rate at which vortices cross
the flow path. The basic physical parameter here is the vortex diffusion constant, D z n/m, as was shown by Ambegaokar et al. [3]. For hydrogen, D & 6 x
10-4 cm2 js. At a driving frequency co, D defines a
diffusion length
r D = )2:: .
D2D w
All KT-vortex pairs with separation r larger than rD cannot equilibrate to the driving oscillating field It is therefore essential that cobe slow so that the diffusion length be larger than the typical pair radius, in order for equilibrium-super- fluidity properties to be accessible.
In the hydrogen context, N,, the total number of adsorbed atoms is necessarily relatively small, which
in turn requires relatively large driving forces. Now, for finite driving fields, a free-vortex nucleation radius
can be defined [3]. The smaller, r c’ the larger the driving
field (the latter is proportional to a relative normal- fluid velocity I Vn - U. I, E (r) ;::t; I being the non- local, static « dielectric constant »). Pairs with r > r,,
are dynamically « ionized ». Again, r,, must be smaller
than rD for a hydrodynamic approximation to work.
Close enough to T,, the dissipation rate is essentially proportional to Drc " [3]. The Cornell results [4] for superfluid 4He on mylar have shown the onset of non-
linear dissipation effects for velocities larger than
about 1 cm/s.
Assume, for hydrogen, driving conditions such that
v. - u. I e(rc) =- 10 cm/s. Then, from equation (4) :
rc ;zt; 5 000 A. If rD is to be larger than rc, the driving frequency (cm = 2 Dlr’ must be smaller than a cut-
off : roo = 5 x 105 s-’. Then the characteristic time
we are after is : T2d = roo 1 = 2 x 10-6 s. Like D, T 0 2d is largely temperature-independent. In section 4,
we shall compare this time scale with the sticking
time is (and with the characteristic recombination
time).
One check should be made at this point, namely that
the vortex-core radius ro is smaller than rc. This is trivial in a helium layer where ro is an atomic length.
Less so for the dilute hydrogen layer we are considering
here. Let us estimate ro as the inverse crossover wave- vector between free-particle behaviour (E = fii2 q2/2 m)
and phonon-like behaviour (w = cq) with a surface- v int ns- 1/2
sound velocity c = m .- s s . Ygnt is a two-dimen-
sional hydrogen-hydrogen integrated triplet state
interaction. Like its bulk counterpart, it is dominated
by hard-core repulsion. Its scale is 1 K (10 A )2.
We get : ro z 1i[4 MVin’n.1- 112, of order 20 A
for ns = 1O 13 at.cm-2, safely smaller than the typical
rc we mentioned above.
3. Sticking.
The sticking coefficient (Xst is defined as the probability
for an atom which hits the surface to adsorb. It has been measured [5] by NMR techniques based on the
small difference in Larmor frequency between bulk and surface (bwo = 2 1t x 49 kHz), and found to be
about 4 % at 200 mK. This modest value, together
with the predicted T1/2 temperature dependence of
ast [6], stems from the strong delocalization of the
incoming H atom with respect to the surface (Fig. 1).
The adsorption energy s. is evacuated through ripplon
emission.
Fig. 1. - Schematic wavefunction profiles for the free and bound states.
At high surface coverage, another channel for
sticking opens up, whereby 8a is converted to transla- tional degrees of freedom of neighbouring adsorbed H I atoms, that is surface-sound emission. By « high
coverage » we mean
which is precisely the condition for surface Bose dege-
neracy. In practice these densities will be around iol3 at . cm - 2, somewhat smaller than
the saturation coverage [2] which brings the chemical potential to zero and therefore marks the onset of Bose
condensation in the bulk gas. Even close to saturation,
the coverage ns as given by the adsorption isotherm is
a thermodynamic mean value around which fluctua-
tions, associated with microscopic sticking/desorption
events, still occur, allowing for a finite sticking proba- bility.
Let us now get an order of magnitude of ast in the
« surface-sound » channel, under two simplifying, but non-restrictive, assumptions :
(i) Perpendicular incidence (otherwise, an angular
average should be performed as in ret [6]).
(ii) Negligible incident energy (this is a low-tempe-
rature approximation : T ej.
We write the sticking coefficient as the sticking pro-
bability w per unit area, normalized to unit incident flux density,
where vth is Maxwell’s thermal velocity (’" T 1/2) and a
is a characteristic interaction range of the order of several angstroms (a3 is the reaction volume for
sticking). In the channel we are considering, the rele-
vant perturbation potential is not the liquid He - to -
H atom potential but the H!-H! triplet state
Kolos-Wolniewicz potential, and
If) and I b >, the free and bound state wavefunctions,
are described in reference [6]. In view of the high-
coverage condition (5), they overlap strongly along the
surface so that w and ast should not depend much upon any further increase in coverage. The matrix element in the golden rule, then, is governed by overlap per-
pendicular to the surface, and actually does not differ
much from the matrix element relevant to the ripplon
channel [6]. Both are of the order of
The one difference lies in the spectral density P2d(8J
of final states at energy e.. Here, any final state consists of two hydrogen atoms receding from each other with translational energy c./2 each. The calculation of
P2d(E) is in essence that of a 2 d-free electron gas :
per unit energy and unit area. This results in a sticking
coefficient
of about 10 % at 100 mK and, again, with the square- root temperature dependence. In view of the uncer-
tainty on the effective value of a, this figure should be
taken with some caution until an experimental check
is available. It nevertheless suggests that the dense
H!-layer channel for sticking might be as efficient as
the ripplon channel.
Let us end up this section with two remarks. It should first be pointed out that, although the surface- sound channel might significantly increase a.,, it will
have little effect on the thermal Kapitza resistance
from Hi gas to liquid helium. The reason is that the translational motion of adsorbed hydrogen atoms is
but weakly coupled to ripplons [7]. Due to this bottle-
neck, the hydrogen layer simply heats up with respect
to the gas. Here, we have implicity assumed this ther- mal imbalance to be small. Furthermore, the sticking
coefficient for dense coverage is governed by the short-
range potential VKW and we do not expect it to be
strongly affected by the onset of superfluidity.
4. The parameters of the adsorbed layer.
In the coverage vs. temperature diagram of figure 2, we plot :
(i) The Kosterlitz-Thouless boundary :
(which in essence is equivalent to the degeneracy
condition A’ n. - 1) with b’ - 0.5. Typically, n. IT =
2 x 1013 cm - 2 at T = 100 mK.
Fig. 2. - Coverage vs. temperature « phase-diagram ». The
full line corresponds to the Grenoble cryogenic capability,
the dashed line to the Kosterlitz-Thouless transition. In
region CD 2d-superfluidity is accessible whereas in region (2)
classical behaviour will be observed
(ii) The Grenoble cryogenic capability [8], ns cryo ’"
T 2/3, obtained from the heat balance (for area A =
0.1 cml) between the cooling power and the recombi- nation energy
Q = 80 NT2 = AL. n: . (ð.Ej2) (12)
where N is the dilute-solution flow rate in mole. s - 1,
AE = 4.8 eV is the energy released in one recombina- tion event, and Ls L--- 2 x 10-24 cm"/s, at a magnetic
1506
field of 7.6 tesla [9, 10], is the surface recombination
rate. The cryogenic relationship Q = 80 NT 2 is a good approximation below 100 mK. Note that in
equation (12) we have left out of the discussion the
question of the thermal boundary resistance from surface to bulk liquid helium.
Obviously, the operating conditions should be chosen inside region (Din figure 2, in order to reach two-dimensional superfluidity. This is not enough, though. Condition (3) must be satisfied too. For a flat
cell of thickness I = 2 V / A, and for ballistic flight, an
atom lifetime in the gas is given by T, (l/Vth) (X 1 .
Then, at operating conditions : T = 200 m& nv =
3 x 1017 cm-3, ns = 4 x 1013 cm- 2, A = 0.1 CM2,
and using relation (2) we get Tg N 4 x 10-’ s which, compared with T 2d = 2 x 10-6 s (Sect. 2), violates inequality (3). Nevertheless, things will rapidly improve
as one gets to lower temperatures since the surface-to- volume population ratio increases fastly, and with it the
sticking time is. For example, at 100 mK, is has increased to 10-4 s. Note that zg and iod are both
much smaller than the recombination time (Lg n s 2)-l ;:z
10-3S. It must be stressed that the latter time cons- tant is by itself rather short, and therefore represents a
severe experimental constraint to the observation of the Kosterlitz-Thouless transition.
Finally, a few words might be added concerning the 3He- 4He substrate. Its main advantage is to enhance
heat transfer via the 3He-quasiparticles. Of course,
both the adsorption isotherm (through eJ and the ripplon-channel sticking coefficient, are different [5].
Correspondingly, iS is affected (Eq. (2)), and sticking
in the surface-sound channel as well although more indirectly (through the I b > bound-state profile, Eq. (8)).
Acknowledgments.
We are grateful to an unknown referee who, on a previous occasion, helped us clarify our view of sticking
under high-coverage conditions, and to B. Castaing for
useful comments.
Note added in proof - After this work was sub- mitted for publication, we heard of an « observabi- lity » criterion comparable to inequality (3), but rela-
tive to spin waves in 2d-H!, by KOELMANN, J., No-
TEBORN, H., DE GOEY, L., VERHAAR, B. and WAL-
RAVEN, J., Phys. Rev. B 32 (1985) 7195.
References
[1] ECKERN, U., J. Low Temp. Phys. 54 (1984) 333.
[2] See for instance SILVERA, I. F., Physica 109-110B (1982) 1499.
[3] AMBEGAOKAR, V., HALPERIN, B. I., NELSON, D. R.
and SIGGIA, E. D., Phys. Rev. B 21 (1980) 1806.
[4] AGNOLET, G., TEITEL, S. L. and REPPY, J. D., Phys.
Rev. Lett. 47 (1981) 1537.
[5] JOCHEMSEN, R., MORROW, M., BERLINSKY, A. J. and HARDY, W. N., Phys. Rev. Lett. 47 (1981) 852.
[6] ZIMMERMAN, D. S. and BEPLINSKY, A. J., Can. J.
Phys. 61 (1983) 508.
[7] ZIMMERMAN, D. S. and BERLINSKY, A. J., Can. J. Phys.
62 (1984) 590.
[8] COLLAUDIN, B., FAURE-BRAC, G., GREENBERG, A. S., GODFRIN, H., HEBRAL, B. and VALLIER, J. C., Proc. ICEC 10, Helsinki, eds. H. Collan, P. Ber- glund, M. Krusius (Betterworth) 1984, p. 255.
[9] HESS, H. F., BELL, D. A., KOCHANSKI, G. P., KLEPP-
NER, D. and GREYTAK, T. J., Phys. Rev. Lett.
52 (1984) 1520.
[10] REYNOLDS, M. W., SHINKODA, I., HARDY, W. N., BERLINSKY, A. J., BRIDGES, F. and STATT, B. W., Phys. Rev. B 31 (1985) 7503.
