Kapitza resistance at the H↓/liquid He interface

2014 In spin-polarized atomic hydrogen, the way to low T and high density is restricted by the Kapitza resistance. This is calculated, together with the corresponding characteristic time, for one channel : from H gas to liquid helium ripplons. J. Physique LETTRES 44 (1983) L-537 L-540 ler mILLET 1983, : Classification Physics Abstracts 67.40 68.10 51.10 34.50 1. In the past three years a set of important results, both experimental and theoretical have been gathered on the physics of spin-polarized atomic hydrogen [1] : recombination and relaxation rates, including the B-2 magnetic field dependence, adsorption energies on liquid 4He and 3He, sticking probabilities [2], etc. The relaxation bottleneck [3] in the population timeevolution has been observed [4], as well as the anisotropy in the surface dipolar relaxation rate [5]. As a result, higher and higher Hl densities have been stabilized on long time scales, at lower temperatures but these achievements seem to be levelling off at about 3 x 101 7 at/cm3 and 200 mK, which is still way out of the Bose-condensation border (16 mK for 1018 cm-3, 75 mK for 1019 cm3 ). A fundamental limitation to achieving lower temperatures, which we discuss in this letter, is the Kapitza thermal resistance RK between H~ gas and walls, which fastly increases at lower T, thus building up a larger and larger temperature difference AT between cold walls and a hot gas due to the continued release of recombination heat (even though it is relaxationally bottlenecked) : about 5 eV per elementary reaction. One is led, therefore, to think of compressing the gas but this brings in a related problem. Even with a small AT, it takes a finite time, iK, to thermally equilibrate the system. rK is given by the product of RK and the gas heat capacity : And for the compression to be roughly isothermal it should proceed on a time scale ’tcomp long compared with r~. The compression time, which of course must remain short on the scale of relaxation time T1, must therefore obey the double inequality Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyslet:019830044013053700 L-538 JOURNAL DE PHYSIQUE LETTRES 2. One can think of various channels for the heat transfer from gas to helium film (3He/4He liquid mixture; see Ref. 5). One is the gas-phonon channel (phonon in liquid helium). We write : here Q is the heat flux, and AT is assumed to be much smaller than T (thereby allowing a linearization procedure). /2~T Ri 1 is proportional to Vth = /2013S2013 an average impact thermal velocity 1 and to a sta1~ o al vth ~ 1tmH Y ( ) tistical integral proportional to T 3 (3-dimensional integration over phonon wave-vector). So, one expects : RK ~ iK ~ T -’~2. This result was obtained long ago by Landau [6] (his « accommodation coefficient » is defined without utb and is therefore proportional to T 3 ). It would yield rather long thermal times below 100 mK. A more efficient channel which we now investigate is the gas-to-ripplon transfer. The principle is the same as for phonons, but the integration is now two-dimensional and the dispersion law is different. Let TG and T L be the gas and liquid helium temperatures, AT = TG 7B ~ T, nH the atomic density of the hydrogen gas. Then t n is the mean number of atoms which hit the helium surface per unit time and unit area. The energy transfer rate Q between gas and liquid is given by where q and roq are the ripplon wave-vector and frequency, nq = [exp nroqlkB TL 1] -1 is the ripplon occupation number, if assumed at the liquid He temperature TL, and K9 is the intrinsic probability for an H atom to be inelastically scattered off a ripplon mode q. A is the total interfacial area. The term i 2 nH Vth K q fi~c~9 n9 thus represents the energy transfer rate from mode q toward to gas. In the limit where the H atom is treated as a classical particle, this rate can be written where u2 is the mean-squared normal velocity of the helium surface. It is then easy to, obtain the expression of Kq by considering the high-temperature case (where nq = kB T/~~) : On the other hand, in the same high-T regime, the total energy of a ripplon mode is where a and p are the helium surface tension (r = 0.15 erg cm ~ for liquid ~He) and mass density ’ ~ 2 = q3 the c illa dis rsion. Thus leading to : coq2 =0 , ap llary persi (1 ) See, for example, Landau and Lifshitz, Statistical Physics. L-539 KAPITZA RESISTANCE AT THE H! Injecting this classical expression into equations 4 and 3, we get in the low-temperature limit ~cvg xMax k B T M~ > 1 XMax= ~B~ / Note that the classical-trajectory approximation we made for H atoms, amounts to neglecting any horizontal momentum transfer. Put another way : to neglecting the hydrogen de Broglie wavelength, ~ as compared with that of the ripplon, ~,r since the only efficient ripplons are those with ~ ~ ~B. These two lengths turn out to be nearly equal at temperatures ~ 100 mK, and their ratio is slowly varying with T. Therefore, although an approximate one, our calculation yields the right order of magnitude for RK. For a perfect classical gas in volume V, the heat capacity is CH = V 3 2 k s nH, and the characteristic thermal equilibration time, defined by equation 1, can be written T/ T/ or, equivalently : r~ ~ 1.6 x 103 T 2013, in seconds, if T in K and 2013 in cm. So, compared Y K A~ ~ A ~ p with the gas-to-phonon channel, we gain one power in temperature, and TK is reasonably short at moderately low T, allowing the double inequality 2 to be easily fulfilled. For example, for a flat cell of width h = 2 VIA = 1 mm, ~ is ~ 40 ms at T = 80 mK. 3. Three remarks should be made at this point. First, from expression 6, the Kapitza resistance is lowered by a factor ~4 1-5 (the ratio of free-surface tensions) for liquid 30e, as compared Y Q 3 U.I 3 ) q ’ p with 4He. Second, it is readily estimated from the capillary dispersion relation, that the penetration depth for the dominant ripplon (?0~ ~ ~ T ) will become comparable with the thickness of the helium film ( ~ 300 A, [1]), only for temperatures below a few mK the effect being, again, more favourable (slightly) for 3He. Finally, the Fermi-liquid viscosity of 3He will ultimately damp off the ripplon modes, but again only for T lower than a few mK. Liquid 3He-4He mixtures [5] (i.e. 3He floating on top of the 6 % solution) are to be favoured, therefore the main reason, though, being that the 3He quasi-particles provide us with an efficient heat transfer channel from ripplons to bulk and walls, as we now discuss. Were it is not for the quasi-particles (q.p.), the only channel on the liquid side would be the ripplon-to-phonon one. This can be shown to become very rapidly unefficient at low T : as T’ (Ref. 7). The reason is that, due to momentum mismatch, two phonons are required to de-excite one ripplon mode (recall for example, the analogous T11 ’" T 7 argument for Raman spinlattice relaxation in solids). For the « gas » of q.p. the thermal resistance can be roughly estimated from equation 6 above, replacing mH by m3 N 6 mH, Vtb by VF (same order of magnitude, at T ~ 50 mK, since T F ^~ 300 mK1 and most important n H by n q.p. ~ n 3 T 1021 CM 3. RK is seen to ~ F be orders of magnitude lower for 3He quasi-particles. So, the total thermal resistance between H-gas and bulk helium is essentially that given by equation 6. Similar arguments hold at the 3He/dilute solution interface (smaller number of q.p.’s, but smaller interfacial tension too). Therefore, the whole thermal transfer problem is largely governed by the results of section 2. L-540 JOURNAL DE PHYSIQUE LETTRES There are other heat transfer channels which we have not discussed, however : for example via sticking to the liquid surface. Sticking coefficients x~ have been measured by NMR techniques [2]. At low coverage ns, aSt was found to be ~ 1.6 x 10-2 for H against liquid 3He. Were this value independent of both T and ns, this channel would dominate at low T, with an incoming energy flux of order (nH vth Ea) ast’ where Ea is the adsorption energy. But the coverage-dependence of ast is not an obvious problem; in a « site-adsorption » model, we expect ast to be very small ( ’" 10-2 exp 2013 for a saturated layer. Although such a model is clearly not acceptable for T the present problem, we think the question is open-especially with respect to volume Bose condensation which precisely requires completion of the adsorption hydrogen layer. On the other hand, the H/q.p. transfer by direct H/3He impact without the mediation of the ripplon bath, seems to be a rather poor channel too : due to the Van der Waals interaction, the incoming H atom interacts with ripplons before hitting the 3He quasi-particle. Using equation 6 and measured values of the relaxation rate G, the Kapitza temperature difference AT can be estimated from equation 3 and the expression Qrec = V(5 eV) Gn’ for the steady-state recombination heat, at nH = 1017 at./cm3 and T = 80 mK for example, we get : e T ~ 80 mK ! this of course is an overestimate since expression 6 for RK assumes AT « T, and it is expected that RK is reduced in the non-linear regime. Still, this figure shows that the Kapitza problem is a serious one at low temperature (more so for AT than for LK, as we have seen). A very recent experiment, performed in Amsterdam [8], gives an upper bound of 15 mK upon AT, at T = 310 mK, B = 5.7 tesla and nH ’" 3 x 1015 at./cm3.