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Kapitza resistance in H↓ gas at limiting surface density
Volume 143, number 9
PHYSICS LETTERS A
5 February 1990
K A P I T Z A R E S I S T A N C E I N H , G A S AT L I M I T I N G S U R F A C E D E N S I T Y Yu. K A G A N , N.A. G L U K H O V , B.V. S V I S T U N O V a n d G.V. S H L Y A P N I K O V I. v. KurchatovInstitute of Atomic Energy, 123182Moscow, USSR Received 24 November 1989; accepted for publication 29 November 1989 Communicated by V.M. Agranovich
The energy exchange between a gas of spin-polarized atomic hydrogen (H~) and a liquid helium surface under the conditions of limiting surface density of adsorbed hydrogen atoms is discussed. It is found that the leading mechanisms of energy exchange are due to capture of gas particles into the condensate of an adsorbed H~ phase. The process accompanied by emission ofa ripplon turns out to be effective only under a strong limitation imposed on the size of the limiting density region. The relations for the rate of energy exchange via the ripplon channel and for the Kapitza resistance at the gas-helium interface differ appreciably from those in the case of a low surface density, discussed earlier. It is shown that for T> 0.1 K the mechanism of adsorption into the surface condensate with simultaneous phonon emission in helium becomes dominant.
1. The energy transfer from H+ gas to the liquid helium surface with limiting density o f a d s o r b e d hydrogen a t o m s is one o f the key p r o b l e m s o f the Bose c o n d e n s a t i o n at high gas densities. As known, the limiting surface density leads to an almost zero adsorption energy (see ref. [ 1 ] ). At the same time, as shown in ref. [2 ], at a low surface density ns the main m e c h a n i s m o f energy transfer is due to the capture o f incident gas particles into an a d s o r p t i o n well on the surface. This process is a c c o m p a n i e d by emission o f a ripplon. On the contrary, the c o n t r i b u t i o n o f a direct inelastic scattering channel turns out to be smaller in virtue o f the p a r a m e t e r (T/EO) 3/2 (~-o being the b i n d i n g energy in the a d s o r p t i o n well). At first sight, it m a y seem that at the limiting surface density it is the inelastic scattering that should be the only m e c h a n i s m o f energy transfer. However, the physical picture proves to be m o r e complicated. At t e m p e r a t u r e s leading to the limiting surface density, the c o n d i t i o n T<
(1)
should take place (Tc, being the t e m p e r a t u r e o f the K o s t e r l i t z - T h o u l e s s transition in the a d s o r b e d gas). In this case t w o - d i m e n s i o n a l quasicondensate arises in the a d s o r b e d gas. U n d e r the c o n d i t i o n ( 1 ) , the correlation radius characteristic o f fluctuations o f the
quasicondensate wave function phase turns out to be so large that the phase fluctuations are, in fact, uni m p o r t a n t , and the b e h a v i o r o f the quasicondensate becomes the same as that o f a real condensate. Then the a d s o r p t i o n transition o f a gas particle to the condensate, a c c o m p a n i e d by the ripplon emission, turns out to be the leading channel o f energy transfer. The corresponding transition m a t r i x elem e n t will be p r o p o r t i o n a l to the condensate wave function g/o- As revealed in our recent p a p e r [3], at the limiting surface density the localization length o f the condensate wave function in the direction perp e n d i c u l a r to the surface r e m a i n s close to the value l = ( 2 m % ) -1/2 (rn being the a t o m i c mass, h = l ) characteristic o f the wave function o f an isolated adsorbed atom. In this sense, the physical picture o f adsorption keeps the features inherent in the case o f ns<
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adsorbed particle to the bulk (at ns<< nso this process has an energy threshold ~ Eo). This creates difficulties in the energy transfer from the ripplon subsystem to the volume of helium. In order to overcome these difficulties one should essentially diminish the size of the limiting surface density region. The free path length of a ripplon with respect to the desorption channel should be larger than the size of this region. The two-scale system for achieving the Bose condensation, where a high gas density should be created in a small space region near the surface (see ref. [ 4 ] ), is precisely the system in which these conditions may be realized. It is worth mentioning that in this case the other problem arising in the kinetics of adsorption into the surface condensate is also resolved. We mean that the gas particles only from a very small phase volume may adsorb, which is predetermined by the conservation laws. The restriction on the size of the limiting surface density region widens this phase volume. In the temperature range T > 0.1 K does the mechanism of direct energy transfer from the incident gas particles to volume phonons of helium (adsorption into the surface condensate with simultaneous emission of a phonon) become dominant. In this case, the restrictions on the size of the limiting surface density region are not necessary. 2. At the surface density of adsorbed hydrogen atoms close to its limiting value, the adsorbed H l phase plays a decisive part in the kinetics of the energy exchange between the bulk H, gas and helium. We consider the adsorbed phase within the approximation of a weak coupling of an H atom with the helium surface, the effective interaction between hydrogen atoms being 0-functional. This approximation corresponds to a limiting case, when the characteristic localization length of an atom in the direction perpendicular to the surface, l = (2meo) -w2, considerably exceeds the radius of interatomic interaction, Ro, as well as the characteristic radius R . of interaction of a hydrogen atom with the helium surface (both the radii are close to 3.5 A (see ref. [5] )). It follows from the condition />>Ro that the interaction between hydrogen atoms has a three-dimensional character. The Hamiltonian of the adsorbed phase may be written as 478
5 February 1990
I:I= f dr(/+(r)[-/X/2m+U°(z)-lt]V(r) + ½0 j dr ~+ ( r ) ~ + (r)~z(r)~/(r).
(2)
Here ~ ( r ) is the field operator of particles,/~ is the chemical potential, 0 = 4na/m is the effective vertex of the elastic interaction of hydrogen atoms, a = 0.72 ,~ [ 6 ] is the scattering length, Uo(z) is the static potential of interaction of a hydrogen atom with the helium surface (the z axis being perpendicular to the surface). Taking into consideration the existence of the quasicondensate in the adsorbed phase, we present (/(r) as a sum of condensate and above-condensate terms: ~ ( r ) = ~u0(r) + (t' ( r ) .
(3)
At ns ~ nso the condition ( 1 ) brings about the correlation radius of the condensate wave function phase,
Re .~ rc(ns U/lT) exp(2nns/mT) (see ref. [7 ], rc being a conventional correlation length which is of the order of l at ns ~ n~o), which turns out to be much larger than the dimensions of the limiting density region, being of practical interest. So, actually, there is a true condensate with wave function ~Uouniform along the surface. On the other hand, under the condition ( 1 ), most of the adsorbed particles are in the condensate (as shown by direct estimates, the fraction of above-condensate particles is ~a/l<< 1 ). Therefore, we may neglect the term ~ ~, 4 in eq. (2) and write the Hamiltonian as /t=Ho
+ f dr{(/+(r)[-&/2m+Uo(z)-l~]~ 9`(r) + ½0~72(z) [4d/'+ (r)~p' (r)
+ ~t'+ ( r ) ~ '+ (r)
+tp' (r)@' (r) ]} , Ho=
f dr {~o(Z) [ - A /2m+ Uo(z)-I~]q/o(Z)
+½0~,~(z)}.
(4)
The Hamiltonian (4) is bilinear and can be reduced to a diagonal form by using the Bogolyubov transformation generalized to an inhomogeneous case (as was done in the theory of superconductivity (see ref. [ 8 ] ) ) :
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¢/' (r)= V -'/2 ~ [Uq~(Z) exp(iq.p)/Sq~ qp
--Vqv(Z ) exp( -iq.p)/~ + ] .
(5)
The functions Uq~(Z) and Vq~(Z) describe the z-dependent state of a quasiparticle with the m o m e n t u m q along the surface (p being the coordinate along the surface) and a set of other quantum numbers u for the motion perpendicular to the surface; Vis the system volume. These functions are orthonormalized by the condition
V - I J- dr (~lqpU~,p, --V~VVq, p. ) exp[i(q--q' )'p] =c~q¥6~,,
(6)
which follows directly from the Bose character of the quasiparticle operators 6¢~ and those of real particles. Substituting eq. (5) into eq. (4), one can easily make sure that the Hamiltonian acquires a diagonal form qP
if the functions Uq~ and Vq~ satisfy the system of equations, 1
2m u ~ ( z ) + eo(z)Uqv(Z)
(7a)
1
2m Vq~(Z) + Uo(z)vq~(z) + 09"2(2) [2Vqv(2) --Uq~,(Z)] = ( -Eqv--qZ/2m+It)Vqv(Z )
proximation of the 8-functional interaction between hydrogen atoms is applicable only to the case of z> Ro, R.. However, at distances Z ~ Ro, R., where I Uol >> ~o, the small value of the adsorption energy Eo allows one to neglect the terms proportional to 09 '2 < ~o in eqs. (7), (8). Thus, these equations prove to be also valid at small distances from the surface. The expressions for 9"o(Z), uq~(z) and Vq~(Z) at z>>Ro, R . can be obtained by omitting the terms with the static potential Uo(z) in eqs. (7), (8) and by introducing, at kthe same time, the boundary conditions
9"0(0)
Uqv(O) v'qv(O) - 9"0(0) - Uq~(0) - Vq~(O)
(7b)
(Uq~ and vq~ are taken to be real). The Schr~Sdinger equation for the wave function 9"ofollows directly from the Hamiltonian Ho (4) with neglect of the above-condensate terms and has the form of the well-known Ginzburg-Pitaevskii-Gross equation
1 9"(~(z) + Uo(z)9"o(Z) + Ug'3(z) =/./9"o(Z) •
2m
(8) It should be noted that, generally speaking, the ap-
1 l"
(9)
Actually, these conditions again correspond to the inequality/>> Ro, R,, which means that only the behavior of the wave function outside the potential well is important for the problem and that there is only one discrete level with the binding energy eo = (2ml2) -t in the well. At ns=nso the chemical potential of the adsorbed phase is/~ = 0 and the solution of eq. (8) for the condensate wave function with the boundary condition (9) and that of 9"o(OO)=0 is [3] /
+ U9"o2(z) [2Uq~(Z) -Vqv(Z)]
= (Eq~-q2/2m+lz)uq~(z) ,
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,,1/2
)
l
(,0,
It should be emphasized that the collective interaction inducing the 9"0 behavior change from the exponential law 9 " o ( Z ) ~ e x p ( - z / l ) (taking place for ns << nso) to the power one (10) at n~ = nso does not change the scale of the localization length. At n~=n~o the motion of elementary excitations along the z axis belongs only to the continuous spectrum corresponding to the nonlocalized states. In this case one may take v to be the momentum along the z axis, k (Eqk= (k2+q2)/2m). If there is a Bose condensate in the bulk and there exists a unified wave function 9"0 for the whole system of H~, it is impossible to divide the system into the volume phase and the adsorbed one. However, the function 9"0 experiences a sharp rise near the surface and at z << ( n o a ) - ~/2 (no being the condensate density in the bulk) it is still determined by eq. ( 10 ) [ 3 ]. Owing to this fact, it is also reasonable to speak 479
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about "localization" of the condensate near the surface. The influence of the bulk condensate on the wave functions of above-condensate particles is negligible under the condition T>> not~. 3. To analyze the energy transfer from the bulk gas to helium, we start with the ripplon mechanism. Under the condition ( 1 ), the process of hydrogen atom capture into the surface condensate with simultaneous emission of a ripplon is predominant. In this case, the resulting energy flux from the gas to the ripplon subsystem is given by the obvious relation j(,)= f
dp
to(p) Wr(P){nqk( T) [ 1 + Np( To) ]
-Np( To) [l + nqk( T) ] , q=p,
(11)
(k2+q2)/2m=to(p) .
~(p)[~t'+(r)+O'(r) ] ,
(13) where
( rq(p)= ~
p 2poo~(p)S
),/2 exp(ip'p)(cP+6-+P)
is the operator of liquid helium surface displace480
ments along the z axis, Opis the annihilation operator of a ripplon with the momentum p, S is the surface area. The presence of a derivative of the static potential Uo(z) in the integrand of eq. (13) implies that the inequality pR. << 1 holds for characteristic momenta of ripplons. Having used eq. (5) for the above-condensate part of the ~u-operator in eq. ( 13 ), in the first order of the perturbation theory, we obtain Wr(p) = 2 n ~, I (il/~i(,'? I f ) I a qk
× c~( ( kZ + q2) / 2 m - t o ( p ) ) -
P | Ok V~(k, p) poo;(p )
0
)<6((k2+q2)/2m-to(p)) ,
(12)
Here W,(p) is the probability of absorption (per unit time) of a ripplon with the m o m e n t u m p, accompanied by simultaneous desorption of a condensate particle to the bulk; t o ( p ) = (a/po)t/2p3/2 is the ripplon frequency, a is the surface tension coefficient, Po is the helium density, To is the surface temperature; nqk(T) and Np(To) are the occupation numbers for gas particles and ripplons, respectively. The relation between the components of the gas particle momentum, parallel (q) and perpendicular (k) to the surface, and the ripplon momentum follows from the conservation laws (12). The first term in the r.h.s. of eq. ( 11 ) describes the adsorption transitions of the gas particles to the condensate with the emission o f a ripplon and the second term is due to the inverse process of desorption accompanied by absorption of a ripplon. The probability W, may be obtained from the Hamiltonian of interaction of hydrogen atoms with ripplons,
I:It~ )t = f drgto(Z) ~
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V,(k,p)=
dz~'o(Z)[Upk(Z)--vpk(z)]
(14) dUo(z)
dz
0
(15) The matrix element ( 15 ) turns out to be insensitive to the form of the potential Uo(z) and depends only on the localization length in t h i s potential, I= (2m~o)- ,/z Indeed, let us subtract eq. (7b) from eq. (7a) and multiply the resulting equation for upk- Vpkby gt~(z). Then, we take a derivative of eq. ( 8 ) with respect to z and multiply it by Upk--Vpk.Finally, subtracting the equation obtained from that for Upk--Vpk, one can easily make sure that the integrand in eq. ( 15 ) does not depend explicitly on Uo and eq. (15) may be transformed to the form
V,(k,p)=-
oo
~m
0
dZq/°(Z)[Urk(Z)--VPk(Z) ]
~mm dz ~'o(Z)Vpk(Z) .
(16)
0
At z << l the dependence of the functions upk and vpk on z is just the same as that of q/o. Hence, the main contribution to the integral (16) comes from distances z > l far from the potential well Uo. Thus, to calculate the integral (16), we can use eq. (10) for ~'o and eqs. (7) for upk and vp~ with Uo = 0 and the boundary conditions (9). At sufficiently small to(p), an approximate solution may be found analytically. For to(p)--,0, the
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conservation laws (12) may be satisfied only under the condition k>>p, which leads to the analytical expressions for Upk and Vpk (not presented here in view of their awkwardness). Using these expressions in calculating (16) and (14), we obtain l~,r(p) ~
i.e. ripplons can still be regarded as weakly damping excitations. At To << T the inverse energy flux due to desorption may be neglected. In this case, taking the ideal gas approximation for the bulk phase in eq. ( 11 ) and using eqs. (17), (18) for Wr(p), we obtain the following expression,
kS(p)p nsol 4co(p) mpo '
k(p) = [ 2mco(p) _ p 2 ] 1/2
( 17 )
j(r) ~
The main part in the energy exchange is played by ripplons with the energy c o ( p ) ~ T. As shown by a straightforward analysis, the theoretical limit (17) may be used only at co(p) < l0 mK. A numerical calculation which is necessary in a large temperature range gives
Wr(p ) = ff'r(p )f(pl) . The function f ( x )
35n _nsol nT3F ( T/eo) , _ 4 a
__
T>> T¢,
~ 35n nsol nT3F ( T / % ) 4 tr
x O . 4 ( T / T c ) 3/2 ,
T~T:,
(19)
where n is the volume density of the gas, To(n) is the Bose condensation temperature. The function F ( x ) is presented in fig. 2 (F--, 1 as x - , 0 ) . The relation between the gas temperature and other parameters of the problem or, otherwise, the Kapitza resistance, may be actually obtained by equating Jtr) tO the value of heat release in the gas. If the temperature Kapitza resistance at the gas-helium interface is small, A T = T - To << T, the desorbing energy flux is close to the adsorbing one and the expression for j(r) may be
(18)
is presented in fig. 1 (f--, 1 as
x-*0). At pl< 1 (this condition is satisfied by ripplons with energies ~<0.5 K) rr ' = wr(p )
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<
1.0 o 0.8
O. 7
He 4
He 3
0.3 0.2
0.1 I
I
i
I
i
I
I
|
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X
Fig. 1. Functionf(x) on 4He and 3He surfaces. 481
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5 February 1990 t"
F
/4rph = J dr [ 1 + ~ ( p ) d / d z ] { [1)r(r) + Uph(r) ]
x [po +~' (r) ] [Jr(r) + Oph(r) ]}. 0.9
Here/~' (r) is the operator of helium density fluctuations, ~r(r) and ~ph(r) are the operators of the particle velocity in the volume and surface waves, respectively. This Hamiltonian can be directly obtained from the general approach (see ref. [ 9 ] ), taking into account the oscillations of the liquid helium surface. As shown by straightforward calculations, the leading channel in the above-mentioned temperature range is the recombination of two ripplons into a phonon. The free path length of a thermal ripplon ( To ~ T) with respect to this mechanism is
0.8
0.7
0.6
0.5
Lrph ~
2xac~ T4 ,
0.4
0.3
0.2
O.l
0.0
I
I
I
I0 -5
10 -~
I0-
Fig. 2. Function surfaces).
3
I
i
b
10 -2
10 - I
X
F(x) (practically the same for both 4He and 3He
obtained by taking a derivative of eq. (19) with respect to T and multiplying it by AT. However, the result obtained requires an additional analysis. The first question is connected with the fate of the ripplons emitted in adsorption of the gas particles. Eqs. (17), (18) yield the value of the ripplon free path length with respect to the desorption channel of decay (i.e. decay with the energy transfer back to the gas ), Lr ~ Vrr ( v = dcn/dp), which is of the order of 1 0 - 2 - 1 0 -3 cm for thermal ripplons at T ~ 10-100 mK. In order to estimate the characteristic time of the energy transfer from the ripplon subsystem to the phonon one, we introduce the ripplon-phonon interaction Hamiltonian in the form 482
where Co is the sound velocity in helium. It is easy to estimate that at T ~ 1 0 - 1 0 0 m K the inequality Lrph>>Lr holds, which becomes still stronger for To << T. It means that for an infinite area of the limiting density region the energy transferred from the gas to the ripplon subsystem (in the adsorption process) should return to the gas with probability close to unity. The energy flux jet) (8) can be transferred completely from the ripplon subsystem to the helium volume only if the characteristic linear dimension of the limiting density region, L, satisfies the condition L<
(20)
This condition allows a ripplon to get out of the limiting density region and, correspondingly, to decay in an ordinary way, transferring its energy to volume phonons. Another problem is connected with the fact that in virtue of the conservation laws ( 12 ), the adsorption into the condensate may occur only from a very small phase volume of particles in the gas. If the conservation laws were satisfied exactly, we should have written the expression ~ r - 1 +Nq(T°) ~ ( ( k 2 + q Z ) / 2 m - ~ ( q ) l Tr(q)
for the sticking coefficient (relative probability of adsorption per collision) of the incident particle with momentum (q, k). The finite size of the limiting
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density region results in a ripplon momentum uncertainty of the order of l / L and in a ripplon energy uncertainty ~ v/L. This leads to a value of the sticking coefficient ~r of the order of L/vzr~ L/Lr. Thus, the same condition (20) leads to the inequality ~r<< 1 which allows one to obtain Wr(p) within the perturbation theory. One more remark should be made. Naturally, under stationary conditions the flux of adsorbing particles is equal to that of desorbing ones. At n~<< nso this imposes an additional condition leading to a certain nonequilibrium value ofn~. At n~= n~o the additionally adsorbed particles cannot be held on the surface and return to the bulk. Hence, in this case the equality of fluxes does not give any additional condition. At n~~ n~o the energy transfer from the bulk to the ripplon subsystem can, in principle, occur without adsorbing gas particles into the condensate, namely, in the channel of purely inelastic scattering. Direct calculations show that in this case the value of the energy flux remains almost the same as that at n~<< n~o to the accuracy of a numerical coefficient. The expression for this flux has an additional small factor ~O.I(T/%) ~/2, as compared to the r.h.s, of eq. ( 19 ). This allows us to consider only the channel with the capture of a gas particle into the condensate in analyzing the ripplon mechanisms. Note that in the case of n~ << n~o the inelastic scattering rate has a small factor (T/%) 3/2. Such a difference is due to a change in the temperature dependence of the energy transfer rate for adsorption in the ns.~ nso case. It should be emphasized that the inequality (20) is also required in the case of the inelastic scattering mechanism. It is interesting to compare eq. (19) with an analogous result of ref. [ 2 ] for the case of n~<< nso. The main difference is that at the limiting density J(~) ( T--,0 ) ~ T 3, while at n~ << n~o we have J(r) ~ T 2. It follows from the fact that in the first case the matrix element for the adsorption process (eqs. (15), (16) ) contains the factor kZ/2m instead of %. This results in the factor of k 4 in the expression for the probability. However, in the adsorption into the condensate, there is no additional integration over the particle momentum along the surface. As a result, there arises an additional factor of T. Thus, the result presented in ref. [2] for the ns~n~o case, which
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is, actually, the extrapolation of the expression for the ns << nso case, is incorrect. The quantitatively correct result (19) differs from that of ref. [ 2 ] by a factor of ~30(T/%)F(T/%) which is close to 0.1 in the temperature range T~ 10-100 mK. The results obtained make it necessary to analyze more accurately the Bose condensation problem in a semi-open system with one magnetic wall (see ref. [ l0 ] ). As can be shown easily, all the results of ref. [10] remain valid, if the density is increased, approximately, by a factor of 3 with decreasing the gas layer thickness d ~ n-~ (this leads to an increase in the Bose condensation temperature Tc ~ n2/3). 4. Let us now consider the channel of the energy transfer from the gas to helium, which is due to the direct interaction of hydrogen atoms with volume phonons. The process of a particle capture into the surface condensate (now with the emission of a phonon) turns out to be dominant again. The interaction Hamiltonian responsible for this process may be written as (see ref. [ 1 ] ).
f dr~o(Z) ~'(r) Uo(z) [ ~ ' ÷ ( r ) + ~ ) ' ( r ) ] , d Po (21)
~' (r)= ~ i(ppo/2Co~)~/2(fv-f + ) exp(ip-r), q
where~ is the annihilation operator of a phonon with momentum p, ~ is the helium volume. The term ~' (r)Uo(z) arises under the condition pR, << 1 as a result of integration of the hydrogen atom interaction with helium surface fluctuations over the helium volume. The conservation laws
q=Pu , (kZ+q2)/2m=cop
(22)
(P~l being the component of the vector p along the surface) lead to the inequality k>> p for the thermal phonons which are most effective in the energy exchange at ns << nso. For temperatures up to ~ 0.5 K this inequality is a fortiori satisfied. Taking into account this inequality and eq. (4) for ~', we obtain the expression for the energy flux from the gas to helium (T>> To):
483
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by a highly inhomogeneous field near the surface [4 ], when n ~ 10 20 cm -3 and T ~ 0 . 3 - 0 . 7 K are the most favourable parameters.
j(ph)= ~ clpdk p k 2 (2•) 3 CoPo 2m np, k V~h(k, Pll )
× (~( kZ / 2 m - c o p ) ,
(23)
Vph (k, PII) = .( d z q/o(Z) Uo(z) [Up,,k(z) - Vp,k(Z) ] • 0
(24) The m a i n c o n t r i b u t i o n to the integral ( 2 4 ) comes from the distances z ~ R . <
T>> Tc,
3× 102 ( nso/ cgm2pol)nT4 ×0.4( T/Tc) 3/2 ,
T<~Tc.
(25)
C o m p a r i n g eqs. ( 2 5 ) a n d ( 1 9 ) , it is easy to see that the p h o n o n a d s o r p t i o n m e c h a n i s m o f the energy exchange becomes the leading one at T > 0. l K even u n d e r the c o n d i t i o n ( 2 0 ) . In particular, it is the p h o n o n m e c h a n i s m that d e t e r m i n e s the heat rem o v a l from H~ to h e l i u m in the idea o f Bose condensation in a small c o m p r e s s e d gas v o l u m e created
484
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5. All the results presented were o b t a i n e d under the c o n d i t i o n Ro/l, R./I<< 1. On real surfaces o f liquid 4He and aHe these ratios are not so small. However, there is no d o u b t that the extrapolation o f the results to the region o f actual p a r a m e t e r s yields a qualitatively correct description o f the p h e n o m e n o n .
References [ 1] I.F. Silvera and V.V. Goldman, Phys. Rev. Len. 45 (1980) 915. [2] Yu. Kagan, G.V. Shlyapnikov and N.A. Glukhov, Pis'ma Zh. Eksp. Teor. Fiz. 40 (1984) 287 [JETP Lett. 40 (1984) 1072]. [3]Yu. Kagan, N.A. Glukhov, B.V. Svistuno and G.V. Shlyapnikov, Phys. Lett. A 135 (1989) 219. [ 4 ] Yu. Kagan and G.V. Shlyapnikov, Phys. Len. A 130 ( 1988 ) 483. [ 5 ] I.B. Mantz and D.O. Edwards, Phys. Rev. B 24 (1979) 4518. [ 6 ] D.G. Friend and R.D. Etters, J. Low Temp. Phys. 39 (1980) 409. [ 7 ] Yu. Kagan, B.V. Svistunov and G.V. Shlyapnikov, Zh. Eksp. Teor. Fiz. 93 (1987) 552 [Sov. Phys. JETP 66 (1987) 480]. [8] P.G. De Gennes, Superconductivity of metals and alloys (Benjamin, New York, 1966). [ 9 ] E.M. Lifshitz and L.P. Pitaevskii, Statistical physics, Part 2 (Nauka, Moscow, 1978 ). [ 10] Yu. Kagan, G.V. Shlyapnikov and N.A. Glukhov, Pis'ma Zh. Eksp. Teor. Fiz. 41 (1985) 197 [JETP Lett. 41 (1985) 238]. [ 11 ] Yu. Kagan and G.V. Shlyapnikov, Phys. Lett. A 95 ( 1983 ) 309.
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K A P I T Z A R E S I S T A N C E I N H , G A S AT L I M I T I N G S U R F A C E D E N S I T Y Yu. K A G A N , N.A. G L U K H O V , B.V. S V I S T U N O V a n d G.V. S H L Y A P N I K O V I. v. KurchatovInstitute of Atomic Energy, 123182Moscow, USSR Received 24 November 1989; accepted for publication 29 November 1989 Communicated by V.M. Agranovich
The energy exchange between a gas of spin-polarized atomic hydrogen (H~) and a liquid helium surface under the conditions of limiting surface density of adsorbed hydrogen atoms is discussed. It is found that the leading mechanisms of energy exchange are due to capture of gas particles into the condensate of an adsorbed H~ phase. The process accompanied by emission ofa ripplon turns out to be effective only under a strong limitation imposed on the size of the limiting density region. The relations for the rate of energy exchange via the ripplon channel and for the Kapitza resistance at the gas-helium interface differ appreciably from those in the case of a low surface density, discussed earlier. It is shown that for T> 0.1 K the mechanism of adsorption into the surface condensate with simultaneous phonon emission in helium becomes dominant.
1. The energy transfer from H+ gas to the liquid helium surface with limiting density o f a d s o r b e d hydrogen a t o m s is one o f the key p r o b l e m s o f the Bose c o n d e n s a t i o n at high gas densities. As known, the limiting surface density leads to an almost zero adsorption energy (see ref. [ 1 ] ). At the same time, as shown in ref. [2 ], at a low surface density ns the main m e c h a n i s m o f energy transfer is due to the capture o f incident gas particles into an a d s o r p t i o n well on the surface. This process is a c c o m p a n i e d by emission o f a ripplon. On the contrary, the c o n t r i b u t i o n o f a direct inelastic scattering channel turns out to be smaller in virtue o f the p a r a m e t e r (T/EO) 3/2 (~-o being the b i n d i n g energy in the a d s o r p t i o n well). At first sight, it m a y seem that at the limiting surface density it is the inelastic scattering that should be the only m e c h a n i s m o f energy transfer. However, the physical picture proves to be m o r e complicated. At t e m p e r a t u r e s leading to the limiting surface density, the c o n d i t i o n T<
(1)
should take place (Tc, being the t e m p e r a t u r e o f the K o s t e r l i t z - T h o u l e s s transition in the a d s o r b e d gas). In this case t w o - d i m e n s i o n a l quasicondensate arises in the a d s o r b e d gas. U n d e r the c o n d i t i o n ( 1 ) , the correlation radius characteristic o f fluctuations o f the
quasicondensate wave function phase turns out to be so large that the phase fluctuations are, in fact, uni m p o r t a n t , and the b e h a v i o r o f the quasicondensate becomes the same as that o f a real condensate. Then the a d s o r p t i o n transition o f a gas particle to the condensate, a c c o m p a n i e d by the ripplon emission, turns out to be the leading channel o f energy transfer. The corresponding transition m a t r i x elem e n t will be p r o p o r t i o n a l to the condensate wave function g/o- As revealed in our recent p a p e r [3], at the limiting surface density the localization length o f the condensate wave function in the direction perp e n d i c u l a r to the surface r e m a i n s close to the value l = ( 2 m % ) -1/2 (rn being the a t o m i c mass, h = l ) characteristic o f the wave function o f an isolated adsorbed atom. In this sense, the physical picture o f adsorption keeps the features inherent in the case o f ns<
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adsorbed particle to the bulk (at ns<< nso this process has an energy threshold ~ Eo). This creates difficulties in the energy transfer from the ripplon subsystem to the volume of helium. In order to overcome these difficulties one should essentially diminish the size of the limiting surface density region. The free path length of a ripplon with respect to the desorption channel should be larger than the size of this region. The two-scale system for achieving the Bose condensation, where a high gas density should be created in a small space region near the surface (see ref. [ 4 ] ), is precisely the system in which these conditions may be realized. It is worth mentioning that in this case the other problem arising in the kinetics of adsorption into the surface condensate is also resolved. We mean that the gas particles only from a very small phase volume may adsorb, which is predetermined by the conservation laws. The restriction on the size of the limiting surface density region widens this phase volume. In the temperature range T > 0.1 K does the mechanism of direct energy transfer from the incident gas particles to volume phonons of helium (adsorption into the surface condensate with simultaneous emission of a phonon) become dominant. In this case, the restrictions on the size of the limiting surface density region are not necessary. 2. At the surface density of adsorbed hydrogen atoms close to its limiting value, the adsorbed H l phase plays a decisive part in the kinetics of the energy exchange between the bulk H, gas and helium. We consider the adsorbed phase within the approximation of a weak coupling of an H atom with the helium surface, the effective interaction between hydrogen atoms being 0-functional. This approximation corresponds to a limiting case, when the characteristic localization length of an atom in the direction perpendicular to the surface, l = (2meo) -w2, considerably exceeds the radius of interatomic interaction, Ro, as well as the characteristic radius R . of interaction of a hydrogen atom with the helium surface (both the radii are close to 3.5 A (see ref. [5] )). It follows from the condition />>Ro that the interaction between hydrogen atoms has a three-dimensional character. The Hamiltonian of the adsorbed phase may be written as 478
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I:I= f dr(/+(r)[-/X/2m+U°(z)-lt]V(r) + ½0 j dr ~+ ( r ) ~ + (r)~z(r)~/(r).
(2)
Here ~ ( r ) is the field operator of particles,/~ is the chemical potential, 0 = 4na/m is the effective vertex of the elastic interaction of hydrogen atoms, a = 0.72 ,~ [ 6 ] is the scattering length, Uo(z) is the static potential of interaction of a hydrogen atom with the helium surface (the z axis being perpendicular to the surface). Taking into consideration the existence of the quasicondensate in the adsorbed phase, we present (/(r) as a sum of condensate and above-condensate terms: ~ ( r ) = ~u0(r) + (t' ( r ) .
(3)
At ns ~ nso the condition ( 1 ) brings about the correlation radius of the condensate wave function phase,
Re .~ rc(ns U/lT) exp(2nns/mT) (see ref. [7 ], rc being a conventional correlation length which is of the order of l at ns ~ n~o), which turns out to be much larger than the dimensions of the limiting density region, being of practical interest. So, actually, there is a true condensate with wave function ~Uouniform along the surface. On the other hand, under the condition ( 1 ), most of the adsorbed particles are in the condensate (as shown by direct estimates, the fraction of above-condensate particles is ~a/l<< 1 ). Therefore, we may neglect the term ~ ~, 4 in eq. (2) and write the Hamiltonian as /t=Ho
+ f dr{(/+(r)[-&/2m+Uo(z)-l~]~ 9`(r) + ½0~72(z) [4d/'+ (r)~p' (r)
+ ~t'+ ( r ) ~ '+ (r)
+tp' (r)@' (r) ]} , Ho=
f dr {~o(Z) [ - A /2m+ Uo(z)-I~]q/o(Z)
+½0~,~(z)}.
(4)
The Hamiltonian (4) is bilinear and can be reduced to a diagonal form by using the Bogolyubov transformation generalized to an inhomogeneous case (as was done in the theory of superconductivity (see ref. [ 8 ] ) ) :
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¢/' (r)= V -'/2 ~ [Uq~(Z) exp(iq.p)/Sq~ qp
--Vqv(Z ) exp( -iq.p)/~ + ] .
(5)
The functions Uq~(Z) and Vq~(Z) describe the z-dependent state of a quasiparticle with the m o m e n t u m q along the surface (p being the coordinate along the surface) and a set of other quantum numbers u for the motion perpendicular to the surface; Vis the system volume. These functions are orthonormalized by the condition
V - I J- dr (~lqpU~,p, --V~VVq, p. ) exp[i(q--q' )'p] =c~q¥6~,,
(6)
which follows directly from the Bose character of the quasiparticle operators 6¢~ and those of real particles. Substituting eq. (5) into eq. (4), one can easily make sure that the Hamiltonian acquires a diagonal form qP
if the functions Uq~ and Vq~ satisfy the system of equations, 1
2m u ~ ( z ) + eo(z)Uqv(Z)
(7a)
1
2m Vq~(Z) + Uo(z)vq~(z) + 09"2(2) [2Vqv(2) --Uq~,(Z)] = ( -Eqv--qZ/2m+It)Vqv(Z )
proximation of the 8-functional interaction between hydrogen atoms is applicable only to the case of z> Ro, R.. However, at distances Z ~ Ro, R., where I Uol >> ~o, the small value of the adsorption energy Eo allows one to neglect the terms proportional to 09 '2 < ~o in eqs. (7), (8). Thus, these equations prove to be also valid at small distances from the surface. The expressions for 9"o(Z), uq~(z) and Vq~(Z) at z>>Ro, R . can be obtained by omitting the terms with the static potential Uo(z) in eqs. (7), (8) and by introducing, at kthe same time, the boundary conditions
9"0(0)
Uqv(O) v'qv(O) - 9"0(0) - Uq~(0) - Vq~(O)
(7b)
(Uq~ and vq~ are taken to be real). The Schr~Sdinger equation for the wave function 9"ofollows directly from the Hamiltonian Ho (4) with neglect of the above-condensate terms and has the form of the well-known Ginzburg-Pitaevskii-Gross equation
1 9"(~(z) + Uo(z)9"o(Z) + Ug'3(z) =/./9"o(Z) •
2m
(8) It should be noted that, generally speaking, the ap-
1 l"
(9)
Actually, these conditions again correspond to the inequality/>> Ro, R,, which means that only the behavior of the wave function outside the potential well is important for the problem and that there is only one discrete level with the binding energy eo = (2ml2) -t in the well. At ns=nso the chemical potential of the adsorbed phase is/~ = 0 and the solution of eq. (8) for the condensate wave function with the boundary condition (9) and that of 9"o(OO)=0 is [3] /
+ U9"o2(z) [2Uq~(Z) -Vqv(Z)]
= (Eq~-q2/2m+lz)uq~(z) ,
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,,1/2
)
l
(,0,
It should be emphasized that the collective interaction inducing the 9"0 behavior change from the exponential law 9 " o ( Z ) ~ e x p ( - z / l ) (taking place for ns << nso) to the power one (10) at n~ = nso does not change the scale of the localization length. At n~=n~o the motion of elementary excitations along the z axis belongs only to the continuous spectrum corresponding to the nonlocalized states. In this case one may take v to be the momentum along the z axis, k (Eqk= (k2+q2)/2m). If there is a Bose condensate in the bulk and there exists a unified wave function 9"0 for the whole system of H~, it is impossible to divide the system into the volume phase and the adsorbed one. However, the function 9"0 experiences a sharp rise near the surface and at z << ( n o a ) - ~/2 (no being the condensate density in the bulk) it is still determined by eq. ( 10 ) [ 3 ]. Owing to this fact, it is also reasonable to speak 479
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about "localization" of the condensate near the surface. The influence of the bulk condensate on the wave functions of above-condensate particles is negligible under the condition T>> not~. 3. To analyze the energy transfer from the bulk gas to helium, we start with the ripplon mechanism. Under the condition ( 1 ), the process of hydrogen atom capture into the surface condensate with simultaneous emission of a ripplon is predominant. In this case, the resulting energy flux from the gas to the ripplon subsystem is given by the obvious relation j(,)= f
dp
to(p) Wr(P){nqk( T) [ 1 + Np( To) ]
-Np( To) [l + nqk( T) ] , q=p,
(11)
(k2+q2)/2m=to(p) .
~(p)[~t'+(r)+O'(r) ] ,
(13) where
( rq(p)= ~
p 2poo~(p)S
),/2 exp(ip'p)(cP+6-+P)
is the operator of liquid helium surface displace480
ments along the z axis, Opis the annihilation operator of a ripplon with the momentum p, S is the surface area. The presence of a derivative of the static potential Uo(z) in the integrand of eq. (13) implies that the inequality pR. << 1 holds for characteristic momenta of ripplons. Having used eq. (5) for the above-condensate part of the ~u-operator in eq. ( 13 ), in the first order of the perturbation theory, we obtain Wr(p) = 2 n ~, I (il/~i(,'? I f ) I a qk
× c~( ( kZ + q2) / 2 m - t o ( p ) ) -
P | Ok V~(k, p) poo;(p )
0
)<6((k2+q2)/2m-to(p)) ,
(12)
Here W,(p) is the probability of absorption (per unit time) of a ripplon with the m o m e n t u m p, accompanied by simultaneous desorption of a condensate particle to the bulk; t o ( p ) = (a/po)t/2p3/2 is the ripplon frequency, a is the surface tension coefficient, Po is the helium density, To is the surface temperature; nqk(T) and Np(To) are the occupation numbers for gas particles and ripplons, respectively. The relation between the components of the gas particle momentum, parallel (q) and perpendicular (k) to the surface, and the ripplon momentum follows from the conservation laws (12). The first term in the r.h.s. of eq. ( 11 ) describes the adsorption transitions of the gas particles to the condensate with the emission o f a ripplon and the second term is due to the inverse process of desorption accompanied by absorption of a ripplon. The probability W, may be obtained from the Hamiltonian of interaction of hydrogen atoms with ripplons,
I:It~ )t = f drgto(Z) ~
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V,(k,p)=
dz~'o(Z)[Upk(Z)--vpk(z)]
(14) dUo(z)
dz
0
(15) The matrix element ( 15 ) turns out to be insensitive to the form of the potential Uo(z) and depends only on the localization length in t h i s potential, I= (2m~o)- ,/z Indeed, let us subtract eq. (7b) from eq. (7a) and multiply the resulting equation for upk- Vpkby gt~(z). Then, we take a derivative of eq. ( 8 ) with respect to z and multiply it by Upk--Vpk.Finally, subtracting the equation obtained from that for Upk--Vpk, one can easily make sure that the integrand in eq. ( 15 ) does not depend explicitly on Uo and eq. (15) may be transformed to the form
V,(k,p)=-
oo
~m
0
dZq/°(Z)[Urk(Z)--VPk(Z) ]
~mm dz ~'o(Z)Vpk(Z) .
(16)
0
At z << l the dependence of the functions upk and vpk on z is just the same as that of q/o. Hence, the main contribution to the integral (16) comes from distances z > l far from the potential well Uo. Thus, to calculate the integral (16), we can use eq. (10) for ~'o and eqs. (7) for upk and vp~ with Uo = 0 and the boundary conditions (9). At sufficiently small to(p), an approximate solution may be found analytically. For to(p)--,0, the
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conservation laws (12) may be satisfied only under the condition k>>p, which leads to the analytical expressions for Upk and Vpk (not presented here in view of their awkwardness). Using these expressions in calculating (16) and (14), we obtain l~,r(p) ~
i.e. ripplons can still be regarded as weakly damping excitations. At To << T the inverse energy flux due to desorption may be neglected. In this case, taking the ideal gas approximation for the bulk phase in eq. ( 11 ) and using eqs. (17), (18) for Wr(p), we obtain the following expression,
kS(p)p nsol 4co(p) mpo '
k(p) = [ 2mco(p) _ p 2 ] 1/2
( 17 )
j(r) ~
The main part in the energy exchange is played by ripplons with the energy c o ( p ) ~ T. As shown by a straightforward analysis, the theoretical limit (17) may be used only at co(p) < l0 mK. A numerical calculation which is necessary in a large temperature range gives
Wr(p ) = ff'r(p )f(pl) . The function f ( x )
35n _nsol nT3F ( T/eo) , _ 4 a
__
T>> T¢,
~ 35n nsol nT3F ( T / % ) 4 tr
x O . 4 ( T / T c ) 3/2 ,
T~T:,
(19)
where n is the volume density of the gas, To(n) is the Bose condensation temperature. The function F ( x ) is presented in fig. 2 (F--, 1 as x - , 0 ) . The relation between the gas temperature and other parameters of the problem or, otherwise, the Kapitza resistance, may be actually obtained by equating Jtr) tO the value of heat release in the gas. If the temperature Kapitza resistance at the gas-helium interface is small, A T = T - To << T, the desorbing energy flux is close to the adsorbing one and the expression for j(r) may be
(18)
is presented in fig. 1 (f--, 1 as
x-*0). At pl< 1 (this condition is satisfied by ripplons with energies ~<0.5 K) rr ' = wr(p )
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<
1.0 o 0.8
O. 7
He 4
He 3
0.3 0.2
0.1 I
I
i
I
i
I
I
|
0.1
0.2
0.3
0.4
0.5
0.6
0.7
X
Fig. 1. Functionf(x) on 4He and 3He surfaces. 481
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PHYSICS LETTERS A
5 February 1990 t"
F
/4rph = J dr [ 1 + ~ ( p ) d / d z ] { [1)r(r) + Uph(r) ]
x [po +~' (r) ] [Jr(r) + Oph(r) ]}. 0.9
Here/~' (r) is the operator of helium density fluctuations, ~r(r) and ~ph(r) are the operators of the particle velocity in the volume and surface waves, respectively. This Hamiltonian can be directly obtained from the general approach (see ref. [ 9 ] ), taking into account the oscillations of the liquid helium surface. As shown by straightforward calculations, the leading channel in the above-mentioned temperature range is the recombination of two ripplons into a phonon. The free path length of a thermal ripplon ( To ~ T) with respect to this mechanism is
0.8
0.7
0.6
0.5
Lrph ~
2xac~ T4 ,
0.4
0.3
0.2
O.l
0.0
I
I
I
I0 -5
10 -~
I0-
Fig. 2. Function surfaces).
3
I
i
b
10 -2
10 - I
X
F(x) (practically the same for both 4He and 3He
obtained by taking a derivative of eq. (19) with respect to T and multiplying it by AT. However, the result obtained requires an additional analysis. The first question is connected with the fate of the ripplons emitted in adsorption of the gas particles. Eqs. (17), (18) yield the value of the ripplon free path length with respect to the desorption channel of decay (i.e. decay with the energy transfer back to the gas ), Lr ~ Vrr ( v = dcn/dp), which is of the order of 1 0 - 2 - 1 0 -3 cm for thermal ripplons at T ~ 10-100 mK. In order to estimate the characteristic time of the energy transfer from the ripplon subsystem to the phonon one, we introduce the ripplon-phonon interaction Hamiltonian in the form 482
where Co is the sound velocity in helium. It is easy to estimate that at T ~ 1 0 - 1 0 0 m K the inequality Lrph>>Lr holds, which becomes still stronger for To << T. It means that for an infinite area of the limiting density region the energy transferred from the gas to the ripplon subsystem (in the adsorption process) should return to the gas with probability close to unity. The energy flux jet) (8) can be transferred completely from the ripplon subsystem to the helium volume only if the characteristic linear dimension of the limiting density region, L, satisfies the condition L<
(20)
This condition allows a ripplon to get out of the limiting density region and, correspondingly, to decay in an ordinary way, transferring its energy to volume phonons. Another problem is connected with the fact that in virtue of the conservation laws ( 12 ), the adsorption into the condensate may occur only from a very small phase volume of particles in the gas. If the conservation laws were satisfied exactly, we should have written the expression ~ r - 1 +Nq(T°) ~ ( ( k 2 + q Z ) / 2 m - ~ ( q ) l Tr(q)
for the sticking coefficient (relative probability of adsorption per collision) of the incident particle with momentum (q, k). The finite size of the limiting
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density region results in a ripplon momentum uncertainty of the order of l / L and in a ripplon energy uncertainty ~ v/L. This leads to a value of the sticking coefficient ~r of the order of L/vzr~ L/Lr. Thus, the same condition (20) leads to the inequality ~r<< 1 which allows one to obtain Wr(p) within the perturbation theory. One more remark should be made. Naturally, under stationary conditions the flux of adsorbing particles is equal to that of desorbing ones. At n~<< nso this imposes an additional condition leading to a certain nonequilibrium value ofn~. At n~= n~o the additionally adsorbed particles cannot be held on the surface and return to the bulk. Hence, in this case the equality of fluxes does not give any additional condition. At n~~ n~o the energy transfer from the bulk to the ripplon subsystem can, in principle, occur without adsorbing gas particles into the condensate, namely, in the channel of purely inelastic scattering. Direct calculations show that in this case the value of the energy flux remains almost the same as that at n~<< n~o to the accuracy of a numerical coefficient. The expression for this flux has an additional small factor ~O.I(T/%) ~/2, as compared to the r.h.s, of eq. ( 19 ). This allows us to consider only the channel with the capture of a gas particle into the condensate in analyzing the ripplon mechanisms. Note that in the case of n~ << n~o the inelastic scattering rate has a small factor (T/%) 3/2. Such a difference is due to a change in the temperature dependence of the energy transfer rate for adsorption in the ns.~ nso case. It should be emphasized that the inequality (20) is also required in the case of the inelastic scattering mechanism. It is interesting to compare eq. (19) with an analogous result of ref. [ 2 ] for the case of n~<< nso. The main difference is that at the limiting density J(~) ( T--,0 ) ~ T 3, while at n~ << n~o we have J(r) ~ T 2. It follows from the fact that in the first case the matrix element for the adsorption process (eqs. (15), (16) ) contains the factor kZ/2m instead of %. This results in the factor of k 4 in the expression for the probability. However, in the adsorption into the condensate, there is no additional integration over the particle momentum along the surface. As a result, there arises an additional factor of T. Thus, the result presented in ref. [2] for the ns~n~o case, which
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is, actually, the extrapolation of the expression for the ns << nso case, is incorrect. The quantitatively correct result (19) differs from that of ref. [ 2 ] by a factor of ~30(T/%)F(T/%) which is close to 0.1 in the temperature range T~ 10-100 mK. The results obtained make it necessary to analyze more accurately the Bose condensation problem in a semi-open system with one magnetic wall (see ref. [ l0 ] ). As can be shown easily, all the results of ref. [10] remain valid, if the density is increased, approximately, by a factor of 3 with decreasing the gas layer thickness d ~ n-~ (this leads to an increase in the Bose condensation temperature Tc ~ n2/3). 4. Let us now consider the channel of the energy transfer from the gas to helium, which is due to the direct interaction of hydrogen atoms with volume phonons. The process of a particle capture into the surface condensate (now with the emission of a phonon) turns out to be dominant again. The interaction Hamiltonian responsible for this process may be written as (see ref. [ 1 ] ).
f dr~o(Z) ~'(r) Uo(z) [ ~ ' ÷ ( r ) + ~ ) ' ( r ) ] , d Po (21)
~' (r)= ~ i(ppo/2Co~)~/2(fv-f + ) exp(ip-r), q
where~ is the annihilation operator of a phonon with momentum p, ~ is the helium volume. The term ~' (r)Uo(z) arises under the condition pR, << 1 as a result of integration of the hydrogen atom interaction with helium surface fluctuations over the helium volume. The conservation laws
q=Pu , (kZ+q2)/2m=cop
(22)
(P~l being the component of the vector p along the surface) lead to the inequality k>> p for the thermal phonons which are most effective in the energy exchange at ns << nso. For temperatures up to ~ 0.5 K this inequality is a fortiori satisfied. Taking into account this inequality and eq. (4) for ~', we obtain the expression for the energy flux from the gas to helium (T>> To):
483
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by a highly inhomogeneous field near the surface [4 ], when n ~ 10 20 cm -3 and T ~ 0 . 3 - 0 . 7 K are the most favourable parameters.
j(ph)= ~ clpdk p k 2 (2•) 3 CoPo 2m np, k V~h(k, Pll )
× (~( kZ / 2 m - c o p ) ,
(23)
Vph (k, PII) = .( d z q/o(Z) Uo(z) [Up,,k(z) - Vp,k(Z) ] • 0
(24) The m a i n c o n t r i b u t i o n to the integral ( 2 4 ) comes from the distances z ~ R . <
T>> Tc,
3× 102 ( nso/ cgm2pol)nT4 ×0.4( T/Tc) 3/2 ,
T<~Tc.
(25)
C o m p a r i n g eqs. ( 2 5 ) a n d ( 1 9 ) , it is easy to see that the p h o n o n a d s o r p t i o n m e c h a n i s m o f the energy exchange becomes the leading one at T > 0. l K even u n d e r the c o n d i t i o n ( 2 0 ) . In particular, it is the p h o n o n m e c h a n i s m that d e t e r m i n e s the heat rem o v a l from H~ to h e l i u m in the idea o f Bose condensation in a small c o m p r e s s e d gas v o l u m e created
484
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5. All the results presented were o b t a i n e d under the c o n d i t i o n Ro/l, R./I<< 1. On real surfaces o f liquid 4He and aHe these ratios are not so small. However, there is no d o u b t that the extrapolation o f the results to the region o f actual p a r a m e t e r s yields a qualitatively correct description o f the p h e n o m e n o n .
References [ 1] I.F. Silvera and V.V. Goldman, Phys. Rev. Len. 45 (1980) 915. [2] Yu. Kagan, G.V. Shlyapnikov and N.A. Glukhov, Pis'ma Zh. Eksp. Teor. Fiz. 40 (1984) 287 [JETP Lett. 40 (1984) 1072]. [3]Yu. Kagan, N.A. Glukhov, B.V. Svistuno and G.V. Shlyapnikov, Phys. Lett. A 135 (1989) 219. [ 4 ] Yu. Kagan and G.V. Shlyapnikov, Phys. Len. A 130 ( 1988 ) 483. [ 5 ] I.B. Mantz and D.O. Edwards, Phys. Rev. B 24 (1979) 4518. [ 6 ] D.G. Friend and R.D. Etters, J. Low Temp. Phys. 39 (1980) 409. [ 7 ] Yu. Kagan, B.V. Svistunov and G.V. Shlyapnikov, Zh. Eksp. Teor. Fiz. 93 (1987) 552 [Sov. Phys. JETP 66 (1987) 480]. [8] P.G. De Gennes, Superconductivity of metals and alloys (Benjamin, New York, 1966). [ 9 ] E.M. Lifshitz and L.P. Pitaevskii, Statistical physics, Part 2 (Nauka, Moscow, 1978 ). [ 10] Yu. Kagan, G.V. Shlyapnikov and N.A. Glukhov, Pis'ma Zh. Eksp. Teor. Fiz. 41 (1985) 197 [JETP Lett. 41 (1985) 238]. [ 11 ] Yu. Kagan and G.V. Shlyapnikov, Phys. Lett. A 95 ( 1983 ) 309.