- Email: [email protected]
Evaporation of liquid 4He; A quantum process
392
Physica 126B (1984) 392-39!) North-llolland, Amslerdam
EVAPORATION OF LIQUID 4HE; A QUANTUM PROCESS A.F.G. WYATT Department of Physics, U n i v e r s i t y o f E x e t e r , Stocker Road, Exeter EX4 4QL, United Kingdom We d e s c r i b e our experiments at low temperatures which show t h a t phonons and rotons can e j e c t atoms from the l i q u i d by the quantum e v a p o r a t i o n process. Other experiments have shown t h a t free atoms i n c i d e n t on the l i q u i d surface have a high p r o b a b i l i t y o f condensing. We then discuss the dynamic e q u i l i b r i u m a t the s a t u r a t i o n vapour pressure a t h i g h e r temperatures T ~ 2 K. We show t h a t the e q u i l i b r i u m number of phonons and rotons can provide the necessary temperature dependent evaporat i o n r a t e i f the average p r o b a b i l i t y f o r quantum e v a p o r a t i o n by an e x c i t a t i o n is I / 3 . I.
INTRODUCTION Recent measurements on the e v a p o r a t i o n of l i q u i d ~He (1,2) t o g e t h e r w i t h o t h e r r e s u l t s on the condensation o f ~He vapour ( 3 - I 0 ) have increased our understanding o f the m i c r o s c o p i c processes i n v o l v e d to a p o i n t where we can cons i d e r the dynamics o f the e q u i l i b r i u m between l i q u i d and s a t u r a t e d vapour a t temperatures T < 2 K. The e q u i l i b r i u m s i t u a t i o n at these temperatures is f a r removed from the non e q u i l i brium c o n d i t i o n s under which d e t a i l e d measurement o f e v a p o r a t i o n have been made. The main questions are whether the two p a r t i c l e quantum e v a p o r a t i o n process (1,2) remains dominant at h i g h e r temperatures (O.l < T < 2 K) and i f so whether i t is able to account f o r the neces s a r i l y high r a t e o f e v a p o r a t i o n needed f o r dynamic e q u i l i b r i u m a t the s a t u r a t e d vapour pressure. We s h a l l see t h a t i t is indeed p o s s i b l e to e x p l a i n the temperature dependent s a t u r a t i o n vapour pressure by the quantum e v a p o r a t i o n proces s. The question of the s a t u r a t e d vapour pressure o f He has been considered f o r many years. It has in the main concentrated on the vapour side o f the i n t e r f a c e and i t is now w e l l e s t a b l i s h e d (7,8) t h a t an i n c i d e n t atom has a p r o b a b i l i t y approaching u n i t y o f condensing i n t o the l i q u i d f o r T < 2 K. However, f o r an atom to leave the l i q u i d i t must acquire a r e l a t i v e l y high energy (7.16 K) compared to kT so i t i s an improbable and h i g h l y temperature dependent event. The energy spectrum of e x c i t a t i o n s extends well beyond 7.16 K (see f i g u r e l ) and i t is probable t h a t atoms acquire the necessary energy from these e x c i t a t i o n s . Atkins ( 3 ) has shown t h a t the condensation r a t e o f 4He vapour could be reasonably w e l l accounted f o r by gas k i n e t i c s and a condens a t i o n c o e f f i c i e n t (m) o f u n i t y , a t T ~ 1.2 K. Osborne (5) measured the v e l o c i t y produced in the vapour by a c o u n t e r f l o w in the l i q u i d 4He and made the i m p o r t a n t o b s e r v a t i o n t h a t atoms in the vapour only exchange momentum w i t h the 0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division
normal f l u i d . As we can i d e n t i f y the normal f l u i d w i t h the e x c i t a t i o n s in the l i q u i d t h i s suggests a d i r e c t c o u p l i n g between the atoms and e x c i t a t i o n s . Osborne quotes an unpublished r e s u l t of Beenaker who found t h a t f l o w in a t h i n f i l m o f 4He in which the normal f l u i d is locked to the s u b s t r a t e , had no i n t e r a c t i o n w i t h the vapour. Hunter and Osborne (6) measured the r e f l e c t i o n of second sound from the l i q u i d vapour i n t e r f a c e and deduced t h a t the condens a t i o n c o e f f i c i e n t .~ > 0.9. The value o f ~ depends on the k i n e t i c theory used in the analysis. This was re-examined by Kessler and Osborne (7) who f i n a l l y quote ~ = 0.9 f o r 1.2
A.F. G. Wyatt / Evaporation of liquid 4He 1.5 K above the binding energy. I t was suggested that atoms were evaporated with this high energy in a single quantum process by the rotons. However, Cole (17) pointed out that the rotons at the minimum have a vanishingly small group velocity which reduces the incident flux on the interface by exactly the same factor as the density of states increases i t , Nevertheless, the idea of a two particle process has proved to be possible. The f i r s t attempt to measure the evaporation of atoms by excitations in the liquid was by Balibar et al (18). They saw no evaporation by b a l l i s t i c phonons but did see a signal which they ascribed to b a l l i s t i c rotons evaporating atoms. However, the signal time suggested a maximum velocity for rotons of 160 ms- l for which no explanation could be given. We have since found that the heater powers used by them were too high to see the effects of b a l l i s t i c rotons. From our recent work (1,2) i t is clear that at least some atoms of 4He leave the liquid state by absorbing a phonon or roton completely. The energy of the excitation goes in overcoming the binding energy to the liquid (7.16 K) and the balance appears as kinetic energy of the atom. These measurements can only be made at low temperatures, T ~ O.l K, where the mean free paths of the excitations are long. This is in contrast to condensation measurements which can be made over a wide temperature range (0.03 - 2 K). Whenthere is equilibrium between the vapour and liquid at higher temperatures we expect a high rate of condensing atoms which must be matched by evaporation. I f the quantum evaporation process is to account for this rate then we must show that the flux of excitations on the interface is s u f f i c i e n t l y high and has the correct temperature dependence. The evaporation rate also depends on the probability that an incident phonon or roton creates a free atom. We consider these points after reviewing the evidence for quantum evaporation at lower temperatures. 2. QUANTUMEVAPORATION 2.1 Phonon-atom Quantum evaporation whereby an excitation in the liquid is annihilated and a free atom is created implies that energy is conserved in the following way b~
= EB + ~2k2/2m
(1)
where ~ is the energy of the phonon or roton, EB is the binding energy of an atom in the l~quid and ~k is the momentum and m the mass of the free atom. The incident phonon or roton perturbs the potential well of the surface atom and causes a transition from a bound state to a free state. To demonstrate the v a l i d i t y of equation l i t is necessary to measure the
393
phonon energy and the kinetic energy of the free atom ~, as EB = 7.16 K is known from bulk latent heat measurements. We do this using the velocity dispersion of the phonons and atoms to distinguish between the broad spectrum of phonons created by the heater. I f the atoms travel a distance ~l and the phonons a distance ~2' then the total signal time is
tt
=
~i/Va + ~21Vg
(2)
where Va is the v e l o c i t y of the atom and VQ is the group v e l o c i t y of the phonons, which i~ a f u n c t i o n of the phonon energy as can be seen in figure I. 24
i - -
E
,
20 17.4-16 13"8->, 12
9~__ o
8-67-7.16 ~
8
4
0
i 0
0.8
i 1.6
wave
vector
I 2.4
(A-')
Figur e I: The dispersion curve for liquid 4He at the SVP and T = 1 K . The three regions of wave vectors are shown.
In figure 2 we show t t , for a particular ~I and ~2, as a function of-phonon energy calculated using dispersion data from neutron measurements (19) and equations l and 2. The minimum in figure 2 shows that the shortest signal time selects out a particular phonon energy. I f ~l is varied with ~l + ~2 constant then the minimum t t occurs for different phonon energies. The measurements are made at T < O.l K where the SVP is so low that the evaporated atom is not scattered and also the phonon mean free path is long, for ~ > wc (~9 K) I20, 21) as there is no scattering by thermal excitations. The phonons are produced by a ~l ~s current pulse in a thin metal film which raises i t s temperature to 2-3 K during the pulse. The atoms are detected by a superconducting Zn bolometer in a magnetic f i e l d and held at a constant temperature with electronic feedback (23,24,25). The time constant of the bolometer is ~l us. The Zn film and substrate is covered with a thin He film on which the atoms condense giving up
304
.I.I< (;. Rq'att / lh'uporatio/~ o./ liquid 41h"
8 ,--
/
65
,= ¢n :"
°# \"\• \
60
E
6 •
=: -o 55
\\\ \
\~"x
,
\,
\.
5
\, \
\,, 50 ~ 8
~
9
.
.
. 10
.
. 11
.
. 12
13
3
energy I K !
30
hal is skown as a funcb:'on 9f p:~ono<' ~vr'~7.~ ,",r ato?~: pat'*; lengt:,. 0 I) /. : ~ ~z r, l iq~':: ,: >at~ lengt/, (~,2) 4.0 ~mn.
k i n e t i c and b i n d i n g energy• The f a s t e s t s i g n a l s are measured, f o r d i f f e r e n t ~l w i t h ~l + L2 cons t a n t ~n an arrangement shown i n s e t in f i g u r e 3. The f i g u r e shows t y p i c a l s i g n a l s .
~
/
=
purified 4He
I !
4He with 3He
/
J
3He 4He 310 -
510
70
time
I:ps)
go
110
L~: Measured phonon-atom signal~ ar functions of time. Inset i:~ sha~n a schemat::~ ]iagram of the experimental arrangement. The times f o r the f a s t e s t s i g n a l s ( t t , m i n ) are shown in f i g u r e 4 t o g e t h e r w i t h t he computed times and i t can be seen t h a t there is good agreement. As a f u r t h e r check on the v a l i d i t y o f the experiment and i t s i n t e r p r e t a t i o n we added enough 3He to the 4He l i q u i d so t h a t a ~.25 monolayer coverage formed on the liquid He ~urface. The 3He are bound to the surface o f ~He (25) by 5.0 K (28). This t o g e t h e r w i t h the d i f f e r e n t mass o f 3He changes the f a s t e s t s i g n a l time a t each L l . In f a c t d i f f e r e n t phonon energies give t t , m i n f o r 3He
phonon- He
40
50
60
time [ p s '
and 4He at the same ':l. With 3He on the surface s i g n a l s due to both 3He and 4He atoms can be seen, as shown in f i g u r e 3. The f a s t e s t signal times f o r 3He are shown in f i g u r e 4 and again there is good agreement w i t h the calcul a t e d times. 2.2. Roton-atoms Rotons have much more momentum than phonons and t h i s suggests a c l e a r way o f d i s t i n g u i s h i n g between them in an e v a p o r a t i o n experiment. I f we assume t h a t the i n t e r f a c e between the l i q u i d and vacuum is t r a n s l a t i o n a l l y i n v a r i a n t then the component o f momentum p a r a l l e l to the surface must be conserved. So we have q sin ~, =
2
\,
l
phonon-3He
k sin :,
(3'I
where q and k are the wave vectors of the roton o r phonon and the atom r e s p e c t i v e l y , :~ and are the angles between the paths and the normal. The angle ¢ can be c a l c u l a t e d in terms of q using equations l and 3, and the d i s p e r s i o n curve. The r e s u l t s are shown in f i g u r e 5. The three regions correspond to the phonon and two roton regions d e f i n e d in f i g u r e I . I t can be seen t h a t f o r some values o f q there is only a l i m i t e d range o f angles of incidence (a) t h a t give an atom angle (~) which is ~90 °. I f no surface e x c i t a t i o n is created or any o t h e r h i g h e r o r d e r process occurs then the e x c i t a t i o n in the l i q u i d w i l l be t o t a l l y r e f l e c t e d . The c r i t i c a l angles o f incidence are shown in f i g u r e 6. I t can be seen t h a t there is a l a r g e range of phonon momenta f o r which there is no c r i t i c a l a n g l e , but at high phonon momenta and f o r a l l the roton range there is one. For low energy phonons, ~ ~ 7.16 K, the evaporated atom has very low energy and momentum and here again the atom cannot match the p a r a l l e l momentum of a phonon o t h e r than at near normal i n c i d e n c e .
A.F. G. Wyatt / Evaporation ofiliquid 4He i
energy (K) 140
~iI
T T T TT I l l
~ITITITI T
e 90~I~°50°40 ° 30 20~ i
90
395
q
i
13.1 lphonon_4H e
~120 12"5t 9"2 100 12-0
~o 1D
17"0
9-5 lto.2
_e
==
30
:
o'.4
o'-8
'1'-2 1'.6 wave vector (/~-')
~o
11~
,o
~o
angle ~ (degrees)
i
~.o
roton-4~/~o-o
15" 1 ~ 3 - 3 14"0
~o
10
h
16.3
10"7
80
S
~14
~.~
Figure 6: The angle (~) that the atom makes to the normal is shown as a function of the wave vector of the 4He excitations for various angles (0) which the excitation makes to the normal. The corresponding excitation energies are shown above.
7: Calculated total times for phonon-atom .atom-f~ton-atom V i signals • uare shown r as efunctions of angle (~) for excitation angle 6 = 15 ° . The corresponding energies of the excitations are given in degrees Kelvin. The path lengths for atomp and excitations are both 6.5 m~.
i
9O
60 o
.-_
ao
6 O0
0"4
018
112 1!6 2!0 wave vector (A-~)
214
~2"8
'
1o
'
go ' 12o ' 18o time (pS /
F i b r e . 8: Phonon-atom and roton-atom signals
Figure 6: The critical angle (ec) between the excitation and the normal shown as a function of excitation wave vector. For quantised evaporation @ must be less than @c"
are shown as functions of time for different atom angZes (t) for excitation angle @ = 13 ° . For ~ = 10, 14, 19 ° the signal is due to phonons and at 33 ° to rotons. Path lengths for atoms and excitations are both 6.5 ~n.
I t is worth noting that these phonons are below the 3pp cut o f f energy. As with phonons the roton v e l o c i t i e s are dispersed, and t t ~i~ corresponds to a p a r t i c u l a r roton energy.~'~h~ v a r i a t i o n of t , with angle t f o r an angle of incidence e = 15°~is shown in figure 7. I t can be seen that phonons evaporate atoms at an angle ~10° regardless of t h e i r energy whereas the d i f f e r e n t roton energies give a wide range of angles ¢. We use both the r e f r a c t i o n e f f e c t and the minimum signal time to d i s t i n g u i s h between phonon-atom and roton-atom evaporation. Typical traces at various angles ~ are shown in figure 8.
These were taken with a bolometer t h a t could be moved around on a radius arm by a superconducting stepping motor. The signal changes shape considerably between the phonon and roton evaporation regions. We would expect t h i s from the range of phonon and roton energies that contribute to the signals. In figure 7 we see that phonons just above the 3pp cut o f f give the fastest signal time. I f the population of t~e phonon states decreases r a p i d l y with energy then the signal at longer times is rapidly deminished, Howewr, the roton s i t u a t i o n is d i f f e r e n t , The fastest signals come from rotons with ~w ~ 14 K and so we expect a higher density
.I. t< (7. Ill Jr: / l-r~q;~,r~lH:,~ :>/liquid 4Ih
396
o f rotons at l o w e r e n e r g i e s which can c o n t r i b u t e a s u b s t a n t i a l s i g n a l at l o n g e r t i m e s . The m i n i mum s i g n a l t o t a l times f o r phonons and rotons agree v e r y w e l l w i t h those c a l c u l a t e d . To examine the a n g u l a r dependence we have i n t e g r a t e d the s i g n a l s at d i f f e r e n t angles o v e r t h e i r f i r s t 17 :,s. This is the time d i f f e r e n c e between the f a s t e s t r o t o n - a t o m and phonon-atom signals. The r e s u l t s are shown in f i g u r e 9.
the c r e a t i o n o f a f r e e atom by an i n c i d e n t e x c i t a t i o n , and s i m i l a r l y f o r the c r e a t i o n oF an e x c i t a t i o n by a condensing atom. On the vapour side o f the i n t e r f a c e we nave the number f l u x d~ a o f f r e e 4He atoms s t r i k i n g the i n t e r f a c e in the ranges Ea t o Ea + dEa, "a t o a + d~a and :a to :a + d;a
dl a
k2sin ........
a
cos . . . . . a . _ d- a d:
8~3h(exp(Ea/kT
6
i
10
20
30 angle
40 ~
50
60
I degrees
2 qi sin i cos i d.:. . . . . . . . . . . . e,i
8:~3(exp(h..,i/kT ~s{,,}L:'
¢
~cz'
~';i?<';','.,':~,(
•
<~z<~z+
dE a
.:4!
where Ea is the e n e r g y , ilk is the momentum and m is the mass o f the ~He atom. We use the B o s e - E i n s t e i n (B-E) d i s t r i b u t i o n f u n c t i o n and we s h a l l see i t is i m p o r t a n t not to approximate t o a Maxwell-Boltzmann d i s t r i b u t i o n . On the l i q u i d side o f the i n t e r f a c e we have the number f l u x d; e i o f e x c i t a t i o n s in the i ti: region of the momentum spectrum (see f i g u r e l , i = I , 2 or 3).
onon-4He
0
a
l)
~7+ =
, .<
u,;
.
The e v a p o r a t i o n due to phonons and rotons can e a s i l y be d i s t i n g u i s h e d . The widths o f the peaks are p a r t l y due t o the l i m i t e d c o l l i m a t i o n ~9 ° o f the g e n e r a t o r and d e t e c t o r . However, from f i g u r e 7 we see the range o f angles o f e v a p o r a t i o n is much l a r g e r f o r rotons than phonons and t h i s is c o n s i s t e n t w i t h the g r e a t e r w i d t h o f the r o t o n - a t o m peak in f i g u r e 9. From these r e s u l t s we can draw s e v e r a l conclusions. High energy phonons and r o t o n s in r e g i o n 3 can be generated by a h e a t e r , t h e y have long mean f r e e paths and they can e v a p o r a t e atoms from the s u r f a c e in a s i n g l e quantum process. A phonon or r o t o n is a n n i h i l a t e d and a f r e e atom i s c r e a t e d w i t h energy and p a r a l l e l momentum conserved• I t is c l e a r t h a t at l e a s t some atoms are e v a p o r a t e d w i t h o u t c r e a t i n g ripplons. We have n o t seen atoms e v a p o r a t e d by r o t o n s in r e g i o n 2. These would need to have t h e i r group v e l o c i t y d i r e c t e d towards the surface and hence t h e i r momentum d i r e c t e d towards the g e n e r a t o r •
3. EQUILIBRIUM BETWEEN LIQUID AND VAPOUR L i q u i d 4He at t e m p e r a t u r e '~,I K w i l l have a high d e n s i t y o f phonons and rotons in a l l regions. There w i l l be a continuous f l u x i n c i dent on the i n t e r f a c e which presumably e v a p o r ates atoms in the same way t h a t i t does under pulse c o n d i t i o n s . We now examine whether i t is p o s s i b l e to account f o r the h i g h l y t e m p e r a t u r e dependent s a t u r a t e d vapour pressure w i t h the s i n g l e quantum process d e s c r i b e d in Section 2. We s h a l l need to i n t r o d u c e p r o b a b i l i t i e s for
d.,
e,!
1'
• d:
e,l
.d . ] ;5)
where h ~ and hg i are the energy and momentum o f the p~onon or r o t o n . The group v e l o c i t y of the e x c i t a t i o n s e x a c t l y cancels the d q / d , fact o r in the d e n s i t y o f s t a t e s (17). In e q u i l i b r i u m the chemical p o t e n t i a l ( ; ) f o r the atoms in the vapour and l i q u i d must be the same• We have put : - 0 in both e q u a t i o n s 4 and 5. We must use the same energy zero f o r the two B-E d i s t r i b u t i o n s in e q u a t i o n s 4 and 5. C o n s i s t e n t w i t h ~ = 0 we r e f e r a l l e n e r g i e s tc the Bose condensate. The energy o f an e x c i t a t i o n is then by d e f i n i t i o n h i and the energy o f an atom in the vapour is Ea
h2k2,'2m + EB
,.61,
where EB is the b i n d i n q enerqy o f the atom to the l i q u i d . The two c o n s e r v a t i o n laws at the i n t e r f a c e r e l a t e angles and e n e r g i e s o f e x c i t a t i o n s in the l i q u i d t o those f o r atoms in the vapour. So f o r a given i n c r e m e n t a l s o l i d angle and energy f o r one momentum r e g i o n o f e x c i t a t i o n s in the l i q u i d , t h e r e is a unlque conjugate i n c r e m e n t a l s o l i d angle and energy f o r atoms in the vapour. E x c i t a t i o n s o n l y e v a p o r a t e atoms at c o n j u g a t e angles and e n e r g i e s and s i m i l a r l y f o r condensing atoms c r e a t i n g e x c i t a t i o n s . This is i l l u s t r a t e d in f i g u r e lOa. We choose to express the phonon and r o t o n f l u x e s in terms o f the c o n j u g a t e f r e e atom angles and e n e r g i e s . From e q u a t i o n s l , 3 and 6, and w r i t i n g h.~.a = Ea we have d i = d , a , d~e, i = d : a i and qi cos ~ e , i d ' e , i = k cos '!'ai d ' a i , s u b s t i t u t i n g i n t o e q u a t i o n s a and 5 we f i n d ;
A.F.G. Wyatt/ Evaporationof liquid4He (a)
condensation c o e f f i c i e n t = 1 at a l l temperatures we see t h a t these p r o b a b i l i t i e s can be independent of temperature. Let P i ( e a , wa) be the p r o b a b i l i t y t h a t an e x c i t a t i o n , in wave vector region i , which is i n c i d e n t on the i n t e r f a c e creates an atom at ea with energy ma" We assume t h a t i t is a f u n c t i o n of angle and energy but not of temperature. The f l u x of atoms created at the conjugate angle and energy is
(b)
i
2/,o.,, (c)
~~
Id~
397
/ i P'
3 i=l Figure 10 : a; Conjugate angles for atoms and excitations from one a wave vector region. ~,: The probability of an excitation evaporating an atom in the three wave vector regions is defined in terms of the conjugate atom angle (Oa). c: The probability of an incident atom creating an excitation in one of the three wave vector regions is defined in terms of the conjugate atom angle. d: The two processes which are required for detailed balance of excitations are shown with their probabilities.
d~e, i P i ( e a , ma)
The s i t u a t i o n is schematically shown in f i g u r e IOb. Let Aj(e a, ma) be the p r o b a b i l i t y that an atom i n c l d e n t on the i n t e r f a c e creates an e x c i t a t i o n in region j as shown in f i g u r e lOc. Then z~A. is the condensation c o e f f i c i e n t f o r the at~mJat angle e. and energy ma The loss of atoms by condensatlon in the incremental i n t e r v a l is .
.
d~a
=
cos e a
8~3(exp(~a +Ei/kT)-
dea d# a d~a
a I)
(7) k 2 sin eai cos eai d#e, i
=
deai d¢ai d~a 8~3(exp(h~[+-EB/kT) - I)
(8)
where the subscript i only reminds us of the q region to which the atom v a r i a b l e s are conjugate and w i l l be omitted from now on. From equations 7 and 8 we see t h a t conjugate f l u x e s , i n c i d e n t on the i n t e r f a c e , are i d e n t i c a l at a l l temperatures in each q region separately. This is an important r e s u l t as i t shows t h a t i f a l l the e x c i t a t i o n s in one q region, which have enough energy and are w i t h i n the c r i t i c a l angle, each evaporate an atom then t h i s e x a c t l y maint a i n s the energy and angular d i s t r i b u t i o n of atoms in the vapour i f a l l atoms i n c i d e n t on the i n t e r f a c e condense. This balance is maintained at a l l temperatures because both atoms and e x c i t a t i o n s obey B-E s t a t i s t i c s . As there are three regions o f e x c i t a t i o n s then the p r o b a b i l i t y t h a t an e x c i t a t i o n does evaporate an atom must be less than u n i t y . However, as the conjugate f l u x e s have the same temperature dependence and the
.
°
3
[ j=l
d~a Aj(e a, ma)
(10)
From the e q u a l i t y of the i n c i d e n t f l u x e s and requirement of dynamic e q u i l i b r i u m we f i n d 3
k2 sin e
(9)
A'(°a' j=l J ~a)
=
3 [ Pi i=l ( ° a ' ~a)
(ll)
We now consider d e t a i l e d balance f o r one of the types of e x c i t a t i o n s in the l i q u i d , say region 1 which is phonons. In t h i s model phonons i n c i d e n t on the i n t e r f a c e e i t h e r specul a r l y r e f l e c t or create free atoms. There must be an equal f l u x of phonons going towards and away from the i n t e r f a c e . Phonons going away from the i n t e r f a c e come from both phonons which are s p e c u l a r l y r e f l e c t e d and those created by condensing atoms. This is shown in f i g u r e lOd, from which we see A1 + (I - PI) = I , hence Ai(e a, Wa)
=
Pi(0a, ~a)
(12)
This r e s u l t is a p r e d i c t i o n of the model and c l e a r l y must be confirmed by measurement. As the condensation c o e f f i c i e n t is n e a r l y u n i t y f o r a l l atoms except those with low energy at glancing angles, we expect [P i = 1 f o r a wide range of energies and angles. The average p r o b a b i l i t y , averaged over conjugate angles, is therefore ~I/3. For an e x c i t a t i o n outside the c r i t i c a l angle P = 0 and i t is s p e c u l a r l y r e f l e c t e d back i n t o the l i q u i d . From the above analysis we see that the simple model can account f o r the d e t a i l e d processes needed to maintain dynamic e q u i l i b r i u m between the l i q u i d and vapour f o r most of the
,1./,] (,. ll'rdlt ' ErdlJ~ruti~m
398
energies and angles. However, there are two extreme and small groups o f atoms in the vapour t h a t do not f i t in w i t h t h i s p i c t u r e . Atoms w i t h energy ~lO K, which w i t h the b i n d i n g energy gives ~17 K, have more energy than the maximum in the e x c i t a t i o n spectrum so they presumably create more than one e x c i t a t i o n . The o t h e r group is around atom energies o f 1.2G K (27). This w i t h the b i n d i n g energy gives an energy of 8.42 K which is below the roton minimum (8.67 K) and so might be expected to create phonons. 5However, the momentum o f t h i s atom is 4.84 lO -2 mkgs-1 w h i l e the momentum o f 1~he corresponding phonon is 4.75 lOmkgs-~ (28). I f t h i s d i f f e r e n c e is real then an atom at g l a n c i n g angles (>80 o to the normal) canno~ create any bulk e x c i t a t i o n s in the l i q u i d ~He. The model also gives the pressure on the i n t e r f a c e which we now consider. When an atom condenses only the p a r a l l e l momentum is taken uQ by the e x c i t a t i o n so t h a t in the normal d i r e c t i o n there is a d i f f e r e n c e between k cos ~i ,a and q cos .,',e which has t o be s u p p l i e d by the surroundings. The ground s t a t e l i q u i d and c o n t a i n e r is given momentum ~(k cos ~!a - q cos ~e) downwards when an atom condenses o r evaporates. I f we consider a p a i r of processes in which an atom at k, Ua, ~a condenses and one at k, (~a and ¢a + ' evaporates then the change in momentum o~ the l i q u i d and e x c i t a t i o n s is 21Qk cos ,.ia, which of course is opposite t o the change in the momentum o f the vapour system. I f the atom s p e c u l a r l y r e f l e c t s from the i n t e r f a c e i n s t e a d of condensing, i t gives the same c o n t r i b u t i o n to the momentum change. The pressure on the l i q u i d is then from equation 7. P
[d,~ Ida, Ida, 2hk c°s 0
0
0
~ k2 sin
c°s'
873(exp (~'~a + EB/kT) - l )
(13) To e v a l u a t e t h i s i n t e g r a l we ignore the - l i n s i d e the brackets which makes a n e g l i g i b l e d i f f e r e n c e and f i n d .
.
.
.
exp h3
,
(14) kT ~
where m is the mass o f the 4He atom. This is the usual expression f o r the s a t u r a t e d vapour pressure. Expressed as an u n c e r t a i n t y in temperature i t is good to 5 lO -4 K up to T ~,. 1.5 K (29). 4, CONCLUSIONS We have reviewed the evidence f o r quantum e v a p o r a t i o n at low temperatures. We then showed t h a t the e q u i l i b r i u m d e n s i t y o f phonons
and rotons at h i g h e r temperatures gives a large enough f l u x i n c i d e n t on the i n t e r f a c e to account f o r the e v a p o r a t i o n rate at the s a t u r a t e d vapour pressure at T 2 K. The f l u x of e x c i t a t i o n s has e x a c t l y the same temperature dependence as the f l u x o f atoms which stems from the f a c t t h a t they both obey Bose-Einstein s t a t i s t i c s . It f o l l o w s from t h i s t h a t the p r o b a b i l i t y of a phonon o r roton e v a p o r a t i n g an atom is temperature independent. From c o n s i d e r a t i o n s of d e t a i l e d balance and from the f a c t t h a t the condensation c o e f f i c i e n t is a p p r o x i m a t e l y u n i t y we found t h a t the average p r o b a b i l i t y is , I / 3 . F i n a l l y we considered the change in momentum of the l i q u i d during e v a p o r a t i o n and condensation and showed t h a t the model gives the usual expression f o r the s a t u r a t e d vapour pressure as a f u n c t i o n of temperature. ACKNOWLEDGEMENTS I would l i k e to acknowledge the immense e x p e r i m e n t a l e f f o r t of my students F.R. Hope and M.J. Baird and valuable discussions w i t h them and D.M. L i v e s l e y , D.V. Osborne and W.F. Vinen. REFERENCES (1) M.J. B a i r d , F.R. Hope and A.F.G. Wyatt, Nature 304 (1983) 325. (2) F.R. Hope, M.J. B a i r d and A.F.G. Wyatt, Phys. Rev. L e t t . 32 (1984) 1528. (3) K.R. A t k i n s , B. Rosenbaum and H. Seki, Phys. Rev. l l 3 (1959) 751. (4) D.V. Osborne, Proc. Phys. Soc. 80 (1962) I03. (5) D.V. Osborne, Proc. Phys. Soc. 80 (1962) 1343. (b) G.H. Hunter and U.V. Osborne, d. Phys. C: Sol. St. Phys. 2 (1969) 2414. (7) W.D. Kessler and D.V. Osborne, J. Phys. C: Sol. St. Phys. 13 (1980) 1571. (8) D.O. Edwards, P.P. Fatouros, G.G. lhas, P.M. Rozinsky, S.Y. Shen, F.M. Gasparini and C.P. Tam, Phys. Rev. L e t t . 34 (1975) i153. (9) V.U. Nayak, D.O. Edwards and N. Masuhara, Phys. Rev. L e t t . 50 (1983) 990. (lO) D.O. Edwards, G.G. lhas and C.P. Tam, Phys. Rev. [5. l~ (In771 3122. ( l l ) P.M. Echenique and J.B. Pendry, Phys. Rev. L e t t . 37 (1976) 561. (12) D.O. Edwards and P.P. Fatouros, Phys. Rev. B. 17 (1978) 2147. (13) W.D. Johnston J r . and J.G. King, Phys. Rev. L e f t . 16 (1966) l l 9 1 . (14) J.G. King, J. McWane and R. T i n k e r , B u l l . of the Am. Phys. Soc. 17 (1972) 38. (15) P.W. Anderson, Phys. L e t t . A. 29 (1969) 563. (16) D.S. Hyman, M.O. S c u l l y and A. Widom, Phys. Rev. 186 (1969) 231. (17) M.W. Cole, Phys. Rev, L e t t . 28 (1972) 1622. (18) S. B a l i b a r , J. Buechner, B. Castaing,
A. k: G. Wyatt / Evaporation of liquid 4He
(19)
(2o) (21 (22) (23) (24)
(25) (2~)
(27)
(2~)
(2~)
C. Laroche and A. Libchaber, Phys. Rev. B 18 (1977) 3096. F.R. Hope, PhD Thesis 'Quantised 4He evaporation', University of Exeter (1983). R.C. Dynes and V. Narayanamurti, Phys. Rev. Lett. 33 (1974) 1195. A.F.G. Wyatt, N.A. Lockerbie and R.A. Sherlock, Phys. Rev. Lett. 33 (1974) 1425. R,A. Sherlock and A.F.G. Wyatt, J. Phys. E" Sci. I n s t r . 16 (1983) 669. R,A. Sherlock and A.F.G, Wyatt, J. Phys. E" Sci. Instr. 16 (1983) 673. R.A. Sherlock, J. Phys. E: Sci. I n s t r . 17 (1984) 386. A.F. Andreev, Sov. Phys. J.E.T.P. 23 (1966) 939. D.O. Edwards and W.F. Saam, Progress in low temperature physics, VoI. VIIA, ed. D.F. Brewer, (North-Holland, Amsterdam 1978) Ch. 4. D.V. Osborne, private communication. W.G. S t i r l i n g , Proceedings of the 75th Jubilee conference on helium-4, ed. J.G.M. Armitage (St. Andrews (1983)). p. 109. F.G. Brickwedde, J. Res. N.B.S. (1960) 644.
399
Physica 126B (1984) 392-39!) North-llolland, Amslerdam
EVAPORATION OF LIQUID 4HE; A QUANTUM PROCESS A.F.G. WYATT Department of Physics, U n i v e r s i t y o f E x e t e r , Stocker Road, Exeter EX4 4QL, United Kingdom We d e s c r i b e our experiments at low temperatures which show t h a t phonons and rotons can e j e c t atoms from the l i q u i d by the quantum e v a p o r a t i o n process. Other experiments have shown t h a t free atoms i n c i d e n t on the l i q u i d surface have a high p r o b a b i l i t y o f condensing. We then discuss the dynamic e q u i l i b r i u m a t the s a t u r a t i o n vapour pressure a t h i g h e r temperatures T ~ 2 K. We show t h a t the e q u i l i b r i u m number of phonons and rotons can provide the necessary temperature dependent evaporat i o n r a t e i f the average p r o b a b i l i t y f o r quantum e v a p o r a t i o n by an e x c i t a t i o n is I / 3 . I.
INTRODUCTION Recent measurements on the e v a p o r a t i o n of l i q u i d ~He (1,2) t o g e t h e r w i t h o t h e r r e s u l t s on the condensation o f ~He vapour ( 3 - I 0 ) have increased our understanding o f the m i c r o s c o p i c processes i n v o l v e d to a p o i n t where we can cons i d e r the dynamics o f the e q u i l i b r i u m between l i q u i d and s a t u r a t e d vapour a t temperatures T < 2 K. The e q u i l i b r i u m s i t u a t i o n at these temperatures is f a r removed from the non e q u i l i brium c o n d i t i o n s under which d e t a i l e d measurement o f e v a p o r a t i o n have been made. The main questions are whether the two p a r t i c l e quantum e v a p o r a t i o n process (1,2) remains dominant at h i g h e r temperatures (O.l < T < 2 K) and i f so whether i t is able to account f o r the neces s a r i l y high r a t e o f e v a p o r a t i o n needed f o r dynamic e q u i l i b r i u m a t the s a t u r a t e d vapour pressure. We s h a l l see t h a t i t is indeed p o s s i b l e to e x p l a i n the temperature dependent s a t u r a t i o n vapour pressure by the quantum e v a p o r a t i o n proces s. The question of the s a t u r a t e d vapour pressure o f He has been considered f o r many years. It has in the main concentrated on the vapour side o f the i n t e r f a c e and i t is now w e l l e s t a b l i s h e d (7,8) t h a t an i n c i d e n t atom has a p r o b a b i l i t y approaching u n i t y o f condensing i n t o the l i q u i d f o r T < 2 K. However, f o r an atom to leave the l i q u i d i t must acquire a r e l a t i v e l y high energy (7.16 K) compared to kT so i t i s an improbable and h i g h l y temperature dependent event. The energy spectrum of e x c i t a t i o n s extends well beyond 7.16 K (see f i g u r e l ) and i t is probable t h a t atoms acquire the necessary energy from these e x c i t a t i o n s . Atkins ( 3 ) has shown t h a t the condensation r a t e o f 4He vapour could be reasonably w e l l accounted f o r by gas k i n e t i c s and a condens a t i o n c o e f f i c i e n t (m) o f u n i t y , a t T ~ 1.2 K. Osborne (5) measured the v e l o c i t y produced in the vapour by a c o u n t e r f l o w in the l i q u i d 4He and made the i m p o r t a n t o b s e r v a t i o n t h a t atoms in the vapour only exchange momentum w i t h the 0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division
normal f l u i d . As we can i d e n t i f y the normal f l u i d w i t h the e x c i t a t i o n s in the l i q u i d t h i s suggests a d i r e c t c o u p l i n g between the atoms and e x c i t a t i o n s . Osborne quotes an unpublished r e s u l t of Beenaker who found t h a t f l o w in a t h i n f i l m o f 4He in which the normal f l u i d is locked to the s u b s t r a t e , had no i n t e r a c t i o n w i t h the vapour. Hunter and Osborne (6) measured the r e f l e c t i o n of second sound from the l i q u i d vapour i n t e r f a c e and deduced t h a t the condens a t i o n c o e f f i c i e n t .~ > 0.9. The value o f ~ depends on the k i n e t i c theory used in the analysis. This was re-examined by Kessler and Osborne (7) who f i n a l l y quote ~ = 0.9 f o r 1.2
A.F. G. Wyatt / Evaporation of liquid 4He 1.5 K above the binding energy. I t was suggested that atoms were evaporated with this high energy in a single quantum process by the rotons. However, Cole (17) pointed out that the rotons at the minimum have a vanishingly small group velocity which reduces the incident flux on the interface by exactly the same factor as the density of states increases i t , Nevertheless, the idea of a two particle process has proved to be possible. The f i r s t attempt to measure the evaporation of atoms by excitations in the liquid was by Balibar et al (18). They saw no evaporation by b a l l i s t i c phonons but did see a signal which they ascribed to b a l l i s t i c rotons evaporating atoms. However, the signal time suggested a maximum velocity for rotons of 160 ms- l for which no explanation could be given. We have since found that the heater powers used by them were too high to see the effects of b a l l i s t i c rotons. From our recent work (1,2) i t is clear that at least some atoms of 4He leave the liquid state by absorbing a phonon or roton completely. The energy of the excitation goes in overcoming the binding energy to the liquid (7.16 K) and the balance appears as kinetic energy of the atom. These measurements can only be made at low temperatures, T ~ O.l K, where the mean free paths of the excitations are long. This is in contrast to condensation measurements which can be made over a wide temperature range (0.03 - 2 K). Whenthere is equilibrium between the vapour and liquid at higher temperatures we expect a high rate of condensing atoms which must be matched by evaporation. I f the quantum evaporation process is to account for this rate then we must show that the flux of excitations on the interface is s u f f i c i e n t l y high and has the correct temperature dependence. The evaporation rate also depends on the probability that an incident phonon or roton creates a free atom. We consider these points after reviewing the evidence for quantum evaporation at lower temperatures. 2. QUANTUMEVAPORATION 2.1 Phonon-atom Quantum evaporation whereby an excitation in the liquid is annihilated and a free atom is created implies that energy is conserved in the following way b~
= EB + ~2k2/2m
(1)
where ~ is the energy of the phonon or roton, EB is the binding energy of an atom in the l~quid and ~k is the momentum and m the mass of the free atom. The incident phonon or roton perturbs the potential well of the surface atom and causes a transition from a bound state to a free state. To demonstrate the v a l i d i t y of equation l i t is necessary to measure the
393
phonon energy and the kinetic energy of the free atom ~, as EB = 7.16 K is known from bulk latent heat measurements. We do this using the velocity dispersion of the phonons and atoms to distinguish between the broad spectrum of phonons created by the heater. I f the atoms travel a distance ~l and the phonons a distance ~2' then the total signal time is
tt
=
~i/Va + ~21Vg
(2)
where Va is the v e l o c i t y of the atom and VQ is the group v e l o c i t y of the phonons, which i~ a f u n c t i o n of the phonon energy as can be seen in figure I. 24
i - -
E
,
20 17.4-16 13"8->, 12
9~__ o
8-67-7.16 ~
8
4
0
i 0
0.8
i 1.6
wave
vector
I 2.4
(A-')
Figur e I: The dispersion curve for liquid 4He at the SVP and T = 1 K . The three regions of wave vectors are shown.
In figure 2 we show t t , for a particular ~I and ~2, as a function of-phonon energy calculated using dispersion data from neutron measurements (19) and equations l and 2. The minimum in figure 2 shows that the shortest signal time selects out a particular phonon energy. I f ~l is varied with ~l + ~2 constant then the minimum t t occurs for different phonon energies. The measurements are made at T < O.l K where the SVP is so low that the evaporated atom is not scattered and also the phonon mean free path is long, for ~ > wc (~9 K) I20, 21) as there is no scattering by thermal excitations. The phonons are produced by a ~l ~s current pulse in a thin metal film which raises i t s temperature to 2-3 K during the pulse. The atoms are detected by a superconducting Zn bolometer in a magnetic f i e l d and held at a constant temperature with electronic feedback (23,24,25). The time constant of the bolometer is ~l us. The Zn film and substrate is covered with a thin He film on which the atoms condense giving up
304
.I.I< (;. Rq'att / lh'uporatio/~ o./ liquid 41h"
8 ,--
/
65
,= ¢n :"
°# \"\• \
60
E
6 •
=: -o 55
\\\ \
\~"x
,
\,
\.
5
\, \
\,, 50 ~ 8
~
9
.
.
. 10
.
. 11
.
. 12
13
3
energy I K !
30
hal is skown as a funcb:'on 9f p:~ono<' ~vr'~7.~ ,",r ato?~: pat'*; lengt:,. 0 I) /. : ~ ~z r, l iq~':: ,: >at~ lengt/, (~,2) 4.0 ~mn.
k i n e t i c and b i n d i n g energy• The f a s t e s t s i g n a l s are measured, f o r d i f f e r e n t ~l w i t h ~l + L2 cons t a n t ~n an arrangement shown i n s e t in f i g u r e 3. The f i g u r e shows t y p i c a l s i g n a l s .
~
/
=
purified 4He
I !
4He with 3He
/
J
3He 4He 310 -
510
70
time
I:ps)
go
110
L~: Measured phonon-atom signal~ ar functions of time. Inset i:~ sha~n a schemat::~ ]iagram of the experimental arrangement. The times f o r the f a s t e s t s i g n a l s ( t t , m i n ) are shown in f i g u r e 4 t o g e t h e r w i t h t he computed times and i t can be seen t h a t there is good agreement. As a f u r t h e r check on the v a l i d i t y o f the experiment and i t s i n t e r p r e t a t i o n we added enough 3He to the 4He l i q u i d so t h a t a ~.25 monolayer coverage formed on the liquid He ~urface. The 3He are bound to the surface o f ~He (25) by 5.0 K (28). This t o g e t h e r w i t h the d i f f e r e n t mass o f 3He changes the f a s t e s t s i g n a l time a t each L l . In f a c t d i f f e r e n t phonon energies give t t , m i n f o r 3He
phonon- He
40
50
60
time [ p s '
and 4He at the same ':l. With 3He on the surface s i g n a l s due to both 3He and 4He atoms can be seen, as shown in f i g u r e 3. The f a s t e s t signal times f o r 3He are shown in f i g u r e 4 and again there is good agreement w i t h the calcul a t e d times. 2.2. Roton-atoms Rotons have much more momentum than phonons and t h i s suggests a c l e a r way o f d i s t i n g u i s h i n g between them in an e v a p o r a t i o n experiment. I f we assume t h a t the i n t e r f a c e between the l i q u i d and vacuum is t r a n s l a t i o n a l l y i n v a r i a n t then the component o f momentum p a r a l l e l to the surface must be conserved. So we have q sin ~, =
2
\,
l
phonon-3He
k sin :,
(3'I
where q and k are the wave vectors of the roton o r phonon and the atom r e s p e c t i v e l y , :~ and are the angles between the paths and the normal. The angle ¢ can be c a l c u l a t e d in terms of q using equations l and 3, and the d i s p e r s i o n curve. The r e s u l t s are shown in f i g u r e 5. The three regions correspond to the phonon and two roton regions d e f i n e d in f i g u r e I . I t can be seen t h a t f o r some values o f q there is only a l i m i t e d range o f angles of incidence (a) t h a t give an atom angle (~) which is ~90 °. I f no surface e x c i t a t i o n is created or any o t h e r h i g h e r o r d e r process occurs then the e x c i t a t i o n in the l i q u i d w i l l be t o t a l l y r e f l e c t e d . The c r i t i c a l angles o f incidence are shown in f i g u r e 6. I t can be seen t h a t there is a l a r g e range of phonon momenta f o r which there is no c r i t i c a l a n g l e , but at high phonon momenta and f o r a l l the roton range there is one. For low energy phonons, ~ ~ 7.16 K, the evaporated atom has very low energy and momentum and here again the atom cannot match the p a r a l l e l momentum of a phonon o t h e r than at near normal i n c i d e n c e .
A.F. G. Wyatt / Evaporation ofiliquid 4He i
energy (K) 140
~iI
T T T TT I l l
~ITITITI T
e 90~I~°50°40 ° 30 20~ i
90
395
q
i
13.1 lphonon_4H e
~120 12"5t 9"2 100 12-0
~o 1D
17"0
9-5 lto.2
_e
==
30
:
o'.4
o'-8
'1'-2 1'.6 wave vector (/~-')
~o
11~
,o
~o
angle ~ (degrees)
i
~.o
roton-4~/~o-o
15" 1 ~ 3 - 3 14"0
~o
10
h
16.3
10"7
80
S
~14
~.~
Figure 6: The angle (~) that the atom makes to the normal is shown as a function of the wave vector of the 4He excitations for various angles (0) which the excitation makes to the normal. The corresponding excitation energies are shown above.
7: Calculated total times for phonon-atom .atom-f~ton-atom V i signals • uare shown r as efunctions of angle (~) for excitation angle 6 = 15 ° . The corresponding energies of the excitations are given in degrees Kelvin. The path lengths for atomp and excitations are both 6.5 m~.
i
9O
60 o
.-_
ao
6 O0
0"4
018
112 1!6 2!0 wave vector (A-~)
214
~2"8
'
1o
'
go ' 12o ' 18o time (pS /
F i b r e . 8: Phonon-atom and roton-atom signals
Figure 6: The critical angle (ec) between the excitation and the normal shown as a function of excitation wave vector. For quantised evaporation @ must be less than @c"
are shown as functions of time for different atom angZes (t) for excitation angle @ = 13 ° . For ~ = 10, 14, 19 ° the signal is due to phonons and at 33 ° to rotons. Path lengths for atoms and excitations are both 6.5 ~n.
I t is worth noting that these phonons are below the 3pp cut o f f energy. As with phonons the roton v e l o c i t i e s are dispersed, and t t ~i~ corresponds to a p a r t i c u l a r roton energy.~'~h~ v a r i a t i o n of t , with angle t f o r an angle of incidence e = 15°~is shown in figure 7. I t can be seen that phonons evaporate atoms at an angle ~10° regardless of t h e i r energy whereas the d i f f e r e n t roton energies give a wide range of angles ¢. We use both the r e f r a c t i o n e f f e c t and the minimum signal time to d i s t i n g u i s h between phonon-atom and roton-atom evaporation. Typical traces at various angles ~ are shown in figure 8.
These were taken with a bolometer t h a t could be moved around on a radius arm by a superconducting stepping motor. The signal changes shape considerably between the phonon and roton evaporation regions. We would expect t h i s from the range of phonon and roton energies that contribute to the signals. In figure 7 we see that phonons just above the 3pp cut o f f give the fastest signal time. I f the population of t~e phonon states decreases r a p i d l y with energy then the signal at longer times is rapidly deminished, Howewr, the roton s i t u a t i o n is d i f f e r e n t , The fastest signals come from rotons with ~w ~ 14 K and so we expect a higher density
.I. t< (7. Ill Jr: / l-r~q;~,r~lH:,~ :>/liquid 4Ih
396
o f rotons at l o w e r e n e r g i e s which can c o n t r i b u t e a s u b s t a n t i a l s i g n a l at l o n g e r t i m e s . The m i n i mum s i g n a l t o t a l times f o r phonons and rotons agree v e r y w e l l w i t h those c a l c u l a t e d . To examine the a n g u l a r dependence we have i n t e g r a t e d the s i g n a l s at d i f f e r e n t angles o v e r t h e i r f i r s t 17 :,s. This is the time d i f f e r e n c e between the f a s t e s t r o t o n - a t o m and phonon-atom signals. The r e s u l t s are shown in f i g u r e 9.
the c r e a t i o n o f a f r e e atom by an i n c i d e n t e x c i t a t i o n , and s i m i l a r l y f o r the c r e a t i o n oF an e x c i t a t i o n by a condensing atom. On the vapour side o f the i n t e r f a c e we nave the number f l u x d~ a o f f r e e 4He atoms s t r i k i n g the i n t e r f a c e in the ranges Ea t o Ea + dEa, "a t o a + d~a and :a to :a + d;a
dl a
k2sin ........
a
cos . . . . . a . _ d- a d:
8~3h(exp(Ea/kT
6
i
10
20
30 angle
40 ~
50
60
I degrees
2 qi sin i cos i d.:. . . . . . . . . . . . e,i
8:~3(exp(h..,i/kT ~s{,,}L:'
¢
~cz'
~';i?<';','.,':~,(
•
<~z<~z+
dE a
.:4!
where Ea is the e n e r g y , ilk is the momentum and m is the mass o f the ~He atom. We use the B o s e - E i n s t e i n (B-E) d i s t r i b u t i o n f u n c t i o n and we s h a l l see i t is i m p o r t a n t not to approximate t o a Maxwell-Boltzmann d i s t r i b u t i o n . On the l i q u i d side o f the i n t e r f a c e we have the number f l u x d; e i o f e x c i t a t i o n s in the i ti: region of the momentum spectrum (see f i g u r e l , i = I , 2 or 3).
onon-4He
0
a
l)
~7+ =
, .<
u,;
.
The e v a p o r a t i o n due to phonons and rotons can e a s i l y be d i s t i n g u i s h e d . The widths o f the peaks are p a r t l y due t o the l i m i t e d c o l l i m a t i o n ~9 ° o f the g e n e r a t o r and d e t e c t o r . However, from f i g u r e 7 we see the range o f angles o f e v a p o r a t i o n is much l a r g e r f o r rotons than phonons and t h i s is c o n s i s t e n t w i t h the g r e a t e r w i d t h o f the r o t o n - a t o m peak in f i g u r e 9. From these r e s u l t s we can draw s e v e r a l conclusions. High energy phonons and r o t o n s in r e g i o n 3 can be generated by a h e a t e r , t h e y have long mean f r e e paths and they can e v a p o r a t e atoms from the s u r f a c e in a s i n g l e quantum process. A phonon or r o t o n is a n n i h i l a t e d and a f r e e atom i s c r e a t e d w i t h energy and p a r a l l e l momentum conserved• I t is c l e a r t h a t at l e a s t some atoms are e v a p o r a t e d w i t h o u t c r e a t i n g ripplons. We have n o t seen atoms e v a p o r a t e d by r o t o n s in r e g i o n 2. These would need to have t h e i r group v e l o c i t y d i r e c t e d towards the surface and hence t h e i r momentum d i r e c t e d towards the g e n e r a t o r •
3. EQUILIBRIUM BETWEEN LIQUID AND VAPOUR L i q u i d 4He at t e m p e r a t u r e '~,I K w i l l have a high d e n s i t y o f phonons and rotons in a l l regions. There w i l l be a continuous f l u x i n c i dent on the i n t e r f a c e which presumably e v a p o r ates atoms in the same way t h a t i t does under pulse c o n d i t i o n s . We now examine whether i t is p o s s i b l e to account f o r the h i g h l y t e m p e r a t u r e dependent s a t u r a t e d vapour pressure w i t h the s i n g l e quantum process d e s c r i b e d in Section 2. We s h a l l need to i n t r o d u c e p r o b a b i l i t i e s for
d.,
e,!
1'
• d:
e,l
.d . ] ;5)
where h ~ and hg i are the energy and momentum o f the p~onon or r o t o n . The group v e l o c i t y of the e x c i t a t i o n s e x a c t l y cancels the d q / d , fact o r in the d e n s i t y o f s t a t e s (17). In e q u i l i b r i u m the chemical p o t e n t i a l ( ; ) f o r the atoms in the vapour and l i q u i d must be the same• We have put : - 0 in both e q u a t i o n s 4 and 5. We must use the same energy zero f o r the two B-E d i s t r i b u t i o n s in e q u a t i o n s 4 and 5. C o n s i s t e n t w i t h ~ = 0 we r e f e r a l l e n e r g i e s tc the Bose condensate. The energy o f an e x c i t a t i o n is then by d e f i n i t i o n h i and the energy o f an atom in the vapour is Ea
h2k2,'2m + EB
,.61,
where EB is the b i n d i n q enerqy o f the atom to the l i q u i d . The two c o n s e r v a t i o n laws at the i n t e r f a c e r e l a t e angles and e n e r g i e s o f e x c i t a t i o n s in the l i q u i d t o those f o r atoms in the vapour. So f o r a given i n c r e m e n t a l s o l i d angle and energy f o r one momentum r e g i o n o f e x c i t a t i o n s in the l i q u i d , t h e r e is a unlque conjugate i n c r e m e n t a l s o l i d angle and energy f o r atoms in the vapour. E x c i t a t i o n s o n l y e v a p o r a t e atoms at c o n j u g a t e angles and e n e r g i e s and s i m i l a r l y f o r condensing atoms c r e a t i n g e x c i t a t i o n s . This is i l l u s t r a t e d in f i g u r e lOa. We choose to express the phonon and r o t o n f l u x e s in terms o f the c o n j u g a t e f r e e atom angles and e n e r g i e s . From e q u a t i o n s l , 3 and 6, and w r i t i n g h.~.a = Ea we have d i = d , a , d~e, i = d : a i and qi cos ~ e , i d ' e , i = k cos '!'ai d ' a i , s u b s t i t u t i n g i n t o e q u a t i o n s a and 5 we f i n d ;
A.F.G. Wyatt/ Evaporationof liquid4He (a)
condensation c o e f f i c i e n t = 1 at a l l temperatures we see t h a t these p r o b a b i l i t i e s can be independent of temperature. Let P i ( e a , wa) be the p r o b a b i l i t y t h a t an e x c i t a t i o n , in wave vector region i , which is i n c i d e n t on the i n t e r f a c e creates an atom at ea with energy ma" We assume t h a t i t is a f u n c t i o n of angle and energy but not of temperature. The f l u x of atoms created at the conjugate angle and energy is
(b)
i
2/,o.,, (c)
~~
Id~
397
/ i P'
3 i=l Figure 10 : a; Conjugate angles for atoms and excitations from one a wave vector region. ~,: The probability of an excitation evaporating an atom in the three wave vector regions is defined in terms of the conjugate atom angle (Oa). c: The probability of an incident atom creating an excitation in one of the three wave vector regions is defined in terms of the conjugate atom angle. d: The two processes which are required for detailed balance of excitations are shown with their probabilities.
d~e, i P i ( e a , ma)
The s i t u a t i o n is schematically shown in f i g u r e IOb. Let Aj(e a, ma) be the p r o b a b i l i t y that an atom i n c l d e n t on the i n t e r f a c e creates an e x c i t a t i o n in region j as shown in f i g u r e lOc. Then z~A. is the condensation c o e f f i c i e n t f o r the at~mJat angle e. and energy ma The loss of atoms by condensatlon in the incremental i n t e r v a l is .
.
d~a
=
cos e a
8~3(exp(~a +Ei/kT)-
dea d# a d~a
a I)
(7) k 2 sin eai cos eai d#e, i
=
deai d¢ai d~a 8~3(exp(h~[+-EB/kT) - I)
(8)
where the subscript i only reminds us of the q region to which the atom v a r i a b l e s are conjugate and w i l l be omitted from now on. From equations 7 and 8 we see t h a t conjugate f l u x e s , i n c i d e n t on the i n t e r f a c e , are i d e n t i c a l at a l l temperatures in each q region separately. This is an important r e s u l t as i t shows t h a t i f a l l the e x c i t a t i o n s in one q region, which have enough energy and are w i t h i n the c r i t i c a l angle, each evaporate an atom then t h i s e x a c t l y maint a i n s the energy and angular d i s t r i b u t i o n of atoms in the vapour i f a l l atoms i n c i d e n t on the i n t e r f a c e condense. This balance is maintained at a l l temperatures because both atoms and e x c i t a t i o n s obey B-E s t a t i s t i c s . As there are three regions o f e x c i t a t i o n s then the p r o b a b i l i t y t h a t an e x c i t a t i o n does evaporate an atom must be less than u n i t y . However, as the conjugate f l u x e s have the same temperature dependence and the
.
°
3
[ j=l
d~a Aj(e a, ma)
(10)
From the e q u a l i t y of the i n c i d e n t f l u x e s and requirement of dynamic e q u i l i b r i u m we f i n d 3
k2 sin e
(9)
A'(°a' j=l J ~a)
=
3 [ Pi i=l ( ° a ' ~a)
(ll)
We now consider d e t a i l e d balance f o r one of the types of e x c i t a t i o n s in the l i q u i d , say region 1 which is phonons. In t h i s model phonons i n c i d e n t on the i n t e r f a c e e i t h e r specul a r l y r e f l e c t or create free atoms. There must be an equal f l u x of phonons going towards and away from the i n t e r f a c e . Phonons going away from the i n t e r f a c e come from both phonons which are s p e c u l a r l y r e f l e c t e d and those created by condensing atoms. This is shown in f i g u r e lOd, from which we see A1 + (I - PI) = I , hence Ai(e a, Wa)
=
Pi(0a, ~a)
(12)
This r e s u l t is a p r e d i c t i o n of the model and c l e a r l y must be confirmed by measurement. As the condensation c o e f f i c i e n t is n e a r l y u n i t y f o r a l l atoms except those with low energy at glancing angles, we expect [P i = 1 f o r a wide range of energies and angles. The average p r o b a b i l i t y , averaged over conjugate angles, is therefore ~I/3. For an e x c i t a t i o n outside the c r i t i c a l angle P = 0 and i t is s p e c u l a r l y r e f l e c t e d back i n t o the l i q u i d . From the above analysis we see that the simple model can account f o r the d e t a i l e d processes needed to maintain dynamic e q u i l i b r i u m between the l i q u i d and vapour f o r most of the
,1./,] (,. ll'rdlt ' ErdlJ~ruti~m
398
energies and angles. However, there are two extreme and small groups o f atoms in the vapour t h a t do not f i t in w i t h t h i s p i c t u r e . Atoms w i t h energy ~lO K, which w i t h the b i n d i n g energy gives ~17 K, have more energy than the maximum in the e x c i t a t i o n spectrum so they presumably create more than one e x c i t a t i o n . The o t h e r group is around atom energies o f 1.2G K (27). This w i t h the b i n d i n g energy gives an energy of 8.42 K which is below the roton minimum (8.67 K) and so might be expected to create phonons. 5However, the momentum o f t h i s atom is 4.84 lO -2 mkgs-1 w h i l e the momentum o f 1~he corresponding phonon is 4.75 lOmkgs-~ (28). I f t h i s d i f f e r e n c e is real then an atom at g l a n c i n g angles (>80 o to the normal) canno~ create any bulk e x c i t a t i o n s in the l i q u i d ~He. The model also gives the pressure on the i n t e r f a c e which we now consider. When an atom condenses only the p a r a l l e l momentum is taken uQ by the e x c i t a t i o n so t h a t in the normal d i r e c t i o n there is a d i f f e r e n c e between k cos ~i ,a and q cos .,',e which has t o be s u p p l i e d by the surroundings. The ground s t a t e l i q u i d and c o n t a i n e r is given momentum ~(k cos ~!a - q cos ~e) downwards when an atom condenses o r evaporates. I f we consider a p a i r of processes in which an atom at k, Ua, ~a condenses and one at k, (~a and ¢a + ' evaporates then the change in momentum o~ the l i q u i d and e x c i t a t i o n s is 21Qk cos ,.ia, which of course is opposite t o the change in the momentum o f the vapour system. I f the atom s p e c u l a r l y r e f l e c t s from the i n t e r f a c e i n s t e a d of condensing, i t gives the same c o n t r i b u t i o n to the momentum change. The pressure on the l i q u i d is then from equation 7. P
[d,~ Ida, Ida, 2hk c°s 0
0
0
~ k2 sin
c°s'
873(exp (~'~a + EB/kT) - l )
(13) To e v a l u a t e t h i s i n t e g r a l we ignore the - l i n s i d e the brackets which makes a n e g l i g i b l e d i f f e r e n c e and f i n d .
.
.
.
exp h3
,
(14) kT ~
where m is the mass o f the 4He atom. This is the usual expression f o r the s a t u r a t e d vapour pressure. Expressed as an u n c e r t a i n t y in temperature i t is good to 5 lO -4 K up to T ~,. 1.5 K (29). 4, CONCLUSIONS We have reviewed the evidence f o r quantum e v a p o r a t i o n at low temperatures. We then showed t h a t the e q u i l i b r i u m d e n s i t y o f phonons
and rotons at h i g h e r temperatures gives a large enough f l u x i n c i d e n t on the i n t e r f a c e to account f o r the e v a p o r a t i o n rate at the s a t u r a t e d vapour pressure at T 2 K. The f l u x of e x c i t a t i o n s has e x a c t l y the same temperature dependence as the f l u x o f atoms which stems from the f a c t t h a t they both obey Bose-Einstein s t a t i s t i c s . It f o l l o w s from t h i s t h a t the p r o b a b i l i t y of a phonon o r roton e v a p o r a t i n g an atom is temperature independent. From c o n s i d e r a t i o n s of d e t a i l e d balance and from the f a c t t h a t the condensation c o e f f i c i e n t is a p p r o x i m a t e l y u n i t y we found t h a t the average p r o b a b i l i t y is , I / 3 . F i n a l l y we considered the change in momentum of the l i q u i d during e v a p o r a t i o n and condensation and showed t h a t the model gives the usual expression f o r the s a t u r a t e d vapour pressure as a f u n c t i o n of temperature. ACKNOWLEDGEMENTS I would l i k e to acknowledge the immense e x p e r i m e n t a l e f f o r t of my students F.R. Hope and M.J. Baird and valuable discussions w i t h them and D.M. L i v e s l e y , D.V. Osborne and W.F. Vinen. REFERENCES (1) M.J. B a i r d , F.R. Hope and A.F.G. Wyatt, Nature 304 (1983) 325. (2) F.R. Hope, M.J. B a i r d and A.F.G. Wyatt, Phys. Rev. L e t t . 32 (1984) 1528. (3) K.R. A t k i n s , B. Rosenbaum and H. Seki, Phys. Rev. l l 3 (1959) 751. (4) D.V. Osborne, Proc. Phys. Soc. 80 (1962) I03. (5) D.V. Osborne, Proc. Phys. Soc. 80 (1962) 1343. (b) G.H. Hunter and U.V. Osborne, d. Phys. C: Sol. St. Phys. 2 (1969) 2414. (7) W.D. Kessler and D.V. Osborne, J. Phys. C: Sol. St. Phys. 13 (1980) 1571. (8) D.O. Edwards, P.P. Fatouros, G.G. lhas, P.M. Rozinsky, S.Y. Shen, F.M. Gasparini and C.P. Tam, Phys. Rev. L e t t . 34 (1975) i153. (9) V.U. Nayak, D.O. Edwards and N. Masuhara, Phys. Rev. L e t t . 50 (1983) 990. (lO) D.O. Edwards, G.G. lhas and C.P. Tam, Phys. Rev. [5. l~ (In771 3122. ( l l ) P.M. Echenique and J.B. Pendry, Phys. Rev. L e t t . 37 (1976) 561. (12) D.O. Edwards and P.P. Fatouros, Phys. Rev. B. 17 (1978) 2147. (13) W.D. Johnston J r . and J.G. King, Phys. Rev. L e f t . 16 (1966) l l 9 1 . (14) J.G. King, J. McWane and R. T i n k e r , B u l l . of the Am. Phys. Soc. 17 (1972) 38. (15) P.W. Anderson, Phys. L e t t . A. 29 (1969) 563. (16) D.S. Hyman, M.O. S c u l l y and A. Widom, Phys. Rev. 186 (1969) 231. (17) M.W. Cole, Phys. Rev, L e t t . 28 (1972) 1622. (18) S. B a l i b a r , J. Buechner, B. Castaing,
A. k: G. Wyatt / Evaporation of liquid 4He
(19)
(2o) (21 (22) (23) (24)
(25) (2~)
(27)
(2~)
(2~)
C. Laroche and A. Libchaber, Phys. Rev. B 18 (1977) 3096. F.R. Hope, PhD Thesis 'Quantised 4He evaporation', University of Exeter (1983). R.C. Dynes and V. Narayanamurti, Phys. Rev. Lett. 33 (1974) 1195. A.F.G. Wyatt, N.A. Lockerbie and R.A. Sherlock, Phys. Rev. Lett. 33 (1974) 1425. R,A. Sherlock and A.F.G. Wyatt, J. Phys. E" Sci. I n s t r . 16 (1983) 669. R,A. Sherlock and A.F.G, Wyatt, J. Phys. E" Sci. Instr. 16 (1983) 673. R.A. Sherlock, J. Phys. E: Sci. I n s t r . 17 (1984) 386. A.F. Andreev, Sov. Phys. J.E.T.P. 23 (1966) 939. D.O. Edwards and W.F. Saam, Progress in low temperature physics, VoI. VIIA, ed. D.F. Brewer, (North-Holland, Amsterdam 1978) Ch. 4. D.V. Osborne, private communication. W.G. S t i r l i n g , Proceedings of the 75th Jubilee conference on helium-4, ed. J.G.M. Armitage (St. Andrews (1983)). p. 109. F.G. Brickwedde, J. Res. N.B.S. (1960) 644.
399