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THE ADSORPTION OF ATOMIC HYDROGEN ON THE SURFACE OF 4He
D. Edwards, I .B. Mantz
To cite this version:
D. Edwards, I .B. Mantz. THE ADSORPTION OF ATOMIC HYDROGEN ON THE SURFACE OF 4He. Journal de Physique Colloques, 1980, 41 (C7), pp.C7-257-C7-265. �10.1051/jphyscol:1980740�.
�jpa-00220178�
JOURNAL DE PHYSIQUE CoZZoque C7, suppZdmclis a! n o 7 , Tome 41, j u i Z Z e t 1980, page C 7 - 2 5 7
T H E ADSORPTION OF A T O M I C HYDROGEN ON T H E SURFACE OF 4 ~ e D.O. Edwards and I .B. Mantz
P h y s i c s Department, The Ohio S t a t e U n i v e r s i t y , CoZwnbus, Ohio 43210, USA.
R6sume.- Un c a l c u l de Mantz e t Edwards donne une v a l e u r minimale de 0,6 K pour 1 1 6 n e r g i e de l i a i s o n d ' u n atome d'hydrogene a l a surface de 4 ~ e 1 iquide. En consequence, l e s p r o p r i e t e s de 1 ' hydrogene p o l a r i s e (Ht) adsorbe, un f l u i d e de Bose bidimensionnel, p o u r r a i e n t S t r e b i e n t 8 t observes experimen- talement. Comme '+He e s t u t i l is 6 en t a n t q u ' e n d u i t p r o t e c t e u r des p a r o i s de c e l l u l e s contenant H+, 1 'a d s o r p t i o n s u r l a surface de '+He peut f a v o r i s e r 1 a recombinaison en mol6cules Hp. Pour une tempe- r a t u r e e t une d e n s i t 6 donnees du gaz, l a q u a n t i t e d'atomes H+ adsorbes d6pend fortement de l ' i n t e r - a c t i o n Ht-H+ a l a surface de 4He. Nous estimons c e t t e i n t e r a c t i o n grdce a une approximation simple, e t trouvons q u ' e l l e l i m i t e l a d e n s i t e atomique en s u r f a c e a une v a l e u r maximale de 1014/cm2. Toute- f o i s , c e t t e d e n s i t 6 d e v r a i t S t r e s u f f i s a n t e pour p e r m e t t r e l ' o b s e r v a t i o n d'une t r a n s i t i o n s u p e r f l u i d e bidimensionnelle, dans des c o n d i t i o n s v o i s i n e s de c e l l e s q u i o n t & t 6 r 6 a l i s B e s experimentalement.
Abstract.- The b i n d i n g energy o f a hydrogen atom t o t h e s u r f a c e of l i q u i d 4 ~ e has been c a l c u l a t e d t o be a t l e a s t 0.6K by Mantz and Edwards. T h i s means t h a t t h e p r o p e r t i e s o f adsorbed s p i n - p o l a r - i z e d hydrogen ( H t ) , a 2D Bose f l u i d , may soon be observed e x p e r i m e n t a l l y . Since 4 ~ e i s used as a n e u t r a l c o a t i n g f o r vessels t o c o n t a i n H+ gas, a d s o r p t i o n onto t h e 4He surface may enhance t h e r a t e o f recombination i n t o Hz molecules. For given temperature and d e n s i t y i n t h e gas, t h e amount o f H+ adsorbed depends s t r o n g l y on t h e H+-Ht i n t e r a c t i o n on t h e 4 ~ e surface. We estimate t h e i n - t e r a c t i o n i n a simple approximation, and f i n d t h a t i t l i m i t s t h e s u r f a c e d e n s i t y t o n o t more than
lo1'+ cm-*. Nevertheless t h i s d e n s i t y should be s u f f i c i e n t f o r a 2D s u p e r f l u i d t r a n s i t i o n t o be observed, under c o n d i t i o n s c l o s e t o those achieved e x p e r i m e n t a l l y .
1. I n t r o d u c t i o n . - The b i n d i n g energy o f atomic f i t t i n g t o v a r i o u s published and unpublished 6 hydrogen t o t h e free surface of l i q u i d He has been 4 experimental and t h e o r e t i c a l data. The b i n d i n g c a l c u l a t e d by several The c a l c u l a t i o n s energy cH i s q u i t e s e n s i t i v e t o v ( r ) , however of Guyer and M i l l e r (GM) and Mantz and Edwards (ME) t h e depth o f t h e minimum i n v ( r ) has r e c e n t l y take i n t o account t h e s t r u c t u r e o f t h e He s u r f a c e 4 been confirmed by Hardy and coworkers7 from d i f f - and a r e v a r i a t i o n a l i n nature. Both c a l c u l a t i o n s usion measurements.
g i v e a p p r e c i a b l e b i n d i n g , w i t h ME o b t a i n i n g t h e I n t h i s paper we have t r i e d t o e x p l o r e some l a r g e r values: cH - 0.6K f o r H, c D - 1.4K f o r D, o f t h e experimental consequencies o f t h e adsorp-
and c T - 1.8K f o r T. We b e l i e v e t h e ME r e s u l t s t o 4
t i o n o f s p i n - p o l a r i z e d hydrogen ( H t ) on t h e He be conservative, i n t h e sense t h a t t h e c a l c u l a t i o n surface. Since a t h i c k He f i l m i s used as a 4 8 was v a r i a t i o n a l , and because most o f t h e o t h e r n e u t r a l c o a t i n g f o r vessels t o c o n t a i n H+, t h e approximations and assumptions t h a t were made were p r o p e r t i e s o f "two-dimensional" ( 2 ~ ) H t can be such as t o reduce t h e estimated b i n d i n g energy. observed a t t h e same time t h a t t h e p r o p e r t i e s o f The most important d i f f i c u l t y i n accepting t h e ME t h e 3D H + Bose f l u i d a r e studied. As e x p l a i n e d r e s u l t s as a genuine lower l i m i t t o t h e b i n d i n g l a t e r i n t h e paper, we b e l i e v e t h a t s u p e r f l u i d i t y was t h e u n c e r t a i n t y i n t h e HeH i n t e r a t o m i c poten- i n t h e 2D H+ f l u i d w i l l be observed before i t i s t i a l v ( r ) . Both ME and GM used a Lennard-Jones seen i n t h e 30 system.
v(r), shown i n Fig. 1, obtained by M i l l e r by 5 A t f i r s t s i g h t i t m i g h t be thought t h a t
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980740
~ 7 - 2 5 8 JOURNAL D E P H Y S I Q U E
p;
HeHe, ]
Fig. 1. The i n t e r a t o m i c p o t e n t i a l s between H+ and Ht, H t and He, and He and He (lower graph). The upper graph shows t h e "difference p o t e n t i a l s " vd and v! d e f i n e d by equations (8) and (17).
t r e a t i n g t h e H t on t h e f i l m and i n t h e vessel as 2D and 3D i d e a l Bose gases should be a s u f f i c i e n t f i r s t approximation. I n t h i s case, i f n i s t h e number p e r u n i t volume i n t h e vessel and N' i s t h e
m* = m ( 3 )
From equation (1) i t i s c l e a r t h a t
v - E ~= -0.6K. (4)
T h i s means t h a t t h e 3D gas, a t temperatures below 0.3K, can never become degenerateor Bose-condensed if t h e 2D gas i s t r e a t e d as i d e a l . For t h i s rea- son we have w r i t t e n t h e chemical p o t e n t i a l f o r t h e 3D gas i n t h e d i l u t e o r Boltzmann form (2).
The d e t a i l e d p r e d i c t i o n s o f t h e i d e a l Bose gas model a r e shown i n F i g . 3, where we observe t h a t f o r a g i v e n n, t h e r e i s always a temperature a t which N'+ m , p r e v e n t i n g t h e chemical p o t e n t i a l
number p e r u n i t area on the He surface, t h e chem- 4 -
I I I I
i c a l p o t e n t i a l (which i s t h e same on t h e s u r f a c e as i n t h e 3D phase) i s
s 2
20: v= -eH+kBT l o g e { l - e x p l - N h /(2nm*kBT) 1 ) (1 )
3 3/2] -
3D: !A= kgT loge [nh /(2rmkgT)
.
( 2 )I
Here m i s t h e w b a r e n1assuof t h e H atom, w h i l e m* -5 0 10 20
d i s t h e e f f e c t i v e mass o f t h e adsorbed H4. The
e f f e c t i v e mass i s enhanced by t h e "backflow" o f
t h e He s u b s t r a t e . 4 However t h i s enhancement Fig. 2. Lower graph: The e f f e c t i v e p o t e n t i a l s f o r s i n g l e I t , Dt o r 3 ~ e atoms a t t h e s u r f a c e o f should be q u i t e small because t h e H4 bound s t a t e , l i q u i d He, c a l c u l a t e d i n Ref. 4. The upper
3 graph shows t h e normalized p r o b a b i l i t y d e n s i t y 42
i n c o n t r a s t w i t h t h a t o f He (see Fig. 2), i s i n t h e H+, D+ and 3 ~ e surface bound s t a t e s . The 4 dashed curve shows t h e d e n s i t y ( i n a r b i t r a r y s i t u a t e d mostly above t h e He l i q u i d . Throughout u n i t s ) o f t h e l i q u i d 4 ~ e .
t h i s paper we t h e r e f o r e assume
from i n c r e a s i n g above -0.6K, t h e assumed value o f -cH. The p r e d i c t i o n s of t h e model a r e c l e a r l y i n - c o r r e c t i n t h i s r e s p e c t b u t i t does make c l e a r t h a t t h e e f f e c t o f hydrogen-hydrogen (H+H+) i n t e r a c t i o n s i n t h e adsorbed 20 system must be v e r y i m p o r t a n t and should be included. A l a t e r p a r t o f t h i s paper i s concerned w i t h t h e e s t i m a t i o n o f t h e s o - c a l l e d
" e f f e c t i v e i n t e r a c t i o n " v S ( 0 ) on t h e surface. We f i n d t h a t , under reasonable experimental conditions,
~ ' ( 0 ) l i m i t s t h e adsorbed d e n s i t y NS t o a maximum o f l o J 4 cm-', and t h a t u can indeed r i s e t o be c l o s e t o zero, so t h a t Bose condensation i n t h e 3D gas can be observed.
Before l e a v i n g t h e i d e a l 2D and 30 gas app- r o x i m a t i o n we should p o i n t o u t t h a t , i n w r i t i n g equations ( 1 ) and (2), we have assumed o n l y one species o f boson t o be present. I n f a c t t h e low- e s t two atomic s t a t e s o f t h e hydrogen atom a r e separated by a very small energy. I n zero magne- t i c f i e l d t h e two l o w e s t s t a t e s a r e t h e ground s t a t e w i t h t o t a l s p i n F=O, and t h e f i r s t e x c i t e d s t a t e w i t h F=l. These a r e separated by t h e hyper- f i n e s p l i t t i n g , a = 67.6 mK. I n h i g h f i e l d t h e two l o w e s t s t a t e s correspond approximately t o
(me = -112, mp = +1/2), w i t h a s e p a r a t i o n 6=2p H+a/2. I n a f i e l d o f 100 kgauss 6 2 54 mK.
P
The r e l a x a t i o n t i m e a t low temperatures from one n u c l e a r s t a t e t o t h e o t h e r i s n o t known, although i t has been estimated by .Siggia9 t o be about 10 minutes. For h i g h d e n s i t i e s t h e two h y p e r f i n e s t a t e s become p a r t o f t h e e x c i t a t i o n spectrum o f t h e Bose f l u i d , as discussed by ~ e r l i n s k y " f o r t h e 3D f l u i d . I n view o f t h e u n c e r t a i n t y i n t h e r e l a x a t i o n t i m e we cannot y e t be c e r t a i n t h a t t h e two h y p e r f i n e s t a t e s w i l l be i n thermal e q u l i b r i u m w i t h each other, so i t seems reasonable t o f o l l o w t h e p r a c t i c e o f some o t h e r authors and discuss
F i g . 3. The 2D surface number d e n s i t y o f H+, NS, versus t h e temperature, i n t h e approximation t h a t b o t h t h e 2D and 3D f l u i d s a r e i d e a l Bose gases.
The curves a r e a t given values of t h e number den- s i t y i n t h e 3D phase, n, and a t given chemical p o t e n t i a l s , u. They were c a l c u l a t e d assuming t h a t t h e b i n d i n g energy t o the surface EH= -0.6K. The curve corresponding t o u = - ~ E H = - .8K d i v i d t h e graph i n t o regions where N s > n ~ / ~ and NScn$33.
As discussed i n t h e t e x t t h e d e n s i t i e s NS and n i n t h i s diagram a r e of e i t h e r hyperfine s t a t e , n o t t h e t o t a l number d e n s i t i e s o f both states.
t h e s t a t i s t i c a l mechanics of each species o f atom separately. I n (1) and ( 2 ) and i n Fig. 3, n and NS t h e r e f o r e represent e i t h e r nl and N; o r n and
2 N;, t h e number d e n s i t i e s o f atoms i n t h e two hyper- f i n e s t a t e s , b u t n o t t h e t o t a l d e n s i t i e s . The t o - t a l d e n s i t i e s ntot=(n +n ), = (N?+I~;). I f
1 2
t h e r e l a x a t i o n t i m e i s s h o r t enough t h e populations o f t h e two s t a t e s a r e r e l a t e d by t h e a p p r o p r i a t e temperature-dependent f a c t o r (see s e c t i o n 5 below).
I n t h e i d e a l gas model t h e two kinds o f atoms a c t independently and, i f t h e d e n s i t i e s i n t h e 3D gas a r e d i f f e r e n t , they w i l l Bose-condense a t two d i f f e r e n t temperatures. This i s c o r r e c t f o r the i d e a l gas model, b u t when i n t e r a c t i o n s a r e taken i n t o account, t h e system must be considered as an i n t e r a c t i n g ml'xture of two kinds o f bosons.
Before c o n s i d e r i n g t h e problem o f hydrogen- hydrogen i n t e r a c t i o n s on t h e He surface, we b r i - 4 e f l y describe t h e ME c a l c u l a t i o n s f o r one Id+ on He. 4
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2. One Ht Atom on t h e Surface o f 4 ~ e . - The t r i a l wave f u n c t i o n f o r 1 H4 atom, a t p o s i t i o n and
4 + +
N-1 He atoms, a t r2...rN, i s
+ + +
IJJ(;~.. .GN) = f ( r l ) I J J ~ ( ~ ~ . . . r N ) . ( 5 ) This i s o f t h e Feynman form; t h e r e a l p o s i t i v e f u n c t i o n I J J ~ i s t h e ground s t a t e o f N 4 ~ e atoms, i n c l u d i n g a f r e e surface. We assume t h e s u r f a c e t o be i n t h e neighborhood of z=O. The t r i a l func- t i o n IJJ has t h e same c o r r e l a t i o n s between t h e H4
4 4
atom and t h e He as between He i n t h e pure 4 ~ e ground s t a t e .
The Hamiltonian f o r one H t and N-1 4 He i s
where
m and m4 a r e t h e H and He masses, and Ho i s t h e 4 Hamiltonian f o r N He atoms. 4 The f u n c t i o n v d ( r ) i s t h e so c a l l e d d i f f e r e n c e p o t e n t i a l : t h e HeH i n t e r a t o m i c p o t e n t i a l v ( r ) minus t h e HeHe poten- t i a l v 0 ( r ) :
This i s i l l u s t r a t e d i n Fig. 1.
The Euler-Lagrange equation which minimizes the energy can be w r i t t e n i n t h e form o f a s i n g l e - p a r t i c l e Schrodinger equation w i t h "wave f u n c t i o n "
) i f t h e f o l l o w i n g s u b s t i t u t i o n i s made:
..-
f G 1 ) = $(;l)/d~@l). (9)
Here p(Fl ) i s t h e number d e n s i t y i n t h e He ground 4 s t a t e :
(Fl ) = N /I$: d?2. . . ~F~/JIJJ: ,;d . . . drN. + (10)
The p r o b a b i l i t y d e n s i t y f o r f i n d i n g t h e H+ atom a t -+ rl i s
/
1 e 12d7,. .(11 1
so 4 r e a l l y does have t h e p r o p e r t i e s o f a wave f u n c t i o n .
The e f f e c t i v e p o t e n t i a l v(F1 ) = V(zl ) which appears i n t h e Schrodinger equation f o r $ depends on various p r o p e r t i e s o f t h e He surface, as ex- 4 p l a i n e d i n ME. I t i s given by
where L4 i s t h e b i n d i n g energy o f He i n i t s ground 4 s t a t e and where we adopt t h e convention ~ ~ 5 " ; ~ ) ~ g12 : g(;l ,F2) e t c . The c o r r e l a t i o n f u n c t i o n g12 i s r e l a t e d t o t h e t w o - p a r t i c l e d e n s i t y i n t h e He 4 ground s t a t e :
The He k i n e t i c energy p e r atom tl=t(;l) 4 i s d e f i n e d
J 2 +
2 -+tl= - M / h 4 ) $o~l I J J ~ dr2.. . d ~ ~ / / " d ? ~ . . .drN. (14) The e f f e c t i v e p o t e n t i a l V1 i s i l l u s t r a t e d i n F i g . 2, which a l s o shows t h e hydroqen bound s t a t e wave f u n c t i o n $(zl ) . The energy i s - E ~ -0.6K.
E x c i t e d hydrogen surface s t a t e s have = .+ +
$(z,)e'k'rl, w i t h f p a r a l l e l t o t h e s u r f a c e and 2 2
energy - E ~+ -fi k /2m.
The t r i a l f u n c t i o n used by Guyer and M i l l e r 3 (GM) i s , i n p r i n c i p l e , more general than ( 5 ) s i n c e i t a l l o w s t h e c o r r e l a t i o n s between H4 and He t o 4
4 4
be d i f f e r e n t from those between He and He. On t h e o t h e r hand, i n o r d e r t o make t h e c a l c u l a t i o n more t r a c t a b l e , GM assumed t h a t t h e c o r r e l a t i o n s were t h e same i n t h e r e g i o n o f t h e s u r f a c e as i n t h e i n t e r i o r o f t h e He l i q u i d . This assumption 4 was n o t made by ME. The two c a l c u l a t i o n s a l s o d i f f e r w i t h r e s p e c t t o o t h e r p r o p e r t i e s o f t h e 4 ~ e surface, t and pl .
3. Two H t Atoms on t h e Surface o f 4 ~ e . - The
problem o f c a l c u l a t i n g t h e i n t e r a c t i o n between 3 ~ e atoms adsorbed on t h e surface o f He was discussed 4 by Edwards, Feder and Nayak." We apply t h e i r approach t o H + by f i r s t c o n s i d e r i n g two s p i n - p o l a r - i z e d hydrogen atcms on t h e surface, represented by a t r i a l wave f u n c t i o n
$o i s t h e He ground s t a t e f o r N atoms and f12 4 stands f o r f(;l, -+ r 2 ) . The Hamiltonian i s Ho+AH w i t h
H H H
I n t h i s expression vd12 = v ~ ( / ; ~ - : ~ / ) and v d ( r ) i s t h e d i f f e r e n c e between the H+-Ht i n t e r a t o m i c po- t e n t i a l v ( r ) (Ref. 12) and t h e He-He p o t e n t i a l H v o ( r ) :
H H
v d ( r ) = v ( r ) - v o ( r ) .
These p o t e n t i a l s a r e i l l u s t r a t e d i n F i g . 1.
An e f f e c t i v e 2 - p a r t i c l e wave f u n c t i o n and Schrodinger equation a r e o b t a i n e d using the sub- s t i t u t i o n :
where o12 i s t h e two p a r t i c l e d e n s i t y i n t h e 4 ~ e ground s t a t e given by (13). The e f f e c t i v e poten- t i a l V12 i n t h e 2 - p a r t i c l e Schrodinger equation i s t h e sum o f t h e e f f e c t i v e p o t e n t i a l s f o r p a r t i c l e s 1 and 2 separately, V1 + V2 = v ( ? ~ ) + v ( ? ~ ) , where V ( r ) i s t h e p o t e n t i a l found by ME; and an a d d i t i o n a l
" i n t e r a c t i o n " A V , ~ :
The l a t t e r i s given by
The l a s t two terms a r e those found i n Ref. 9 f o r 3 ~ e on He; t h e f i r s t two a r i s e because o f t h e 4 d i f f e r e n c e s i n t h e i n t e r a t o m i c p o t e n t i a l s between H+ and He. I n t h e second term t h e t r i p l e c o r r e l a - t i o n f u n c t i o n i n t h e He ground s t a t e i s d e f i n e d by 4
I n t h e l a s t term t h e k i n e t i c energy t / l ) i s d e f i n e d by
The expression f o r A V i n (20) i s t o o compli- ~ ~ cated t o be u s e f u l . However, we have s t u d i e d i t s asymptotic behavior when and F2 a r e f a r above t h e l i q u i d ( b u t n o t n e c e s s a r i l y f a r from each o t h e r ) . When t h e two i m p u r i t y atoms a r e helium i s - otopes ( o f any mass) then t h e terms i n vd and vd H must be o m i t t e d and one f i n d s :
aV12 " V 012 (adsorbed 3 ~ e ) . (23)
For two H+ atoms, i n c l u d i n g t h e d i f f e r e n c e poten- t i a l ,
fl12-* (vd12 H + Vo12) = vY2 (adsorbed H+), (24)
j u s t as one m i g h t expect. T h i s suggests a simple approximation: s i n c e t h e hydrogen "wave f u n c t i o n "
4 i n F i g . 2 i s m o s t l y q u i t e f a r above t h e l i q u i d
JOURNAL DE PHYSIQUE
surface, we rep1 ace AV1 by t h e "bare" i n t e r a t o m i c p o t e n t i a l v12: H
I n what f o l l o w s we adopt (25) as a working hypo- t h e s i s t o p r e d i c t t h e number d e n s i t y N' i n t h e adsorbed phase.
4. E s t i m a t i o n o f t h e Surface S c a t t e r i n g Amplitude vS(0).- To c a l c u l a t e tHe chemical p o t e n t i a l i n t h e 2D s u r f a c e phase of H4 we need t h e s u r f a c e s c a t t - e r i n g amplitude ~ ' ( 0 ) o r s u r f a c e " e f f e c t i v e i n t e r - action".13 To o b t a i n vS(0) we c o u l d t r y t o s o l v e t h e 2 - p a r t i c l e Schrodinger equation f o r 012 and o b t a i n t h e ground s t a t e energy o f two H+ atoms on a 4 ~ e s u r f a c e o f area A. T h i s energy i s - 2 r H + v S ( 0 ) / ~ . I n s t e a d we use an approximate wave f u n c t i o n and c a l c u l a t e t h e e x p e c t a t i o n value o f t h e energy, u s i n g t h e approximation (25) f o r AV12. An obvious
choice i s 012 = corresponding t o
f12 = flf2 i n t h e N-body t r i a l f u n c t i o n (15). T h i s leads t o
p r o v i d e d t h a t $ has been normalized (as i n Fig. 2) so t h a t :
However, s i n c e we a r e assuming t h a t AV12 = v12 and H t h a t t h e H atoms a r e m o s t l y above t h e l i q u i d sur- face, a b e t t e r choice f o r 012 i s perhaps 0 1 0 2 ~ , where g12 i s t h e c o r r e l a t i o n f u n c t i o n i n t h e 3D H+ H gas i n t h e d i l u t e l i m i t . T h i s corresponds t o f12 = f l f 2 G i n t h e N-body t r i a l wave f u n c t i o n (15). The s c a t t e r i n g amp1 i t u d e becomes
By changing t o center-of-mass and r e l a t i v e co- o r d i n a t e s and performing t h e i n t e g r a t i o n s w e o b t a i n :
Here
which i s t h e 3D s c a t t e r i n g amplitude i n 3D H4 gas, and
so t h a t
F(0) i s the average p r o b a b i l i t y d e n s i t y i n t h e H4 bound s t a t e ; we o b t a i n i t s v a l u e using t h e r e s u l t s o f ME.
The 3D s c a t t e r i n g amplitude V(0) i s propor- t i o n a l t o t h e s c a t t e r i n g length14 a:
We f i n d a = + 0.88i and V(0) = 7.4 x e r g cm 3 from t h e Monte C a r l o c a l c u l a t i o n s o f E t t e r s , Danilowicz and palmer15 f o r t h e ground s t a t e energy o f H4 gas as a f u n c t i o n o f t h e d e n s i t y n. I n c l u d i n g t h e l a r g e s t value o f d e n s i t y c a l c u l a t e d by E t t e r s e t a l , n = 3 x lo2' cm 3 , t h e energy was found t o f i t t h e f i r s t f o u r terms i n t h e expansion g i v e n by T.T. wu16 very w e l l . S u b s t i t u t i n g i n (29) g i v e s
It i s i n t e r e s t i n g t o compare t h i s w i t h t h e e x p e r i -
3 4
mental value17 f o r He on He:
5. Discussion: 2D S u p e r f l u i d i t y i n Adsorbed H+.-
To f i r s t o r d e r i n t h e i n t e r a c t i o n t h e chemical po- t e n t i a l s i n t h e i n t e r a c t i n g , d i l u t e 2D and 3D gases a r e :
The function P ~ ~ ( ~ , T ) i s the chemical potential in an ideal 3D gas of number density n. I t i s zero below the Bose-Einstein condensation temp- erature Tc, where
The subscript i = 1 o r 2 in (36) and (37) r e f e r s t o the two nuclear spin s t a t e s so t h a t si = 0,
= 6 = 2~ H + 4a ' 54mK a t ti = 100kG.
P
I f the relaxation time i s short enough and the two spin s t a t e s a r e in thermal equilibrium then p1=p2; i f not, presumably t h e i r number d e n s i t i e s can be varied independently by s u i t a b l y preparing t h e sample.
The r e s u l t s of applying (36) and (37), assuming t h a t p 1 = u 2 , a r e shown in Fig. 4. In t h i s graph we have plotted the t o t a l number den- s i t i e s NS =
N;
+ N; and n = n, + n2. The mostimportant difference from Fig. 3 i s t h a t now t h e density on the surface never r i s e s above
NS 2 1.1 x l o T 4 cm-', and t h a t t h e chemical po- t e n t i a l does reach values near p=O. Of course the d e t a i l e d numerical r e s u l t s in Fig. 4 depend r a t h e r strongly on the value of the surface bind- ing energy, cH = 0.6K, t h a t we have assumed. I t must be remembered t h a t the calculations of ME indicate t h a t 0.6K i s a lower l i m i t f o r the bind- ing energy. However, t h e value of t h e maximum density on the surface, NS = E H / ~ S ( ~ ) , depends
Fig. 4. Similar t o Fig. 3, but including H + - H t
~ n t e r a c t i o n s t o f i r s t order. The number d e n s i t i e s Ns and n in t h i s figure a r e t h e t o t a l number den- s i t i e s including both hyperfine s t a t e s , assuming t h a t they a r e in thermal equilibrium with each other. The l i n e marked T!:~) i s the upper l i m i t f o r the 2D superfluid t r a n s i t i o n temperature f o r adsorbed H+ atoms in the lower hyperfine s t a t e . I t was calculated from N i using (41) with bl=O.
only l i n e a r l y on c H .
For t h e ranges of n and NS i l l u s t r a t e d i n Fig.
4 the e f f e c t of t h e term nV(0) i n the 3D gas i s very small, although the corresponding term N'v'(o) = N ~ F ( O ) V ( O ) has a l a r g e e f f e c t on the 2D gas. This i s because the "equivalent" density per unit volume in the adsorbed phase F(O)N' becomes q u i t e large. For a 2D density of 10 14 cm-2 3
roughly the l a r g e s t value a t t a i n a b l e f o r the con- d i t i o n s shown i n Fig. 4, the equivalent density per unit volume F(O)N' i s loz1 Substituting F(O)N' f o r n in equation (38), the equivalent Bose t r a n s i t i o n temperature Tc would be 1.6K! This leads us t o conjecture t h a t adsorbed H may be superfluid a t d e n s i t i e s and temperatures which a r e already experimentally accessible. The 20 tran-
s i t i o n temperature should follow the universal Kosterlitz-Thouless equation:
where i s the superfluid mass per unit area j u s t
S C
JOURNAL DE PHYSIQUE
(2D). Equation (39) i s below the t r a n s i t i o n a t Tc
not, of course, enough t o predict the 2D t r a n s i - t i o n temperature; i t gives only an upper l i m i t , since p S C 5 NSm.
To predict TtPD) we need t o know the r e l a t i o n between pSC and NS. For HI. on He i t seems rea- 4 sonable t o suppose t h a t the superfluid density a t T=O i s the t o t a l density of the HI.:
Therefore t o c a l c u l a t e T ! ~ ~ ) we need the r a t i o between pS(0) and p S C , which we define a s (1 + b ' ) E pS(0)/pSC. Then
For thin superfluid films i t i s found t h a t due t o the thermal excitation of bound vortexpal'rs, and f o r temperatures not too f a r below T r D ) :
The dimensionless constant b i s not universal.
Bishop and ~ e p p y ' ~ found the number b t o be about 5, f o r a He film having a Tc of -lK. Nelson and 4
~ o s t e r l i t z ' ~ calculated b-0.5 from the XY model.
Extrapolation of (42) t o T=O (where i t i s not c o r r e c t ) indicates t h a t b'-b. I f b ' - 5 , (41 ) gives T ! ~ ~ ) - 0.1 K f o r NS - l 0 l 4 c i 2 . However a real- i s t i c value of b' can only be obtained from a microscopic model of t h e H t film, taking i n t o account the contribution of the elementary exci- t a t i o n s , including the bound vortex p a i r s , t o t h e normal f l u i d density. In Fig. 4 we show the upper l i m i t f o r T!:~) i n the 2D HI. f l u i d (corresponding t o the t r a n s i t i o n i n the lower hyperfine s t a t e ) , obtained by putting bl=O in (41). Since we have assumed thermal equilibrium between the two hyper- f i n e s t a t e s , u, = u 2 , the superfluid t r a n s i t i o n
does not occur in f l u i d 2 because i t s density
N;
i s never large enough.
As Fig. 4 shows, unless b' i s very large, i t seems t h a t the 20 t r a n s i t i o n T!:~) i s experiment- a l l y more accessible than t h e 30 Bose condensation.
For example, the 3D t r a n s i t i o n f o r n=1018 i s a t Tc = 16mK.
In conclusion we point out t h a t t h e equiva- l e n t H 4 d e n s i t i e s F(O)N' discussed here, which a r e not g r e a t e r than -10 2 1 would be s t a b l e acc- ording t o Berlinsky's c r i t e r i o n l O , in t h e bulk f l u i d a t O K in moderate magnetic f i e l d s . Using Berl insky' s formula, Lantto and ~ i e m i n e n ~ ' have calculated t h a t the minimum f i e l d f o r s t a b i l i t y a t lo2' i s about 40kG.
Acknowledgements
Thanks a r e due t o t h e U.S. National Science Foundation f o r financial support (Grant number DMR 7901073-01); and t o T. Greytak, C.V. Heer, V.U. Nayak, B . R . Patton and W.F. Saam f o r i n t e r - e s t i n g and useful discussions.
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