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Submitted on 1 Jan 1980

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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�

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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

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~ 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)

.

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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

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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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C 7 - 2 6 0 JOURNAL DE PHYSIQUE

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,. .

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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 .

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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

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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 :

(8)

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 most

important 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

(9)

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.

References

1. Stwalley, W.C., see footnote 15 in reference 4.

2. De Simone, C. and Maraviglia, B., Chem. Phys.

Letters 60 (1979) 289.

3. Guyer, R.A., and Miller, M . D . , Phys. Rev. Lett.

42 (1979) 1754.

-

4. Mantz, I.B., and Edwards, D.O., Phys. Rev. B20

(1979) 4518.

5. Miller, M.D., Phys. Rev. B18 (1978) 4730.

6. Certain, P. , J . Chem. Phys. 3 (1976) 3063;

Toennies, J.P., Welz, W . , and Wolf, G . , Chem.

Phys. Lett. 3 (1976) 5.

7. Hardy, W . N . , e t a l , ( t h i s conference).

8. S i l v e r a , I.F., and Walraven, J.T.M., Phys. Rev.

Lett. 44 (1980) 164; Walraven, J.T.M. and

(10)

S i l v e r a , I.F., Phys. Rev. L e t t . 44 (1980) 168.

Siggia, E., unpublished. We a r e g r a t e f u l t o Professor T. Greytak f o r i n f o r m a t i o n about t h i s c a l c u l a t i o n .

B e r l i n s k y , A.J., Phys. Rev. L e t t . 39 (1977) 359.

Edwards, D.O., Feder, J.D., and Nayak, V.U., Quantum F l u i d s and S o l i d s (1977), e d i t e d by Trickey, S.B., Adams, E. D., and Dufty, J.W., Plenum, New York, p. 375.

Kolos, W., and Wolniewicz, L., J. Chem. Phys.

43 (1965) 2429; Chem. Phys. L e t t . 24 (1974) -

457.

See Edwards, D.O., Shen. S.Y., Eckardt, J.R., Fatouros, P.P., and Gasparini, F.M., Phys. Rev.

812 (1975) 892, where v ' ( ~ ) i s d e f i n e d f o r

-

3 ~ e on t h e He surface. 4

Landau, L.D., and L i f s h i t z , E.M., S t a t i s t i c a l Physics (Addison-Wesley, 1974) p. 235.

E t t e r s , R.D., Danilowicz, R.L., and Palmer, R.W., J. Low Temp. Phys. 33 (1978) 305.

Wu, T.T., Phys. Rev. 115 (1959) 1390.

Edwards, D.O., and Saam, W.F., i n Progress i n Low Temperature Physics, Vol. V I I A, e d i t e d by D. F. Brewer (North-Holland, 1978) 300.

Bishop, D.J., and Reppy, J.D., Phys. Rev.

L e t t . 3 (1978) 1727.

Nelson, D.R., and K o s t e r l i t z , J.M., Phys. Rev.

L e t t . 2 ( 1 977) 1201.

Lantto, L.J., and Nieminen, R.M., J. Low Temp. Phys. 27 (1979) 1.

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