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SMALL CLUSTERS OF PURE AND ADMIXED H ↑ , 3He ↑ , T ↑ , AND 4He ATOMS
T. Lim, S. Nakaichi, Y. Akaishi, H. Tanaka
To cite this version:
T. Lim, S. Nakaichi, Y. Akaishi, H. Tanaka. SMALL CLUSTERS OF PURE AND ADMIXED H ↑ , 3He ↑ , T ↑ , AND 4He ATOMS. Journal de Physique Colloques, 1980, 41 (C7), pp.C7-189-C7-190.
�10.1051/jphyscol:1980731�. �jpa-00220168�
JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6rnent au n o 7 , Tome 41, j u i Z Z e t 1980, page C 7 - 1 8 9
SMALL CLUSTERS. OF PURE AND ADMIXED H + ,
3 ~ e +, T+ AND
4 ~ eATOMS
T.K. L i m , S. Nakaichi*, Y. ~ k a i s h i * and H. Tanaka
*
DrexeZ U n i v e r s i t y , Philadelphia, Pa. 19104, USA.
X Hokkaido U n i v e r s i t y , Sapporo 060, Japan.
Resume.- Nous rapportons l e s r d s u l t a t s de c a l c u l s v a r i a t i o n n e l s p r e c i s du type Faddeev-UPE en v u t de determiner l l e x i s t e n c e de p e t i t s amas d'atomes H + , 3 ~ e + , T+ e t 4 ~ e en composes p u r s ou mixtes d deux ou t r o i s dimensions. On montre que
i) l e s amas ( 4 ~ e ) N - ( ~ + ) ne peuvent e x i s t e r que s i N e s t s u p e r i e u r d 2 0 .
n .
ii) l e s compos6s ( T + ) , e x i s t e n t pour des v a l e u r s de n 5 p a r t i r de 3 . iii) l e s amas bosoniques ( 3 ~ e ) e x i s t e n t pour d e s n n s u p e r i e u r s 5 6.
Abstract.- The r e s u l t s of a c c u r a t e v a r i a t i o n a l and e x a c t Faddeev-UPE c a l c u l a t i o n s t o determine t h e e x i s t e n c e o f small c l u s t e r s of admixed and pure H + , 3 ~ e + , T+ and 4 ~ e atoms i n two and t h r e e dimen- s i o n s a r e r e p o r t e d . Evidence i s found t h a t
i) ( 4 ~ e ) N - (Hf) can e x i s t only i f N i s l a r g e r than 2 0 . ii) (Tf), systems e x i s t with n as small a s 3 , and iii) bosonic c3He) c l u s t e r s e x i s t f o r n g r e a t e r than 6.
1NTROJIUCI'ION.- The physics of small c l u s t e r s o f rare-gas atoms has received well-deserved a t t e n - t i o n r e c e n t l y . /1-4/ Ue have p a r t i c i p a t e d f u l l y i n t h i s resurgence o f i n t e r e s t i n t h i s a r e a of r e s e a r c h using a s t o o 1 s . a v a r i a t i o n a l technique c a l l e d ATMS and Faddeev theory. Thus we have a l - ready t r e a t e d t h e 3-, 4- and
5-
atom systems of pure and admixed 3 ~ e and He, and e s t a b l i s h e d 4 t h e e x i s t e n c e of a number o f t h e s e molecules when t h e y i n t e r a c t p a i r w i s e through r e c e n t l y proposed phenomenological helium-helium p o t e n t i a l s . /2/Now with t h e r e a l i z a t i o n t h a t systems of spin- p o l a r i z e d hydrogen and 3 ~ e atoms e x h i b i t quantum behavior even more pronounced than He 4
/5/
wehave decided t o t u r n o u r a t t e n t i o n t o t h e study o f t h e s e spin-polarized quantum systems. I n t h i s paper we r e p o r t t h e r e s u l t s o f o u r c a l c u l a t i o n s t o determine t h e e x i s t e n c e o f s m a l l c l u s t e r s o f admixed and pure Ht
,
3 ~ e ! , Tt,
and He atoms i n 4 two a n d - t h r e e dimensions. O u r s t u d y of t h e s e molecules i n reduced dimensions stems from t h e suggestion of Lantto and Nieminen /6/ that 2 .systems be i n v e s t i g a t e d more thoroughly s i n c e wall-surface e f f e c t s o f t h e containment v e s s e l may be s i g n i f i c a n t i n determining t h e s t a b i l i t y o f H f .
THE ATMS ISiTHOD AED FADD2EX-UPZ THEORY
.-
Ue l a y o u t h e r e s h o r t d e s c r i p t i o n s of t h e two methods we have used i n o u r i n v e s t i g a t i o n s .The ATHS Method; T h i s v a r i a t i o n a l method was devised by two of u s ( ~ k a i s h i and ~ a n a k a ) with o t h e r c o l l a b o r a t o r s f o r use i n t h e study of t h e few-body systems of n u c l e a r physics./ ?/ The tech- nique, l a b e l l e d ATPIS by u s , d i f f e r s from t r a d i - t i o n a l ways of c o n s t r u c t i n g t r i a l f u n c t i o n s i n t h a t i t s operandi i n c l u d e s t h e d e r i v a t i o n of two-body c o r r e l a t i o n f u n c t i o n s which a r e i n c o r - p o r a t e d e x p l i c i t l y i n t o t h e f i n a l v a r i a t i o n a l wavefunction. Used w i t h t h e Temple formula, ATNS can y i e l d an a c c u r a t e lower bound f o r t h e bind- i n g energy. The i n t e g r a l s which appear in o u r eva- l u a t i o n of t h e energy e x p e c t a t i o n v a l u e a r e com- puted w i t h a quasirandom method a f t e r a s u i t a b l e transformation o f v a r i a b l e s . Me took 50 000 sam-
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980731
JOURNAL DE PHYSIQUE
p l i n g p o i n t s t o b e s u r e t h a t t h e i n t e g r a l s a r e converged.
The Faddeev-UPE Method: I n t h i s method t h e cou- pled i n t e g r a l equations a r i s i n g from t h e Faddeev formulation of t h e few-body problem a r e solved through t h e device of representing t h e two-body l o c a l p o t e n t i a l by a s e r i e s of separable terms.
I n p r i n c i p l e , t h i s technique i s exact. However, t r u n c a t i o n of t h e partial-wave expansion and of t h e number of terms i n t h e separable expansion a s w e l l a s t h e l i m i t a t i o n s on t h e number of qua- d r a t u r e points i n t h e computation of i n t e g r a l s render Faddeev-WE l e s s than exact.
RESULTS.- Our r e s u l t s can be summarized a s f o l l - ows a
i ) We have performed a de%ailed and systematic calculation of t h e ( 4 ~ e ) ~ - ( H t ) systems and searched, i n p a r t i c u l a r , f o r t h e maximum value of
1
= fi2/14,, f o r which each of t h e s e systems has a bound s t a t e . Table 1 i l l u s t r a t e s our r e s u l t s f o r Lennard-Jones p o t e n t i a l s .Table
1
-
2D 3D
N ? , = ( ~ t ) i n ~ ~ ~ N
qnax!~?
i nu2
4 1 8 4 23
6 l&l 6 2321
8 1&2 8 243-2
I t i s obvious t h a t a s N i n c r e a s e s , t h e r e is l i t t l e change i n t h e value of ?),,(Ht ) i n 2D. I n 3D, t h e change i s barely discernible. The extracted values of
lmax
(Hf ) a r e very f a x from t h e physical value of 47.73 KA2. The t r e n d i n o u r r e s u l t s l e a d s us t o believe t h a t N has t o be a t l e a s t 20 before a bound molecule w i l l appear. O u r c a l c u l a t i o n s indi- c a t e t h a t t h i s conclusion is n o t a l t e r e d by t h e in- clusion of more atoms of Ht.i i ) ( ~ t ), systems e x i s t f o r n a s small a s
3.
I n f a c t , f o r 3D, we have found binding energies of 0.042 K and 0.243 K f o r t h e ground s t a t e s of t h e n = 3 and n = 4 systems respectively. I n add- i t i o n , t h e r e is an excited s t a t e in (Tt)3 which we o r i g i n a l l y suspected t o be an Efimov s t a t e . / 8 / However it f a i l s t o manifest t h e c h a r a c t e r i s t i c s expected of one.iii) When we assume t h a t (%e), systems i n 3D a r e bosonic, we f i n d t h a t a t n = 8, t h e r e is a bound molecule with an energy of 0.38 K. It ap- pears t o us a s i f a bound molecule may be formed a t n = 7. What t h i s suggests t o us i s t h a t t h e fermionic ( 3 ~ e ) n systems w i l l not be bound f o r n any l e s s than about 12.
Acknowledgments: Our work was supported i n p a d by NSF Grant No. PHY-7819375 and i n p a r t by t h e I t o Science Foundation. One of us (TU) i s g r a t e f u l t o t h e I n t e r n a t i o n a l Programs Office of t h e NSF f o r a t r a v e l grant,
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