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STABILITY OF SPIN-POLARIZED ATOMIC
HYDROGEN AGAINST RECOMBINATION
REACTIONS
J. Yuan, T. Lim
To cite this version:
JOURNAL DE PHYSIQUE CoZZoque C7, suppZgment a72 n o 7 , Tome 41, juiZZet 1980, page C 7 - 3 9
STABILITY OF SPIN-POLARIZED ATOMIC HYDROGEN AGAINST RECOMBINATION REACTIONS
J.M. Yuan and T.K. LimDrexeZ U n i v e r s i t y , PhiZade Zphia, Pa. 19104, USA.
Rdsum6.- NOUS passons b r i s v e n e n t e n revue l e s e t u d e s s u r l a s t a b i l i t 6 de 11hydrog6ne atomique p o l a r i s d par r a p p o r t allx c o l l i s i o n s de "quenching". Des t r a v a u x a n t e r i e u r s r d v s l e n t l ' i m p o r t a n c e d e s r 6 a c t i o n s de recombinaison 1 t r o i s corps en t a n t que m6canismes de d e s t r u c t i o n . Une estima- t i o n a n t d r i e u r e de l ' o r d r e de grandeur d e c e t a u x de r e a c t i o n 1 t r o i s c o r p s c o n d u i t 1 p r 6 d i r e que l a d e n s i t 6 d'un gaz s t a b l e de HI. n e p e u t d6passer 1017 atomes p a r cm3 s u r une durde de un j o u r . Une d i s c u s s i o n des approximations f a i t e s sugg6re que l a d e n s i t 6 r d e l l e p o u r r a i t 8 t r e de deux o r d r e s de grandeur s u p 6 r i e u r e l c e t t e e s t i m a t i o n .
A b s t r a c t . - S t u d i e s on t h e s t a b i l i t y of s p i n - p o l a r i z e d atomic hydrogen a g a i n s t c o l l i s i o n a l quenching a r e b r i e f l y reviewed. Previous work r e v e a l s t h e importance of t h e three-body recombination r e - a c t i o n a s a d e s t r u c t i v e mechanism. An order-of-magnitude e s t i m a t e of t h i s three-body r e a c t i o n r a t e made p r e v i o u s l y l e a d s t o a maximal s t a b l e d e n s i t y equal t o 1 0 ' ' atoms/cm f o r a d u r a t i o n of a .b.y.
A d i s c u s s i o n of approximations used s u g g e s t s t h a t t h e r e a l d e n s i t y could be a couple of o r d e r s of magnitude h i g h e r .
INTRODUCTION.- Hydrogen atoms w i t h e l e c t r o n i c s p i n p o l a r i z e d by an i n t e n s e magnetic f i e l d c o n s t i t u t e a Bose g a s which should e x h i b i t quantum p r o p e r t i e s more pronounced t h a n ' ~ e . /1,2/ T h i i a r i s e s from t h e f a c t t h a t s p i n - p o l a r ized hydrogen atoms (H+) have s m a l l masses and i n t e r a c t mainly v i a weakly a t t r a c t i v e b 3 ~ + i n t e r a c t ion. A major c u r r e n t i s s u e i n t h i s a r e a o f r e s e a r c h i s t h e s t a b i l i t y of H+ a t a workable d e n s i t y f o r a s u f f i c i e n t l y long time. Various d e s t r u c t i v e mechanisms /3-8/ have been considered and it appears a t t h e p r e s e n t time a l l but t h e three-body c o l l i s i o n a l quenching r e - a c t i o n can b e e f f e c t i v e l y suppressed by proper ex- p e r i m e n t a l c o n d i t i o n s .
Quenching o f H + v i a two-body and three-body c o l l i s i o n s was s t u d i e d by J o n e s e t a 1 /8/. T h i s work, done about twenty y e a r s ago h a s s t i m u l a t e d
r e s e a r c h a c t i v i t y t h a t has c o n t r i b u t e d t o o u r understanding o f H+. However s e v e r a l of t h e i r c o n c l u s i o n s a r e i n c o r r e c t . The l i f e t i m e o f H + , T , a g a i n s t 2-body c o l l i s i o n s t h e y o b t a i n e d i s about 3
lo-'
t o s e c f o r a d e n s i t y of atoms/cm.
T i s f a r t o o small a s was p o i n t e d o u t by Stwal- l e y /9/. Using quantum c o n s i d e r a t i o n s h e argued t h a t t h e s p i n - f l i p mechanism due t o n o n a d i a b a t i c t r a n s i t i o n i s n o t important and suggested t h a t H+
i s s t a b l e a g a i n s t 2-body c o l l i s i o n s f o r B / T - - > I o ~ Gauss/K. This r e v e a l s t h e importance o f t h e t h r e e - body recombination r e a c t i o n i n determining t h e
16 s t a b i l i t y of H+. The l i f e t i m e T"4x10 T-" s.ec
e s t i m a t e d by J o n e s e t a l . based on b a r r i e r pene- t r a t i o n is again i n c o r r e c t and t h i s time f a r t o o l a r g e . T h i s a r i s e s from t h e i r assumption t h a t hy- drogen molecules a r e always formed i n t h e ground
JOURNAL DE PHYSIQUE
v i b r a t i o n a l s t a t e
/lo/.
T h i s i s not t r u e accor- ding t o t h e Franck-Condon p r i n c i p l e and t h i s p o i n t becomes c l e a r i n a simple c a l c u l a t i o n /11/ we made.An a c c u r a t e c a l c u l a t i o n of t h e three-body r e - combination r e a c t i o n r a t e p r e s e n t s a new challenge t o molecular r e a c t i o n dynamicists. The d i f f i c u l - t i e s involve t h e r e p r e s e n t a t i o n o f t h e i n i t i a l s t a t e of t h r e e f r e e p a r t i c l e s , treatment of non- a d i a b a t i c t r a n s i t i o n between t h e s t a t e s involved, e x i s t e n c e of many open f i n a l s t a t e s and unavail- a b i l i t y of an a c c u r a t e upper e l e c t r o n i c p o t e n t i a l energy s u r f a c e . Because of t h e s e d i f f i c u l t i e s , i n our r e c e n t work /11/ we aimed a t an order-of-mag- n i t u d e c a l c u l a t i o n based on p h y s i c a l arguments. R e s u l t s of t h i s work w i l l be r e p o r t e d here. We begin with a b r i e f review of t h e treatment o f Jones e t a l . about 2-body c o l l i s i o n s and S t w a l l e y ' s
argument a g a i n s t it. Comments on t h e f a c t t h a t t h e a c t u a l d e n s i t y could be h i g h e r t h a n our e s t i m a t e and a n i n t e n s e magnetic f i e l d could reduce t h e t h r e e body recombination r a t e of HI. c o n s i d e r a b l y , conclude t h e r e p o r t .
CALCULATION.- The s t r a t e g y we took is t o c a l c u l a t e t h e three-body recombination r e a c t i o n r a t e k
3
as a product o f two f a c t o r s . The f i r s t f a c t o r
k (GK) d e s c r i b e s t h e r a t e t h a t t h r e e HI. atoms come
3
i n t o t h e i n t e r a c t i o n region a t t h e same time. The second f a c t o r P e s t i m a t e s t h e p r o b a b i l i t y t h a t an e l e c t r o n i c s p i n f l i p s each time t h r e e Hf atoms a r e c l o s e t o g e t h e r . k3(GK) can be obtained i n two ways. Based on p u r e l y geometric arguments k (GK) is pro-
3
p o r t i o n a l t o t h e product o f t h e average thermal v e l o c i t y and t h e f i f t h power of t h e average i n t e r - atomic d i s t a n c e , a s given by Smith's formula /12/. kj(GK) obtained t h i s way
is
1 x cm6/atom2 s e c,
i f t h e c r o s s i n g d i s t a n c e o f 8 bohr i n a mag- n e t i c f i e l d o f 100 kG i s used. This v a l u e is6 2 reasonably c l o s e t o K (H) = 1 x cm /atom
3
s e c , t h e experimental value f o r t h e termolecular recombination r a t e of hydrogen atoms a t 77OK. The l a t t e r v a l u e w i l l be used.
To e x p l a i n how we estimated P , l e t us b r i e f l y review t h e t r e a t m e n t by Jones e t a l . f o r two H I . atoms. In t h e Born-Oppenheimer approximation one f i r s t s o l v e s t h e e l e c t r o n i c p a r t o f t h e Schroe- d i n g e r equation o f t h e f i e l d - f r e e Hamiltonian f o r f i x e d n u c l e a r c o n f i g u r a t i o n s . This g i v e s u s t h e
X'C+ and b3z: p o t e n t i a l curves of concern here. g
In a magnetic f i e l d t h e b
3 ~ i
l e v e l is s p l i t i n t o t h r e e curves. The 14 = -1 component c r o s s e s witht h e X LC+ curve. Now t h i s c r o s s i n g becomes an g
avoided one when t h e h y p e r f i n e coupling term i s included. This means t h a t i n a low energy c o l l i - s i o n between 2Hf atoms t h e atoms can follow t h e a d i a b a t i c p o t e n t i a l curve and come c l o s e . t o each o t h e r t o form a H ~ ( X ' C + ) . Based on t h i s semiclas-
.
g
s i c a l argument, i t seems t h a t H 2 molecules should
j
be formed with a p p r e c i a b l e r a t e a s p r e d i c t e d by ,
Jones e t a l . F o r t u n a t e l y , a s was f i r s t pointed o u t , by Stwalley, t h i s process w i l l not t a k e p l a c e f o r t h e r e a r e no v i b r a t i o n a l l y e x c i t e d s t a t e s of hydro-
t
gen molecule (Hz) w i t h i n t h e thermal energy range considered (O.I°K). But t h i s s p i n - f l i p mechanism h a s t o be considered i n three-body c o l l i s i o n s . This i s because t h e f i n a l s t a t e s of two-body c o l l i - s i o n s of Hf a r e d i s c r e t e and t h o s e of t h r e e Hf c o l l i s i o n s a r e i n t h e continuum. I n o t h e r words, t h e t h i r d atom i n t h e three-body c o l l i s i o n c a s e can c a r r y away t h e excess energy o f d i s s o c i a t i o n and permit t h e f i r s t two atoms t o combine i n t o a bound H molecule. O f course, t h e r e a c t i o n p r o b a b i l i t y
2
s p e c i f i c v i b r a t i o n a l s t a t e n 2 c a n b e e s t i m a t e d . The g e n e r a l i z e d Landau-Zener Model /13, 1 4 / y i e l d s
2 2 ~ ( n ~ ) = 2 n l ~ ~ ~ 1 l < ~ ~ l n ~ > / / *vIsl-s21 (1) where H12 i s t h e h y p e r f i n e c o u p l i n g m a t r i x e l e m e n t between t h e i n i t i a l and f i n a l e l e c t r o n i c s t a t e s ; c l i s t h e i n i t i a l t r a n s l a t i o n a l energy between t h e
t
two Hf atoms which recombine i n t o H2.
v
is t h e ve- l o c i t y o f t h e t h i r d atom r e l a t i v e t o t h e c e n t e r o f mass o f t h e o t h e r two a t t h e c r o s s i n g p o i n t and s . i s t h e s l o p e of t h e i t h p o t e n t i a l s u r f a c e a l o n g a d i r e c t i o n p e r p e n d i c u l a r t o t h e c r o s s i n g seam. The o v e r l a p p i n g f a c t o r , <c11n2>, i n Eq. (1) p r e d i c t s t h a t most o f t h e hydrogen m o l e c u l e s formed a r e i n h i g h l y v i b r a t i o n a l l y e x c i t e d s t a t e s . F u r t h e r c o l -f.
i-
l i s i o n s o f H Z w i t h H+ c a n b r i n g H z down t h e v i b r a - t i o n a l l a d d e r and r e l e a s e its i n t e r n a l energy. From Eq. (1) it i s a l s o c l e a r t h a t t h e a s s u m p t i o n of J o n e s e t a l . t h a t H 2 a r e formed s o l e l y i n t h e ground s t a t e w i l l g i v e an e x t r e m e l y s m a l l s p i n - f l i p p r o b a b i l i t y . Summing P(n2) o v e r n 2 and assuming t h a t a c o m p l e t e s e t o f s t a t e s is summed o v e r , we
8 -40 6 o b t a i n e d P = 5 x 10-
,
s o t h a t k3 = 5 x 1 0 cm / atom2 s e e,
which y i e l d s a d e n s i t y p-
1017 atoms/ cm3 f o r a d u r a t i o n o f a b o u t a day. Because o f t h e a p p r o x i m a t i o n s i n t r o d u c e d , f o r example, t h e semi- c l a s s i c a l method o f c a l c u l a t i n g t h e e l e c t r o n i c t r a n - s i t i o n p r o b a b i l i t y , o u r e s t i m a t e o f t h e d e n s i t y c o u l d b e s e v e r a l o r d e r s o f magnitude t o o low. One d i r e c t consequence of t h i s model is t h a t an i n t e n s e m a g n e t i c f i e l d (e.g., above 4 4 0 kG) c o u l d b e ex- t r e m e l y e f f e c t i v e i n i n h i b i t i n g c o l l i s i o n a l quench- i n g p r o c e s s e s , b e c a u s e t h e p o t e n t i a l a t t h e c r o s s - i n g seam i s t h e n much h i g h e r t h a n t h e t h e r m a l e n e r - g i e s o f H+.R e f e r e n c e s and Comments
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