THEORETICAL ISSUES CONCERNING THE STABILITY OF ELECTRON SPIN-POLARIZED HYDROGEN

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THEORETICAL ISSUES CONCERNING THE STABILITY OF ELECTRON SPIN-POLARIZED

HYDROGEN

W. Stwalley, Y. Uang, R. Ferrante, R. Webeler

To cite this version:

W. Stwalley, Y. Uang, R. Ferrante, R. Webeler. THEORETICAL ISSUES CONCERNING THE

STABILITY OF ELECTRON SPIN-POLARIZED HYDROGEN. Journal de Physique Colloques,

1980, 41 (C7), pp.C7-27-C7-31. �10.1051/jphyscol:1980705�. �jpa-00220142�

JOURNAL DE PHYSIQUE CoZZoque C7, suppZ6mer.t ^ctr* n o 7 , Tome 41, j u i Z Z e t 1980, page C 7 - - 2 7

THEORETICAL ISSUES CONCERNING THE STABILITY OF ELECTRON SPIN-POLARIZED HYDROGEN

W.C. Stwalley, Y.H. Uang, R.F. Ferrante and R.W.H. Webeler

U n i v e r s i t y o f Iowa, Iowa C i t y , Iowa 52242, USA.

Resume.- Des atomes d'hydrogene dont le spin de l'electron est polarise dans un fort champ magneti- que

(

"hydrogene pol aris6" , ^symbol is6 par H+) devraient rester un fluide quantique gazeux metastable jusqu'au zero absolu. Cependant i l existe de nombreux processus susceptibles de detruire cette me- tastabilite. A forte densite

(?

1021/cm3) Berlinsky et al ont suggere qu'un mecanisme de destruc- tion par ondede spin coherente emp&che la stabilite. A plus faible densite (suffisante toutefois pour donner lieu

2

la condensation de Bose) des mecanismes de reconbinaison en phase gazeuse et en surface devraient jouer un rdle primordial. Une question cruciale concerne la limite d'explosion pour H+, c'est-&-dire a partir de quelle densite un seul retournement de spin electronique declen- chera une recombinaison massive, avec echauffement et destruction du recipient (peut-etre un bon

"detecteur" pour des processus hautement interdits)

?

La seconde question cruciale est la suivante:

quels evenements declencheront de telles explosions si 1 'on suppose qu'on est au-dessous du seuil

?

Nous allons presenter le point 00 nous en sommes actuellement dans la compr@hension de ces deux problemes cruciaux (en donnant des calculs de collision a deux corps d'une grande precision) et nous en montrerons les consequences pour la conception des appareillages experimentaux (par exem- ple effets des inhomogeneites de champ, enduits d'helium sur les surfaces, elimination des impure- tes de deuterium, etc.).

Abstract.- Hydroien atoms with their electron spin polarized in a strong magnetic field ("spin- aligned hydrogen

;

symbolized H+) should remain a metastable gaseous quantum fluid even at abso- lute zero. However, there are many possible ways of destroying this metastability. At high density

( 2

1021/cm3) Berlinsky et al. have suggested that a coherent spin-wave destruction mechanism pre-

vents stability. At lower densities (high enough for Bose-Einstein condensation, however), gas phase and surface recombination mechanisms should be of primary concern. One critical question is essen- tially the explosion limit for H+, i.e. at what density will a single electronic spin flip event trigger massive recombination, heating and destruction of the sample (perhaps a good "detector"

for highly forbidden processes)

?

The second critical question is what events will trigger such explosions assuming one is above the limit

?

We shall present our current understanding of these two critical questions (including high accuracy two-body coll ision calculations) and their imp1 ica- tions in experime~ltal design (e.g. field inhomogeneity effects, he1 ium coating of surfaces, removal of deuterium impurities, etc.) .

1. INTRODUCTION.- Hydrogen atoms with their elec- tron spin polarized in a strong magnetic field ("spin-a1 igned hydrogen"; symbolized

H+)

should re- main a metastable gaseous quantum fluid at low tem- perature, even at absolute zero. One primary goal of current studies is to achieve Bose-Einstein con- densation, which requires a density of ?1019/cm3 for reasonable temperatures and magnetic fields 11-31. The rough densities of interest for our considerations here are summarized in Figure

1.

The low densities present in polarized atomic beams are of historical interest only. The MIT /4/ and University of British Columbia 151 magnetic reso- nance experiments demonstrated sizeable densities at

4

K. The recent Amsterdam experiment (0.27 K, 7 T) 161 demonstrated clearly the effect of the

magnetic field and was apparently strongly polar- ized in contrast to the 4 K magnetic resonance ex- periments. It is thus reasonable to assume that there are no unanticipated mechanisms giving rise to instability (on the time scale of minutes at least) at densities of

-1014

!4/cm3. At high den- sities (?lo2' H+/cm3), it has been suggested 17-81 that a spin wave instability will destroy H+. How- ever, since in the near term Bose-Einstein conden- sation rather than solidification is being sought, we shall only consider lower densities where the spin wave instability should not be crucial. One critical question in our view is at what density (for a pure

H t

sample under given magnetic field/

temperature conditions with inert walls) a "thermal explosion" occurs (sudden heat release, as in tri-

3

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980705

JOURNAL DE P H Y S I Q U E

F i g u r e 1. Hydrogen d e n s i t i e s o f i n t e r e s t . d e n s i t y (H+/cm3) B = 10 T, T = 0.1 K

l o z 2 -

s o l i d ( P = 50 atm.) spin-wave i n s t a b i l i t y

- l o z 0 -

BE condensation

f l i p p i n g a s p i n than i s an H atom! Accurate close- coupled s c a t t e r i n g c a l c u l a t i o n s /14-15/ f o r v a r i o u s M ( = ms, + mIA + ms, + mIB), J ( p a r t i a l wave), 3

( p a r i t y ) channels i n d i c a t e t h e cross s e c t i o n magni- tudes given i n Table I. Note t h a t i n a11 cases

1 0 l 8

- -

e x p l o s i o n l i m i t ?

I

l o 1 = -

-

1014

-

I

Amsterdam 1980 MIT, UBC 1979

1O1O

-

i n t e n s e H+ beam

l o 8

-

Stern-Gerlach Ag+ beam

-

l o 6 -

densest i n t e r s t e l l a r clouds

t i u m - s o l i d hydrogen / 9 - I T / ) ? Thus i n t h i s manu- s c r i p t we w i l l b r i e f l y survey i n s t a b i l i t y mechan- isms i n general and then t u r n t o t h e d e t a i l e d k i n e - t i c model o f an atomic recombination event we a r e i n the process o f developing. While we do n o t y e t have an e s t i m a t e o f t h e e x p l o s i o n l i m i t , e.g. f o r H+ a t 0.1

K

and 10 T, we hope t o attempt such an estimate soon. I t might be noted t h a t t h i s l i m i t may be o f considerable importance since, f o r exam- p l e , ESR experiments would be d e s t r u c t i v e o f t h e sample above t h e 1 im i t .

2. SURVEY OF INSTABILITY MECHANISMS.- One o f us (WCS) has p r e v i o u s l y surveyed i n s t a b i l i t y mechan- isms i n d e t a i l /12-13/. Recently t h e D i m p u r i t y problem has been examined q u a n t i t a t i v e l y /14/ and i t has been estimated t h a t , f o r B = 1 0 T and T = 0.1 K, a D atom i s r o u g h l y 10" more e f f e c t i v e i n

Table I. Magnitudes o f v a r i o u s two body c o l l i s i o n cross s e c t i o n s ( i n a2 u n i t s ) f o r i s o t o p i c H+-H+

c o l l i s i o n s

(B

= 10 T?.

process exampl e* magni tude

I. e l a s t i c s c a t t e r i n g

-4, 4; -4, 0> +

-lo2

[ a l s o reso- -4, 4; -4, o> nances and Ram-

sauer-Townsend minima]

11. i n e l a s t i c s c a t t e r i n g ( s p i n f l i p s ) A. l e - , 1N -4, 4; -4, 0> +

-lo-' - l o - "

-4, 4;

15,

-1> [above 13.4 K t h r e s h o l d ]

B. 2e-, 2N -4, 4; -4, 0> + - 1 0 - ' ~ ~ [ a b o v e 26.8 4 -4;

,

1 K t h r e s h o l d ] C. 2N -4, 4; -4, o> +

-4, -4; -$

,

1>

*

(-lmS1, mIH. ms2, mID> b a s i s )

accurate piecewise s p l i n e r e p r e s e n t a t i o n s o f t h e ab i n i t i o p o t e n t i a l s (xlrt and b3r:) o f Kolos and Wol-

9

niewicz are used /20/. These two-body c o l l i s i o n r e s u l t s are a l s o being used t o e s t i m a t e low temper- a t u r e t r a n s p o r t c o e f f i c i e n t s (e. g. t h e v i s c o s i t y /16/ needed f o r e s t i m a t i n g an NMR T, r e l a x a t i o n t i m e /3/), and a l s o t h e E/N values as a f u n c t i o n o f d e n s i t y f o r t h e nonideal Bose (H+) /17-18/ and Fermi (Df) gases /19/.

The three-body c o l l i s i o n e f f e c t s a r e v e r y

im-

p o r t a n t t o examine i n depth, b u t very d i f f i c u l t t o c a l c u l a t e r e l i a b l y and we have n o t y e t attempted t o do so.

Surface e f f e c t s are minimized by having a weakly i n t e r a c t i n g surface, presumably helium. The i m p o r t a n t q u e s t i o n i s t h e b i n d i n g energy ^E o f H t o an He surface. Estimates /13, 21-23/ o f E have a l l been i n t h e range 0-2 K, b u t t h i s small range c o r -

responds t o a tremendous range o f surface coverage.

For example, a t 0.1 K w i t h a c e l l o f area 6 cm2 and volume 1 cm3, we e s t i m a t e -10% o f t h e H atoms wi 11 be on t h e s u r f a c e f o r E = 1 K, b u t o n l y -0.001% i f

E = 0.1 K. A h i g h s u r f a c e d e n s i t y , o f course, c o u l d l e a d t o a s p i n wave i n s t a b i l i t y /7-8/ even though t h e gas phase d e n s i t y was low enough.

I t m i g h t be noted t h a t t h e r e a r e several ways o f t r a p p i n g i o n s and atoms a t low temperature w i t h - o u t w a l l s /24-26/, b u t t h e r e are c u r r e n t l y no con- v e n i e n t l a s e r s f o r t h e case o f Hf ( f o r L i + o r Be

+ +,

such experiments m i g h t be p o s s i b l e ) .

Some f i n a l i n s t a b i l i t i e s i n c l u d e magnetic f i e l d g r a d i e n t s (which l e a d t o serious d e n s i t y gra- d i e n t s a t very low temperatures) and t h e thermal leakage r e c e n t l y discussed /27/ ( n o t a problem a t

l o o

T/K).

3. THE MICROSCOPIC RECOMBINATION EVENT AND THE THERMAL EXPLOSION LIMIT.- An extremely important and d i f f i c u l t q u e s t i o n i n designing experiments on Hr and o t h e r e l e c t r o n s p i n p o l a r i z e d atoms i s whe- t h e r a s i n g l e s p i n f l i p o r s i n g l e recombination w i l l d e s t r o y t h e b u l k system (explode) o r whether t h e heat from a recombination can be d i s s i p a t e d w i t h o u t f u r t h e r recombinations occuring. I n t h i s s e c t i o n we o u t l i n e o u r c u r r e n t understanding o f t h e i m p o r t a n t k i n e t i c s involved. I n Table 11, we l i s t what we f e e l a r e t h e important steps which must be understood. For now we w i l l consider a p u r e l y gas phase system, although i t w i l l be apparent t h a t s u r f a c e / s i z e c o n s i d e r a t i o n s w i l l be a l s o q u i t e im- p o r t a n t as a means o f d i s s i p a t i n g recombination en- ergy. The i n i t i a l s p i n f l i p ( s t e p 1 ) produces t h e

" a c t i v e " species H+ more o r l e s s i n t h e way a c h a i n r e a c t i o n i n v o l v i n g f r e e r a d i c a l s may be i n i t i a t e d t h e r m a l l y ( o r i n many o t h e r ways). T h i s H+ atom has b a s i c a l l y o n l y two f a t e s : t h e reverse o f step 1, and recombination step 2. A v e r y important fea- t u r e o f t h i s choice i s t h a t i t i s d e n s i t y depen-

Table 11. Microscopic K i n e t i c s o f a Recombination Event.*

1. e l e c t r o n s p i n f l i p

k ,

H+

+

H+

2

^H+

+

H I

-

13 K 4- ¹

2. recombination

3. v i b r a t i o n a l d e a c t i v a t i o n

4. t h e r m a l l y produced s p i n - f l i p

k h ~ A.

H.f

+ H+ ⁺ Hr

+

H+

*

Underlined species a r e produced t r a n s l a t i o n a l l y hot, n o t o n l y i n comparison t o an -0.1 K bath b u t a l s o i n comparison t o t h e 13 K e n d o e r g i c i t y o f step 1.

dent. L e t

N I

be t h e " i n p u t " number o f s p i n f l i p s needed f o r a s i n g l e recombination event t o occur on t h e average. Then i n our k i n e t i c scheme

where n i s t h e d e n s i t y o f HI.. The r e l a t i v e values o f k-, and k, determine what i s meant by low den- s i t y and h i g h d e n s i t y w i t h regard t o recombination.

A t low d e n s i t y , NI = k-,/(k,n) and most s p i n f l i p s do

not

y i e l d recombination events (NI i s much l a r g e r than u n i t y ) . A t h i g h d e n s i t y , NI approaches u n i t y , so n e a r l y every s p i n f l i p d i r e c t l y y i e l d s a recom- b i n a t i o n event. We can estimate k-, f a i r l y w e l l from our d e t a i l e d s c a t t e r i n g c a l c u l a t i o n s /14-15/

t o be -1 0 - l 4 cm3/(atoms-sec) . The recombination r a t e constant k, i s much more d i f f i c u l t t o e s t i - mate a t these very low temperatures. Room tempera- t u r e k, values are u s u a l l y i n t h e range

cm6/(atom2-sec), so we estimate k,

- .

^Thus

i n t e r m e d i a t e d e n s i t y where k-, = k,n corresponds t o 1021+2 atoms/cm3. Hence i n approaching Bose-Ein- s t e i n condensation d e n s i t i e s from below, i t i s un- l i k e l y t h a t s p i n f l i p s w i l l u s u a l l y produce reconi- b i n a t i o n events d i r e c t l y .

C 7 - 3 0 JOURNAL DE PHYSIQUE

A second q u e s t i o n o f importance i n d e c i d i n g upon an estimate o f t h e explosion l i m i t i s how many s p i n - f l i p s N a r e produced ( " o u t p u t " ) by a s i n g l e

0

reco!nbination event. A t f i r s t blush, i t might be expected t h a t r e l e a s i n g t h e 52000 K H, bond energy i n a low heat c a p a c i t y medium a t 0.1 K would be s u f f i c i e n t f o r a dramatic e x p l o s i o n even a t low d e n s i t i e s . However, we f e e l t h e energy r e l e a s e me- chanism i s s u f f i c i e n t l y slow and t h e s p i n f l i p cross sections s u f f i c i e n t l y small t h a t t h e answer t o t h i s question i s n o n t r i v i a l . Looking again a t Table 11, we see t h a t t h e recombination should usu- a l l y y i e l d o n l y 210 K i n e n d o e r g i c i t y ( n o t 52000 K) since t h e 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 species i s by f a r the most l i k e l y t o be formed ( f o r s i m p l i c i t y we i g n o r e r o t a t i o n ) . Moreover, t h i s energy i s shared by t h e two t r a n s l a t i o n a l l y h o t products H, ( v = 14) and H+. These species should f a i r l y r a - p i d l y d i s s i p a t e t h e i r thermal energy ( t o below t h e 13.4 K s p i n f l i p t h r e s h o l d ) and probably n o t pro- duce a s p i n f l i p ( r e c a l l from Table I t h a t t h e s p i n f l i p cross s e c t i o n above t h r e s h o l d i s l e s s than a p a r t per m i l l i o n o f t h e gas k i n e t i c cross sec- t i o n ! ) . However, l a r g e r q u a n t i t i e s o f energy can be released i n t h e subsequent v i b r a t i o n a l r e l a x a - t i o n steps 3. Because o f t h e low temperature o f t h e bath, we expect Av = 1 r e l a x a t i o n s t o dominate, b u t t h e v i b r a t i o n a l spacings become q u i t e l a r g e (6000 K f o r AG%). The magnitude o f these k, ( v ) r a t e s should be small, b u t they should be c a l c u l a - b l e . The determination then o f No ( t h e number o f s p i n f l i p s produced per recombination event) i s d i f f i c u l t . A f i r s t step i s t o consider t h e r e l a t e d number

N;

(E), t h e number o f s p i n f l i p s occuring, e.g. when a p a r t i c l e o f h i g h k i n e t i c energy E i s i n t r o d u c e d i n t o a b a t h o f 0.1 K H+. We expect t h a t N' i s a s t r o n g l y ( e x p o n e n t i a l l y ? ) i n c r e a s i n g func-

0

t i o n o f E and we p l a n molecular dynamics s i m u l a t i o n

o f i t i n c o l l a b o r a t i o n w i t h Dr. Arnold Karo o f Law- rence Livermore Laboratory. We hope t h i s w i l l pro- v i d e a reasonable estimate o f t h e e x p l o s i o n l i m i t f o r H+. C l e a r l y , No >> NI i s a runaway "chain branching" s i t u a t i o n corresponding t o thermal ex- p l o s i o n which we want t o avoid. We r e a l i z e t h a t we a r e i n c o r p o r a t i n g c l a s s i c a l and continuum concepts i n t o what i s e s s e n t i a l l y a microscopic quantum event, b u t f e e l i t i s a useful f i r s t s t e p toward understanding t h i s complex t o p i c .

4. ACKNOWLEDGMENTS.- Acknowledgment i s made t o t h e Donors o f t h e Petroleum Research Fund, administered by t h e American Chemical Society, and t o t h e Na- t i o n a l Aeronautics and Space A d m i n i s t r a t i o n f o r support o f t h i s work.

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