THE MAGNETIZATION AND DENSITY OF SPIN POLARIZED ATOMIC HYDROGEN

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THE MAGNETIZATION AND DENSITY OF SPIN POLARIZED ATOMIC HYDROGEN

J. Walraven, I. Silvera

To cite this version:

J. Walraven, I. Silvera. THE MAGNETIZATION AND DENSITY OF SPIN POLARIZED ATOMIC HYDROGEN. Journal de Physique Colloques, 1980, 41 (C7), pp.C7-147-C7-149.

�10.1051/jphyscol:1980723�. �jpa-00220160�

JOURNAL DE PHYSIQUE CoZZoque C7, suppl&ment au n o 7 , Tome 41, j u i z l e t 1980, page C 7 - 1 4 7

THE MAGNETIZATION AND DENSITY OF S P I N POLARIZED ATOMIC HYDROGEN

J.T.M. Walraven and I. F. S i l v e r a

Natuurkundig Laboratoriwn, U n i v e r s i t e i t van Amsterdam, V a l c k e n i e r s t r a a t 65, 1018 XE Amsterdam, Pays Bas

R6sume.- L ' a i m a n t a t i o n e t l a d e n s i t e d ' u n gaz d'hydrogene p o l a r i s e (H4) p l a c e dans un champ magne- t i q u e dependent fortement de l a presence de g r a d i e n t s magnetiques. C e t t e p r o p r i e t e e s t d i s c u t e e dans l e cadre de c o n d i t i o n s experimentales r e a l i s t e s , rendant m a n i f e s t e l e c a r a c t e r e s p e c i f iq u e de Bose-Einstein du gaz. Dans une c e r t a i n e mesure, l a comparaison e n t r e ces p r e d i c t i o n s e t l ' e x p e r i e n c e e s t p o s s i b l e .

Abstract.- The magnetization and d e n s i t y d i s t r i b u t i o n o f s p i n p o l a r i z e d atomic hydrogen i n a magne- t i c f i e l d i s v e r y s e n s i t i v e f o r t h e presence of magnetic f i e l d g r a d i e n t s . T h i s f e a t u r e i s discussed i n r e l a t i o n t o r e a l i s t i c experimental c o n d i t i o n s , accounting f o r t h e s p e c i f i c Bose-Einstein charac- t e r o f t h e gas. To a l i m i t e d e x t e n t comparison w i t h experiment i s possible.

Gaseous s p i n p o l a r i z e d atomic hydrogen (Hs) ^{Bz(z) = B - o} - ^{B Z ~}

(1) has been shown /1, 2/ t o be s t a b l e a g a i n s t recom-

b i n a t i o n i n a small (1 c c ) open ended c o n t a i n e r w i t h w a l l s covered w i t h s u p e r f l u i d He. Several 4 minutes a f t e r l o a d i n g t h e c e l l a t 270 m i l l i - K e l v i n i n a magnetic f i e l d o f 7 Tesla t h e presence o f H+

c o u l d be demonstrated by breaking t h e He-film, so t h a t t h e atoms recombined and t h e r e s u l t a n t heat o f recombination was observed. I n these experiments a f i e l d dependent d e n s i t y decay was observed and a t t r i b u t e d t o thermal leakage (escape) o f t h e atoms o u t o f t h e c e l l .

I n t h i s paper we discuss some o f t h e conse- quences o f f i e l d inhomogeneities f o r t h e study o f

. . BZ(p) = BO + $so2

where BZ(z) and BZ(p) are t h e a x i a l and r a d i a l f i e l d d i s t r i b u t i o n s near t h e c e n t e r of t h e magnet. Bo i s t h e maximum a x i a l f i e l d and B = B o / z i where zo i s defined by L Z ( z = zo) = 0. I n our solenoidal f i e l d , when f i t t o t h e q u a d r a t i c form, zo = 5 1 mm. The d e n s i t y d i s t r i b u t i o n i s then e a s i l y c a l c u l a t e d by r e q u i r i n g t h e chemical p o t e n t i a l (u) t o be a con- s t a n t over space. Neglecting i n t e r a c t i o n s we f i n d f o r d e n s i t i e s smaller than t h e Bose- E i n s t e i n con- densation (BEC) d e n s i t y ( n < n c ) :

=

a

h 3 Q = 1 ( 2 )

Q 5 / 2 H+ a t low temperatures. For t h i s purpose we w i l l

where n(B) i s t h e d e n s i t y a t f i e l d B, g i s t h e consider t h e geometry o f t h e magnetic f i e l d present-

( n u c l e a r ) degeneracy f a c t o r , h i s t h e thermal l y b e i n g used i n our experimental apparatus. The

wavelength, pB i s t h e Bohr magneton and k i s f i e l d i s produced by a conventional superconducting

Boltzmann's constant. The magnetization i s simply c o i l w i t h a 1% inhomogeneity over a sphere o f 1 cm

p r o p o r t i o n a l t o t h e d e n s i t y

fi

1;)

d e n s i t i e s where i s much s m a l l e r than t h e i n t e r - magnetic f i e l d d i s t r i b u t i o n has t h e shape o f a saddle

p a r t i c l e spacing and using t h e q u a d r a t i c approxima- p o i n t i n space which i s expressed i n t h e q u a d r a t i c

f i e l d approximation by: t i o n f o r t h e f i e l d , equation 4 2 ) can be w r i t t e n as

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980723

JOURNAL DE PHYSIQUE

Z

=

where n ( z ) and n ( ~ ) a r e t h e a x i a l and r a d i a l d e n s i t y d i s t r i b u t i o n s near t h e c e n t e r o f t h e mag- n e t .

The a x i a l d e n s i t y d i s t r i b u t i o n thus has a Gaussian shape w i t h h a l f - w i d t h

A z = z0 ( X T ~ ~ ~ B ~ ) ~ ( 4 )

I f we assume t h e atoms t o be confined t o t h e a x i s o f t h e magnet c o i l by means o f a c y l i n d r i c a l tube we can c a l c u l a t e t h e a x i a l and r a d i a l homo- g e n e i t y i n t h e d e n s i t y over t h i s tube as a func- t i o n o f t h e temperature. Some c h a r a c t e r i s t i c values are given i n t a b l e 1 f o r B = 10 Tesla and a tube o f r a d i u s po = 1.8 mm.

E v i d e n t l y a t low temperatures t h e H+ i s h i g h l y l o c a l i z e d near t h e c e n t e r o f t h e magnet and pressed t o t h e w a l l s o f t h e confinement c e l l .

There a r e two important consequences o f eq. ( 2 ) . I n the f i r s t p l a c e t h e t o t a l number o f atoms t h a t can be " s t o r e d " i n t h e f i e l d i s l i m i t e d , i . e . t h e r e i s a s a t u r a t i o n d e n s i t y which i s determ- i n e d by t h e (steady s t a t e ) d e n s i t y i n zero f i e l d .

Table 1

Secondly, i f t h e zero f i e l d d e n s i t y i s reduced t o zero, t h e d e n s i t y i n t h e f i e l d r e g i o n should a l s o

decay t o t h i s value w i t h a t i m e constant determined by thermal leakage.

I n order t o describe b o t h f e a t u r e s i t i s u s e f u l t o i n t r o d u c e t h e concept o f t h e magnetic compression f a c t o r c

c(Bo-B) 5 n(Bo)/n (B) = exp IpB(B0-B)/kT} (5) This f a c t o r enables us t o c a l c u l a t e t h e s a t u r a t i o n d e n s i t y i n t h e c e n t e r o f t h e magnet, n e g l e c t i n g recombination, once t h e (steady s t a t e ) d e n s i t y i s known a t a given f i e l d B. The i m p l i c a t i o n s o f t h i s concept f o r experiments a r e evident. H-atoms should be i n j e c t e d i n a 4 ~ e covered s t a b i l i s a t i o n c e l l a t as low a f i e l d Bi as p o s s i b l e i n o r d e r t o f u l l y e x p l o i t magnetic compression. For T =0.5 K and AB E B -B. = 10 T, c= 7 x10 whereas f o r 5

0 1

AB = 5 T we o n l y f i n d c ~ 8 3 0 . For T 0.1 K and AB 5 T, c = 4 x1014 i n d i c a t i n g t h a t under these c o n d i t i o n s t h e s a t u r a t i o n d e n s i t y i s n o t e a s i l y reached enabling t h e study o f more i n t r i n s i c l i m i t i n g e f f e c t s i n an open ended geometry.

The r a t e o f thermal leakage i s b a s i c a l l y determ- i n e d by t h e d e n s i t y i n t h e r e g i o n z = rr where r e - combination s t a r t s t o become important ( i . e . f o r z zr we assume t h e H e - f i l m n o t t o be t h i c k enough t o prevent s u r f a c e recombination from o c c u r r i n g . To a f i r s t approximation t h e r a t e o f leakage

dN/dt

-

T = 4 c V e f f / K 7 ~ where Veff = N/no i s t h e e f f e c t i v e v o l ume o f t h e c e l l , K an e f f e c t i v e C 1 ausing f a c t o r /3/, 7 = ( 8 k ~ / n m ) ~ t h e average v e l o c i t y o f t h e atoms and A t h e c r o s s - s e c t i o n a l area o f t h e confinement tube. For a tube o f u n i f o r m cross-sec- t i o n Veff = A zo ( n k ~ / u ~ ~ ~ ) ~

= n 42 K-I czo ( ~ / / Y ~ B ~ ) ~ ( 6 )

I n c r e a s i n g t h e d e n s i t y t o values where t h e i n t e r - p a r t i c l e s e p a r a t i o n becomes comparable t o t h e thermal wavelength, degeneracy e f f e c t s become im- p o r t a n t and w i l l a l s o show up i n t h e d e n s i t y p r o f i l e as l o n g as H - H i n t e r a c t i o n s remain n e g l i g i b l e . I n f i g . 1 we show d e n s i t y p r o f i l e s as c a l c u l a t e d from eq. ( 2 ) f o r n << nc and n = n c ' A narrowing o f t h e d i s t r i b u t i o n as t h e c r i t i c a l d e n s i t y i s approached i s c h a r a c t e r i s t i c o f t h e Bose gas. Beyond nc BEC may r e s u l t i n a sharp peaking up o f t h e a x i a l d e n s i t y d i s t r i b u t i o n a t t h e f i e l d maximum although f o r r e a l i s t i c d e n s i t i e s (temperatures) i n t e r a c t i o n s between t h e atoms a r e

pENSllY PROFILE OF H I

10 20 3 0

d e n s i t y l i m i t : El(n) = an

-

d3.

Requiring u = (a/an) { nEl(n) ) t o be a constant over t h e system y i e l d s f o r t h e c h a r a c t e r i s t i c h a l f w i d t h due t o i n t e r a c t i o n s

azint= Z ~ ( ~ U ~ ~ / P ~ B ) 9 ( 7 )

I n o r d e r t o observe t h e e f f e c t s o f t h e s t a t i s t i c s on t h e d e n s i t y p r o f i l e we must r e q u i r e az/azint>> 1 a t t h e Bose condensation d e n s i t y . Using equation ( 4 ) , (7) and nc = 2 . 6 1 2 / ~ 3 we f i n d az/azint = 1.8 T ' ~ , where T i s t h e temperature. T h i s means t h a t although H+ can be prepared as a very weakly i n t e r a c t i n g system i n comparison t o 4 ~ e , temperatures i n t h e ( s u b ) m i l l i k e l v i n regime a r e probably r e q u i r e d t o enable observation o f these i n t e r e s t i n g e f f e c t s . This concept o f smearing o f t h e wave f u n c t i o n s as a r e s u l t o f t h e i n t e r a c t i o n s i s w e l l s t u d i e d f o r He 4 /5/. Another /6/ way o f s t a t i n g t h e requirements f o r observable e f f e c t s o f BEC on t h e d e n s i t y d i s - t r i b u t i o n i s t h a t t h e c h a r a c t e r i s t i c h e a l i n g l e n g t h , d e f i n e d by io = (8n an)-$, i s much l a r g e r than a c h a r a c t e r i s t i c dimension o f t h e wave f u n c t i o n . Here DISTANCE FROM CENTER (arb.units1- a i s t h e s-wave s c a t t e r i n g l e n g t h (a = 0.72

8

Fig. 1 : Density p r o f i l e s as a f u n c t i o n of t h e d i s - For = 1U16 (Tc = co .0.18 tance from t h e c e n t e r o f t h e magnet.

whereas a c h a r a c t e r i s t i c dimension f o r t h e wave expected t o l a r g e l y broaden t h i s e f f e c t . The e x t e n t functions i s 6.8 11m /2/.

o f t h i s broadening can be estimated u s i n g t h e c a l c u l - We g r a t e f u l l y acknowledge t h e f i n a n c i a l support a t e d /4/ ground s t a t e energy p e r atom i n t h e low o f t h e S t i c h t i n g voor Fundamenteel Onderzoek d e r

Materie.

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