PROSPECTS FOR LOW TEMPERATURE H MASERS USING LIQUID HELIUM COATED WALLS

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PROSPECTS FOR LOW TEMPERATURE H

MASERS USING LIQUID HELIUM COATED WALLS

W. Hardy, M. Morrow

To cite this version:

W. Hardy, M. Morrow. PROSPECTS FOR LOW TEMPERATURE H MASERS USING LIQ- UID HELIUM COATED WALLS. Journal de Physique Colloques, 1981, 42 (C8), pp.C8-171-C8-179.

�10.1051/jphyscol:1981819�. �jpa-00221714�

page 68-171

PROSPECTS FOR LOW TEMPERATURE H MASERS USING L I Q U I D H E L I U M COATED WALLS

W.N. Hardy and M. Morrow

Department of Physics, University o f B r i t i s h CoZwnbia, Vancouver, B.C., Canada V 6 4 1W5

A b s t r a c t . - Recent e x p e r i m e n t s have d e m o n s t r a t e d t h a t a t o m i c hydrogen g a s c a n b e s t o r e d f o r l o n g p e r i o d s o f t i m e a t low t e m p e r a t u r e s u s i n g l i q u i d helium c o a t e d c o n t a i n e r s . T h i s a r t i c l e summarizes t h e p r o p e r t i e s o f t h i s g a s r e l e - v a n t t o t h e o p e r a t i o n o f a c r y o g e n i c maser, and t h e n makes a p r e l i m i n a r y a n a l y s i s o f t h e e x p e c t e d performance o f s u c h a maser.

I. I n t r o d u c t i o n . - A maser o p e r a t e d a t t e m p e r a t u r e s n e a r 1 K would have a number of n a t u r a l a d v a n t a g e s o v e r t h e s t a n d a r d room t e m p e r a t u r e maser, t h e most o b v i o u s of which a r e i n c r e a s e d d i m e n s i o n a l s t a b i l i t y o f t h e microwave c a v i t y a n d reduced thermal n o i s e . O t h e r a d v a n t a g e s i n c l u d e a g r e a t l y reduced c r o s s - s e c t i o n f o r t h e s p i n - exchange b r o a d e n i n g and a n a t u r a l c o m p a t a b i l i t y w i t h c o o l e d e l e c t r o n i c s . U n t i l v e r y r e c e n t l y , t h e l a c k of a s u i t a b l e w a l l c o a t i n g t o c o n t a i n t h e H atoms h a s p r e c l u d e d much p r o g r e s s i n t h i s d i r e c t i o n . The b e s t s o l i d s u r f a c e , f r o z e n m o l e c u l a r hydrogen, h a s been shown by Crampton ( 1 ) t o b i n d I3 atoms w i t h a n e n e r g y of a b o u t 40 K, and i s u n s u i t a b l e f o r o p e r a t i o n n e a r 1 K ( a l t h o u g h i t may b e u s e f u l f o r somewhat h i g h e r t e m p e r a t u r e s ) . R e c e n t l y , however, t h e f e a s i b i l i t y o f u s i n g l i q u i d Helium c o a t e d w a l l s h a s been e s t a b l i s h e d a s a by-product of t h e e f f o r t s by s e v e r a l groups t o o b t a i n h i g h d e n s i t i e s o f a t o m i c H a t low t e m p e r a t u r e s w i t h t h e aim o f o b s e r v i n g Bose-Einstein c o n d e n s a t i o n . The Amsterdam roup, u s i n g a l a r g e magnetic f i e l d t o c o n f i n e t h e H atoms t o o n e end o f a l i q u i d 'He c o a t e d v e s s e l , were a b l e t o s t a b i l i z e 1014 atoms/cm3 f o r p e r i o d s of o r d e r 500s a t 0.3 K ( 2 ) . S u b s e q u e n t l y , g r o u p s a t MIT ( 3 ) and C o r n e l l ( 4 ) , u s i n g s i m i l a r t e c h n i q u e s , a l s o o t a i n e d s t a b i l i z a t i o n . D e n s i t i e s a s h i g h as 1017/cm3 were r e p o r t e d ( 3 ) , ( 5 ) . I n d e p e n d e n t l y , w o r k e r s a t t h e U n i v e r s i t y o f B r i t i s h Columbia developed a method u s i n g a p u l s e d d i s c h a r g e in a c l o s e d c e l l t o produce and c o n f i n e a n a t o m i c H g a s ( 6 ) . The c l o s e d geometry a l l o w e d e x p e r i m e n t s t o b e c a r r i e d o u t i n z e r o m a g n e t i c f i e l d , and p u l s e d magnetic r e s o n a n c e a t t h e z e r o - f i e l d h y p e r f i n e r e s o n a n c e was u s e d t o s t u d y t h e atoms ( t r a n s i t i o n 1+3 i n f i g u r e 1 ) . T h i s t e c h n i q u e h a s produced a w e a l t h of i n f o r m a t i o n on t h e low t e m p e r a t u r e p r o p e r t i e s o f a t o m i c hydrogen g a s and i t s i n t e r a c t i o n w i t h t h e l i q u i d h e l i u m c o a t e d w a l l s .

Output The b u l k of t h i s a r t i c l e i s de- Cavity v o t e d t o a d e s c r i p t i o n of t h e a f o r e -

mentioned r e s u l t s , from t h e p o i n t of view of t h e i r r e l e v a n c e t o a c r y o g e n i c Storage

F:E- ^.fi

ib

I I

t o t h e 1420.405 MHz t r a n s i t i o n ( 7 ) . H-Beam We w i l l n o t c o n c e r n o u r s e l v e s h e r e w i t h t h e problem o f p r o d u c i n g t h e ( 9 ) ( b ) s t a t e s e l e c t e d beam, b u t w i l l assume F i g . 1: ( a ) Ground s t a t e h y p e r f i n e l e v e l s t h a t a n a p p r o p r i a t e f l u x of atoms i n of a hydrogen atom i n a magnetic f i e l d . t h e a p p r o p r i a t e h y p e r f i n e s t a t e i s (b) S t a n d a r d maser c o n f i g u r a t i o n c o n s i d e r e d i n c i d e n t on t h e e n t r a n c e h o l e o f t h e i n t h i s p a p e r . s t o r a g e bulb. T y p i c a l p a r a m e t e r s f o r

a room t e m p e r a t u r e maser would b e a

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981819

C8-172 JOURNAL DE PHYSIQUE

minimum atom f l u x Imi, o f 2 ~ 1 0 ~ ~ s e c - ~ , a b u l b s t o r a g e t i m e of 0 . 3 s e c and a l o a d e d c a v i t y Q, QL, of a b o u t 30,000. F o r c o n v e n i e n t comparison we have i n mind t h a t t h e c r y o g e n i c maser o p e r a t e s under a p p r o x i m a t e l y t h e same c o n d i t i o n s .

11. Low Temperature P r o p e r t i e s o f Atomic H Gas.- The d e n s i t y of H atoms i n a maser s t o r a g e b u l b is t y p i c a l l y l o Y atoms p e r cm%t t h r e s h o l d . Recent e x p e r i m e n t a l and t h e o r e t i c a l r e s u l t s have g i v e n a f a i r l y c o m p l e t e p i c t u r e o f t h e low t e m p e r a t u r e be- h a v i o u r of s u c h a g a s when e n c l o s e d by l i q u i d He w a l l s . I n p a r t i c u l a r , t h e decay t i m e of t h e g a s d e n s i t y due t o r e c o m b i n a t i o n i s e x t r e m e l y l o n g , b e i n g o f o r d e r many months a t t h e a p p r o p r i a t e maser o p e r a t i n g t e m p e r a t u r e s and d e n s i t i e s . T h e r e f o r e one i s d e a l i n g w i t h a system t h a t i s i n a w e l l d e f i n e d s t a t e o f m e t a s t a b l e e q u i l i b r i u m . I t c a n be t r e a t e d a s a p e r f e c t t h r e e d i m e n s i o n a l g a s i n e q u i l i b r i u m w i t h a p e r f e c t two d i m e n s i o n a l g a s c o n s i s t i n g o f t h e atoms on t h e l i q u i d helium w a l l s . A g i v e n atom s p e n d s most o f i t s t i m e i n t h e b u l k g a s ( i n t r a n s i t a c r o s s t h e b u l b ) , t h e n i s t r a p p e d from t i m e t o t i m e o n t o t h e s u r f a c e where i t becomes a temporary member o f t h e 2-D g a s . The f r a c t i o n x o f t i m e a n atom s p e n d s on t h e s u r f a c e i s c o n t r o l l e d by t h e b i n d i n g e n e r g y t o t h e s u r f a c e , Eg, s u c h t h a t f o r s m a l l x

where is i s t h e a v e r a g e t i m e a n atom spends o n t h e s u r f a c e i n a g i v e n s u r f a c e r e s i - dency and 'fB i s t h e a v e r a g e t i m e between s u c h e v e n t s ( 8 ) . A/V is t h e s u r f a c e t o volume r a t i o o f t h e b u l b and A = h/ (2nmkT) i s t h e t h e r m a l d e B r o g l i e wavelength.

On t h e o t h e r hand, t h e a v e r a g e t i m e between c o l l i s i o n s w i t h t h e w a l l s i s g i v e n by

- 4R

T C = - _{34 (2)}

where t h e a v e r a g e v e l o c i t y C = ( B k ~ l n m v = 1 . 4 5 T' x l ~ ~ c m / s e c 'K f o r H and R i s t h e r a d i u s of t h e s p h e r i c a l b u l b . The c o n n e c t i o n between 7, and TB i s v i a t h e s t i c k i n g p r o b a b i l i t y , a , d e f i n e d by -

where, i n c o n t r a s t t o most s o l i d s u r f a c e s , a i s r a t h e r small f o r H c o l l i d i n g w i t h l i q u i d h e l i u m s u r f a c e s . Thus under g i v e n c o n d i t i o n s o f t e m p e r a t u r e and a r e a t o volume r a t i o , t h e a v e r a g e h i s t o r y o f a hydrogen atom i n t h e b u l k i s d e t e r m i n e d by

two p a r a m e t e r s Eg and a . (a=0.04 f o r 4 ~ e and 0.016 f o r 3 ~ e ( 1 0 ) ) .

The hydrogen maser o p e r a t e s on t h e F=0, MF=O ~f F = l , MF=O h y p e r f i n e t r a n s i t i o n , ( t r a n s i t i o n 1-3 i n f i g u r e I ) , and t h e u l t i m a t e performance of t h e maser i s l i m i t e d by t h e s h i f t i n g and b r o a d e n i n g o f t h i s l i n e by i n t e r a c t i o n s o f t h e H atoms w i t h t h e m s e l v e s , w i t h t h e s u r f a c e and w i t h g a s e o u s He atoms.

Spin-exchange c o l l i s i o n s between H atoms b r o a d e n s (T2 p r o c e s s ) a n d s h i f t s t h e t r a n s i t i o n f r e q u e n c y , and g i v e s a l i f e t i m e t o atoms i n t h e u p p e r s t a t e (TI p r o c e s s ) . At t e m p e r a t u r e s and d e n s i t i e s a p p r o p r i a t e t o H maser o p e r a t i o n , s p i n exchange p l a y s a r o l e o n l y through c o l l i s i o n s between atoms i n t h e b u l k g a s and t h e s u r f a c e c o l l i - s i o n s may b e n e g l e c t e d . One c a n t h e n w r i t e (9)

where n ~ i s t h e number d e n s i t y of H atoms, 6, =

v

The main e f f e c t o f t h e s u r f a c e i s t o s h i f t and b r o a d e n t h e r e s o n a n c e through t h e h y p e r f i n e s h i f t , A,, a n atom e x p e r i e n c e s when i t i s on t h e s u r f a c e . A s i m p l e t r e a t m e n t g i v e s ( s e e f o r example r e f . 1 0 )

and

where 9=2n ?,As i s t h e a v e r a g e e x t r a p h a s e s h i f t a n atom a c c u m u l a t e s d u r i n g o n e s u r f a c e r e s i d e n c y . I n f a c t a t t h e t e m p e r a t u r e s of i n t e r e s t h e r e , where $ 2 < < 1 , t h e d e t a i l s o f t h e model a r e u n i m p o r t a n t and we have

e f f e c t s . F i r s t , t h e a v e r a g e h y p e r f i n e f r e q u e n c y i s s h i f t e d by a n amount p r o p o r t i o n a l t o t h e vapour d e n s i t y

d f ~ = B ~ e "He (7)

which i s u s u a l l y r e f e r r e d t o a s a b u f f e r g a s " p r e s s u r e " s h i f t . (Broadening of t h e resonance by t h e same p r o c e s s i s c o m p l e t e l y n e g l i g i b l e ) . Second, t h e mean f r e e p a t h of t h e H atoms i s no l o n g e r l a r g e compared t o t h e s t o r a g e b u l b dimensions, a s i t u a t i o n v e r y d i f f e r e n t from that o f room t e m p e r a t u r e o p e r a t i o n .

( i ) S i n Exchan e.- From e q u a t i o n s 4a and 4b, one-san s e e t h a t t h e spin-exchange p a r a m e b r s of d i z e c t importance are t h e q u a n t i t i e s ov, and f i j

.

(11)

F i g . 2

Spin-exchange c r o s s - s e c t i o n t i m e s r e l a t i v e v e l o c i t i e s . Dashed lines r e p r e s e n t n e g a t i v e v a l u e s .

BfS: Theory of B e r l i n s k y and S h i z g a l (9)

A: Theory of A l l i s o n (11)

A: Experimental Er,

D e s a i n t f u s c i e n e t a1.(12') 0: Experimental-X+Vr,

D e s a i n t f u s c i e p

z t

x: Experimental-A+vr, Crampton e t a L (12) 0: Experimental ovr, Hardy,

e l a l . (6)

t o 10K. One s e e s t h a t 5; d r o p s a l m o s t 3 o r d e r s o f , m a g n i t u d e a s t h e t e m p e r a t u r e i s reduced from 300K t o l K , i n d t h e n f a l l s s l o w l y a s fi below 0.4K

( o

( i i ) Recombination i n Helium Coated C e l l s (T<lK).- For a l l t e m p e r a t u r e s and d e n s i t i e s and f o r b o t h jHe and q ~ e t h e hydrogen d e n s i t y n~ h a s been found ( 6 , 8 , 1 0 ) t o f o l l o w a

second o r d e r r a t e e q u a t i o n d n ~ / d t = - ~ n g . The v a l u e s f o r t h e recombination r a t e c o n s t a n t K, p l o t t e d a s Log ~ f i v s 1/T a r e shown i n f i g u r e s 3 and 4 f o r 4 ~ e and 3 ~ e , r e s p e c t i v e l y . A t t h e h i g h t e m p e r a t u r e s i d e , recombination i s dominated by t h e t h r e e body p r o c e s s i n t h e g a s , H+H+He -+ H2+He. As t h e vapour p r e s s u r e of t h e helium g a s d e c r e a s e s w i t h t e m p e r a t u r e , t h i s p r o c e s s becomes n e g l i g i b l e , and t h e s u r f a c e recom- b i n a t i o n p r o c e s s , H+H+wall -+ H2+wall t a k e s o v e r . For low d e n s i t i e s , t h e s u r f a c e a d s o r p t i o n i s o t h e r m i s g i v e n by eqn. 1: nS = nHh exp(EB/kBT). The t o t a l r a t e equation i n c l u d i n g b o t h p r o c e s s e s t h e n besomes

d n H / d t = - k n ~ ~ n ~ - Xvs (A/v) h2nH exp (2EB/kBT) 2

.

C8-174 JOURNAL DE PHYSIQUE

F i g . 3 F i g . 4

Recombination r a t e f o r H atoms i n a 4 ~ e Recombination r a t e f o r H atoms i n a 3 ~ e c o a t e d c e l l . s o l i d l i n e : A/V = 4 cm-I c o a t e d c e l l . s o l i d l i n e : A/V = 4 cm-' dashed l i n e : s c a l e d t o A/V = 0.4 cm-l dashed l i n e : s c a l e d t o A/V = 0.4 cm-l Here vs = (32kB~/3mnH)li i s t h e a v e r a g e r e l a t i v e v e l o c i t y i n t h e 2-dim a n a l o g o f a c o l l i s i o n c r o s s s e c t i o n and A and V a r e t h e s u r f a c e a r e a and t h e volume o f t h e c e l l , r e s p e c t i v e l y . nHe i s t h e d e n s i t y o f t h e s a t u r a t e d h e l i u m v a p o r , a n d k t h e t h r e e body r a t e c o n s t a n t . For 4 ~ e we have d e t e r m i n e d t h e c o n s t a n t s t o b e (6,8)

0

X = 0.20 A a n d ER _- = 1 . 1 5

*

For 3 ~ e

0

X = 0.18 A ER _- = 0.39K and k = 0 . 1 2 x 1 0 - ~ ~ c m ~ s e c - ~ f o r T = 0.5K ( 1 0 ) .

The h i g h t e m p e r a t u r e r a t e s a r e o f c o u r s e , i n d e p e n d e n t of t h e s u r f a c e t o volume r a t i o A/V, whereas t h e low t e m p e r a t u r e r a t e s s c a l e a s A/V. The dashed l i n e s i n f i g u r e s 3 and 4 are t h e s c a l e d r e s u l t s f o r A/V = 0.4 cm-l, a p p r o p r i a t e t o a 1 5 cm d i a m e t e r b u l b ( i n s t e a d o f t h e v a l u e 4.0 cm-l a p p r o p r i a t e t o t h e e x p e r i m e n t s ) . One s e e s t h a t f o r a helium c o a t e d s t o r a g e b u l b w i t h a r e p r e s e n t a t i v e hydrogen d e n s i t y o f 1 0 ~ c r n - ~ u n d e r masing c o n d i t i o n s , t h e r e c o m b i n a t i o n l i f e t i m e o f t h e atoms s h o u l d b e of o r d e r 0.5 y e a r f o r 4 ~ e a t 0.52 K and 0.8 y e a r f o r 3 ~ e a t 0.2K.

( i i i ) H y p e r f i n e S h i f t . - J u s t a s t h e r e c o m b i n a t i o n p r o c e s s changes from a b u l k p r o c e s s t o a s u r f a c e p r o c e s s a s t h e t e m p e r a t u r e i s lowered, t h e mechanism f o r t h e h y p e r f i n e s h i f t c h a n g e s from a b u l k p r e s s u r e s h i f t a t h i g h t e m p e r a t u r s t o a w a l l s h i f t a t low t e m p e r a t u r e s . I f t h e s e c o n t r i b u t i o n s had o p p o s i t e s i g n s , t h e n e t s h i f t would p a s s t h r o u g h z e r o . I t i s f o r t u n a t e t h a t i n f a c t t h e y have t h e same s i g n

( n e g a t i v e ) , s o t h a t t h e magnitude o f t h e n e t s h i f t goes t h r o u g h a minimum. T h i s a f f o r d s some i n s e n s i t i v i t y t o t e m p e r a t u r e i f we o p e r a t e t h e maser a t t h i s minimum.

The a b s o l u t e h y p e r f i n e s h i f t h a s been measured from 0.15 t o 1 . 3 K f o r a 4 ~ e c o a t e d c o n t a i n e r (8) and from 0.06 t o 1.0 K f o r a 3 ~ e c o a t e d c o n t a i n e r ( 1 0 ) . These r e s u l t s a l l o w a d e t e r m i n a t i o n o f t h e b u l k p r e s s u r e s h i f t c o e f f i c i e n t BHe, t h e s u r f a c e s h i f t As a n d a l s o a n i n d e p e n d e n t d e t e r m i n a t i o n of EB, t h e s u r f a c e b i n d i n g energy.

Over t h e t e m p e r a t u r e r a n g e o f i n t e r e s t t h e d a t a i s w e l l r e p r e s e n t e d by

w h e r e ⁼ -49kHz, BHe = -11.8 1 0 - ' * ~ z cm3 and EB = 1.15 K f o r 'He. ~ h edenSity,nHe ~ ~ e

The t e m p e r a t u r e dependence o f t h e 1 3 ~ e g a s d e n s i t y i s more c o m p l i c a t e d , and o n e h a s t o u s e t h e vapour p r e s s u r e t a b l e s ( 1 3 ) ; f o r rough c a l c u l a t i o n s 9 . 8 4 ~ 1 0 ~ ~ ~ ~ / ~ x exp(-2.352/T) c a n b e u s e d ( e r r o r <7%). W e n o t e t h a t t h e v a l u e f o r EB, 0.43 K, i s somewhat d i f f e r e n t from t h e v a l u e 0.39 K o b t a i n e d from t h e r e c o m b i n a t i o n d a t a . T h i s i s p r o b a b l y due t o a t e m p e r a t u r e dependence of A , b u t i s of no consequence h e r e a s we a r e s e e k i n g simply t o p a r a m e t r i z e t h e d a t a .

F i g . 5: P r e d i c t e d h y p e r f i n e s h i f t vs.

t e m p e r a t u r e f o r a 1 5 cm. d i a m e t e r b u l b c o a t e d w i t h 3 ~ e a n d 4 ~ e .

I n f i g u r e 5 t h e e x p e c t e d h y p e r f i n e s h i f t f o r a 1 5 cm d i a m e t e r s t o r a g e b u l b f o r b o t h 3 ~ e i s p l o t t e d a a i n s t t e m p e r a t u r e and He wal;ls.

f

t h e r e s p e c t i v e minima: 0.52 K f o r 4 ~ e a n d 0.20 K f o r 3 ~ e . The c o e f f i c i e n t i n Af = f m i n

+

f o r 4 ~ e and 3 ~ e r e s p e c t i v e l y , f o r a 1 5 cm b u l b .

Measurements (10) a t l o w e r temp- e r a t u r e s o f b o t h Af and T2 f o r t h e hy- p e r f i n e l i n e h a v e a l l o w e d a measurement o f t h e s t i c k i n g p r o b a b i l i t y a , d e f b e d by E q u a t i o n 3 . T h i s i n t u r n a l l o w s u s t o c a l c u l a t e t h e temperature-dependence o f 'Is and t h e r e f o r e 4 = 2nAsrS. I n t h i s way T2 can b e p r e d i c t e d f o r h i g h t e m p e r a t u r e s where i t i s t o o l a r g e t o b e measured. We o b t a i n , f o r t h e w a l l c o n t r i b u t i o n , Tf= 568 s e c . f o r 4 ~ e w a l l s a t 0.52 K and T2 = 295 s e c . f o r

3 ~ e w a l l s a t 0.2K; t h e Tp 's due t o c o l l i s i o n s w i t h t h e g a s e o u s helium a r e much l o n g e r s t i l l .

( i v ) D i f f u s i o n o f H i n Helium Gas F i g u r e 6 shows t h e low tempera- t u r e d i f f u s i o n c r o s s - s e c t i o n f o r H i n 4 ~ e measured by Hardy e t a l . ( 6 ) u s i n g spin-echo t e c h n i q u e s a t t o 1420 MHz t r a n s i t i o n . Also i n c l u d e d a r e t h e c a l c u l a t i o n s o f Jochemsen and Hardy (unpubished) f o r s e v e r a l o f t h e b e s t H-He p o t e n t i a l s . One o b t a i n s a v e r y

F i g . 6: Measured v a l u e s o f t h e d i f f u - s i o n c r o s s - s e c t i o n f o r H i n ' ~ e compar- ed t o t h e second o r d e r a p p r o x h t i o n i n Chapman-Enskog t h e o r y c a l c u l a t e d f o r t h r e e r e c e n t l y a v a i l a b l e p o t e n t i a l s . The s o l i d l i n e i s a n e m p i r i c a l poten- t i a l developed by Jochemsen and Hardy, and t h e upper and lower c u r v e s are t h o s e o f Das e t a l . , and S c o l e s , r e s p e c t i v e l y . ( s e e 1 4 and r e f e r e n c e s t h e r e i n ) .

C8-176 JOURNAL DE PHYSIQUE

s t r o n g t e m p e r a t u r e dependence below 2 K t h a t is s e n s i t i v e t g t h e p a r t i c u l a r p o t e n t i a l chosen. However, one c a n p r e d i c t t h a t QD w i l l b e a b o u t 1 2 a t 0.52 K t o w i t h i n 15% o r s o . Using t h e Helium vapour d e n s i t y a t t h i s t e m p e r a t u r e o n e f i n d s t h e mean f r e e p a t h X = ( n H , ~ ~ ) - l % 1 . 4 cm, which i s r a t h e r s h o r t . For 3 ~ e t h e r e i s , up t o now, n e i t h e r t h e o r y n o r e x p e r i m e n t . I f we g u e s s QD % 5 i 2 a t 0.2 K we o b t a i n X = 3.2 cm, a l s o q u i t e s h o r t . T h i s means t h a t t h e motion o f t h e H atoms i n t h e b u l b i s l a r g e l y d i f f u s i v e , i n c o n t r a s t t o t h e s i t u a t i o n i n room t e m p e r a t u r e masers. U n l e s s , f o r ex- ample, t h e p r e s s u r e o u t s i d e t h e b u l b i s r e d u c e d by d i f f e r e n t i a l pumping, t h i s seems

t o r u l e o u t t h e u s u a l t y p e of s t a t e s e l e c t i o n which r e q u i r e s t h e f o r m a t i o n of a beam of atoms.

111. Performance of a Cryogenic Maser.- I n t h i s s e c t i o n we c o n s i d e r t h e p e r f o r m a c e of t h e c l a s s i c a l maser c o n f i g u r a t i o n ( 7 ) shown i n f i u r e 1 o p e r a t e d a t t h e optimum

9

t e m p e r a t u r e s 0.5K f o r 4 ~ e c o a t e d w a l l s and 0.2K f o r He w a l l s .

( i ) O p e r a t i n g C o n d i t i o n s . - T h i s d i s c u s s i o n c a n b e enormously s i m p l i f i e d a t t h e o u t s e t by n o t i n g t h a t t h e q u a l i t y p a r a m e t e r , q , d e f i n e d by K l e p ~ a e t a 1 ( 1 5 ) , which i s e s s e n t i a l l y t h e r a t i o of s p i n exchange damping t o r a d i a t i o n damping, i s a b o u t 3 o r d e r s of magnitude s m a l l e r a t 0.5 K ( k e e p i n g t h e same c a v i t y Q and b u l b s i z e ) due t o t h e r e d u c t i o n i n

ovr

Under t h e assumption t h a t t h e s t a t e s e l e c t i o n is p e r f e c t , ( o n l y atoms i n t h e s t a t e F = 1, MF = 0 e n t e r t h e b u l b ) , t h e power r a d i a t e d by t h e atoms i s g i v e n by (14)

The t h r e s h o l d f l u x , Ith, i n a t o m s l s e c i s r e l a t e d t o PC by Ith = P c / % w w i t h

Here Vc is t h e c a v i t y volume, uo i s t h e Bohr magneton, QL i s t h e l o a d e d c a v i t y Q,and

,,

b u l b c a v i t y = n1 'cfVb (12)

where 0 ' i s t h e u s u a l f i l l i n g f a c t o r i n magnetic r e s o n a n c e . The d e n s i t y nH o f hy- drogen atoms i n t h e b u l b can b e c a l c u l a t e d from t h e f l u x I a n d t b v i a

I n T a b l e I a few v a l u e s o f PC, I t h a n d (nHlthres-

7x109 4x106 h o l d a r e g i v e n a s a func-

7x107 4 x 1 0 ~ t i o n o f t h e s t o r a g e t i m e

320 6 . 6 ~ 1 0 - l 9 7x1 0 4 x 1 0 ~ t b , assuming Q -30,000,

t.-

n=3. _, V,=l. 3x10 cm3 and

..

v b = l .8x103cm3. One p r o b a b l y w a n t s t o u s e tb & 3 s e c . b e c a u s e o f t h e r a t h e r s l o w t h e r m a l motion in t h e b u l b . R e f e r r i n g t o T a b l e I we s e e t h a t f o r r e a s o n a b l e maser o u t p u t power ( i . e .

<

1 0 - l 3 w a t t s ) one i s a l w a y s w e l l above t h r e s h o l d .

( i i ) Accuracy.- I t seems l i k e l y t h a t t h e f r e q u e n c y a c c u r a c y o f t h e maser w i l l b e l i m i t e d 6 y t h e a b i l i t y t o measure t h e e f f e c t i v e s u r f a c e a r e a of t h e s t o r a g e b u l b . T h i s assumes t h a t b e c a u s e of t h e p u r i t y and u n i f o r m i t y of t h e f i l m , t h e f r e q u e n c y s h i f t a t t h e mlnimum s h o u l d b e q u i t e r e p r o d u c e a b l e f o r a g i v e n v a l u e o f A/V, and t h e r e f o r e a c c u g a t e l y m e a s u r e a b l e . A f u r t h e r a d v a n t a g e o f t h e h e l i u m f i l m (which c a n be of o r d e r 200A t h i c k ) i s t h a t i t w i l l t e n d t o smooth o u t t h e s u r f a c e i r r e g u l a r i t i e s of t h e b u l b . These c o n j e c t u r e s w i l l have t o b e demonstrated however, s i n c e inhomo- geneous p e r t u r b a t i o n s s u c h a s e l e c t r o n s t r a p p e d above t h e h e l i u m s u r f a c e may prove i m p o r t a n t .

( i i i ) S t a b i l i t y . - There a r e a number o f f a c t o r s t h a t c o n t r i b u t e t o f l u c t u a t i o n s i n t h e o u t p u t f r e q u e n c y o f t h e maser. We f i r s t d e a l w i t h t h e i n f l u e n c e of e l e c t r i c a l n o i s e on t h e "short-term" s t a b i l i t y o f t h e maser and t h e n d i s c u s s l a t e r t h e v a r i o u s s o u r c e s of l o n g e r term f l u c t u a t i o n s .

Thermal and A m p l i f i e r Noise.- The fundamental l i m i t a t i o n on t h e e l e c t r i c a l n o i s e i s t h e t h e r m a l n o i s e of t h e microwave c a v i t y i t s e l f , which c o n t r i b u t e s a f r a c t i o n a l r m s f r e q u e n c y f l u c t u a t i o n ( 7 ) .

a t o m i c r e s o n a n c e ("1ine"Q). F o r t h e c r y o g e n i c maser wkere t h e w i d t h of t h e a t o m i c l i n e i s g e n e r a l l y dominated by t h e l i f e t i m e i n t h e b u l b , ~ ~ > n f , t ~ . I n a d d i t i o n t o t h i s n o i s e t h e r e i s t h e n o i s e added by t h e a m p l i f i e r system, which l e a d s t o a n r m s phase f l u c t u a t i o n A @ = ( ~ T ~ B I P ~ ) ' ~ where TN = system n o i k e t e m p e r a t u r e r e f e r r e d t o t h e i n p u t o f a m p l i f i e r , Po i s t h e power d e l i v e r e d from maser i n t o t h e a m p l i f i e r , and B is t h e bandwidth o f t h e measuring system. T h i s l e a d s t o a f r a c t i o n a l rms d e v i a - t i o n o f

These two c o n t r i b u t i o n s have been d i s c u s s e d i n d e t a i l by V e s s o t e t a l . (16) who ex- p l o r e some o f t h e consequences o f low t e m p e r a t u r e o p e r a t i o n . They n o t e t h e d i s t i n c - t i o n between Po a n d Pb and s u g g e s t o v e r c o u p l i n g t h e c a v i t y t o r a i s e Po t o Pb.

We p o i n t o u t h e r e t h a t i n f a c t , t h e T i n e q u a t i o n 1 4 i s n o t t h e c a v i t y temper- a t u r e b u t something l i k e

T = . Tc/Qc + T ~ / Q e x t (16)

1 / Q c + l / Q e x t

where Qc and Q a r e t h e unloaded c a v i t y Q and c o u p l i n g Q r e s p e c t i v e l y

( ~ ~ - ~ = ~ ; l t ~ ; $ ~ F ; ~ a n d TL i s t h e t e m p e r a t u r e "seen" l o o k i n g towards t h e a m p l i f i e r

.

I f t h e r e i s a n i s o l a t o r , a s shown i n f i g u r e 7, t h e n TL = TI; o t h e r w i s e TL i s d e t e r m i n e d by t h e n o i s e e m i t t e d (17) CAVITY ISOLATOR by t h e a m p l i f i e r (Note t h a t TL i s n o t

C- -

- H A M P L I F I E R ~ " ~

- -

Tc (NOISE =TN, o n e c a n p u t t h e e f f e c t i n d i c a t e d i n e a u a t i o n 1 6 t o a d v a n t a g e : - by s t r o n g - o v e r c o u p l i n g (Qext << Qc) , T + TL = TI F i g . 7: Placement of i s o l a t o r between

when u s i n g a n i s o l a t o r . Thus o n e can c a v i t y and a m p l i f i e r .

r e d u c e T by l o w e r i n g only t h e tempera- t u r e o f t h e i s o l a t o r .

Combining t h e r e s u l t s of e q u a t i o n s 1 4 and 1 5 a n d u s i n g Qk=nfotb we c a n w r i t e t h e n e t f r a c t i o n a l f l u c t u a t i o n a s

where, f o r s i m p l i c i t y we have t a k e n Pb = Po. A s d i s c u s s e d by V e s s o t e t a l . ( 1 6 ) , f o r s h o r t T , Af/f v a r i e s a s r - l and c r o s s e s o e r t o a T-% dependence a t l o n g a v e r a g - i n g times. The c r o s s - o v e r o c c u r s when r = Zt: T~ B. L e t u s t a k e T = 0.5K, TN=~OK

T-

( a c h i e v a b l e now w i t h c o o l e d GaAs FET a m p l i f i e r s ) , B = 6Hz, and ~ b = 1 0 s e c . The c r o s s - o v e r t h e n o c c u r s f o r c = 3x103sec a t which p o i n t Af/fo = 3 . 5 ~ 1 0 - ' a f o r a typ- i c a l maser power o f 1 0 - l 3 w a t t s . T h i s i s l i k e l y t o b e c o m p l e t e l y masked by s y s t e - m a t i c d r i f t s , a n d t h e r e f o r e , f o r most p r a c t i c a l p u r p o s e s , t h e maser s t a b i l i t y is dominated by a m p l i f i e r n o i s e a t s h o r t T and d r i f t a t l a r g e T . A c o n v e n i e n t form f o r t h e former c o n t r i b u t i o n i s

- = - A f I. ( ~ T ~ B I P , ) ~

f o 2 r f 0 T (18)

which h a s t h e n u m e r i c a l v a l u e 1 x 1 0 - 1 4 / r f o r t h e v a l u e s of t h e p a r a m e t e r s assumed above.

Thermal F l u c t u a t i o n s . - I t seem t o b e g e n e r a l l y a c c e p t e d t h a t much of t h e long term d r i f t i n t h e room t e m p e r a t u r e masers i s t h e r e s u l t of s m a l l t e m p e r a t u r e v a r i a t i o n s which c a u s e t h e c a v i t y r e s o n a n t f r e q u e n c y , f c , t o f l u c t u a t e . The maser o u t p u t f r e q u e n c y , f o , i s c o r r e s p o n d i n g l y changed by a n amount ( 7 ) .

Low t e m p e r a t u r e o p e r a t i o n h a s s e v e r a l a d v a n t a g e s . F i r s t l y , t h e t h e r m a l expansion o f t e c h n i c a l m a t e r i a l s i s e x t r e m e l y low. For example, copper h a s a = 1 . 6 x ~ O - ' ~ T ( K - ' ) f o r T < 1 K ; a t 0.5K, a i s more t h a n f i v e o r d e r s of magnitude s m a l l e r t h a n a t room t e m p e r a t u r e . The e f f e c t s a r e c o m p l e t e l y n e g l i g i b l e a s c a n b e s e e n from a n example: f o r 6 T = ~ o - ~ K ( e a s i l y o b t a i n e d a t 0.5K) and QL/Q = 3 x 1 0 ~ / 1 0 ~ ,

R

C8-178 JOURNAL DE PHYSIQUE

Efo/fo ^Q, 10-18. Here we have chosen a v e r y modest v a l u e f o r Qk. The o t h e r advan- t a g e s o f low t e m p e r a t u r e o p e r a t i o n a r e t h a t QR c a n b e made l a r g e r and QL s m a l l e r , which f u r t h e r r e d u c e s t h e p u l l i n g e f f e c t s ; a l t h o u g h t h e r m a l e x p a n s i o n e f f e c t s may b e of no consequence, i t may b e i m p o r t a n t t o r e d u c e t h e e f f e c t o f o t h e r mechanical o r e l e c t r i c a l i n s t a b i l i t i e s .

A more i m p o r t a n t e f f e c t o f t e m p e r a t u r e f l u c t u a t i o n s comes v i a t h e t e m p e r a t u r e dependence of t h e h y p e r f i n e s h i f t shown i n f i g u r e 5. Using t h e r e s u l t s of s e c t i o n I I ( i i i ) , o n e f i n d s t h a t a v a r i a t i o n of ~ o - ~ K changes f o from t h e minimum v a l u e by 2 . 7 x 1 0 - ~ Hz and 1 0 . 7 x 1 0 - ~ Hz f o r 4 ~ e and 3 ~ e w a l l s r e s p e c t i v e l y .

Spin-Exchange S h i f t . - I t was s e e n i n s e c t i o n I I ( i ) t h a t a l t h o u g h t h e s p i n - exchange b r o a d e n i n g becomes v e r y s m a l l a t low t e m p e r a t u r e s , t h e spin-exchange s h i f t does n o t , and t h e l a t t e r may be a c o n t r o l l i n g f a c t o r i n t h e a c c u r a c y and s t a b i l i t y of t h e c r y o g e n i c maser. I n t h e f o l l o w i n g , we e v a l u a t e t h e s h i f t g i v e n by e q u a t i o n 4b a s a f u n c t i o n o f t h e o p e r a t i n g p a r a m e t e r s . From e q u a t i o n 1 3 , f o r powers w e l l above t h r e s h o l d , t h e hydrogen d e n s i t y , nH, i s g i v e n by

For ( ~ 3 - P I ) , we u s e t h e r e s u l t o f Crampton ( 1 8 ) , m u l t i p l i e d by a f a c t o r 2 b e c a u s e o f o u r a s s u m p t i o n t h a t o n l y one, r a t h e r t h a n two, h y p e r f i n e s t a t e s a r e i n t h e beam. D e f i n i n g t h e d i f f e r e n c e between t h e t r a n s i t i o n f r e q u e n c y a n d t h e c a v i t y t u n i n g a s (Aw/2n), t h i s y i e l d s

where PC i s g i v e n by e q u a t i o n (11) and

For t h e c i r c u m s t a n c e s a p p r o p r i a t e t o t h i s d i s c u s s i o n , we may n e g l e c t ( A W T ~ ) ~ compared t o 1 and a l s o d r o p t h e spin-exchange c o n t r i b u t i o n s t o t h e r e l a x a t i o n r a t e s . Equation (21) t h e n s i m p l i e s t o

1 (23)

s o t h a t s u b s t i t u t i n g e q u a t i o n s (23) and (20) i n t o e q u a t i o n (4b) g i v e s

For t h e s t a n d a r d maser c o n f i g u r a t i o n d i s c u s s e d above, one f i n d s

PC = 6.67 x 10-14/t; in w a t t s s o t h a t f o r P t; >> 6.67 x 1 0 - l 4 w a t t s e c 2 t h e s p i n - exchange s h i f t a p p r o a c h e s t h e l i m i t

i n d e p e n d e n t of power. The (tg)-' dependence may o f f e r a means of l i m i t i n g t h e e f f e c t of t h e spin-exchange f r e q u e n c y s h i f t i n t h e c r y o g e n i c maser.

I V . D i s c u s s i o n . - At t h e p r e s e n t l e v e l o f u n d e r s t a n d i n g , t h e r e e x i s t s c o n s i d e r a b l e promise f o r a v e r y h i g h performance c r y o g e n i c H maser. Although we have c o n c e n t r a t e d i n t h i s p a p e r on t h e p o s i t i v e a s p e c t s t h e r e a r e a l s o s p e c i a l problems a s s o c i a t e d w i t h low t e m p e r a t u r e o p e r a t i o n . F o r Z ~ e c o a t e d w a l l s o n e n e e d s a t l e a s t a c o n t i n u -

o u s l y r u n n i n g 3 ~ e r e f r i g e r a t o r ; f o r 3 ~ e w a l l s , a 3 ~ e / 4 ~ e d i l u t i o n r e f r i g e r a t o r would b e r e q u i r e d . Achievement o f l o n g term u n a t t e n d e d o p e r a t i o n w i l l undoubtedly r e q u i r e a n i n s p i r e d e n g i n e e r i n g e f f o r t .

Another s p e c i a l problem i s a s s o c i a t e d w i t h t h e s l o w d i f f u s i v e motion of t h e atoms. The c r o s s - r e l a x a t i o n o f t h e F=l, MF'O l e v e l w i t h t h e MF= k1 l e v e l s w i l l be d i f f i c u l t t o a v o i d a t v e r y low b i a s f i e l d s . For example, a t 300K, t h e " r a t t l e

At low t e m p e r a t u r e s , t h e motion i s d i f f u s i v e w i t h a t i m e c o n s t a n t o f o r d e r R ~ / ~ D where D i s t h e d i f f u s i o n c o n s t a n t . T h i s a m o u n t s t o 2.6 msec f o r t h e 4 ~ e c o a t e d w a l l s and 1.8 msec f o r 3 ~ e . The d e t a i l s have n o t y e t been worked o u t b u t i t would a p p e a r t h a t o n e h a s t o b i a s t h e maser s o t h a t t h e s e p a r a t i o n o f t h e F = l s u b l e v e l s i s m u c h g r e a t e r t h a n t h e e f f e c t i v e " r a t t l e frequency", r a t h e r t h a n t h e i n v e r s e . T h i s r e n d e r s t h e maser more s e n s i t i v e t o magnetic f i e l d f l u c t u a t i o n s a n d inhomo- g e n e i t i e s . F o r t u n a t e l y , t h e p o s s i b i l i t y e x i s t s of u s i n g s u p e r c o n d u c t i n g s h i e l d s , which a r e q u i t e e f f e c t i v e i n e l i m i n a t i n g f i e l d f l u c t u a t i o n s .

R e f e r e n c e s

1. CRAMPTON, S. B . , J. d e P h y s i q u e 41 (1980) C7-249.

2. SILVERA,I. F. and WALRAVEN, J.T.M., Phys. Rev. L e t t .

44

3. CLINE, R. W . , GREYTAK, T. J., KLEPPNER, D., and SMITH, D. A., J. de P h y s i q u e 4 1 (1980) C7-151.

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