TECHNlCn

u. s. DEPARTMENT

OF COMMERCE

National Bureau

of Standards

NOTE 387

NATIONAL BUREAU OF STANDARDS

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UNITED STATES DEPARTMENT OF COMMERCE Maurice H. Stans, Secretarly

NATIONAL BUREAU OF STANDARDS B Lewis M. Branscomb, Director

e TECHNICAL NOTE 387 ISSUED MARCH 1970

Not. Bur. Stand. (U.S.), Tech. Note 387, 13 pages (March 1970) CODEN: NIBTNA

Hydrogen Spin Exchange Freqdency Shifts

Helmut Hellwig

Atomic Frequency and Time Standards Section Time and Frequency Divisian Institute for Basic Standardb

National Bureau af Standards Boulder, Colorado 80302

NBS Technical Notes are designed to suppljement the Bureau's regular publications program. Thgy provide a means for making ovailoble scientific daya that are of transient or limited interest. Technical lhotes may be listed or referred to in the open literature.

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TABLE O F CONTENTS

ABSTRACT . . . . . . . . . . . . . . . . . . . . . .

1 . 2 .

3 .

4 .

5 .

6 .

7 .

INTRODUCTION . . . . . . . . . . . . . . . . . . .

SPIN EXCHANGE FREQUENCY SHIFT . . . . . . . . .

CONDITIONS FOR THE HYDROGEN STORAGE BEAM TUBE . . . . . . . . . . . . . . . . . . . .

CONDITIONS FOR THE HYDROGEN MASER OSCILLATOR

TUNING O F THE HYDROGEN MASER . . . . . . . . .

CONCLUSIONS . . . . . . . . . . . . . . . . . . .

REFERENCES . . . . . . . . . . . . . . . . . .

P a g e

1

1

2

3

6

9

11

13

iii

HYDROGEN SPIN EXCHANGE FREQUENCY SHIFTS

Helmut Hellwig

Frequency shifts due to hydrogen spin exchange collisions a r e discussed for the hydragen m a s e r and the hydrogen s torage beam tube. of spin exchange and cavity pulling in the hydrogen m a s e r a r e evaluated with emphas is on frequency e r r o r s introduced by s tandard tuning procedures . that a n automatic cavity tuning sys tem based on a var ia t ion of the linewidth does not introduce a frequency e r r o r . In contrast , manual tuning procedures based on an interpolation of measu remen t s a t different cavity sett ings do not yield a perfect compensation of spin exchange effects. However, the frequency e r r o r be- come s only significant a t unnecessar i ly la rge cavity offsets.

The combined effects

It is found

The smal lness of the spin exchang,e shifts and the near absence of cavity pulling in a hydrogen s torage beam tube a r e expected to pe rmi t a determination of spin exchange shifts by varying the bejam intensity with a precis ion adequate for stabil i ty and accu racy f igures of o r be t te r .

Key Words: hydrogen m a s e r ; m a s e r tuning; spin exchange.

Frequency pulling; hydrogen beam tube;

1. INTRODUCTION

Collisions between radiating hydrogen a toms cause a broadening

of the resonance line a s well a s a frequency shift. An important feature

of the hydrogen m a s e r i s the automatic cancellation of the spin exchange

frequency shift by cavity pulling i f the hydrogen m a s e r tuning method

i s based on varying the hydrogen relaxation t ime. The mos t widely

used method is the beam intensity modulation technique (Vanier and

Vessot, 1964).

A need was fel t to d i scuss and recheck th i s r e su l t . An additional

impetus for reevaluating th i s aspect of the hydrogen m a s e r was provided

by the fact that only one shor t discussion of spin exchange and cavity

pulling is published in the open l i t e ra ture (Crampton, 1967) although the

resul t is widely used in exper iments with the hydrogen m a s e r (Vanier

and Vessot , 1964). F u r t h e r m o r e , a quantitative es t imate of the spin

exchange frequency shift t o be expected in the proposed hydrogen s torage

beam tube was needed (Hellwig, 1970) .

1

The following discussion s t a r t s f rom bas ic equations for spin

exchange effects given by Bender (1963).

what f rom the one given by Crampton (1967); in par t icu lar a detailed

discussion of the cavity pulling i n combination with spin exchange shifts

i s given, and the frequency e r r o r s associated with maser tuning tech-

niques a r e evaluated.

The t rea tment differs some-

2. SPIN EXCHANGE FREQUENCY SHIFT

2 The shifted frequency differs f r o m the unperturbed t ransi t ion

frequency kt by a n amount proportional t o the ra te of spin exchange 0 phase shifting V and to the population difference (P22-P44) of the two

m = 0 leve ls . The relationships a r e given by (19) , (22),and (24) in

Bender ' s a r t i c l e (1963) F

0

Every publication giving a frequency determinat ion of the hydrogen m a s e r r e l i e s on the automatic cancellation of the spin exchange shift by cavity pulling when the maser i s tuned.

Throughout this paper is just called "frequency" although, i n a s t r i c t s e n s e , i t is the angular frequency,

2

2

- K2 t ( u - c c ' ~ ) ~

8p2 K r

p22-p44 - [ K2 + (u- kl')'] [ 2 t U / P ] + ~

( 3 )

The symbols have the following meaning: cr' = running frequency;

n = atomic density; 7 = relative atomic velocity; R = collision

p a r a m e t e r ;

of spin exchange coll isions; 6 = rf magnetic field p a r a m e t e r ; r = ra te

of escape of a toms f rom the bulb.

re1 A(R) = phase shift ; K = total relaxation constant; U = r a t e

K and B can be expressed as

K = r - t i U (4)

where 1

average magnetic rf field i n the bulb region.

relaxation p rocesses other than escape f rom the bulb and spin exchange

a r e not included in this discussion.

= magnetic dipole moment of the trqnsit ion, and (Hz)b =

It mus t be noted that 0

3 . CONDITIONS FOR THE HYDROGEN $TORAGE BEAM TUBE I

The hydrogen s torage beam tube (Hellwig, 1 9 7 0 ) i s a proposed

new device which potentially combines the adqantages of the hydrogen

m a s e r with those of a beam tube.

of Ramsey ' s original s torage beam proposal qnd h is "bounce box"

experiments with ces ium (Kleppner, Ramsey,bnd F je l s t ad , 1956) which

led to the development of the hydrogen m a s e r l(Kleppner, Goldenberg,

and Ramsey, 1 9 6 2 ) .

I ts concept1 is a logical extension

,

I

3

The hydrogen s torage beam tube r e sembles essent ia l ly the

configuration of a hydrogen m a s e r , however it is operated well below

osci l la t ion threshold, and a n ex te rna l microwave signal i s injected into

the cavity.

a t ransi t ion i s monitored with a second se lec tor magnet and a beam

de tec tor in a fashion quite s imi l a r to the s tandard operational principle

of a n atomic beam tube.

The number of a toms leaving the bulb which have undergone

Of the var ious a spec t s of the hydrogen s torage beam tube the

one which i s of importance to the d iscuss ion presented h e r e i s the n e a r

absence of cavity pulling.

approximately a pulling which is proport ional to the r a t io of the linewidths

of cavity and a tomic t rans i t ion squared (Holloway and Lacey, 1964) whereas

the pulling in a m a s e r is d i rec t ly proport ional to this ra t io as can be seen

f r o m (20) below.

beam tube can be expected to be reduced by many o r d e r s of magnitude

a s compared to the hydrogen m a s e r (Hellwig, 1970).

F o r pass ive beam tube devices one obtains

As a resu l t the cavity pulling in the hydrogen s torage

In the following we shal l e s t ima te the spin exchange frequency

shift to be expected in a hydrogen s torage beam tube.

We a s s u m e that beam intensi t ies of Id’ a toms p e r second a t 3 s torage t imes of about 1 s shal l not be exceeded. F o r these conditions

the spin exchange line broadening is small compared to the linewidth due

to the s torage t ime. We therefore m a y use the approximations

3 F o r a bulb of IO3 cm3 volume this cor responds to a n a tomic density of about 10’ a toms p e r c m . 3

4

W e fur thermore r e s t r i c t ourse lves to microwdve driving frequencies

well within the linewidth 6 L! = 2K of the transit ion. We m a y then use

the approximation

W - a'<< K.

With the above approximations (3) simplifies t O

The power of the exciting microwave signal wiel approximately be

chosen a s

B l r - - _ - r 2 '

Bender (1963) evaluated numerical ly thk integral in ( 2 ) , and

gave a s a resul t

- V x 13n aO2 n v re1

(7)

where a i s the Bohr radius (a = 0. 529 X IO-' c m ) . After combining

( I ) , (6) , (7),and (8) we obtain with 7 0 0

= 2 X IO5 cm/s re1

0-9- o = k n 0

where k, -2. 6 X c m 3 / s , and n is meabured in cm-30

(9)

5

The spin exchange frequency shift is proportional to the atomic den-

s i ty n , and i t s value for the a s s u m e d highest density of n = IO9 a t o m s / c m 3

i s w’ - o x -2 . 6 X s-‘ o r a fractional frequency shift of about

3 X According to (9) this frequency shift i s l inear with the atomic

density and can therefore be measu red conveniently by varying the beam

intensity. It has to be noted that a var ia t ion of the beam intensity causes

only negligible cavity pulling due to the near -absence of this effect in the

beam tube device a s was pointed out previously.

0

4. CONDITIONS FOR THE HYDROGEN MASER OSCILLATOR

The condition for self- sustained m a s e r oscil lations can be writ ten

a s

P = P rad abs

where the radiated power i s given by

d P 2 2 P = n V h a 7 rad b

and the absorbed power by

V --

C P = (H2)

abs Q %

(11)

/dt the rate of dP22 V i s the bulb volume, V the cavity volume,

emis s ion f r o m one a tom, and (H”) the average squared rf magnet ic

field in the cavity. The rate of emis s ion was given by Bender (1963)

a s

b C

6

dp22 2 B 2 K dt

[ K2 t (CL' - o:')" ] [ 2 t 'c

The combination of (10) to (13 ) yields

- - 8p" K r

[ K' t ( w - ~ i l ) ' ] [ 2 t U / r ] +

2 fl in (14) can be substituted f rom (5). The f i

by Kleppner and Vessot e t a l . (19.65) is g ' iver

We introduce the p a r a m e t e r A ' which contai

given m a s e r configuration

Equations (14) to (16) combine with ( I ) , (3) ,a

[It(= - uo = - n K V

7

88" K r1 f

(13)

C r (H2) C

ing factor 7 ' a s defined

'Y

s only constants for a

I ( 5 ) to

Inserting (8) into (17) leads to

If we r e s t r i c t the frequencies to a sma l l range around 0' we obtain

with C = 13 2 - v A ' T a ~ re1

Equations (17) and (18) do not contain explicitly the beam intensity;

instead, the spin exchange frequency shift is proportional to the total

linewidth 6 w = 2K. At f i r s t the resu l t (19) i s surpr i s ing because one

w,) 4 0 for 6ua 4 6wE where 6 w is the l ine- would expect (0' - width due to escape f r o m the bulb alone (vanishing contribution of spin

exchange effects).

spin exchange effects.

for self-sustained m a s e r oscil lations and i s therefore only valid under

these conditions. In o r d e r to real ize a n oscillating m a s e r in the limit

6 wa according t o (1 8).

a

E

However, (18) does not pe rmi t the l imit of vanishing

The equation was der ived using the condition (10)

4 6 wE, the p a r a m e t e r has to approach zero,and (wl - "0) + .O

A' re la tes to the m a s e r oscil lation p a r a m e t e r q, which was defined by Kleppner and Vessot e t a l . (1965).

4

8

5. TUNING O F THE HYDROGEV MASER

The pulling of the m a s e r output frequericy by a cavity which is

detuned f r o m the a tomic t rans i t ion frequency is one of the e a r l i e s t

d i scover ies i n m a s e r theory (Gordon, Zeiger, and Townes, 1955), and

is quantitatively given by (Shirley, 1968)

Fi w c w a + 6 UaWc =

6 wc -t 6 wa

where 6w

respect ively, and wa i s the a tomic t ransi t ion ,frequency.

i s included by substituting w' for . We alslo substi tute (Q = loaded

cavity Q)

and 6c", are the linewidths of cavity and t ransi t ion l ine, C

Spin exchange

a

and obtain

w' can be expres sed f r o m (18), and (21) then r e a d s a f t e r some s imple

r e grouping

9

If we use the approximations

(22) simplif ies to

L' C

C ) * cL'-cL' = - 2Q K ( & , -by - - c 0 2 Q

C o w

Equation (23) i s the s tandard "tuning equation" for the hydrogen m a s e r

(Vanier and Vessot, 1964). If the m a s e r is tuned s o that a var ia t ion

of K does not cause a change in the m a s e r output frequency then the

m a s e r frequency i s equal to the unperturbed atomic t ransi t ion f r e -

quency (w = wo), and the

spin exchange frequency

f r o m (23).

cavity i s detuned by an amount equal to the

shift (w - u = C w / 2 Q ) a s can be seen c o C

The e r r o r commit ted in tuning the m a s e r in this fashion can be

evaluated f r o m the above approximations. Omitting K in the denom-

inator of the "pulling factor" in (22) does not introduce any e r r o r in

the tuning procedure. The omiss ion of the l a s t , K-dependent, t e r m

in (22) s e e m s to introduce a frequency e r r o r .

of this t e r m can be rewri t ten using (19)

The K-dependent p a r t

(q)y w - w OK t C K ) a =(TtcJ ,

The constant C in this expression does not introduce an e r r o r in the

above tuning method. The K-dependent p a r t ( w - w ) / K approaches z e r o

as the m a s e r i s tuned (0' wo).

e r r o r is introduced by a tuning procedure which uses the var ia t ion of K.

0 As a resu l t no appreciable frequency

10

This i s only t rue for exper iments where the 4 a s e r is actually se t to

oscil late on the t ransi t ion frequency w a s for example in automatic

s e r v o tuning sys t ems (Hellwig and Pannaci , 1667; and Vessot, Levine,

Mueller,and Baker , 1967). Frequently, howeqer , the m a s e r frequency

i s only indirectly de te rmined using two o r mope cavity sett ings a t which

K i s var ied (Vanier and Vessot, 1964).

calculated using a l inear interpolation between the measu red var ia t ions

of the m a s e r output frequency.

troduced which corresponds to the variation d i t h K of the l a s t t e r m in

(22) . This t e r m can be wri t ten as -C

then approximately a s C

The m a s e r constant C has typically a value qf C z IO-".

the e r r o r has an appreciable magnitude only if fa i r ly large cavity off-

s e t s and large K-variations a r e used.

corresponding to a fractional m a s e r frequencv offset of

produce a f ract ional e r r o r of about

0

I

l

The + a s e r frequency is then

In this case aifrequency e r r o r i s in-

. The e r r o r follows (a- K

b K, where 1LZ K i s the var ia t ion of K.

Fortunately,

(0- K2

F o r exkmple a cavity setting

IO-'' will

IOd3 i f L$K= K, 1 s-'.

6. CONCLUSIONS

The discussion given in this report indlicates that an automatic

cavity servo control based on varying the l ineyidth of the atomic t r an -

si t ion yields an exact tuning of the m a s e r .

shift will be perfect ly compensated by a propeir cavity detuning.

manual tuning procedure in which the m a s e r tbning i s found by in te r -

polating between measu remen t s a t diflerent cavity sett ings will introduce

in general a systematic e r r o r due to an incomblete compensation of

frequency shifts. However, this e r r o r only has a n appreciable magnitude

i f unnecessar i ly la rge cavity offsets a r e used.l

The spin exchange frequency

A

I

11

It was pointed out that the automatic compensation of spin

exchange and cavity pulling effects , a s descr ibed by ( 2 3 ) , i s only

valid for a n oscillating m a s e r .

In a passive device such as the proposed hydrogen s torage

beam tube the frequency shift due to spin exchange i s present . The

shift i s l inear with the beam intensity (a tomic density in the storage

bulb) and can be m e a s u r e d and cor rec ted for by varying the beam

flux. for

typical beam intensit ies and s torage t imes . Thus only a modera te

measu remen t prec is ion and a modera te control of the beam flux i s

n e c e s s a r y i f s tabil i t ies and accurac ies of

of a passive device (Hellwig, 1970).

The magnitude of the shift i s smal l ; of the o rde r of

a r e the design goal

12

7. REFER.ENCES

Bender, P. L. (1963), Effects of hydrogen-hydrogen exchange colli-

sions, Phys. Rev. 132, 2154-2158. - Crampton, S. Bo (1967), Spin-Exchange shifts in the hydrogen m a s e r ,

Phys. Rev. - 158, 57-61.

Gordon, J. P . , H. J. Zeiger , and C. H. Towries (1955), The m a s e r -

new type of microwave amplifier, frequency standard, and

spectrometer , Phys. Rev. - 99, 1264-1274,

Hellwig, H. (1970), the hydrogen storage beam tube, a proposal

for a new frequency standard.

Metrologia - 6, No. 2 , April 1970.

Accepte(d for publication in

Hellwig, H., and E. Pannaci (19671, Automatic tuning of hydrogen

m a s e r s , Proc . IEEE, 55, 551-552. - Holloway, J. H . , and R. F. Lacey (1964), Faators which limit the

accuracy of cesium atomic beam frequeincy standards, P roc .

CIC, Lausanne, 317-331.

Kleppner, D o , H. C. Berg, S o B. Crampton, N. F. Ramsey,

R. F. C. Vessot, H. E. P e t e r s , and J. Vanier (1965), Hydrogen-

m a s e r principles and techniques, Phys. Rev. - 138, A972-A983.

Kleppner, D . , H. M. Goldenberg, and N. F. qamsey (1962), Theory

of the hydrogen mase r , Phys. Rev. 126, 603-615. - Kleppner, D . , N. F. Ramsey, and P. Fjelstad (1958), Broken atomic

beam resonance experiment, Phys. Rev. Let ters 1, 232-233. - Shirley, J. H. (1968), Dynamics of a simple m s e r model, American

J. Phys. 36, 949-963. - Vanier, J . , and R. F. C. Vessot (1964), Cavity’tuning and pressure

dependence of frequency in the hydroge4 mase r , Appl. Phys.

Let ters 4, 122-123. - Vessot, R. F. C . , M. Levine, L. Mueller, and M. Baker (1967), The

design of an atomic hydrogen maser system for satell i te ex-

periments, Proc . 21st Annual Symposium on Frequency

Control, For t Monmouth, N. J. 512-542.

13

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