NITED STATES ARTMENT OF MMERCE

BLICATION

-^°' C<V

NBS TECHNICAL NOTE 387

U.S.

ARTMENT

OF

SMMERCE

National

Bureau

Standards

Hydrogen Spin

Exchange Frequency

Shifts
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NBS TECHNICAL NOTE 387

ISSUED MARCH 1970

Nat. Bur. Stand. (U.S.), Tech. Note 387, 13 pages (March 1970)

CODEN: NBTNA

Hydrogen Spin Exchange Frequency Shifts

Helmut Hellwig

Atomic Frequency and Time Standards Section

Time and Frequency Division

Institute for Basic Standards

National Bureau of Standards

Boulder, Colorado 80302

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide

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of transient or limited interest. Technical Notes may

be listed or referred to in the open literature.

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TABLE OF CONTENTS

Page

ABSTRACT 1

1. INTRODUCTION 1

2. SPIN EXCHANGE FREQUENCY SHIFT 2

3. CONDITIONS FOR THE HYDROGEN STORAGE

BEAM TUBE 3

4. CONDITIONS FOR THE HYDROGEN MASER OSCILLATOR 6

5. TUNING OF THE HYDROGEN MASER 9

6. CONCLUSIONS 11

7. REFERENCES . 13

in

HYDROGEN SPIN EXCHANGE FREQUENCY SHIFTS

Helmut Hellwig

Frequency shifts due to hydrogen spin exchange

collisions are discussed for the hydrogen maser and

the hydrogen storage beam tube. The combined effects

of spin exchange and cavity pulling in the hydrogen

maser are evaluated with emphasis on frequency errors

introduced by standard tuning procedures. It is found

that an automatic cavity tuning system based on a variation of the linewidth does not introduce a frequency

error. In contrast, manual tuning procedures based on

an interpolation of measurements at different cavity

settings do not yield a perfect compensation of spin

exchange effects. However, the frequency error becomes only significant at unnecessarily large cavity

offsets.

The smallness of the spin exchange shifts and the

near absence of cavity pulling in a hydrogen storage

beam tube are expected to permit a determination of

spin exchange shifts by varying the beam intensity

with a precision adequate for stability and accuracy

figures of 1Q~ 14

or better.

Key Words: Frequency pulling; hydrogen beam tube;

hydrogen maser; maser tuning; spin exchange.

1. INTRODUCTION

Collisions between radiating hydrogen atoms cause a broadening

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

of the hydrogen maser is the automatic cancellation of the spin exchange

frequency shift by cavity pulling if the hydrogen maser tuning method

is based on varying the hydrogen relaxation time. The most widely

used method is the beam intensity modulation technique (Vanier and

Vessot, 1964).
A need was felt to discuss and recheck this result. An additional

impetus for reevaluating this aspect of the hydrogen maser -was provided

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

pulling is published in the open literature (Crampton, 1967) although the

result is widely used in experiments with the hydrogen maser (Vanier

and Vessot, 1964). Furthermore, a quantitative estimate of the spin

exchange frequency shift to be expected in the proposed hydrogen storage

beam tube -was needed (Hellwig, 1970).

The following discussion starts from basic equations for spin

exchange effects given by Bender ( 1 9 63 ) . The treatment differs somewhat from the one given by Crampton ( 1 9 67 )

; in particular a detailed

discussion of the cavity pulling in combination with spin exchange shifts

is given, and the frequency errors associated with maser tuning techniques are evaluated.

2. SPIN EXCHANGE FREQUENCY SHIFT

2 The shifted frequency a/' differs from the unperturbed transition

frequency tc by an amount proportional to the rate of spin exchange

phase shifting V and to the population difference (p - p ) of the two

m = levels. The relationships are given by (l9), (22), and (24) in F

Bender's article ( 1 9 63)

^-

a

.o

= -i V (P 22 .p

44 ) (!)

CO

/

V = 2irnv , / {sin£(R))RdR (2) rel

Every publication giving a frequency determination of the hydrogen

maser relies on the automatic cancellation of the spin exchange shift

by cavity pulling when the maser is tuned.

2 Throughout this paper a) is just called "frequency" although, in a

strict sense, it is the angular frequency.
p D

K' P tli£^£T (3) 22" P 44

~ PK [K2 + (co- co') 2

] [2 + U/r] +

The symbols have the following meaning: CO = running frequency;

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

rel

parameter; Z^(R) = phase shift; K = total relaxation constant; U = rate

of spin exchange collisions; R - rf magnetic field parameter; r = rate

of escape of atoms from the bulb. K and 8 can be expressed as

K = r + i U (4)

2 h

where \i - magnetic dipole moment of the transition, and (H ) = average magnetic rf field in the bulb region. It must be noted that

relaxation processes other than escape from the bulb and spin exchange

are not included in this discussion.

3. CONDITIONS FOR THE HYDROGEN STORAGE BEAM TUBE

The hydrogen storage beam tube (Hellwig, 1970) is a proposed

new device which potentially combines the advantages of the hydrogen

maser with those of a beam tube. Its concept is a logical extension

of Ramsey's original storage beam proposal and his "bounce box"

experiments with cesium (Kleppner, Ramsey,and Fjelstad, 1956) which

led to the development of the hydrogen maser (Kleppner, Goldenberg,

and Ramsey, 1962).
The hydrogen storage beam tube resembles essentially the

configuration of a hydrogen maser, however it is operated well below

oscillation threshold, and an external microwave signal is injected into

the cavity. The number of atoms leaving the bulb which have undergone

a transition is monitored with a second selector magnet and a beam

detector in a fashion quite similar to the standard operational principle

of an atomic beam tube.

Of the various aspects of the hydrogen storage beam tube the

one which is of importance to the discussion presented here is the near

absence of cavity pulling. For passive beam tube devices one obtains

approximately a pulling which is proportional to the ratio of the linewidths

of cavity and atomic transition squared (Holloway and Lacey, 1964) whereas

the pulling in a maser is directly proportional to this ratio as can be seen

from (20) below. As a result the cavity pulling in the hydrogen storage

beam tube can be expected to be reduced by many orders of magnitude

as compared to the hydrogen maser (Hellwig, 1970).

In the following we shall estimate the spin exchange frequency

shift to be expected in a hydrogen storage beam tube.

We assume that beam intensities of 10 atoms per second at

, 3 storage times of about 1 s shall not be exceeded. For these conditions

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

to the storage time. We therefore may use the approximations

U/r < < 1 and Kjk r .

For a bulb of 10 cm volume this corresponds to an atomic

density of about 10 atoms per cm3

.
We furthermore restrict ourselves to microwave driving frequencies

well within the linewidth 5 W = 2K of the transition. We may then use

the approximation

CO - CO' < < K.

With the above approximations (3) simplifies to

ZZ 44

2r2 + 8/32 2 + 8 (P/vf

The power of the exciting microwave signal will approximately be

chosen as

r 2

Bender (1963) evaluated numerically the integral in (2), and

gave as a result

V^ I37ra 3 nv (8) rel

where a is the Bohr radius (a = 0. 529 X 10 ' cm). After combining

(1)> (6), (7),and (8) we obtain with v 1 = 2 X 1 5 cm/s

rel

C0

! - C0n

= kn (9)

where k^ -2. 6 X lO"12 cm3 /s, and n is measured in cm"3

-A
The spin exchange frequency shift is proportional to the atomic density n, and its value for the assumed highest density of n = 109 atoms /cm3

is to' - LC' ^ -2. 6 X 10~" s" 1 or a fractional frequency shift of about

3 X 10 -1. According to (9) this frequency shift is linear with the atomic

density and can therefore be measured conveniently by varying the beam

intensity. It has to be noted that a variation of the beam intensity causes

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

beam tube device as was pointed out previously.

4. CONDITIONS FOR THE HYDROGEN MASER OSCILLATOR

The condition for self- sustained maser oscillations can be written

as P = P (10) rad abs

where the radiated power is given by

dp

P , == nV nO>-~- (11) rad b dt

and the absorbed power by

V

P k

= -£-5T- <h2 >

' (12 > abs U 877 c

V is the bulb volume, V the cavity volume, dp /dt the rate of

b c ZZ

emission from one atom, and (H2

) the average squared rf magnetic

field in the cavity. The rate of emission was given by Bender (1963)

as
dP

22 ZB

3 K

[Ka +(60-w')a

] [2 + U/r] + —-^

The combination of (10) to (13) yields

(13)

l67rt,QV, £

3 [Ka +(co-ai')5

] [2 + U/r] +

8 ^ K

= — nK . r V <H2

> c c

3 P in (l4) can be substituted from (5). The filling factor 77' as defined

by Kleppner and Vessot et al. ( 1 9 65) is given by

z b b , r?' = . (15)

(H2

> V

c c

We introduce the parameter \' which contains only constants for a given maser configuration

fi

l6rr/i

3

7?' Q

(16)

Equations (14) to ( 1 6) combine with (l), (3),and (5) to

2
Inserting (8) into (17) leads to

a — , CO- CO' 2 co'-co = -Urr^Q v

rel

\' [l+(——) ]K . (18)

If we restrict the frequencies to a small range around co' we obtain

with C = 13tt a

2 v J 1 rel

CD'-co = -CK . (19)

Equations (l7) and (l8) do not contain explicitly the beam intensity;

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

linewidth 6 CO = 2K. At first the result (l9) is surprising because one

3. would expect (co ! - CO^) -* for 6 CO -> 6 CO-,., where 6 CO,., is the linea E E

width due to escape from the bulb alone (vanishing contribution of spin

exchange effects). However, (18) does not permit the limit of vanishing

spin exchange effects. The equation was derived using the condition (10)

for self-sustained maser oscillations and is therefore only valid under

these conditions. In order to realize an oscillating maser in the limit

4

6 CO ~* 6 CO , the parameter \' has to approach zero,and (co1 - CO ) -* a E

according to (l 8).

4

X

1 relates to the maser oscillation parameter q, which was defined

by Kleppner and Vessot et al. ( 1965) .
5. TUNING OF THE HYDROGEN MASER

The pulling of the maser output frequency by a cavity which is

detuned from the atomic transition frequency is one of the earliest

discoveries in maser theory (Gordon, Zeiger, and Townes, 1955), and

is quantitatively given by (Shirley, 1968)

6 co co + 6 co co - (20)

6 co + 6 co c a

where fiCO and 5co are the linewidths of cavity and transition line,

c a respectively, and CO is the atomic transition frequency. Spin exchange

is include

cavity Q)

included by substituting co' for co . We also substitute (Q = loaded

cL

COc

6 CO = -pr- and 6 co = 2K

c Q a

and obtain

W

-'W' =

(co /2Q)+K H" W,) ' Ul)

c

CO

1 can be expressed from (18), and (2l) then reads after some simple

regrouping

<*> /,.,.. \ 2 CO- 01, =

fa =

(c0c /2Q) + K l^c-%-T5 C (fir) ]

(22)
If we use the approximations

CO —c— > > K and CO- CO1 < < K

(22) simplifies to

2Q °-c co-co_ = — K(o -w_ -tptC) . (23)

co c 2Q

Equation (23) is the standard "tuning equation" for the hydrogen maser

(Vanier and Vessot, 1964). If the maser is tuned so that a variation

of K does not cause a change in the maser output frequency then the

maser frequency is equal to the unperturbed atomic transition frequency (co = 01 ), and the cavity is detuned by an amount equal to the

spin exchange frequency shift (co -CO. = C co /2Q) as can be seen

c c from (23)

The error committed in tuning the maser in this fashion can be

evaluated from the above approximations. Omitting K in the denominator of the "pulling factor" in (22) does not introduce any error in

the tuning procedure. The omission of the last, K-dependent, term

in (22) seems to introduce a frequency error. The K-dependent part

of this term can be rewritten using (19)

(**) -PR' (=?«; (24)

The constant C in this expression does not introduce an error in the

above tuning method. The K-dependent part (co-co )/K approaches zero

as the maser is tuned (co -* CO ). As a result no appreciable frequency

error is introduced by a tuning procedure which uses the variation of K.

10
This is only true for experiments -where the maser is actually set to

oscillate on the transition frequency oo as for example in automatic

servo tuning systems (Hellwig and Pannaci, 1967; and Vessot, Levine,

Mueller, and Baker, 1967). Frequently, however, the maser frequency

is only indirectly determined using two or more cavity settings at which

K is varied (Vanier and Vessot, 1964). The maser frequency is then

calculated using a linear interpolation between the measured variations

of the maser output frequency. In this case a frequency error is introduced which corresponds to the variation with K of the last term in

(22). This term can be written as -C — . The error follows

I - 'V

3 then approximately as C 773— ^ K, where AK is the variation of K.

The maser constant C has typically a value of C ^ 10" . Fortunately,

the error has an appreciable magnitude only if fairly large cavity offsets and large K-variations are used. For example a cavity setting

corresponding to a fractional maser frequency offset of 10 -1 will

produce a fractional error of about 10 -13 if AK^Ka. 1 s~ .

6. CONCLUSIONS

The discussion given in this report indicates that an automatic

cavity servo control based on varying the linewidth of the atomic transition yields an exact tuning of the maser. The spin exchange frequency

shift will be perfectly compensated by a proper cavity detuning. A

manual tuning procedure in which the maser tuning is found by interpolating between measurements at different cavity settings -will introduce

in general a systematic error due to an incomplete compensation of

frequency shifts. However, this error only has an appreciable magnitude

if unnecessarily large cavity offsets are used.

11
It was pointed out that the automatic compensation of spin

exchange and cavity pulling effects, as described by (23), is only

valid for an oscillating maser.

In a passive device such as the proposed hydrogen storage

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

shift is linear with the beam intensity (atomic density in the storage

bulb) and can be measured and corrected for by varying the beam

flux. The magnitude of the shift is small; of the order of 10~13 for

typical beam intensities and storage times. Thus only a moderate

measurement precision and a moderate control of the beam flux is

necessary if stabilities and accuracies of 10 -14 are the design goal

of a passive device (Hellwig, 1970).

12
7. REFERENCES

Bender, P. L. (1963), Effects of hydrogen-hydrogen exchange collisions, Phys. Rev. 132 , 2154-2158.

Crampton, S. B„ (1967), Spin-Exchange shifts in the hydrogen maser,

Phys. Rev. j_58, 57- 6l.

Gordon, J. P., H. J. Zeiger, and C. H. Townes (1955), The masernew 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. Accepted for publication in

Metrologia 6, No. 2, April 1970.

Hellwig, H. , and E. Pannaci (1967), Automatic tuning of hydrogen

masers, Proc. IEEE, 55, 551-552.

Holloway, J. H. , and R. F. Lacey (1964), Factors which limit the

accuracy of cesium atomic beam frequency standards, Proc.

CIC, Lausanne, 317-331.

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

R. F. C. Vessot, H. E. Peters, and J. Vanier (1965), Hydrogenmaser principles and techniques , Phys. Rev. 138, A972-A983.

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

of the hydrogen maser, Phys. Rev. 126, 603-615.

Kleppner, D. , N. F. Ramsey, and P. Fjelstad (l 958) , Broken atomic

beam resonance experiment, Phys. Rev. Letters 1, 232-233.

Shirley, J. H. (l968), Dynamics of a simple maser model, American

J. Phys. 36, 949-963.

Vanier, J., and R. F. C. Vessot (1964), Cavity tuning and pressure

dependence of frequency in the hydrogen maser, Appl. Phys.

Letters^, 122-123.

Vessot, R. F. C. , M. Levine, L. Mueller, and M. Baker (1967), The

design of an atomic hydrogen maser system for satellite ex- periments, Proc. 2lst Annual Symposium on Frequency

Control, Fort Monmouth, N. J. , 512-542.

13
It was pointed out that the automatic compensation of spin

exchange and cavity pulling effects, as described by (23), is only

valid for an oscillating maser.

In a passive device such as the proposed hydrogen storage

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

shift is linear with the beam intensity (atomic density in the storage

bulb) and can be measured and corrected for by varying the beam

flux. The magnitude of the shift is small; of the order of 10~13 for

typical beam intensities and storage times. Thus only a moderate

measurement precision and a moderate control of the beam flux is

necessary if stabilities and accuracies of 10~ 14 are the design goal

of a passive device (Hellwig, 1970).

12
7. REFERENCES

Bender, P. L. (1963), Effects of hydrogen-hydrogen exchange collisions, Phys. Rev. 132, 2154-2158.

Crampton, S. B. (1967), Spin-Exchange shifts in the hydrogen maser,

Phys. Rev. 158, 57-6l.

Gordon, J. P., H. J. Zeiger, and C. H. Townes (1955), The masernew type of microwave amplifier, frequency standard, and

spectrometer, Phys. Rev. 9% 1264-1274.

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

for a new frequency standard. Accepted for publication in

Metrologia _6, No. 2, April 1970.

Hellwig, H. , and E. Pannaci (1967), Automatic tuning of hydrogen

masers, Proc. IEEE, 55, 551-552.

Holloway, J. H. , and R. F. Lacey (1964), Factors which limit the

accuracy of cesium atomic beam frequency standards, Proc.

CIC, Lausanne, 317-331.

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

R. F. C. Vessot, H. E. Peters, and J. Vanier (1965), Hydrogenmaser principles and technique s , Phys. Rev. 138, A972-A983.

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

of the hydrogen maser, Phys. Rev. 126, 603-615.

Kleppner, D. , N. F. Ramsey, and P. Fjelstad (1958) , Broken atomic

beam resonance experiment, Phys. Rev. Letters 1, 232-233.

Shirley, J. H. (l968), Dynamics of a simple maser model, American

J. Phys. 36, 949-963.

Vanier, J., and R. F. C. Vessot (1964), Cavity tuning and pressure

dependence of frequency in the hydrogen maser, Appl. Phys.

Letters jl, 122-123.

Vessot, R. F. C. , M. Levine, L. Mueller, and M. Baker (1967), The

design of an atomic hydrogen maser system for satellite ex- periments, Proc. 2lst Annual Symposium on Frequency

Control, Fort Monmouth, N. J. , 512-542.

13

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