TECHNlCn NOTE 387
Transcript of TECHNlCn NOTE 387
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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
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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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