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NBS TECHNICAL NOTE 660

*^eA u of *

U.S. DEPARTMENT OF COMMERCE/National Bureau of Standards

Molecular Beam Tube Frequency Biases

Due To Distributed Cavity Phase Variations

C.2-
NATIONAL BUREAU OF STANDARDS

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ib/D

4- ate

00

VtsO Molecular Beam Tube Frequency Biases

Due To Distributed Cavity Phase Variations

Stephen Jarvis, Jr.

Time and Frequency Division

Institute for Basic Standards

U.S. National Bureau of Standards

Boulder, Colorado 80302

TeakiuGaJ note ac> LbO

°*CM Of *

U.S. DEPARTMENT OF COMMERCE, Frederick B. Dent, Secretary

NATIONAL BUREAU OF STANDARDS Richard W Roberts Director Issued January 1975
Library of Congress Catalog Card Number: 74-600205

National Bureau of Standards Technical Note 660

Nat. Bur. Stand. (U.S.), Tech Note 660, 43 pages (Jan. 1975)

CODEN: NBTNAE

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 20402

(Order by SD Catalog No. 013.46:660) $1.00
CONTENTS

PAGE

I INTRODUCTION 1

II RAY TRACING 5

III THE CAVITY FIELD 8

IV THE TRANSITION PROBABILITY 11

V APPLICATIONS 18

VI CONCLUSIONS 21

Vli APPENDIX 23

Viii REFERENCES 2 6

in
MOLECULAR BEAM TUBE FREQUENCY BIASES DUE TO

DISTRIBUTED CAVITY PHASE VARIATIONS

For atomic beam frequency standards, an analysis

is described for estimating the frequency bias

due to distributed cavity phase difference over

finite beam widths, and for estimating the resulting inaccuracy in power shift and beam

reversal experiments. Calculated atomic trajectories and simplified rf-field distributions are

used, as well as certain assumptions about beam

tube alignment. The results are applied to one

of the present NBS primary time & frequency

standards and a shorter tube geometry.

One conclusion is that beam reversal experiments

are not necessarily much more accurate than power

shift experiments and that the use of both methods

(plus the use of pulse techniques) is desirable.

Key words: Accuracy evaluation; Atomic beam frequency

standards; Cavity phase shift.

IV
I. INTRODUCTION

The primary sources of error in state-of-the-art Ramsey-type

cesium beam frequency standards, which are approaching a part in 10* ^ accuracy, are the second-order Doppler shift (DS2) and the effect (PD)

of a phase lead 6 of the rf -field in the second cavity over the first.

An experimental method, based on the pulse technique [1,2,3] and

theoretical methods using measured Ramsey resonance curves at different power levels [4,5 ] ; have been reported which predict the relevant

velocity distribution P(V) of detected atoms. From p(V), it is an easy

matter to compute pseudo-velocities V (b, V __ ) and V (b, V ,__ ) coefficients depending on the power parameter b and the modulation

width for line -center servoing from which the biases due to D S2 and PD

are:

where V

ceg

is the cesium transition frequency, c the speed of

light, L is the cavity separation and £ the width of each cavity. Since

V /2TTL is typically about 10 Hz/radian, a milliradian phase difference P 1 ? 6 generates a bias v__ of 10 -i

to V . Even with careful cavity adiust- PD ces 7 J ment before assembly, control of 6 to levels significantly lower than a milliradian is not at present practical, so that the value of 6 obtained

in the beam machine at any particular time must be inferred by other

means. Two techniques are currently employed. In the first, the

beam tube is designed to permit both forward and reversed beam

operation. For forward operation, the bias is

F F I ,

'- ,7 F

V - V — -=> 7 V * k

where Jv is other sources of bias. For reversed operation, the bias is
(We must anticipate that (V V ) result from different velocity distri.

butions in the two operating modes). The difference

FK ft ft

is found by comparison with a very stable source (its accuracy is

irrelevant), and thus

from which V is easily computed for each mode. For example,

where

^ - v, /(*:* k

k

) -

/» Z

5

This bias estimate is limited in accuracy by the measurement A /2 , the average DS2 bias, and the (yet) unknown bias:

1 2

In the second technique, two power levels are used, (b , b ).

(The velocity distributions may be slightly different for these cases;

this must be considered in the calculation of the (V.TV" ), or the twovelocity distribution approach used) [2]. Again, measuring:
A /*

s v

n ~ va

we find:

a,j

- (vL~i - v/,-,j- Ccr

- &*)

3rU V'-V R

P P

Then;

/

f3 ~ ^' ' * J i? ' r DSa

where

-*-- v'/(vp ~ v/y r

With care in the choice of power levels and sufficiently wide velocitydistributions ^| may be as small as about 3, but in any case, it will

amplify the errors in A V , and h

1

, by this factor. This method

has the advantage of being applicable to any beam machine and of generating redundancy (for three or more power levels), but suffers from

the amplification factory. A combination of both methods (plus use of

the pulse techniques) will be the best approach towards a comprehensive accuracy evaluation.

Our purpose in this paper is to estimate the amplification factor^ <^ and the (PV) bias A> (b, v ), due to non-uniformity (phase variations)

of the rf-field in the portions of the cavities traversed by the atomic

beam. These effects must be expected to be the more important in

accuracy evaluations the shorter the beam tube.

The estimate involves three components. In Section II, a ray-trace technique is described which is intended to model the beam

tube under consideration and generates velocity distribution moments

over the cavity windows. In Section III, a calculation is described

which provides an estimate of the relevant rf-magnetic field in the

microwave cavity with finitely conducting walls; only a simple
rectangular guide is considered. In Section IV, the atomic transition

probability is derived, and the total bias computed for sinusoidal or squarewave modulation by averaging over the atomic beam.

In Section V, the application of these methods to specific

systems is discussed.
II. RAY TRACING

Referring to figure 1, let jp = (Y , Z , Y Z ) be the

parameter vector of atoms whose trajectories have coordinates

(Y , Z ) at the center of the first cavity, (Y , Z ) at the second,

where p = is assumed to be on the line of centers of the cavityapertures. We consider those atoms with velocity V, and assume all

trajectories make small angles with the line of centers. Let cO be

the rf-fiald angular frequency, and let P (ip, V, <«_'-) be the transition

probability for one of these atoms over its passage through the two

cavities and the uniform C-field region between the cavities.

Consider atoms emitted either "spin up" (j = 1) or "spin down"

(j = 2). If transition occurs, these atoms contribute to the detector

signal the flux element:

where (?; — C if the path fails to pass from emitter to detector, and

otherwise, p

-/" is the emitter source strength, dependent on the position

on the emitter face and the launch angles. If transition did not occur,

the contribution to the detected flux element is:

U

(v, $>)(/- Pr ,(tpX^))jlP <<K

where p

"" has properties analogous to

•4

t

Integrating over the cavity openings $-(/p) and the velocity, the

total flux at the detector is:

^ (-) - J*> v

p, d?> v)L'e G*(v tP) Pn

fjp . IA J c £<>) '

neglecting a term independent of uu , and writing

U Op , v) Q Yv P) - Z Of - f ),
where Ot-,(P> V)is an emitter source strength and Q (V,/ft ) a beam tube E

form function for unit source strength. (The Jacobian of the transformation from initial to final values of (Y, Z, Y, Z) through deflecting

magnets where force is independent of X is unity. ) For simplicity, we

may take P

Tr (p, V) to be the Maxwellian emission distribution, independent

of rp .

With the intention of expanding P ( V, 60 ) to second order in /to, pq

the ray trace computation generates the velocity dependent moments:

/

.

(n « / / r)

I

2

?'it?

I

y 2

%3 *

c^ 7

J

Finally, defining

we can write:

(v) - t Jv) P E (V)

where p are the appropriate expansion coefficients of / ( n V <-C>J^
The computer program generates the q (V) for several intervals

(m = 1, M) in the detector plane, so that the effect of detector location

may be studied.
III. THE CAVITY FIELD

The microwave structure is assumed to be a very symmetric

U-shaped rectangular wave guide shorted at the ends and driven at the

center. It propagates only the TE -mode, each arm having the ideal

complex harmonic field structure

EL (ir) & = ^w. '-—- ^

,rv r (y / yj ^

l IK „ where IK is a unit vector in the z -direction, Ya is the cavity wave

number and y an arbitrary phase chosen to make lEF vanish at the

short y = -y

n

-

In beam machines, the apertures in the cavity arms which

permit the atoms to pass through are usually elaborated with features

which include transverse "approach sections" and horizontal

(y-direction) fins to reduce field leakage in the x-direction. These

complications make calculation of the exact rf field nearly impossible

even in the case of perfectly conducting walls.

We have examined only the relatively simple case in which the

cavity apertures are absent, and the wall conductivities are very high,

dependant only on the perimeter position (independent of y). For the

magnetic field-independent transition in cesium defining the atomic

second, only the rf magnetic field parallel to the C -field, which we

shall take in the z -direction, is important in causing transitions.

We shall solve the harmonic field equations:

V y V x IE - £'* (E - o

!_

subject to the wall skin depth conditions

en x E -= y ff/rr ) ^t . *± 0+ O'/f^T >

H - - ~J— v x (E

i/w

where Iri is the outward normal, ^ ('^sJ the skin depth at the wall point

/A-", (independent of y), and 'l~~i the tangential magnetic field on the wall, T

The wall condition can be written

m x EL ~ ^

Cf^ J ^Va (e) >
where:

i- /

where the real skin depth is

ff,n )- ^-,/-^

= .?. 7 o c (/ c)

~ ;? /ST c ^

5 -1 -1 For o =5.92 x 10 ohm cm for copper, the free space cesium transition

wave length \ = 3.264 cm, we obtain:

Wave guide dimensions are assumed to be a = 2.2 86 cm (0.9") and

b = 1. 016 cm (0.4"). (X-band).

We take the electric field to be of the form

ELf (in) =- c^ (x^z) ^—c ry ^ £L a (V ) = C^ Cxj i ) ^c~^ ry t J=3 OhS = <^j 6c, <?) ^-^ ry s

Ca (xyj-)^ ea Cx) ^^1(^1- c)e + a] ,

where the <P,f2iare assumed to have series expansions in a mean skin

depth 6 , while <E and B are to be determined and are constants of

order 5

.

The important magnetic field component H {it

J

) is found to within

a proportionality constant from:

f-~13

Cir) = ,<p^ i y I e^

x

Cx ) + t (/ / e. (x ) ) }

-

^~*c [ (ZT 1- f) 2 +

ft J .
The complex propagation constant Y = Y n

- i & is determined by the a na 1 y s i s

.

The calculation is extremely lengthy, and requires an analysis

to second order in 5 to determine all the first order quantities. It will

not be reproduced here.

An end-short condition for terminal skin depth 8 is easily

e applied to the result by replacing sin 8 y by sin8(y + y ) and applying

one of the wall conditions at yy = -y , which determines y^(S ) . A V y e second wall condition is ignored; in fact, the skin depth distribution

must be of the correct form on the end wall to be consistent with this

single mode analysis.

The real and imaginary parts of H (ff) are determined from a computer program where the primary input is constant values

(5 , 8 , 8 , 8 , 8 ) for the skin depths on the four walls and shorting

surface. (In fact, the parametric variation of skin depth by cosine

series coefficients is included, but such effects must be expected to be

small; in any case, such coefficients are unknown).

Finally, coefficients of the real and imaginary parts of H (if

)

are generated for least squares best fit to the form:

H3 Cr) - r, (yz) * x r^(y^) + x

J ^Cy

^

y

Yj(yj)^ rj + rjy + <?/* * -;/ /

*

y- rj v* * rj -3

about each cavity center lt^ = point on the line of centers of the cavity

apertures. The program also gives the expansion coefficients for the

remaining field components as well.

It must be emphasized that this relatively simple calculation

may not represent adequately the true field in the complicated cavity

structure near the cavity apertures, but we shall be able to determine

which are the critical coefficients in the field expansion.

10
IV. THE TRANSITION PROBABILITY

The quantum mechanical probability amplitudes C (t), C (t) for a two-state system perturbed by a uniform magnetic field in the

z -direction and a parallel oscillating field may be written:

f - ^ C.

ic.3

+ a A

c H2

lie.

Na o C--

where E E are the unperturbed frequencies associated with the

energy states, H a frequency associated with the uniform field, and b

a constant proportional to the. oscillating field amplitude (the "power

parameter"). The field /"7V CfP) fjin. each cavity j = 1,2 is related to

the complex field H /-/_ calculated in Section III by:

4 A

Ih.

I*, 'J x H£

fa t) - fa LH* 'fa) <^ fa t , x) * /fa 'faU^fa / , z,

where .sH>. and A- are chosen so that

and:

J?/c*)~ />

-£/ (o) - <fa\ We can readily compute from the coefficients for the field U-/'1 fa) (~J^ Qh))

in Section III the representation

y + CJ.Z~t CJ 2 / f Cj y*)

+ X (c\J

l * Q J

/- X c

y

' )

3 fa in which C. -O. A similar expansion holds for -A , but is not required.

We assume that the atom has a trajectory through each cavity:

A- Vs

j

' y = y + v y^ s > T - 2 +

fax

V2 <r

11
where, assuming the atom enters the first cavity at time t = 0, S - T- ^

in the first cavity, S- t- (Tfj-t) in the second, with T*-» ^A/ 7~~^- *-/\/m g From the representation of the field components in terms of the c , we

can express the time -dependent functions n

f

pi,

(3

1 ^

f s

3

1 2

pJ

*- s~ J2 I)' J ?

( <3 where the coefficients /^.- are small, associated with the cavity

resistivity and field amplitude gradients in each cavity.

The uniform field causes the energies of the states to be

modified. The unitary diagonalizing matrix

A -

OC H uS ^r-£,)l

where:

p £°*- ^J p K

1 o< -u<*,-E,y+H:' ' p "1

<u^ *J = J- ' o? generates the following system for the eigenstates

4.

For the very weak C-fields employed in beam machines,

^/fe3-£')« f j and A is essentially the unit matrix; the only important

effect of H is to replace (E , E ) by (uu , U) ). Thus, we consider

only the system:

b1 00

Cs

* )f'D

D

Qb

o

H,

12
Putting: A _ i </>

2 2 which does not effect the state probabilities, (i.e., |D | = |F | ): \-6

c>

I

Introducing in the j cavity (j =1,2),

Hj (,r(f)) = - yf;^. C^t^Yj)- ,< (*«, (^ t r V, J,

and putting:

/; (t) = - ^ f - y, ^ ^ = n - y,- ^

where <?v is composed of terms with oscillating frequency 2uu . These

lead to the Bloch-Siegert Shift [ 6 ] f which we shall not consider here.

We introduce the real variables:

C- F,F, , V-Fa Fa t R'FF=*F,i=yS-i(F,K-Fa F-l

U* (/= l

i //t/= O- l/j <J <*£(/ + ui)

> v £ (/- w),

and derive the real equations

where Q &

(The sign of X is reversed from Ramsey's [ 7 ] usage). The effect of

6. / is merely a rotation in the (R, S) subspace. With the initial data

at t = : /c*/- u- /, i/-y?= X- o,

13
so that W = 1, we can write after passage throught the first cavity:

I

< 5 - SA

- 5. .

over cavity one with 6 =0. Over the C -field region, b = 0, we have

W\ (I c o ) Cu

I nT+ v

Over the second cavity, we have (8 = 0):

7f

We need finally the transition probability at t = T + 2t : p - JC *\ , v(T-f it) * i(l- W(T* Q-e)\

Then:

I a o

V jo ai^/J ^xns)\ H jo

The cavity equations may be written:

Si x y - - 3 £V a

'

If this sytem is iterated once, treating the right-hand-side as a small

perturbation, we can write for cavity j : X n

c / /

y

14
where the matrix *°i

perturbation matrix. With ^1 - I A t >' h ,

? - '/"o h s&-^'v <r; z"

s the same for each cavity, and I is a small

ft 1 c-l

3(A f 4-C,\^ «r, z?) ^/-qU^Mr <-fAL6-c. (T<Z CiC

Then to the first order in I , the transistion probability is

found from: ,

,

-*oc- kk -K-K).

Line centering by the modulation technique is modeled by

putting

X - Ji ' A F Ct) Mc ft I where p(t) is a symmetric periodic function, usually a square -wave or

a sinusoidal wave, X, ,^^ is a modulation amplitude and X

' is a small

^ MOD r adjustable offset. V'" is varied so that the detector signal is also a symmetric periodic function as sensed by a symmetric linear filter.

Assuming that 8 , I, and X

'"* are all small quantities, the asymmetric

part of P is as follows:

pq

(X= ^MOD here):

15
where (O, E) refer to antisymmetric and symmetric parts in X respectively.

MOD'

We also assume a modulation amplitude small enough that terms

in (\,nr./b) may be ignored. To this order, we calculate:

f

1 A r 3

y

lo 10

2 C

where in each cavity j 4

o

o,

loJ+ J

( -**?** K

K-l

ryj^H-,^ * ^a£* ^

)

r 9ik K*

and

t y =

i>K

-P S

c2

-2f

The frequency bias V and the second order Doppler correction

13 are introduced by writing:

A*-- ?r^/il^j

16
Then integrating P over velocity and the permissible atom trajectories, and carrying out the linear filtering associated with

modulation, a null result is obtained when the frequency bias V

satisfies:

Z- (CK^fj * c*aKt,3,)y 1

**/ . where the C are the -/t_ {^p) expansion coefficients for cavities j = (1,2),

and the U ^7 y^oft ) ar e defined in the Appendix.

Calling

we obtain for the bias:

where we can now identify the terms used in the discussion of bias

estimation in Section I :

and (-yfy.) is the bracketed term.

A computer program has been written which accepts the velocity

distributions q

m(V) of Section II, the //-/-field coefficients l^j of

Section III, a set of field level parameters [ b j , and a single modulation frequency width V_ , __ . For each detector location m, it computes MOD

the 0~. (b, V, ,__ ), and taking pairs of field coefficient sets, constructs * MOD * * p

the associated constants C- . It presents the PV bias^(b), and for

pairs (b , b ), presents the power shift error multipliers of the

coefficients ( c . c ) as well as the power shift-induced error for that

field pair.

17
V. APPLICATIONS

The procedures described here have been applied to two systems,

NBS-5 and a shorter off-line beam tube (Tube B). However, it must be

born in mind that the NBS-5 results are relevant to the true system only

if two assumptions hold. First, the ray trace must adequately match the

system. This may be checked reasonably well for p (V) which is just

the detected velocity distribution, but not at all for the other distributions

(p , . . . , p ) which are much more critically dependent on system and

2 . .1 5 cavity alignment. Second, the simple wave guide model must be

adequate to represent the complicated structure of the cavity apertures.

For a given structure, however, if all skin depths are increased by a factor x, then all C^ , and hence the PV bias -A, are increased by the

factor x. Skin depths are expected to be known to (and have a time

dependent change of) only a few percent.

We have considered only three rf-field types. In the first, the

skin depth is uniformly 6. 82 x 10 cm, corresponding to copper. In

the second, skin depths are uniformly 10% higher, while in the third,

only two adjacent walls are 10% higher, which should represent a worst

case for non-uniformity in this simple model.

Figure 2 shows the correct velocity distribution for NBS-5, as

determined by measurement and confirmed by Ramsey curve inversion.

Also shown are ray trace results for a detector of height 0.2 cm,

centered 0. 3 cm below the line of centers for M= 2, 0. 3 cm above the

line of centers for M = 4. All these curves are normalized to unit

height. Neither ray trace curve matches the experimental curve well;

the M = 2 curve has a better center of gravity, while the M = 4 curv-3

has the proper location for the maximum. These ray traces were done

directly from the beam tube design value, and no effort was made to

adjust them to improve the match.

Figure 3 shows results for an off-line geometry with an interaction region length L = 0. 5m. Both detectors, M = 1 and M = 3, are

located on the opposite side of the line of centers from the emitter,

with the M = 1 detector further away. The strongest signal is detected

at the M = 3 location, while the M = 1 is very weak, and exhibits

"shot noise" of the limited number of rays examined.
In table 1 we list the most important rf -field expansion coefficients Cr{ and field derivative at jp = which they represent for the

three rf -field types we have examined. In table 2, we show the

corresponding multipliers (

.- v^+2 ,C n

' + 2(? ^ or ^e fcwo cavity fields at

nominal modulation amplitude (equal to resonance half -width) for three

values of b bracketing the optimum value. When the field type is

assigned to each cavity (j = 1,2), each product £.-, L',"-> ,. 7 , c-^ U~n+ 2c

is a component of the PV bias /H,{h, v

A/rnn )« K = 1 for the NBS-5

model; K = 2 for the shorter tube (Tube B). S = 1 is when

the beam tube geometry is completely symmetric about y = 0; S = 2

is the case in which emitter and detector are shifted 0. 1 cm on

opposite sides of the symmetry axis y = 0, and show the effect of

misalignment. (A similar effect is produced by non-uniform emission

or detection in y). M is the particular detector location number

(varied in z).

Referring to £3 (&#), even though cfy and d^ may be quite large,

as long as the cavity aperture is centered (in z) in the wave guide face

and the z-dimension of the cavity opening (here 0. 95 for K = 1, 0.508

for K = 2) is not too large, this term is not important.

Referring to C ( <3y ), we see that the unavoidable rf -field

phase gradient in the propagating direction can be a major source of

PV bias for relatively small beam asymmetries in y, whether due to

physical misalignment of components or non-uniform beam intensity.

The values ( O" <_'-_ ) for K = (1, 2) appear to be roughly proportional

to the slope / of the misalignment assumed. This error can be

reduced by narrowing the cavity aperture widths (here 1.27 cm) at a proportionate loss in detected signal.

-4

The field curvature terms (c , c ) are in the 1 x 10 range for

4 6

K = 1, but as high as 2 x 10 for the shorter tube. These terms should

drop quadratlcally with decreasing aperture dimensions.

The terms due to C and C which modify the transition probability by introducing an apparent frequency variation in the cavity

have very substantial coefficients, ( cr^ ^ c^ L

) and ( ^7 <-"} s

), for

the two cavities, but are fortunately of such amplitudes and signs as

to cancel identically when the cavities and their fields are identical.

For the variations in cavity fields considered here, these terms are

less than 1 x 10 ~ 4

, but x-asymmetries in cavity construction could

perhaps produce large biases because of these terms. It is worth

19
noting, however, that the (a

g

, a,

6

, cuO depend only on p, (V)

the directly determinable velocity distribution; in a cavity

design in which these biases were dominant, power shift measurements could be used to estimate (C., C

fi ), and some correction applied.

Tables 3 through 10 are final results for these cases.

Over a set of relevant power parameter values { b \

, VQ and

V (b, v mn) are shown; these numbers are independent of which

rf-field types are used. For the cavity 1/cavity 2 rf-field

type designations shown, the PV biasJiis given, and for power

parameter pairs, (b , b ), the resulting PV power-shift bias

v/ = V- i U * v„ d/J f- (l- <l ) I (li ~ 'mod A

due only to distribution of phase over cavity-apertures, is

-3 given in Hertz. (For cesium, a 1.0 x 10 Hz bias is approxi13 mately 1 part in 10 ) . These numbers can be considered to

represent the uncertainty values which are encountered under

the conditions stated.

Let us now consider a few specific examples. Consider the

case of NBS-5 (K=l) operated between b = 20000 s (near

optimum for M = 4) and b = 2 b (6 dB higher power) . With

perfect symmetry in y (S = 1; tables 3 and 4), we find PV

-3 -3

power shift biases of 0.6 x 10 Hz for M = 2 and 0.4 x 10

Hz for M =4. The corresponding PV beam reversal biases,

1/2 (h

F

+ h

R

) , at b" are 0.2 x 10~ 3

Hz for M = 2 and 0.0 x 10~ 3

Hz for M = 4.

For the asymmetric case (S = 2; tables 5 and 6), the PV

-3 -3

power shift biases are 1.1 x 10 Hz for M = 2, and 1.0 x 10

Hz for M = 4 . The corresponding PV beam reversal biases at

_o _3

b are 0.8 x 10 Hz for M = 2 and 0.6 x 10 Hz for M = 4.

20
We should remember that these biases do not include the

DS2 bias (which can be very accurately calculated) , and must

be considered to be uncorrectable uncertainties at this time.

Consider now the shorter Tube B (K = 2) operated between

b~ = 14000 s~ (near optimum for M = 3) and b

+

= 23000 s"

1

, (4.3 db higher power). With perfect symmetry in y ( S = 1:

Tables 7 and 8) ,we find PV power shift biases of 25. x 10 Hz

-3

for M = 1 and 2.5 x 10 Hz for M = 3 , a very striking difference. The corresponding PV beam reversal-biases at b are

0.8 x 10~ Hz for M = 1 and 0.5 x 10~ Hz for M = 3.

For the asymmetric case (S = 2; tables 9 and 10) , the PV

-3 -3

power shift biases are 17 x 10 Hz for M = 1 and 3.0 x 10

Hz for M = 3. The corresponding PV beam reversal biases are

-3 -3

5.8 x 10 Hz for M =1 and 2.8 x 10 Hz for M = 3.

It is interesting to note that the power shift results

are not always as bad relative to the beam reversal results

as might have been expected from the values of the amplification factors -4/ which are easily computed from the V (b) values

in the respective cases: for NBS-5 (K=l) , A- = 5 for M = 2, and

^ = 9 for M = 4, while for Tube B (K = 2) , -4y = 13 for M = 1

and M =3. This is due to the high degree of correlation of

PV biases at various power levels for a fixed configuration.

VI. CONCLUSIONS

Under rather severe assumptions on cavity symmetry and

ray-tracing validity, estimates of the PV bias uncertainty

(due to distributed field phase and amplitude in the cavities)

have been obtained for power shift measurements on NBS-5 and a

shorter beam tube, and tabulated in tables 3 through 10. With

proper care in alignment and emitter fabrication, and optimum

21
choice of power levels, the PV bias uncertainty in power

-13 shift experiments c with NBS-5 should be less than 1 x 10 v ces

,

It is important to note from these tables that the apparent advantage of beam reversal techniques may be substantially

negated because of the fact that, in beam reversal, we must

average two uncorrelated PV biases -A and Jl> , while for

power shift measurements, even though the amplification factor

& may exceed 3, the biases Jru and Jk, are correlated, and the

resulting bias uncertainty may even be less than the average

{Jl- + A )/2. This fact confirms the view that a combination

of both methods (plus the use of pulse techniques) is the best

approach toward a comprehensive accuracy evaluation.

22
VII. APPENDIX

With z = 2ttL V /V, we define the linear servo filter; MOD

over X(t) = X, ,__ p(t). For square-wave modulation: mod r v

' "

For sinusoidal modulation, ^ r

J*f*)- z^O)- ?;&)[,

where J , J are Bessel functions.

With x = b T, we define

stSk

* - 3 y , ^?J-^-v «u % * 27 jL 6) •

Then we define the ^ • ( £ V |/\

^= -^\? *

J),

> f ^p£- -« -'* ) )* ,

23
where we have dropped the terms of order (^/L)

Then designating: ^ <F(v)> - ^

v F Cv)t

the ^y . (b, v,,^^) are as follows:

1 MOD

«.« <^p* * zr <i* (?+-?*)>

^+

(-4

^/a

e ^

<^*f* * / CT<- <f*~ <P^

7

<v~/y

24
I / z<±

t~l JY

c4 ar

= ^*<^eA ^-^*<a,,T >

25
VIII. REFERENCES

1. Hellwig, H., Jarvis, S., Glaze, D. J., Halford, D., Bell,

H. E., "Time Domain Velocity Selection Modulation as a Tool to Evaluate Cesium Beam Tubes", Proc . 27th Annual

Frequency Control Symp., Ft. Monmouth, pp. 357-366, (1973^

2. Howe, D. A., Bell, H. E., Hellwig, H., DeMarchi , A.,

"Preliminary Research and Development of the Cesium Tube

Accuracy Evaluation System," 28th Annual Symposium on

Frequency Control, 19 74.

3. Glaze, D. J., Hellwig, H., Allan, D. W. , Jarvis, S. Jr.,

Wainwright, A. E. , Bell, H. E. , "Accuracy Evaluation and

Stability of the NBS Primary Frequency Standards, IEEE

Transactions on Instrumentation and Measurement, IM-2 3,

1974.

4. Jarvis, S. Jr., "Determination of Velocity Distributions

in Molecular Beam Frequency Standards," Metrologia, 10 , 3, pp. 87-98, September, 1974.

5. Daams, H. , "Corrections for Second Order Doppler Shift

and Cavity Phase Error in Cesium Atomic Beam Frequency

Standards," IEEE Transactions on Instrumentation and

Measurements, IM-23, 1974.

6. Shirley, J.H., "Some Causes of Resonant Frequency Shifts

in Atomic Beam Machines," J. Appl. Phys., 34^ 4,

783-788, April, 1963.

7. Ramsey, N.F., "Molecular Beams," Oxford University Press,

London, 195 6.

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'N35-'J4A (REV. 7-73)

U .S. DEPT. OF COMM.

BIBLIOGRAPHIC DATA

SHEET

1. PUHLK A I ION OR RKPORT NO.

NBS TN-660

2. (rov't Accession

No.

3. Recipient's Accession No.

4. TITLE AND SUBTITLE

MOLECULAR BEAM TUBE FREQUECY BIASES DUE TO DISTRIBUTED

CAVITY PHASE VARIATIONS.

5, Publication Date January 1975

6. Performing Organization Code

7. AUTHOR(S)

STEPHEN JARVIS, JR.

8. Performing Organ. Report No.

9. PERFORMING ORGANIZATION NAME AND ADDRESS

NATIONAL BUREAU OF STANDARDS, Boulder Labs.

DEPARTMENT OF COMMERCE

WASHINGTON, D.C. 20234

10. Project/Task/Work Unit No. 2770246

11. Contract/Grant No.

12. Sponsoring Organization Name and Complete Address (Street, City, State, ZIP) 13. Type of Report & Period Covered

14. Sponsoring Agency Code

15. SUPPLEMENTARY NOTES

16. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here.)

For atomic beam frequency standards, an analysis is described

for estimating the frequency bias due to distributed cavity phase

difference over finite beam widths, and for estimating the

resulting inaccuracy in power shift and beam reversal experiments.

Calculated atomic trajectories and simplified rf-field distributions are used, as well as certain assumptions about beam tube

alignment. The results are applied to one of the present NBS

primary time & frequency standards and a shorter tube geometry.

One conclusion is that beam reversal experiments are not

necessarily much more accurate than power shift experiments and

that the use of both methods (plus the use of pulse techniques)

is desirable .

17. KEY WORDS (six to twelve entries; alphabetical order; capitalize only the first letter of the first key word unless a proper name; separated by semicolons)

Accuracy evaluation; Atomic beam frequency standards; Cavity phase shift

18. AVAILABILITY [xj Unlimited

~2 For Official Distribution. Do Not Release to NTIS

J Order From Sup. of Doc, U.S. Government Printing Office Washington, D.C. 20402, SD Cat. No. C13 _ Order From National Technical Information Service (NTIS)

Springfield, Virginia 22151

19. SECURITY CLASS

(THIS REPORT)

21. NO. OF PAGES

43

UNCLASSIFIED

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(THIS PAGE)

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"?6-A916 ilGPO 1974—677-237/1233 REGION NO. 8