DETAILED DESIGN 

OF A 

VARIABLE VOLUME HYDROGEN MASER 

PETER 0, CERVENW 

REPRODUCED BY. C US Department of Commerce 

National Technical Information Service 

Springf~eld, V~rg~n~a 22161 
i TABLE OF CONTENTS 

PAGE 

EI Frquency Error 

TV Systematic Freyt.teilc-)- Error 

V Filling F2ctor 20 
INTR0'3UCTf ON - REV I E'il OF PERTI:.IErjT I4ASER EQUATI0;IS -- 

;::?-a a hydi-ogsn aton apprczc52s tile ::a? Is of the m3ser btr15, the 

o./erlzp of the elx'ii-on clouds of tke atom 2nd the :;a71 cause the at~m 

to rev2i-se its velocity 2nd .m\re a>iay. The \cave ftincii~n ovei-?(;p is 

ii~ir2~i?d znd 60 15 the ~ner~y[l]. Thc ~et ph~se shift thus ci.::cted is '1 

1 given by $ = 5 / hE(t)dt \;i;zrs ~t is "Lz du;zticn of the collision, (1.1) 

At C AE "L2 charige in hypei-i-inz &iiercjr. Thc coi-i-zspnding si~ift Sn the z,ser 

1- @ 

cL-.-ac.,?_ -- - i i ~~+:.,ncy is gj by ;f = Zx .-.--- .,I:, 3 I- is the coflfsfon r~te. - 

- ,!A - -manetic dipole nozsnt of thz electrron in the H zt3n = 9.2549 x 

70-21 ~r~s/~auss 

Tt = total tin? 

Vc = ~01~r2 of the 6:ser cavity 

Q . = q~ality fsctoi- of the c-vity 

ftot = total flux of H atoms 

I = flux of atox in the right excited state. 

Far naser operatfon, cne nust haye q < .772. 

In ord2r to tune the n3ser accurct2ly, the ratio of the high flux to 7~w 

J= 11 ux I-zs~nance 1 ine ;iiiths n!,~st b? as large as possib1.e. The theoretical 

3ne also definzs -- I - - Ye 1 

Q-= Vb '1 

iihenever a change of thz bulb v0iua2 is involved, it is found useful to 

talk about a quantity B dcfinsd 2s B = 

'I - 

The. (4) and (-) subscripts in the volur;ie V i-zi'er to the lalye (open) ad 

small (closed) configurations, respectively. In the varizble volume hyd,-ogen 

mser shown sche~3tically in Figure 1, there is a freg;c;ency shift dlis to t3e 

chznze in the volune. It is our b0p~ to &sign the systsn so that this shift 

will be less than one part in 10'~. 
CHAPTER 1-1 

IDEidTIFICATION OF DESIGN PARAMETERS 

The design parameters are: 

(1) Sha~e of the Bulb: The bulb will be inade up of t:.;o parts: 

cne fixed and the other variable. The fixed part is going to be cylindrical. 

It will have an opening into a variable volune that will be a truncated cone. 

- (2) q or Quality Parameter: I his \.rill determine whether or note mser 

action will occur. 

(3) Tuning Factors: R: flux tuning fzctor 

0: variable volurre tuning factor 

(4) Material to be Used: FEP Teflon has been exmined (see n3xt 

section) and found quite sui tajle. 

(5) Operational Temperature to Off-Set the Ilall Shift: This teiiipzrature is believed to be approximately lOOOC and will have to be investigated 

experim~ntal 1 y once the maser is bui 7 t. 

Study of Teflon as !{all Material 

'A2 have used a 1 square foot piece of Teflon in order to study its 

vacuuin behavior and suitability for use as wall raterial for the nsw maser 

design. 

The sample was crumpled to simulate operational conditions and placed 

in a v~cuum ~i-ranifold. \,!hen the 20 L/sec Noble Vacron purnp ad sample :.;ere 

heated to 117'~~ the pressure reached 8 x lo-' Torr after three days. 

The punp and sanple were then transf2rred to an environczntal chzz5er 

~h2r-s th2 teizp2rztui-e was maintained at 125" for one day. The te~perature 

nas th~n raised to 131'~ for another day. At approxinately this te~psrature, 

FEP Teflon is known to change phase. 
I 

I The tomperatul-e was then grzCual7y decreased to 126'~ for one day. The 1 

annealing process took six days at which stage the pressure was 1.8 x 10-j1 

Torr at room temperature. When opened, the sample had not changed its 

physical shape or dimmsi~ns and was judged suitable for use in the ne:J xascr. 

Case 1: The K.1S noise is indepznlent of the line-viidth so that 62FH = 62FL 

62Ftuned+ + 62Ftun2d- = ti2FO (111.17) 

>le obtain after substituting 

which becomes, after one defines 6F = aFH = 6FL 

Case 2: The Ri4S noise is proportional to the line-width 

~ssuiriin~ the line-width to be inversely proportional to the 

corresponding vol ume, 
and 

62FH- = R-262FL- = 

Finally,defining 6F = 6FL, gives 
CHAPTER IV 

SYSTEI.IATIC FREQUENCY ERROR 

The aim is the study of localized variations in the concentration of 

excited atom 2nd how this is affected by the shape of the- container (the 

rase of surface to vo71ine in psrticular-). This yields 3 frequei?cy shift 

kthich can be used to develop criteria that can be used to design the vzriable 

volune part of the xaser. 

The principal assuxption made is to consider that the ualls 

act as a sink on the excited atox, This effect is directly proportional to 

the !$a11 area snd is large corpred to any other ccntribution. 

After exhaustive conputer nodeling, it is found thzt in order to obtain 

an upper bound on the frequency error, that part of the external volu~e fsrthest 

av:ay from the fixed v01u:qe contributes the most if it is consider?d to incf utiz 

the sharp tapered part up to the knee bend in the volune, Refst-ring to Figure 

2, this would include the region marked (11) up to Cl. The other region considered 

is iiiarked (I) on the sane figure. The fixed vuluce Vo is cc;lh?cte4 to the 

var!'ah!e VO!U~P_ via Cog a CO~TIIR~GS~~QR part which consists of a hole. 

This is shown scheratically in the following section. 
VARLABLE VOLUNiE EXTERNAL STORAGE BULB 

FIGURE 2 - VARIABLE VOLUJIE EXTERNAL STORAGE BULB 
IV. 1 33iro Region Model 

WVO REGlON MODEL 

REGION (I) REGION (If) 
Let the n's refer to the density of excited ators in the rzgion undzr / 

consideration. The S's refer to an equivalent p&ping speed due to interaction with the walls and the f's are co~minication terns. . At equilibrium, n = n fm 

in re1 zxation escape 

and 

Then 

and 

Thus 

(IV.1) 

(IV .2) 

Assuming that the comnunication terns (or C3) are large co~pared to the 

S's 
But 

and 

(IV -9) 

After substitution and elimination of terrrs in which are considered 

snall compared to - S, the result is: 

C 

If r(A) is the collision rate with the wall and n(A) is the density of 

excited atoms near an elerrent of wall area d4, the wall shift can be written 

using (1.2) r 

where is the total nurher of atons in voluq~! V and $(A) is the zverage 

phase shift in area dA. 

Then I- 

(IV -11) 
The larqest contribution to AVW cows from the smallest volume 

configuration which we call V,, 

one has 

i = regions (0), (I), (11) 

IV.2 First Approxi-ation 

Suppose 

Khen the phzse shift 4 is constant over the region (i) 

i 

4 

4 *A 4 +A q, =+A 

0 11 2 2 total 

:!hen the bulb tem?erature is such that to zerotif appraxin~tion the 

wal I shift vanishes, 

(IV .?3) 

(IY .15) 

(IY .16) 

(rv J7) 

(IV. 18) 

(IV. 19) 

(IV. 20) 

(IV. 21 ) 

After substitution for the E's in term of the S's and C's 

(IV. 22) 
Anothzr way of writing this is 

IV. 3 Second Approxirration (Using the Proper ;)- 

Let us c3a?ute f-accurately in terns of n , n 2nd n . This gives 

0 1 2 

Also-by definition of the average 

(IV ,231 

(IY .24) 

(IY. 25) 

(IV ,263 

(IV .27) 

(IV .28) 

(IV .29) 
But 

J J - 

A6 +A$ +A+ =+A 

00 11 2 2 total 

(IV. 32) 

(IV .33) 

At the nu1 1, 7 = o (IV. 34) 

(IV. 34) 

After substitution for the Ek the ~xpression is 

where 

(IV. 36) 

Th2 wall siift error czn be written 

Since = o and hence 

With reference to Figure 4 for the rzzning of sore of the no~znclature, 

(IV. 39) 

let us describe so-e details of the co-prrtztions. The vo1 uses labelled V , V 

7 2 

and V are shown in the same Figure 2nd the corresponding areas will have the szre 

3 

subscript (i .e., A refers to the area of volune V etc). 

2 2 
S@f-lEPv,/IATIC OF ONE HALF OF THE VARIABLE, VOI-UME - - 

FIGURE 4 - SCHEI'JATIC OF ONE HALF OF THE VARIABLE VOLUME 
So one obtains 

(IV. 40) 

(IV .47) 

(IV .42) 

(IV .43) 

A =r(R3+~4-a) +T + ZiR L -ir~~+Zii2 D (IV. 44) 

2 1 3 4 3 4 
For the equivalent pu-?ing sp~zd we use the expression 

to9etkr with forr-~l a (IV. 33) obt3-i qed in the second a?proxi-?tion, 
CHAPTER V 

FILLIHG FACTOR 

r~c = V H 

- b < z> bulb 

V <HL> cavity 

C 

where H = 1 

< z'bulb V 

(V.2) 

b 

and <Hz> 

cavity = "*2dv C 

= " ("2 + dV . 

s cavity 

For operation in the TE ode [3]: 

01 1 

H z = $)sin (F) (Y .A) 

0 

H r =(ia ~d ) J~*(S 

cos (;z) - 

01 a 

07 

= - (xa -\ J f r) cos bz) -

S dl 1 07a 

O1 / 

where a = cavity radius 

d = cavity length 

S = 3.8317 (first zero of J ) 

0 1 1 
If the cavity squatness is described by 

Then 

where p =d so, 2g2+x2 

It is found that the optimum val ue for - p = 0.450 

a 

when one writes 

Then one can define the new factor U 

Using the formulae put forth in section IV one can obtain the systematic 

frequency shift (section 10 in the program listed in Appendix A). Using the two 
new values of q obtained when the variable volume is in the open and closed 

configurations and substituting into equation (111.13) gives the statistical 

error for different values of U (iection 12 of the prugram 1 isting - Appendix A). 

Such that 

Ye have plotted the statistical and systematic frequency errors as a 

function of L , the external volune length for g = 2, 3 and 4 for U = 7.5 and 2. 

3 

These plots are shown in Figures 5, 6 and 7. The values of g = 3 or 4 for U = 1.5 

or 2 and L n, 10 cn give error values that are acceptable for the present work. 

3 



CHAPTER VI 

ACTUAL DESIGN - COUCLUSIOUS 

Referring to Figure 2 for the neaning of the various quantities, the 

di-rensions for a trial configuration are 7 isted below. 

A. Fixed Vol unz 

The cavity is to be rad2 of aluminum and the bulb will be cylindrical 

with a radius equal to .45 that of the cavity. The raxinum length of the 

cavity can be about 30" (limited to the length of Penal loy sheets comsrcial fy 

available) and is equal to the length of the bulb. 

g is the ratio of length to radius of the cavity. For g = 4 = d - 

a 

d x .45 cavity 

a 

bul b (VI.7) 

So a = d x .45 - 33 x 2.54 x .45 = 8.6 crn 

bulb 4 4 

is the radius of the bulb. 

For a nassr stable to one part in lo1'+, the dimnsions for the vari zble 

vol urns can be: 

V = 6250 cc (for g = 4) q' = .442 

0 

R = 7cn; H =3 cs; R = 71 cm; R = 13 cm 

2 3 4 

D = .5 cn; 9D = .5 cn 

Bulb :-'ateri a1 : FEP Teflon. 
2, I0 C PROGRAH Hi-iI118 DATE:lZ-JAN-77 

2.20 C FOR k VARIABLE VOL* CONTAINING A CUT-OFF IN THE 

2*30 C NkRROW PART* 

2-40 C 

2.50 C THIS FPOGRAH ALSO HAS CHANGING ETA FRIHE 

2.60 C FROGRAM HODIFICATION DATE: 17-JAN-77 

8.90 C CTHE FOLLOWING SECTION SETS UF ALL THE DEFAULT VALUES3 

lO*lO C CTHIS PART CALCULATES T#LSsfSTEHATIC ERROR3 

16.11 C 

10.20 S L3=IiS L2=2Si 

10-23 S Vl=PIt(E3"2#iL2-L3)f.3333XR332%H) 

LC, 25 S tf2=FIX (RJ-2$L3-. 3333L38 iF;3-Z+(R4-11) -2tfiZs (R4-1i > 1 1 

ifi.zo S V3=PIw(LZ-L3jX(R4"2-R3^2) 

16 I 22 5 Vn=VOf VliVZSV3 

3G4Z3 S C1=2XFI*k3$iL2-L3) 

15.35 5 C2=Pla+! (R3-2-H3-2 j 

16.40 5 CG=Plx(R2-2) 

10.50 5 ~1=~Pl~~3~FSQT(R3n2iHH21+PI~CH332-R222)~ 

10*60 S A2=FSQTi(R4-D-R3)"2fL3-2> 

10.61 S A2=iRS+R4-D>%A2+2%R4PL3 

10+62 5 AZ=AZfR4-2-(R4-Dj-2 

10.53 S A2=FI+A2 

10.65 S A3=PI%(2SR4%(~2-L~i+(R44~-R33~)) 

10+70 5 ZZ=(A1+R2fR3)/CO 

ii,.75 S ZX=W1*(AlfA3)XZZ+W2fi;2$(ZZfA2,~'C2) 

10. $":, 5 Z"r'=-CJEYf;4ZX/(S~Fr$'lii) 

Lc*E5 S ZR=ML~I>~*.ZZ~U~~(A~~A~~A~ZZ+CA~+A~~/'CI 1 

10.95 S ZS=-~JEZb;.*ZR/(8%FI$Vii) 

13.92 f ~ZS-ZY~i0,?5~10.95~IGC?7 

- 1;,95 = SE=ZSt:G 10-98 

- - icl"T SEzrJ 

13.45 S SE=SE/lc42E09 

:id.54: D 3.i 
11-05 C CCALCULATION OF E7 

11.06 C 

11.10 S VN=R4"2%L2+L3/3x(R3n2+(R4-I~)-2tR3$(fi4-I~)) 

11-12 S VN=FI* (VblfR3-2*H/3) 1 m- c11.30 S VP=VOfVN QR~G~~J&% P>cGr $3- 

11.50 S E=VP/VH POOR QUAFST~ 

11.70 S QA=VF/VO 

11 t40 S EX=O 

L1.95 11 12 

12110 C CCOHFUTE THE STATISTICAL ERROR ON THE FREQUENCY USING 

12t20 C THE HEW VALUE OF E WITH THE ERROR FORMULA FROM HMD013 

12-30 C 

12.40 S EPx.5 

12*50 S C=.213669ES03/EP 

12t52 S C=C/U 

12.60 S OO=C*8/35000; 

12*62 S R=QOLVPi/VO 

12.70 S SO=3-6fiQ+Q"2;1 <-S@)12.72r12.72iR 

12+72 S SO=FAES(SQ) 

12.73 S J=FSOT(SQ) 

lZ*EO S H?i=(l+Q+J)/(I+Q-J) 

12,81 S O=QO%VP/VO 

12t82 S SL!=1-6*RfR-2;1 (-SQ)12t83t12t83iQ 

12+83 S J=FSQT(SRI 

12+84 S RP=(ltGiJ)/(ltQ-J) 

12.85 S G=(F-2*(RPn2tI))/((RF'-1)-2) 

12,90 S H=(R?f-2Sli/((RM-1)^2) 

12.95 S ER=FSRT(GfH)/(B-l) 

12.49 S 'f=Ez 

EINSERT AT THIS LOCATION A SUITABLE PLOTTING PROGRAH3