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First results from a multiple-microwave-cavity search for dark-matter axions

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F irst R e s u lts fro m a M u ltip le-M icro w a v e-C a v ity Search for
D a r k -M a tte r A x io n s
BY
D arin Shaw n K inion
B .S. Physics
C am b rid g e, MA 1994
D ISSE R T A T IO N
S u b m itte d in p a rtia l satisfaction o f th e req u irem en ts for th e degree of
D O C T O R O F P H IL O S O P H Y
in
A pplied Science
in th e
O F F IC E O F G R A D U A T E STU D IES
o f th e
U N IV E R SIT Y O F C A L IF O R N IA
D avis
A pproved:
I
C o m m ittee in C h arg e
2001
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UMI Number 3019020
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Daxin S h aw n K inion
M ay 2001
A p p lie d Science
F ir s t R e su lts fro m a M u ltip le -M ic r o w a v e -C a v ity S ea rch
fo r D a r k -M a tte r A x io n s
A b s tra c t
R esu lts are presented fro m a se a rc h for g alactic halo axions a t Law rence
L iverm ore N ational L ab o rato ry (L L N L ). T h e ex p erim en t, a c o lla b o ratio n in­
volving LLNL, th e M assach u setts In s titu te o f Technology, th e U n iv e rsity of
F lo rid a , F erm ilab, a n d Law rence B erk eley N atio n al L ab o rato ry has b een ta k ­
ing d a ta since 1995 w ith s e n sitiv ity to K im -S h ifm ann-V ainshtein-Z akharov
(K SV Z ) m odel axions.
T h is ex p erim en t utilizes th e m icrow ave-cavity axion d e te c tio n schem e
p ro p o sed by P ierre Sikivie. P rev io u s searches have all used single m icrow ave
cavities as th e detector; th e re s u lts p re sen te d in th is thesis a re th e first
fro m a m icrow ave-cavity axion d e te c to r using m u ltip le m icrow ave cavities
w ith th e o u tp u t powers co m b in ed co h eren tly .
K SVZ m odel ax io n s w ith
3.3639 < m a [//eV] < 3.3642 h av e b e e n excluded w ith 90% confidence as
m a jo r co n stitu en ts of our g a lac tic h alo .
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Acknowledgment s
I have been very fo rtu n a te to b e p a rt of a rem arkable te a m h u n tin g for th e axion.
I w ould like to th a n k m y su p erv iso r, K arl van B ibber, for his high stan d ard s, insightfidness, and unw avering o p tim ism in the face o f adversity. Leslie Rosenberg
was prim arily responsible for g e ttin g m e sta rte d in th is business, a m ove I have yet
to re g re t, even in th e to u g h e st o f tim es.
T h e top quality work o f E d Daw in th e com m issioning an d anlavsis of th e single­
cav ity experim ent has m a d e m y w ork m uch easier. M ore im p o rtan tly , he helped
m a in ta in m y sanity d u rin g th e B oston to Liverm ore tra n sitio n .
T h e com m issionning o f th e m ultiple-cavity array w ould not have been as suc­
cessful w ithout th e efforts o f th e local crowd, C hris H agm ann, Steve A sztalos, and
W olfgang Stoeffl, who o ften h a d answ ers to questions I d id n ’t know th a t I had.
Progress in the piezoelectric a c tu a to rs was helped im m easu rab ly b y th e efforts of
D ennis M oltz and Jam es Pow ell.
I w ould like to th a n k th e m em b ers of th e H igh-E nergy Physics group a t LLNL
for providing a stim u la tin g w ork environm ent. P a rtic u la r th a n k s go to M arshall
M ugge for his num erous p e p -ta lk s.
It has also been m y p le a su re to w ork w ith Jo h n C lark e’s group a t UC Berkeley.
I have learned a tre m e n d o u s a m o u n t from b o th M arc-O livier A n d re an d M ichael
M uck.
Finally, I wish good lu ck to th e n e x t generation of axion h u n te rs, D anny Yu and
ii
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Ja so n K night.
W ork perform ed u n d e r th e auspices of th e U .S. D e p a rtm e n t o f Energy by th e
U niversity of C alifornia Law rence Liverm ore N atio n al L a b o ra to ry u nder co n tract
W -7405-ENG-4S.
iii
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Contents
1
I n tr o d u c tio n
1
2
T heory
5
5
2.1 A xions and Q C D ....................................................................................................
2.1.1 P e rtu rb a tiv e QCD a n d th e U ( l ).4 P r o b l e m .....................................
6
7
2.1.2 T h e © - v a c u u m ...........................................................................................
2.1.3 T h e S trong-C P P r o b l e m ...................
S
2.1.4 T h e P eccei-Q uinn S o lu t i o n ......................................................................
9
2.1.5 A xion P rop erties an d C o u p lin g s ............................................................... 10
2.2 A xions and C o s m o lo g y .............................................................................................. 13
2.2.1 O verview and H istorical D ev elo p m en t o f C o s m o lo g y ........................13
2.2.2 E a rly T ests of C osm ology ..........................................................................IS
2.2.3 E x p e rim en ta l Tests of th e S ta n d a rd C o s m o l o g y ...............................21
2.2.4 A ccounting of M a tte r in th e U n iv e r s e .................................................... 24
2.2.5 T h e A xion as a CDM c a n d i d a t e ...............................................................33
2.2.6 A llow ed A xion M ass R an g e a n d T w o -P h o to n C oupling . . . . 36
2.3 C av ity D e tec tio n of A x io n s .......................................................................................39
2.3.1 C y lin d rical R esonant C a v i t i e s .................................................................. 40
2.3.2 E x p e c te d Signal Pow er fro m A xion C o n v e rs io n .................................. 44
2.3.3 Pow er C om bining M u ltip le C a v i t i e s ....................................................... 49
2.3.4 N u m erical R esonant F req u en cy a n d F orm F acto r C alcu latio n . 52
2.3.5 E x p e c te d Scan R a t e .................................................................................... 55
3
A p p a r a tu s
58
3.1 T h e M a g n e t ...................................................................................................................60
3.1.1 C o n stru ctio n ...................................................................................................60
3.1.2 O p e r a t i o n ......................................................................................................... 62
3.1.3 P e rf o rm a n c e ......................................................................................................64
3.2 In se rt C r y o g e n ic s ........................................................................................................ 67
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3.3
3.4
3.5
3.6
3.7
3.S
3.9
C avities an d T uning R o d s ....................
6S
P iezoelectric T uning and C oupling M e c h a n is m ..............................................79
3.4.1 T h e P iezoelectric Effect .............................................................................79
3.4.2 M icro P u lse Cryogenic A c t u a t o r s ...........................................................81
3.4.3 P iezoelectric A c tu ato r Im p e d a n ce M e a su re m e n ts...............................82
3.4.4 P iezoelectric Tuning M e c h a n is m .............................................................. 87
3.4.5 P iezoelectric C oupling M e c h a n is m ...........................................................90
3.4.6 P iezoelectric D riving E le c tro n ic s .............................................................. 92
3.4.7 H eatin g from Piezoelectric A c t u a t o r s ....................................................97
C ryogenic E le c tro n ic s ............................................................................................... 98
3.5.1 W ilkinson Pow er C om biner ..................................................................... 98
3.5.2 H F E T A m p lif ie r s ...........................................................................................99
T ransm ission a n d Reflection M easu rem en t E le c tro n ic s ..............................105
R eceiver E le c tro n ic s .................................................................................................106
3.7.1 Im age-R eject M ix e r ..................................................................................... 107
3.7.2 C ry stal F ilte r and Second M ixing S ta g e ...............................................114
3.7.3 A udio Frequency F F T .............................................................................. 115
3.7.4 H igh-R esolution F F T ..................................................................................117
M iscellaneous I n s tr u m e n ta tio n ............................................................................U S
C o m p u te r S y s t e m .................................................................................................... 118
4
R e s u lts
119
4.1
Pow er C om bining T e s t ..........................................................................................120
4.2
S canning T e s t ........................................................................................................... 125
4.3
D a ta S c a n ...................................................................................................................130
4.3.1 D a ta C om bining .........................................................................................131
4.3.2 Single-bin Pow er Excess S e a r c h ............................................................ 135
4.3.3 Six-bin Pow er Excess S e a r c h ................................................................... 139
4.4
C o n c l u s io n s ............................................................................................................... 142
5
F u tu re D ir e c tio n s
A
d c S Q U ID A m p lifiers
149
A .l S u p e rc o n d u c tiv ity O v e r v ie w ............................................................................... 151
A .2 Josephson Ju n ctio n s ............................................................................................. 152
A .2.1 R C S J C ircu it M o d e l ..................................................................................153
A .3 SQ U ID B a s i c s ........................................................................................................... 156
A .4 dc SQ U ID s as R F A m p lifiers............................................................................... 161
A .5 dc SQ U ID M e a s u re m e n ts ...................................................................................... 163
144
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A .6
C onclusion
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List of Figures
.1
.2
.3
.4
.5
.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
2.14
A xion C ouplings ........................................................................................................11
Scale F acto r E volu tio n in a M atter-D o m in ated U n iv e rs e .............................17
E x p e c ted K ep lerian R o ta tio n C urve for a G alactic D i s c ............................ 28
M easured R o ta tio n C u rv e for th e M ilky W ay G a l a x y ................................ 29
C alcu lated R o ta tio n C u rv e for a F ou r-C o m p o n en t M ass M odel . . . . 30
E x p e c ted T h e rm a liz e a A xion L ineshape ......................................................... 35
A llowed A xion M ass an d C o u p lin g s ....................................................................38
P rim ak o ff P ro d u c tio n o f A x i o n s ...........................................................................40
R ig h t-circu lar C av ity G e o m e t r y ...........................................................................41
O n e-p o rt C av ity E q u iv alen t C i r c u i t ....................................................................45
S chem atic o f a n N -P o rt Pow er C o m b i n e r /D iv id e r ....................................... 50
T w o -p o rt C av ity E q u iv alen t C i r c u i t ................................................................... 51
C av ity M ode T ran sfer F u n c t i o n ...........................................................................51
C alcu lated L o ngitu d in al E lectric Field for th e TM oio M ode o f a Cir­
cu lar C av ity w ith a Single A lu m in a T u n in g R o d ............................................52
2.15 C alcu la te d L on g itu d in al E lectric F ield for th e TM oio M ode o f a Cir­
cu lar C av ity w ith a M etal R o d ............................................................................... 53
2.16 Form F a c to r o f th e C av ity w ith a Single A lu m in a R o d ................................54
2.17 Form F a c to r o f th e C av ity w ith a Single M etal R o d ................................... 54
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
F our-cavity A xion D e te c to r L a y o u t ................................................................... 59
C a lcu la te d z-com po n en t o f a Solenoidal M agnetic F i e l d ...............................61
M agnetic F ield R eg u latio n C ir c u it......................................................................... 62
D aily M o d u latio n o f th e M agnetic F ield .........................................................65
M agnet C able R esistan ce versus Shed T e m p e r a t u r e ................................... 66
B asic C o m p o n en ts o f a 1.4 K L am b d a-R efrig erato r ..................................... 67
K nife-edge Seal for C av ity E n d - c a p s ..................................................................... 69
Pow er T ransm issio n M easu rem en t for an E m p ty C a v i t y ............................ 70
Pow er R eflection M easu rem en t for th e M ajo r P o rt o n am E m p ty C avity 71
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3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
3.19
3.20
3.21
3.22
3.23
3.24
3.25
3.26
3.27
3.28
3.29
3.30
3.31
3.32
3.33
3.34
3.35
3.36
3.37
3.38
3.39
3.40
3.41
3.42
3.43
3.44
3.45
3.46
G e o m e try o f a C av ity w ith a Single T u n in g R o d ............................................72
M ode M ap of th e C av ity w ith a Single A lu m in a R o d ............................... 74
M easu rem en t of TMoio freq u en cy versus tu n in g ro d a n g l e ..........................75
M ode M ap o f th e C av ity w ith a Single M etal R o d .......................................76
T y p ic a l M ode C rossing R e g io n ................................................................................77
M easu rem en t of TMoio u n lo ad ed Q versus fr e q u e n c y .................................... 78
M icropulse L-104 C ryogenic A c t u a t o r ................................................................. 81
P iezo electric E q u iv alen t C i r c u i t ............................................................................82
T y p ica l R esonance as C a lcu la te d from th e E q u iv alen t C i r c u i t ...................83
M easu red P iezoelectric R esonance a t 4.2 K ...................................................... 83
M easured P iezoelectric D ielectric C o n s t a n t .............................................. . 84
C irc u it for M easuring th e Im p ed an ce o f th e P ie z o elec tric A c tu ato rs . 85
M easured P iezoelectric Im p e d a n ce at 300 K ....................................................S6
M easured P iezoelectric Im p e d a n ce a t 77 I v ....................................................... 86
M easured P iezoelectric Im p e d a n ce a t 4.2 K ....................................................... 86
P iezo electric T uning M e c h a n is m ............................................................................ST
M easured CVV R o ta tio n S peed versus F requency a n d A pplied V oltage 89
M easured CCVV R o ta tio n Speed versus F requency a n d A pplied V oltage 89
P iezo electric C oupling M e c h a n is m ........................................................................ 90
M easu rem en t of T M 0io coupling versus a n te n n a in sertio n d e p th . . . 91
B lock D iagram o f P iezo D riv in g E lectronics ...................................................94
P iezo electric P ow er-am plifier C ircu it D i a g r a m ............................................... 95
P iezo electric T an k C i r c u i t .......................................................................................95
S m all S tep H is to g ra m ................................................................................................. 96
W ilkinson Pow er C o m b in e r.......................................................................................99
B alan ced A m plifier D e s i g n .....................................................................................101
G ain a n d Noise T e m p e ra tu re o f th e F irst C ryogenic A m plifier . . . . 102
G ain an d Noise te m p e ra tu re of th e second cryogenic am plifier . . . . 103
C o m bined G ain an d N oise T e m p e ra tu re of th e C ascad ed Cryogenic
A m p lif ie r s .....................................................................................................................104
M easured Noise T e m p e ra tu re o f th e N R A O A m p l i f i e r s ........................... 104
S etu p for C av ity T ran sm issio n an d R eflection M e a s u r e m e n t s .................105
R eceiver E le c tro n ic s ................................................................................................... 106
G ain o f th e M IT E Q P o s t- A m p lif ie r ...................................................................107
Im age R eject M i x e r ................................................................................................... 108
S etu p for M easuring Im ag e R e j e c t i o n ............................................................... 109
M easu rem en t of M IT E Q Im ag e-R eject M i x e r ................................................. 110
Im age R ejection an d In se rtio n Loss of th e M IT E Q IR M a t 10.7 M Hz 111
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3.47
3.48
3.49
3.50
3.51
3.52
Im age R ejection a n d Insertion Loss o f th e M IT E Q IRM a t 30 M Hz . I I I
M easu rem en t of R eb u ilt Im age-R eject M i x e r ................................................ 112
Im age R ejectio n a n d Insertion Loss of th e R eb u ilt IR M a t 30 M Hz . . 112
M easurem ent of R eb u ilt Im age-R eject M i x e r ................................................ 113
Im age R ejection a n d Insertion Loss o f th e R eb u ilt IRM a t 30 M Hz . .1 1 3
C ry stal F ilte r T ran sfer F u n c t i o n s ......................................................................114
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
4.21
4.22
S etup for Pow er C om bining T e s t ......................................................................... 120
R esults o f a Pow er C om bining T e s t ................................................................... 122
R esults o f a Pow er C om bining T e s t ................................................................... 122
R esults o f a Pow er C om bining T e s t ................................................................... 123
R esults o f a Pow er C om bining T e s t ................................................................... 123
R esults o f a Pow er C om bining T e s t .........................................................
124
B lock D iag ram of Piezo Feedback Schem e
................................................ 125
Frequency D eviations in Scan T est .................................................................. 126
Pow er C om bining Efficiency in Scan T e s t ....................................................... 127
Frequency Shift A utocorrelation for Fixed S tep S i z e .................................. 128
Frequency Shift A utocorrelation for F eedback-C ontrolled S tep Size . . 129
R aw F F T Pow er S p e c tr u m ....................................................................................131
N orm alized F F T Pow er S p e c tr u m ......................................................................132
N orm alized F F T Pow er S pectrum w ith R adio P e a k .................................. 132
F F T F it R e s i d u a l .....................................................................................................133
Noise T e m p e ra tu re C o r r e c tio n .............................................................................134
K SV Z Pow er a n d Noise L e v e l .............................................................................136
Single-bin Search D a t a .......................................................................................... 137
Single-B in E xclusion P lo t from F our-C avity R u n n i n g .............................. 138
Six-bin Search D a t a ................................................................................................. 140
Six-B in E xclusion P lo t from F our-C avity R u n n i n g ......................................141
M icrow ave C av ity A xion Search E xclusion P lo t ......................................... 143
5.1
5.2
5.3
M ulti-p o st C a v i t y .....................................................................................................145
M u lti-p o st T uning ................................................................................................. 146
T h e Lowest T h re e T M Modes of th e P ro to ty p e M u lti-P o st C av ity . . 146
A .l
A.2
A .3
A.4
A .5
dc SQ U ID A m plifier L a y o u t ................................................................................ 150
Josephson J u n c tio n an d RCSJ M o d e l .............................................................. 152
V oltage A cross a C urrent-B iased Josephson J u n c t i o n ...............................154
dc SQ U ID C u r r e n t s ................................................................................................. 156
V -$ C h arac te ristic s o f a dc S Q U I D .................................................................. 158
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A .6
A .7
A.S
A .9
A .10
A .11
A .12
A .13
A. 14
A .15
I-V C h arac te ristic s o f a dc SQ U ID ...................................................................1-59
M agnetic H ysteresis in a dc S Q U I D ...................................................................160
F lux-biasing o f a d c S Q U I D .................................................................................161
Square W asher dc SQ U ID a m p l i f i e r .................................................................. 163
SQ U ID G ain M easu rem en t S e t u p ...................................................................... 164
G ain of a S ix-T urn d c S Q U I D ............................................................................. 164
M icrostrip R eso n an t F requency Versus Coil L e n g t h ...................................165
Layout o f V aracto r C i r c u i t .................................................................................... 166
V aractor T uning o f a dc S Q U I D ......................................................................... 167
dc SQ U ID Noise M e a s u re m e n t............................................................................. 16S
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±
Chapter 1
Introduction
T h e S ta n d a rd M odel o f P a rtic le P hysics has w ith sto o d n e arly four decades of ex­
p e rim e n ta l scrutiny. O ne o f th e o p en questions is w hy th e p re d ic te d violation o f
charge co n ju g atio n tim es p a rity (C P ) sy m m etry in stro n g in te ra c tio n s is not seen
in e x p erim en t. T h is is com m only called th e S trong-C P P ro b lem .
In 1977, R o b e rto Peccei a n d H elen Q uinn proposed a so lu tio n to th e S trong-C P
problem th a t involved a new , lig h t, n e u tra l pseudoscalar p a rtic le n a m e d th e axion
a fte r a b ra n d of la u n d ry d e te rg e n t. T h e original m odel p re d ic te d an axion m ass of
~ 100 keV w ith sufficiently stro n g couplings to ensure th e ir d e te c tio n in accelerato rand re a cto r-b ase d ex p erim en ts. T h ese axions were quickly ru le d o u t.
Soon, new m odels w ere c o n stru c te d w ith very light (m a ~ 1f i e V ) axions, possess­
ing couplings so feeble th e y w ere referred to as ‘in visible.’ O bviously, th ese m odels
were m uch b e tte r news for th e o rists th a n for e x p erim en talists.
A m a jo r b re a k th ro u g h o ccu rred in 1983 when P ie rre Sikivie solved th e conun­
d ru m of how to d e te c t d a rk m a tte r axions w ith vanishingly sm all couplings[l]. T his
tech n iq u e relies on th e P rim ak o ff effect, i.e. th e tw o -p h o to n in te ra c tio n o f a pseu­
doscalar, w here one of th e p h o to n s is v irtu a l. Here, axions w ould convert in to a weak
m icrow ave signal in a stro n g m a g n e tic field - a sea o f v irtu a l p h o to n s - th e process
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being re so n a n tly enhanced by a high-Q c av ity resonator. T h e signal corresponds
to th e to ta l energy of th e axion, m ass plus k in etic, and being n o n -relativ istic, th e
signal is e x p ec te d to be nearly m o n o c h ro m a tic , A E a/ m a ~ 10- 6 . A m a jo r cav eat is
th a t th e c av ity m ust be tu n e d to th e freq u en cy corresponding to th e unknow n, and
a rb itra ry , axion m ass. For th is reason, th e cav ity m u st be tu n a b le .
T w o pilot experim ents w ere p erfo rm ed in th e la te 19S0’s, one by a R ochesterB rookhaven-F erm ilab (RBF)[2] c o lla b o ratio n , an d a n o th er by a group a t th e U ni­
v ersity o f F lo rid a (UF)[3]. B o th e x p e rim e n ts u tilized sm all-volum e su p erco n d u ctin g
solenoidal m ag n ets (up to S tesla, over a v o lu m e o f several liters), a n d h etero ju n ctio n b ased am plifiers in th e 4-20 K noise te m p e ra tu re range. D espite falling sh o rt by two
to th re e orders of m ag n itu d e in pow er sen sitiv ity , th ese e x p erim en ts provided a val­
id a tio n o f th e cavity-search te ch n iq u e, an d p rovided m uch p ra c tic a l ex perience for
fu tu re ax ion searches, including th e one I d escribe in this thesis.
M y th e sis describes m y w ork as p a rt o f an ongoing search for axions a t Law rence
L iverm ore N atio n al L aborato ry (L L N L ). In 1995, a co llab o ratio n involving LLNL,
th e M assach u setts In s titu te o f Technology (M IT ), th e U niversity of F lo rid a (U F ),
F erm ilab , and Lawrence B erkeley N atio n al L ab o rato ry (LB N L) began an axion
search w ith th e goal of achieving se n sitiv ity to K im -S chifm an-V ainshtein-Z akharov
(K SV Z ) m odel axions. T h is search u tiliz e d a m uch larger m a g n et volum e along
w ith b e tte r am plifiers th a n th e prev io u s ex p erim en ts. As re p o rte d in [4], o u r goal
was achieved m aking us th e first axion search to reach cosm ologically in te re stin g
sen sitiv ity .
All o f th e searches described so far em ployed a single m icrow ave cav ity w hich
filled th e m a g n et volum e. B y e arly 1999, we h ad covered m o st o f th e m ass (fre­
quency) ran g e o f th e single cavity.
E x p lo rin g higher frequency regions requires
sm aller cav ities. However, p lacin g a single, sm aller cavity in to th e m ag n et w ould
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be inefficient.
Since th e axion signal is co h eren t over th e volum e o f th e cav ity
(Ad ~ 10 — 1000 m ), it is possible to pow er com bine th e signal from m u ltip le cav­
ities a n d m a in ta in effective use o f th e m a g n e t volume[5, 6]. In M arch 2000. we
achieved th e first results from a fo u r-cav ity axion d etecto r.
I have been involved w ith th is g ro u p since th e com m issioning o f th e original,
single-cavity search. As w ith m ost research ers in sm all collaborations, I was heavily
involved in all phases of th e com m issioning a n d o p eratio n of the single-cavity ex p er­
im en t. O f p a rtic u la r im portance, I was p rim a rily responsible for th e dev elo p m en t
and im p le m e n ta tio n o f th e d ata-ac q u isitio n (D A Q ) system which en ab led us to op­
e ra te th e e x p erim en t w ith well over 90% liv e-tim e, largely u n a tte n d ed . T h is freed
o u r lim ite d m anpow er from m any d a y -to -d a y op eratio n s to co n cen trate on analysis
and th e developm ent of fu tu re im p ro v em en ts.
T h is w ork was p re p a rato ry for m y role leading th e com m issioning o f th e first
axion search em ploying m ultiple cav ities. T h is technique is v ital for ex ten d in g th e
reach o f c av ity axion searches.
A m a jo r technological achievem ent was th e first
im p le m e n ta tio n of piezoelectric a c tu a to rs in a com bined cryogenic a n d high-field
en v iro n m e n t. T his developm ent will e n a b le us to o p e ra te fu tu re d etecto rs a t m uch
lower te m p e ra tu re s .
M y th esis describes th e com m issioning a n d first results from th e four-cavity
axion d e te c to r a t LLNL. C h ap ter T w o gives a b rie f outline of b o th th e p a rtic le
physics of th e axion and its possible significance to cosmology, p a rtic u la rly its role
as a n o n -b ary o n ic d a rk m a tte r c a n d id a te . T h e ex p erim en tal a p p aratu s is d escrib ed
in d e ta il in C h a p te r T hree, w ith em p h asis on th e developm ent of th e piezoelectric
a c tu a to rs . R esu lts are given in C h a p te r Four. T his was p rim arily an engineering
ru n , b u t we still g en erated a m ean in g fu l exclusion lim it. Finally, in an ap p en d ix , I
d escrib e som e o f m y recent work developing a new ty p e of m icrowave am plifier based
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4
on S uperconducting Q u a n tu m In terferen ce Devices (SQ U ID s). T h is technology will
en a b le a tru ly definitive ax io n search , i.e. one th a t w ill be able to discover o r rule
o u t axions of th e m o st p e ssim istic a x io n -p h o to n coupling, even sh o u ld th e y only
c o n stitu te a fraction o f th e h alo density.
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5
Chapter 2
Theory
T h is c h a p te r provides a n overview o f th e th e o ry o f axions, b eg in n in g w ith th e ir origin
in Q C D . N e x t, I discuss th e c u rre n t s ta te o f cosm ology a n d th e ax io n ’s possible role
in th e d a rk -m a tte r p ro b lem . Finally, I d escribe th e p rin cip les o f th e m icrowavecavity d e te c to r used in th is axion search ex p erim en t.
2.1
Axions and QCD
Q u a n tu m C h ro m o d y n am ics (Q C D ) is th e th e o ry of th e stro n g in teractio n s, which
describes th e color force w hich binds q u a rk s an d gluons in to co lo r-n eu tral hadrons.
It successfully describes th e observed h a d ro n s p e c tra a n d th e observed p h enom ena
of confinem ent a n d a sy m p to tic freedom . T h e color force b etw een tw o quarks goes
to zero a t very sm all sep a ra tio n , so th e y b eh av e essen tially as free p articles; this
is called a sy m p to tic freedom . O n th e o th e r h a n d , se p a ra tin g th e m in to two free
quarks requires a n in fin ite am o u n t o f energy. B ecause o f th is confining p ro p e rty of
th e color force, q u ark s a re alw ays b o u n d in co m p o u n d o b je c ts.
A field th e o ry w hich is b o th confining a n d a sy m p to tic a lly free m u st be nonabelian. T h e n o n -ab elian n a tu re o f Q C D m akes ex act so lu tio n s n early im possible.
In an a b elian th e o ry such as Q u a n tu m E lec tro d y n a m ic s (Q E D ), th e gauge bosons
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6
(pho to n s for Q ED ) a re not charged, so th ey do not in te ra c t. T h e gauge bosons in
Q C D are th e gluons w hich c arry th e color charge and h av e self-interactions. T his is
im p o rta n t for explaining confinem ent an d a sy m p to tic freedom , b u t also m akes th e
field equations very difficult to solve.
2.1.1
Perturbative QCD and the U(1) a Problem
C alculations in Q CD are ty p ically p e rtu rb a tiv e . i.e. th e sp e c tru m o f particles and
in teractio n s is found by ex p an d in g th e fields around th e g ro u n d s ta te , or vacuum .
P e rtu rb a tiv e solutions in QCD a re found using th e L agrangian den sity Cpert
= Y ,* l(iD - m ,W , - -F l„ F ?
/
^
(2.1)
w here th e first te rm describes th e q u a rk fields ibf, f d en o tes flavor, and F*u is the
gluon field tensor. M ore details can be found in [7].
>Cpert is ad eq u ate for p e rtu rb a tiv e calculations, b u t it is a n incom plete description
of Q C D . Cpert is in v arian t u n d e r global axial tra n sfo rm a tio n s described by e1''50. If
Q C D was described only by Cpert-, th e n it would also possess th is sym m etry, called
U{1) a - This would lead to p a rity d o u b lets in th e h ad ro n s p e c tra w hich are not seen
experim entally, im plying th a t th e U (1 ) a sy m m etry is sp o n tan eo u sly broken.
Spontaneous sy m m e try b reak in g (SSB) is a com m on p h en o m en o n in Lagrangian
field theories. T h e L agrangian a n d th e vacuum sta te need not possess th e sam e
sy m m etry . A sy m m e try of th e L agrangian not possessed by th e vacuum s ta te is
said to be spontaneously broken. A consequence of SSB of an ex act sy m m etry is
th e existence of a m assless p a rtic le called a G oldstone boson. If th e sy m m etry was
n o t ex act, th e associated p a rtic le is referred to as a p seudo-G oldstone boson and
will have a sm all m ass.
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If U ( 1).4 w ere sp o n tan e o u sly broken, th e pseudo-G oldstone boson w ould be a
light pseudoscalar, w hich sh o u ld have been d e te c te d . T h e rf has th e rig h t q u a n tu m
num bers ( J 71- = 0- ) to be th is p a rtic le , b u t its m ass m ^ = 958 M eV is m u ch too
heavy. T his is called th e U (1 ) a problem .
2.1.2
The 0-vacuum
t ’ Hooft solved th e ( J ( I ) a p ro b lem by p o stu la tin g th a t U(1) a is broken anom alously.
Anom alies o ccu r in field theo ries w hen classical sy m m etries are broken by q u a n tu m
effects. T h e U (1 ) a sy m m e try is broken by th e A dler-B ell-Jackiw (A B J) anom aly,
often called th e tria n g le o r chiral a n o m aly [8]. T h e consequence o f th is a n o m aly is
th e presence of a second te rm C@ in th e Q CD L agrangian
( 2 .2 )
where g is th e coupling c o n sta n t an d F “ denotes th e dual of th e gluon field stre n g th
tensor.
0 becom es a p a ra m e te r o f th e theory, 0 < 0 < 2 tt. T his te rm c a n be
expressed as a to ta l d e riv a tiv e so it is often ignored. Its n o n -p ertu rb ativ e effects,
however, are very im p o rta n t.
T h e origin o f th e a n o m alo u s te rm is th e com plex vacuum stru c tu re o f n o n -ab elian
field theories such as Q C D . T h e vacuum s ta te o f Q C D consists of an infinite n u m b e r
of degenerate vacua w hich are topologically d is tin c t. T h is m eans th a t one v acuum
can not be tra n sfo rm e d in to a n o th e r by a continuous tran sfo rm atio n . T hese vacua
are labelled by an in te g e r n called th e topological w inding num ber. W hen tu n n e lin g
solutions betw een th e d e g en e ra te vacua ex ist, th e tru e vacuum becom es a lin e a r
superposition of th e m
|© ) =
CO
£
eln0jn)
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(2.3)
s
|0 ) is called th e 0 -v a c u u m which a su p erp o sitio n o f th e d eg en e ra te vacua |n) .
T h e first tu n n e lin g solutions discovered w ere solitons, localized in b o th sp ace
an d tim e . B ecause o f th e ir in stan tan eo u s n a tu re , th e y w ere called in stan to n s[9 ].
O th e r solutions have since been found, som e o f w hich are n o t localized, b u t in s te a d
have an infinite sp a tia l e x te n t. These configurations e x p lain why Co can n o t b e
ignored in p e rtu rb a tio n theory, to tal d erivatives can only b e neglected as su rface
te rm s w hen th e fields fall off sufficiently rap id ly a t infinity.
D espite its n o n -p e rtu rb a tiv e n atu re, C q can be in c o rp o ra te d into p e rtu rb a tiv e
calculations th ro u g h th e m odification o f th e v acu u m s ta te .
In stead o f e x p an d in g
a b o u t th e n o rm a l vacuum , calculations can b e perfo rm ed b y ex panding a b o u t th e
0 -v a c u u m .
C& is often called th e b a re -0 term ; th e re is a n o th e r te rm arising from th e electrow eak secto r w hich gives a second te rm oc F F .
It is o fte n convenient to w ork
in th e basis w here th e q u ark m ass m a trix M is diagonal. T h is is accom plished by
perform ing a chiral ro ta tio n on the q u ark fields ib, b u t th is ro ta tio n also suffers fro m
th e A B J anom aly. T h is is in co rp o rated in to C o by ch an g in g th e 0 p a ra m e te r
0<?2
= 32 ^F rF ^
(2 ‘4)
w here 0 = 0 + a rg (d e t M ) .
2.1.3
The Strong-CP Problem
W ith C q . th e U ( l) ^ p ro b lem is solved, b u t a n o th e r p ro b lem tak es its place. C q vi­
olates ch arg e-conjugation tim es p arity (C P ) sy m m etry . A n observable consequence
of th is C P v io la tio n is an electric dipole-m om ent of th e n e u tro n dn , w hich is c alc u ­
la te d to be dn ~ 8 x 1O-16|0 | e-cm[10]. T h e e x p e rim e n ta l b o u n d is c u rre n tly \dn \ <
12 x 10-26 e -c m [ll].
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9
C onsistency betw een th e o ry and e x p erim en t requires |0 | < 10-1°.
W hen an
a rb itra ry p a ra m e te r such as © m u st b e fin e-tu n ed , th e re is a n atu raln ess problem in
th e theory. In th is case, th e problem is co m p o u n d ed b y th e fact th a t it is not a single
p a ra m e te r w hich is fine-tim ed, b u t th e co m b in a tio n o f tw o independent q u an tities
arising from different sectors o f th e S ta n d a rd M odel. T h e sm allness of 0 is referred
to as th e stro n g -C P problem .
2.1.4
The Peccei-Quinn Solution
P a ra m e te rs such as 0 w hich m u st be fin e-tu n ed a re o ften indications of add itio n al
sy m m e trie s in th e theory. In th is sp irit, in 1977 R o b e rto Peccei and Helen Q uinn
p o s tu la te d th a t th e sm all value of 0 could b e e x p lain ed by th e SSB of a new global
U (l) sy m m e try called th e P eccei-Q uinn o r U (1 ) p q sym m etry[12]. Soon th ereafter,
W einberg and W ilczek in d ep en d en tly p o in ted o u t th a t th e re would be a pseudoG oldstone boson asso ciated w ith th e SSB [13]. T h is new pseudoscalar p article was
called th e axion.
T h e U (1) pq sy m m e try suffers from th e sam e ch iral anom aly, adding yet an o th er
te rm C^xion to
C
a x io n
D
= -4 f a . f , F £ UF* + in t e r a c t i o n s
JPQ olTT
(2.5)
w here <pa is th e axion field, / pq is th e scale of th e U (1 )p q SSB, and .4 is a m odeld e p en d e n t c o n sta n t.
Som e o f th e im p o rta n t axion in te ra ctio n s will be described
in th e n e x t sectio n . 0 is also m odified to ta k e th is te rm in to account 0 = 0 +
a rg (d e t M ) C q a n d th e first te rm o f C axi0n form a p o te n tia l for th e axion field. T he m inim um
o f th is p o te n tia l is
< <Pa > =
+ a rg (d e t M ) )
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(2.6)
10
A t this value of 4>a, th e coefficient o f th e C P-vioIating te rm s oc F F is zero, solving
th e strong-C P p ro b lem . 0 is forced to th e C P-conserving value dy n am ically as the
axion field relaxes to th e m inim um o f its p o ten tial. T h e rem a in in g problem is now
ex p erim en tal d e tec tio n o f th e axion.
2.1.5
Axion Properties and Couplings
T his section sum m arizes th e im p o rta n t properties a n d couplings o f axions. All of
these depend on th e a rb itra ry an d unknow n sy m m e try b reak in g scale f p Q of the
U (1 )p q
sym m etry.
T h e m ass and couplings o f th e axion are derived u sin g th e techniques of current
algebra[l4]. T h e m ass o f th e axion is re la te d to / pq by
U m lty / m um d
107 G e V
= 7 — TKFr — T
r ~ 0.6 e V — — —
f p Q / N (m u + m d)
f PQ/ iV
w here f ^ sa 93 M eV is th e pion decay co n stan t, m „ is th e pio n m ass,m u,
(2. /)
are the
up and down q u ark m asses, an d N is th e color anom aly, a non-zero in teg er related
to th e axion coupling to color.
All of th e axion couplings are m odel dep en d en t, a n d inversely p ro p o rtio n al to
f pq- T h e v ariety of m odels can be sep a ra te d into two g en eric classes depending on
th e tree-level couplings to sp in -1 /2 ferm ions shown in F ig u re (2.1) and described by
th e in teractio n L agrangian
£afj = ig a jffa flsf
( 2 .8 )
w here gaj j is a m o d el-d ep en d en t coupling co n stan t . H a d ro n ic axion m odels have no
tree-level coupling to charged leptons, only to quarks. G ra n d U nified T h eo ry (G U T )
axion m odels have sim ila r q u a rk an d lep to n couplings. T h e tree-level coupling to
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11
Y
►
a
9aff
►9m
—
y
F igure 2.1: Feynman diagram for the axion tree-level coupling to two fermions (left) and
anomalous coupling to two photons (right).
color is re q u ire d for axions to solve th e stro n g -C P p roblem so it is p resen t, a t som e
level, in all m odels.
T h e im p o rta n t in teractio n for m icrow ave-cavity d etec to rs is th e effective twophoton coupling show n in Figure (2.1). T h e co u p lin g is th ro u g h th e A B J anom aly
w here th e in te rm e d ia te loop is quarks for h ad ro n ic m odels a n d e ith e r quarks or
leptons for G U T m odels. T he in teractio n L ag ran g ian is
Ax-rr = 9a~n4>aE - B
(2.9)
w here th e coupling constan t ga^ is given by
a / ' 2 tt f E
2 4 + 2N
al'2-n
( E
9arn ~ f p Q / N \ N ~ 3 1 + J = f p Q/ N \ N ~
w here z = ^
\
}
)
« 0.5, a is the fin e-stru ctu re c o n sta n t, a n d E is th e electro m ag n etic
an o m aly re la te d to th e axion coupling to charged lep to n s. In h ad ro n ic m odels, E =
0, w hile m o st G U T m odels have
= §.
T h e d ecay tim e of th e axion in to two p h o to n s is given by
r « „ == 6.8 x I024 s
e
c
[ ( # - 1 . 9 5 ) /0.T2]
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(2. 11)
12
For axions w ith m asses relev an t for m icrow ave-cavity d e te c to rs ( m a = 0 ( 1 yreV))
Ta~n = O (1 0 54 sec). T h e age o f th e universe is t0 ~ 1018 sec. so these light axions
are q u ite sta b le . A w ay to en h an ce th is process is g iven in th e next section.
T h e m o st w idely used h ad ro n ic axion m odel is th e K im -S hifm an-V ainshteinZ akharov (K SV Z ) m odel[15]. T h e re are a v ariety o f m o d els in this class d ep en d in g
u p o n w hich q u a rk s a re coupled to axions. I alw ays refer to th e sim plest K SV Z m odel
w here th e ax ion only couples to a single heavy q u ark .
T h e tra d itio n a l m o d el for G U T axions is th e D in e-F isch ler-S red n ick i-Z h itn itsk y
(D FS Z ) m o d el[l6 ]. T h e m o st com m on D FSZ m odel has a significantly sm aller tw op h o to n coupling th a n th e K SVZ m odel, an d th is e x p e rim e n t has not y e t reach ed
sufficient se n sitiv ity to d e tec t th e m .
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13
2.2
Axions and Cosmology
T h is sectio n describes th e so-called sta n d a rd m odel o f cosmology, along w ith th e
role o f th e axion as a c an d id a te for th e non-baryonic d a rk m a tte r in th e universe.
2.2.1
Overview and Historical Development of Cosmology
T h e m a th e m a tic a l foundation o f s ta n d a rd cosm ology is th e F reid m an n -R o b ertso n W aiker (FRVV) cosmological m odel[17]. T h is m odel assum es th e universe is hom oge­
nous a n d isotropic. T hese a ssu m p tio n s were originally m ade to sim plify th e form
of E in ste in ’s E quation (2.13). b u t th e y have proven to b e rem arkably a c c u ra te on
sufficiently large distance scales ( ~ 50 M pc).
T h e g eom etry of a hom ogenous an d isotropic space is described by th e R o b ertso n VValker (RW ) m etric.
dr2
+ r 2(dd2 + sin2 9d<p2) | = g ^ u d x ^ d x 1'
1 - kr2
d r 2 = d t2 — R 2(t)
( 2 . 12 )
R ( t ) is th e scale p aram eter, ( t , r , 9 . o ) are com oving coordinates w ith 0 < r < 1, a n d
k is th e cu rv atu re signature w ith values o f —1 ,0 ,1 for spaces w ith negative, zero, or
p o sitive sp a tia l curvature.
Cosm ological m odels are b ased on solutions of th e E in stein E quation
G^ = R
71 - Ag^u = 8 ~ G T ^
R p Uis th e Ricci tensor, 7Z
th e
is th e Ricci scalar,
(2.13)
is th e stress-energy ten sor, G is
g ra v ita tio n a l constant an d A is th e infam ous cosm ological co n stan t originally
in serted by E instein to allow a s ta tic universe solution.
A n in finite variety of m odel universes can be c o n stru cte d by varying th e a b u n ­
dances o f th re e basic co n stitu en ts: m a tte r, rad iatio n , an d vacuum energy. A ll of
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14
th e se can be ap p ro x im ated as perfect fluids w ith a sim p le e q u atio n of s ta te relatin g
th e ir den sity p to th e ir pressure p, p = wp. w is a co n sta n t equal to 0 for m a tte r, i
for ra d ia tio n , an d -1 for vacuum energy a sso ciated w ith A. T h e stress-energy ten so r
for a perfect fluid is given by
(2.14)
Ti0 = 0
Too = p (t)
For th e RW M etric
Ron = —3
R
R
(2.15)
Rij = —
R
A2
. k
1- 2 ------b 2 — 9ij
R
R2
R2
(2.16)
11= -6
—
—
J l.
R + R2 + R2
(2.17)
F irst, consider th e 0-0 com ponent of th e E in stein E q u atio n Goo = SttTc
oo
R2
k
SttG
A
R2 + R2 ~ ~1TP + J
(2.18)
W h en A = 0 th is reduces to th e F ried m an n E q u a tio n . T h e q u a n tity H = f is called
th e H ubble p a ra m e te r and m easures th e r a te of expansion.
Its present value is
d e n o ted by H q = j g . S ubsript ’O’ alw ays refers to th e present value of a p a ra m ete r.
T h e red u ced H ubble p a ra m ete r h is defined by Ho = lOOh k m /se c /M p c . U sing th e
definition o f H gives
rr2
k2
S ttG .
+ ^ 2 — ~ 3 ~ ( ^ + A\)
pA =
A
8 <rG
In tro d u cin g th e density p a ra m e te r 12 = ^ P'- w here p c =
(2.19)
is th e critical d en sity
sim plifies th e F riedm ann eq u atio n to
k2
H 2R 2
=
n
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(2 .2 0 )
15
T h is e q u a tio n d em onstrates th e c o n n ectio n betw een th e c u rv a tu re sig n a tu re k and
th e en erg y density of th e universe. W h en
= 1, k = 0 an d th e u n iv erse is flat,
hence th e im p o rtan ce o f th e c ritical d e n sity pc.
T h e re is only one indep en d en t i-i c o m p o n en t o f th e E in stein E q u a tio n .
2|
+ ^
+ J r = -S"-GP + A
C o m b in in g (2 .IS) and (2.21) gives a n ex p ressio n for
(2.21)
R
R
4 t~G
A
R = - — [> + 3 P ) + 3
T h e d e celeratio n p aram eter q is defined as q = —^
<2 -22)
T h e reaso n for this
n a m e is obvious when th e scale fa c to r is expressed in a Taylor series a ro u n d t 0
R ( t)
— R 0 + Ro(t — to) + —R a (t — to)2 + • • •
=
Ro(l + H 0(t - t0) - \ H 2q0{t - to)2 + - - •)
(2.23)
T h e scale facto r is re la ted to a m o re d ire c tly observable p a ra m e te r, th e cosm o­
logical red sh ift z given by
- = ^ i _ l
^em
=
(2.24)
ISobs
T h e su b sc rip ts ’obs’ and ’e m ’ refer to th e w avelengths and frequencies a t o b serv atio n
a n d em ission. T h e w avelengths change w ith th e scale facto r so r is sim p ly
re la ted
to th e scale factors at observation a n d em ission.
!+ * = £*•
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(2.25)
16
B o th d ista n ce s a n d tim es are expressible in te rm s o f z , th e e x ac t relatio n s dep en d on
th e p a rtic u la r cosm ological m odel. In sertin g (2.25) in to (2.23) a n d in v ertin g gives
th e re la tio n betw een tim e an d red sh ift for sm all z (t « £0)
T h e p ro p e r d ista n c e D p to an o b je ct in te rm s o f
is found by in te g ra tin g (2.12)
over a ra d ia l geodesic (d r = 0 )
rt° cdi
DP
Jt
R
[Robs
rRobs
dr
-jR^m \ / l — k r 2
cz
___
Hq
A th ird im p o rta n t e q u atio n is fo und from p. = 0 co m p o n en t o f th e conservation
law for th e S tress-E n erg y T ensor T°.£ = 0 w here th e sem i-colon denotes th e covariant
derivative
! ( , * • ) = -P ±.t m
(2,28)
P lugging th e eq u atio n o f s ta te in to (2.28) a n d solving th e differential eq u atio n
gives th e re la tio n betw een den sity a n d scale fa c to r.
p (t) cc R ( t ) ~ 3il+w)
(2.29)
T h e d en sity o f m a tte r scales as Z2- 3 , ra d ia tio n as Z2- 4 , an d th e d en sity of vacuum
energy re m a in s c o n sta n t. I will focus on m o d el universes co n tain in g only m a tte r
an d vacuum energy.
W ith th e infinite v ariety of possible m o d el universes, it is im p o rta n t to iden­
tify m e asu ra b le q u a n titie s w hich can d ifferen tiate betw een th e m . O ne of th e m o st
im p o rta n t a sp e c ts of a cosm ological m odel is th e tim e ev o lu tio n o f th e scale facto r
R ( t). A n a ly tic solutions of (2.13) a re possible w hen one co m p o n en t dom inates.
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17
k=
k= 0
o
k = +l
Big Bang
Big Crunch
Time
F igure 2.2: Scale factor evolution in a m atter-dom inated universe.
Figure (2.2) shows th e evolution of th e scale fa c to r for a m a tte r-d o m in a te d (M D )
universe w hich is e ith e r open, flat, or closed. All o f th e geom etries have R (0) = 0,
th is is referred to as th e Big B ang.
A closed (fio > 1) universe collapses in a
Big C runch, a flat (fio = 1) universe asy m p to tica lly sto p s expanding, an d an open
(Ho < 1) universe e x p an d s forever. Since m a tte r obeys th e strong-energy condition
w > —
t he g eo m etry and u ltim a te fate o f th e universe is d eterm in ed by a single
p a ra m e te r k or equivalently flo- T h a t is w hy so m uch effort is being p u t in to an
ac c u ra te m easu rem en t o f fioIn a universe w ith only vacuum energy, th e solutions describe th e so-called de
S itte r space. For a flat, vacuum d o m in ated universe, th e scale facto r evolution is
described by
R ( t) oc eHot
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(2.30)
IS
H ence, a vacuum d o m in a te d universe undergoes a n e x p o n e n tia l expansion. T h is is
im p o rta n t in in flatio n ary scenarios m entioned la te r.
V acuum energy does n o t satisfy th e strong-energy c o n d itio n , so th e connection
betw een f2o an d th e u ltim a te fa te o f th e universe is lo st. A u n iv erse w ith f l m > 1 will
ex p an d forever if th e vacuu m energy te rm comes to d o m in a te w hile H > 0. Sim ilarly,
a universe w ith f lm < 1 will collapse w ith a n eg ativ e cosm ological co n stan t. T h e
u ltim a te fate o f th e u niverse d ep en d s on th e co m p o sitio n o f th e energy, a n d not ju s t
th e overall density. p \ is co n sta n t while pm cc R ~ z so a n im m easu rab le vacuum
energy to d a y could still som ed ay d o m in ate th e universe.
A n im p o rta n t observable is th e expansion age o f th e u n iv erse t0 which is given
by
(2.31)
For a flat universe w ith b o th m a tte r and vacuum energy, (2.20) and (2.29) give
(2.32)
In sum m ary, th e six m o st im p o rta n t p a ram eters for d escrib in g th e universe are
Ho,qo,to,Qo, firm a n d f l ^ . Significant progress in m e a su rin g th e m has been m ad e
in th e last decade, b u t first I will briefly describe th e h isto ric a l developm ent of
cosm ological m e asu re m e n ts.
2.2.2
Early Tests of Cosmology
In 1916, S lipher n o tic e d t h a t galaxies tend to sy ste m a tic a lly red -sh ift as opposed to
blue-shifting. T h is im plies th a t galaxies are receding fro m us in all directions. In
th e la te 1920’s E dw in H u b b le q uantified th is resu lt a n d was th e first to correctly
s ta te th e law governing th e expansion of th e universe. H u b b le ’s law sta te s th a t th e
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19
recession v elo city of a g alax y is p ro p o rtio n a l to th e d ista n c e urec = H 0r w here H q is
th e sam e H u b b le p a ra m e te r defined earlier. T h e v alu e o f Ho has proven difficult to
m easu re a cc u ra te ly . H u b b le’s o rig in al value was 600 k m /s e c /M p c , which has since
been revised dow nw ard alm ost an o rd e r o f m a g n itu d e . A fte r SO years, th e re is still a
10%
u n c e rta in ty in Ho w hich tra n s la te s into u n c e rta in tie s in all o th e r cosm ological
p a ra m ete rs.
E arly e s tim a te s o f th e age of th e universe w ere m a d e b y d e te rm in in g th e age
of th e o ld est g lo b u la r clusters. M ost o f th e h eav y e lem en ts in th e u n iv erse w ere
created inside s ta rs an d d u rin g su p ern o v a explosions.
T h e earliest s ta rs form ed
in th e u n iv erse sh ould be very m e ta l-p o o r c o m p a re d to y o ung s ta rs su ch as o u r
sun. T h u s, th e oldest globular c lu sters are selected as th o se w hich co n tain m o stly
m e ta l-p o o r s ta rs.
An e s tim a te o f th e c lu ster age is d e te rm in e d from th e H ertzsp ru n g -R u ssell (H -R )
diag ram . A n H -R d ia g ra m is a p lo t o f effective te m p e ra tu re versus lu m in o sity for
stars in a p a rtic u la r clu ster. U sing s te lla r m odels, th e age o f th e c lu ste r can be
e stim a te d fro m th e tu rn -o ff p o in t for th e m ain -seq u en ce sta rs. T h ese e stim a te s give
a n age o f 10-20 G y r for th e clu sters, a n d assum ing th e y fo rm ed w ith in 1 G y r o f th e
Big B ang, a s im ila r e stim a te for i 0[18].
In th e m id -1 9 3 0 ’s, F ritz Zwicky p ro v id ed th e first ev id en ce for one of th e biggest
questions in cosm ology today, m issing m ass in th e u n iv erse. Z w icky n o ted th a t rich
clusters o f galax ies should behave as w ell-relaxed g ra v ita tin g sy stem s, an d th a t th e
virial th e o re m co u ld be used to e s tim a te th e ir to ta l m ass from th e m e asu re d velocity
dispersion. T h e m ass in th e lu m inous m a tte r could b e e s tim a te d by co u n tin g sta rs.
W h en carefu l m e asu re m e n ts were m a d e , it was discovered th a t th e m ass in ferred
from th e v iria l m e th o d was an o rd e r o f m a g n itu d e la rg e r th a n could b e a cc o u n te d
for in lu m in o u s m atter[1 9 ]. T his was th e first e v id en c e for th e ex isten ce o f large
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20
q u an tities o f d a rk m a tte r.
P e rh a p s th e m ost im p o rta n t discovery for cosm ology since th e H ubble expansion
took place in 1964 w hen Penzias an d W ilson discovered th e 3 K cosm ic m icrow ave
background (C M B ). T his is a very isotropic rad iatio n background w ith a nearly
perfect b lack-body energy sp e c tru m . N ot only did th is v alid ate th e isotropy and
h om ogeneity assum ptions of FR W cosmology, it was also a p re d ic ted relic of a
h o t Big B ang. Som e of th e im p o rtan c e of th e CM B to m o d ern cosm ology will be
discussed la te r.
As th ese early pieces of th e puzzle were being p u t to g eth er, it becam e obvious
th a t so m eth in g was fu n d am en tally w rong. F irst, th e re was th e age p roblem , it is
nearly im possible for th e universe to b e as old as it is w ith o u t being e x ac tly flat,
th a t is
Q —1
rv—6 0
n I <^ i10
. T h e fi = 1 solution in FR W cosm ology is u n stab le, if f I
is slightly less th a n 1 , th e universe should b e essentially em pty. Likewise, if f2 is
slightly la rg e r th a n 1 it should have recollapsed long ago. It is possible for f20 to
b e ex actly 1 by chance, b u t th e fine-tuning involved poses a n atu raln ess problem
sim ilar to th e one in th e S trong-C P problem .
A second problem was th e isotropy o f th e CM B. T h e finite age of th e universe
leads to horizons specified by th e d istan ce light could trav el in th e age of th e universe.
T h e CM B was actu a lly too isotropic, regions w hich were never in causal c o n tac t still
have th e sa m e te m p e ra tu re .
An eleg an t solution to b o th o f th ese problem s is th e idea o f inflation, first pro­
posed by G uth[20].
T h e basic id ea is th a t o u r universe u n d erw en t a p erio d of
e x p o n en tial grow th. T h e en tire universe could have grow n from a tin y piece w hich
was w ith in cau sal co n tact, so th e horizon p roblem is solved. Secondly, as seen from
E q u atio n (2.20), as th e scale fa c to r grows exponentially, 0, is d riv en to w ard 1 , solving
th e age, o r flatness problem .
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21
A dding inflation to th e FR W cosm ological m odel gives th e so-called s ta n d a rd
m odel of cosmology. T h e universe began fro m a very hot and dense Big B an g . At
som e point it u n d erw en t a p erio d o f in flatio n . F lu ctu atio n s in th e p re-in flatio n uni­
verse were enlarged a n d becam e th e seeds for s tru c tu re form ation. In th e m e an tim e,
th e universe continues to ex p an d an d cool. D etails of stan d a rd cosm ology can be
found in [2 1 ].
2.2.3
Experimental Tests of the Standard Cosmology
T h e developm ent o f cosm ology a n d p a rtic le physics has allowed physicists to m odel
th e universe to v ery early tim es, an d m ake som e very specific p red ictio n s ab o u t
it. M eanw hile, ex p erim en talists have begun to constrain th e various cosm ological
p aram eters sufficiently to begin ru lin g o u t m a n y o f th e m odels.
T h e value of 0 m is e stim a te d using Z w icky’s virial m ethod on g alactic clu sters.
T h e inferred values a re consisten tly a ro u n d 0.3[22]. This is not consistent w ith th e
inflationary scenario unless th e re is a sm o o th com ponent w ith f lsmooth ~ 0.7. T he
obvious c a n d id a te is th e cosm ological c o n sta n t A.
Einstein referred to it as his
g reatest b lu n d e r so th e re was som e relu c tan c e to re-invoke it.
T h a t changed in 1998 w hen two groups re p o rte d the results of th e ir stu d ie s of
high-redshift ty p e la supem ovae. T y p e la supernovae result from th e explosion of a
carbon-oxygen w h ite d w a rf w hose m ass exceeds th e C handrasekhar b o u n d . W h ite
dwarfs are very d ense stars w hose g ra v ita tio n a l collapse is balanced by electro n
degeneracy pressure. T h e y have th e p ecu liar p ro p e rty th a t th e ir volum e is inversely
prop o rtio n al to th e ir m ass. Above th e C h an d ra se k h a r m ass, th e radius ap p ro ach es
zero. Before th is h a p p en s, a th erm o n u clear ru n aw ay occurs, o b literatin g th e s ta r.
T h e h y p o th esized scenario for a ty p e la supernova is th a t a C -0 w h ite d w arf
accretes m ass from a n earb y co m panion u n til th e C handrasekhar lim it is reach ed
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a n d th e s ta r explodes. Since th e exploding sta rs have sim ilar m asses ( ~ 1.4 M q),
th e explosions release sim ila r am o u n ts of energy ( ~ 1051 erg) fueled b y th e fusion
o f carb o n and oxygen in to 56 Ni. In a d d itio n , an em pirical relatio n h a s been found
b etw een th e m a x im u m lu m in o sity a n d th e ra te a t which th e su p ern o v ae decrease in
brightness. T h e fixed en erg y release m akes th e m useful as s ta n d a rd can d les, ob jects
w hose lum inosity is in d e p e n d e n t o f evolution. T h e only draw back to th e ir use is
t h a t th e u n d e rstan d in g o f th e m rem ain s em pirical.
Tw o team s used ty p e la supernovae to m easu re qo and b o th re p o rte d a very
su rp risin g result, qo < 0, th e exp an sio n o f th e universe is accelerating[23].
m easurem ents involve th e lu m in o sity d istan ce cPL =
The
w here L is th e lum inosity
a n d J- is th e m easu red flux. T h e re la tio n betw een cLl an d th e red sh ift z involves th e
m e tric , so objects w ith know n lu m in o sity a n d red-shift such as ty p e la supernovae
can probe th e g e o m etry o f th e universe.
As m ore su p em o v ae w ere m easu red , it b ecam e possible to c o n strain Qm an d £l\.
T h e results are co n siste n t w ith th e d y n am ical resu lt Qm ~ 0.3 an d
~ 0.7 for th e
best-fit flat universe m o d el.
T y p e la su p em o v ae a re also used to e stim a te th e H ubble p a ra m e te r. N earby
(low-z) supernovae a re used as s ta n d a rd candles an d H 0 is d e te rm in e d from th e
correlatio n betw een th e ir d ista n ce an d recession velocity. So far, th is m e th o d has
p ro d u ced th e m ost a c c u ra te m easu rem en t o f th e H ubble p a ra m e te r, Ho = 6 S ± 2
± 5 km /sec/M pc[24].
T h e age of th e u n iv erse e stim a te d from th e b est-fit flat universe u sin g th e latest
value o f Ho is 14.2 =b 1.7 G y r, co n sisten t w ith th e e stim ates from th e ages o f globular
clusters.
T h e v ersatility o f ty p e la supernovae for m easu rin g cosm ological p a ra m e te rs is
rivaled only by th e cosm ic m icrow ave background. T his b ackground o f ra d ia tio n
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is a n ecessary a rtifa c t of a hot B ig B an g . W h en th e early universe was very ho t
a n d d e n se , ra d ia tio n and m a tte r w ere tig h tly coupled. P h o to n s w ere able to ionize
h y d ro g en so electrons and p ro to n s w ere n o t b o u n d . T hese ch arg ed p articles in te r­
a c te d w ith th e photons an d th e y w ere all in th e rm a l eq u ilib riu m . T h e ra d ia tio n
q u ick ly sm o o th e d any m a tte r d e n sity flu c tu a tio n s, so th e early universe was hig h ly
iso tro p ic.
W h e n th e te m p e ra tu re cooled su ch th a t n e u tra l hydrogen co u ld form , th e u n i­
verse qu ick ly becam e tra n sp a re n t to ra d ia tio n an d it decoupled from m a tte r. T h is is
c alled re c o m b in atio n . A fter re c o m b in atio n , th e rad iatio n ju s t cooled as th e universe
e x p a n d e d . Today, th is rad iatio n form s th e C M B which has a te m p e ra tu re of 2.7277
± 0 . 0 0 2 K[25].
A fte r a correction for a d o p p ler sh ift caused by our m o tio n in th e cosm ic rest
fra m e , th e energy sp ec tru m is h ig h ly iso tro p ic an d w ithin u n c e rta in ty p erfectly
b la c k -b o d y in n a tu re . T he CM B is a sn ap -sh o t o f th e universe a t reco m b in atio n .
Its sm o o th n e ss reflects th e isotropy o f th e un iv erse a t th e tim e. Since flu ctu atio n s in
th e e a rly universe were only sm o o th ed over th e horizon d istan ce dfj. a t sufficiently
sm all a n g u la r scales th e re should b e an iso tro p ies in th e C M B . T h e an g u lar size o f
th e a n iso tro p ie s is a m easure o f dfj a t re c o m b in atio n and is sen sitiv e to th e value o f
floA n iso tro p ies in th e CM B are c h a ra c te riz e d by th e corresponding te m p e ra tu re
flu c tu a tio n s e x p an d ed in th e sp h erical h arm o n ics.
AT
^
= £ « « » . V?” (0 ,* )
T h e a n g u la r scale of th e an iso tro p ies is specified by th e m u ltip o le index I.
(2.33)
The
a n iso tro p y caused by th e e a r th ’s m o tio n in th e cosm ic rest fram e has I = 1 a n d is
called th e dip o le anisotropy.
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24
T h e Cosm ic B ackground E x p lo rer (C O B E ) sate llite was th e first e x p erim en t to
d e te c t sm all-scale anisotropies in th e C M B [26]. Its purpose was to p erfo rm a full sky
m a p , so its an g u lar resolution was lim ite d . E x p erim en ts lim ited to sm all-angle (large
t) an isotropy m easurem ents w ith h ig h er an g u lar resolution have b een perform ed
rec en tly w ith balloon-borne d etecto rs. T h e B O O M E R A N G an d M A X IM A groups
have recen tly re p o rte d a peak in th e pow er sp ec tru m neax I = 200.
T his peak
is called th e first acoustic peak a n d was caused by b ary o n -p h o to n oscillations a t
th e la st sca tte rin g surface a t reco m b in atio n . Its position in th e pow er sp e c tru m is
co n sistent w ith fi 0 = 1[27, 28].
A lthough I only discuss th e use o f th e CM B to d eterm in e th e global g eo m etry
o f th e universe, it has m any o th e r th e o re tic a l uses, see for ex am p le [29]. T h e lack of
significant anisotropy on an g u lar scales up to I ~ 2 0 0 , is one of th e m o st pow erful
lim its in constraining m any cosm ological ideas a n d m odels. T h e re la tiv e sizes an d
locations of th e various acoustic p e ak s will be valuable as b o th th e th e o ry an d
m easu rem en t of th e m continue to advance.
In sum m ary, no m easu rem en t to d a te is inconsistent w ith th e h o t Big B ang
m odel. Tw o independent m e asu rem en ts p oint to a flat universe, co n sisten t w ith in­
flation. T h e re is incontrovertible ev id en ce for an expanding universe, an d increasing
evidence th a t th e expansion is acceleratin g .
2.2.4
Accounting of Matter in the Universe
T h e CM B a n d supernova resu lts th a t Oo
~ 1 leads to th e next q u estio n in cos­
mology, th e c o n stitu tio n of m a tte r a n d energy. A ssum ing th e supernova resu lts are
a cc u ra te , a t least tw o-thirds of th e c ritic a l d en sity is in som e form o f sm o o th energy
com ponent. W h at th is d ark energy consists o f is still a question for th e th eo rists.
A se p a ra te problem is th a t it is n o t possible to ad eq u ately account for th e m a tte r
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25
th a t c o n stitu tes th e rem a in in g th ird o f th e critical density. T h is p roblem was h in te d
a t by Zwicky in th e 1930’s, a n d his work on th e s ta r co u n ts places an u p p e r lim it
on th e ab u n d an ce o f lum in o u s m a tte r a t fl[UTn < 0.01 [30]. N early all o f th e m a tte r
in th e universe m u s t be d a rk , and d eterm in in g th e n a tu re o f th e d ark m a tte r is a
m a jo r effort.
A reasonable a ssu m p tio n is th a t th e re are sim ply a lo t o f planets, b u rn ed -o u t
stars, and in te rs te lla r gas th a t can not be seen by telescopes. All of those o b jects
are know n to exist; n o th in g exotic is required.
T his sim ple e x p la n a tio n is refuted by B ig B ang N ucleosynthesis (B B N ), a th eo ry
for th e form ation o f light elem en ts in th e early universe. T h e relativ e ab u n d an ce
of th e various light e lem en ts (H ,D , 3 H e, 4 H e ,'L i, 7 Be) p ro d u ced in th e early universe
depends p rim arily o n th e bary o n -to -p h o to n ratio . C o m b in in g this ra tio w ith th e
inferred photon n u m b e r from th e CMB te m p e ra tu re gives th e baryon n u m b er, or
equivalently, th e b a ry o n ab u n d an ce.flg [3 1 ].
M easuring th e p rim o rd ia l abundances of th e light elem en ts is co m p licated by
stellar processes w hich e ith e r create or d estro y th em . D e u te riu m is th e m ost useful
ex p erim en tal c o n stra in t because stellar processes only d e stro y it. T h e ab u n d an ce
of D is m easured by th e ab so rp tio n o f q u asar light by an in terv en in g gas cloud
containing it. T h e p red ic tio n s o f BBN axe consistent w ith th e m easu red p rim o rd ial
D abundance, as w ell as th e abundances o f th e o th e r light elem en ts in a sm all in terv al
of baryon -to -p h o to n ra tio consistent w ith Q b ~ 0.05[32].
T his result has m a jo r im plications for th e d ark m a tte r.
O nly a b o u t 15-20%
of th e m a tte r in th e universe is m ade up o f baryons. T h e re axe two d a rk m a tte r
problem s in cosm ology, (i) w h at axe th e d a rk baryons (fig >
th e n a tu re of th e n o n -b ary o n ic d ark m a tte r (Om
an d (ii) w h a t is
O s)? T h e baryonic d a rk m a tte r
is likely diffuse gas a n d th e d ark stars a n d p lan ets m e n tio n e d earlier. T h e axion, a
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
26
very light, no n -b ary o n ic p a rtic le is a serious c a n d id a te to m ak e up p a rt, if not all,
o f th e non-baryonic d a rk m a tte r.
T h ere are o th e r c a n d id a te s for th e n o n -baryonic d a rk m a tte r , a n d th e re is m o u n t­
ing evidence th a t one has a lre a d y b een d e tec te d . S u p e rsy m m e try theories predict
th e existence o f a m assive, s ta b le p a rtic le called th e L ig h test S u p ersy m m etric P a r t­
n e r (L SP), th is a n d o th e r u n s ta b le su p ersy m m e tric p a rtic le s a re com m only called
W eakly In te ra c tin g M assive P a rtic le s o r W IM Ps[33]. A second p o ssib ility is th a t
neutrinos, long th o u g h t to b e m assless, m ay in fact b e m assive. If th is is th e case,
th e y would be th e first fo rm o f n o n -b ary o n ic d a rk m a tte r to b e d e te c te d . Evidence
for neutrino m asses com es fro m o scillations b etw een n e u trin o species. R esults from
th e S uper-K am iokande e x p e rim e n t a re co n sisten t w ith a non-zero m ass for m uon
n e u trin o s[34]. As will be seen, th e re are reasons w hy n e u trin o s are n o t expected to
be a significant c o n trib u tio n to
N eutrinos are d is tin c t from axions an d W IM P s as d a rk -m a tte r can d id ates (aside
from being d e te c te d ). N e u trin o s a re re la tiv istic a n d as such a re described as hot
d a rk m a tte r (H D M ). A xions a n d W IM P s b o th h av e sm all velocity dispersion and
are form s of cold d a rk m a tte r (C D M ). T h e d istin c tio n is im p o rta n t for stru c tu re
form ation. H D M q u ickly sm o o th s th e d en sity p e rtu rb a tio n s w hich lead to g rav ita­
tio n a l collapse, in h ib itin g s tru c tu re fo rm atio n . C D M , how ever, collects in regions of
g ra v ita tio n a l in sta b ility , fueling th e grow th o f s tru c tu re .
All of th e ev id en ce so fa r is for a d a rk -m a tte r p ro b le m on th e cosm ological scale,
b u t if it is to b e d e te c te d locally, th e re should b e som e e x p e c ta tio n th a t it exists
here. T he first clue is th e e x p e c te d scenario o f s tru c tu re fo rm atio n .
M odels o f s tru c tu re fo rm a tio n a tte m p t to re p ro d u c e th e observed d istrib u tio n
o f galaxies. T h e re a re tw o basic scenarios o f s tru c tu re fo rm atio n , top-dow n and
b o tto m -u p . T op-dow n fo rm a tio n p o stu la te s th a t sm a lle r s tru c tu re s re su lt from th e
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
b reak-up of larger ones. T h is occurs in universes d o m in ated by H D M . B o tto m -u p
form ation, w hich occurs in C D M o r all-baryonic universes, p o stu la tes th a t larger
stru c tu re s axe b u ilt from m ergers of s m aller objects.
B aryonic a n d HD M d o m in a te d universes are not able to rep ro d u ce th e observed
d istrib u tio n of galaxies and clusters. F o r an all-baryonic universe w ith in itia l den­
sity p e rtu rb a tio n s lim ited by C M B isotropy, th e re is insufficient s tru c tu re form ation
in th e tim e-scale set by th e age o f th e universe. HDM m odels lead to m ore largescale stru c tu re th a n is observed. CD M m odels are m ore c o m p atib le w ith observa­
tions. CDM acts as seeds for th e s tru c tu re s to form , speeding th e process over th e
all-baryonic case, allow ing sufficient s tru c tu re form ation w ith su ita b ly sm all initial
density p e rtu rb a tio n s [35].
T h e m ost successful m odel of s tru c tu re form ation is th e m ixed m odel, often
called ACDM[36]. O bserved large, sm o o th stru c tu re s are not rep ro d u cib le in CDM
m odels so a sm o o th energy-com ponent is m ixed in. Since CD M is th e seed for galaxy
form ation, it should be a b u n d a n t in galaxies, including our own. A ssum ing th a t th e
th re e neutrinos have nearly d eg en erate m asses, as suggested by recent ex p erim en ts,
A CDM m odels place a lim it o f m „ < 0.6 eV[37]. N eutrinos are ferm ions, so th e ir
n u m b e r d ensity in galaxies is lim ited by th e P auli exclusion principle, placing a lower
lim it on th e n e u trin o m asses th a t can acco u n t for all o f th e g alactic d a rk m a tte r. A
slightly m ore strin g en t boun d com es fro m th e fact th a t n eu trin o s are dissipationless
a n d as such th e ir m axim um p hase space d en sity can not increase a fte r th e y becom e
g rav ita tio n a lly bound. T rem ain e an d G u n n calculated th a t in o rd er for neu trin o s
to account for all of th e galactic d ark m a tte r, m u ~ 100 eV[3S]. C u rre n t d a ta is
consistent w ith th e abu n d an ce o f n eu trin o s Q.v being sim ilar to th a t o f lum inous
m a tte r, f 2„ ~ Qlum = 0 ( 0 . 0 1 ).
D irect evidence for galactic d a rk m a tte r com es from th e m easu red ro ta tio n curves
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
•28
,-0.5
>
s
Radius
F igure 2.3: Expected Keplerian rotation curve for a galactic disc with the
for R > Rdisc-
behavior
o f galaxies, including our own. Because baryonic m a tte r is dissipative, lu m in o u s
m a tte r in spiral galaxies tends to lie in discs w ith radii ~5-10 kpc a n d thicknesses
~ 0 .5 - l kpc. If this were the e x ten t of th e m a tte r in th e galaxy, th e ro ta tio n curve
w hich shows ro ta tio n velocity v rot versus g a lac to c en tric distance R w ould be K ep­
lerian as shown in Figure (2.3). O utside th e disc, th e ro tatio n curve should fall as
l
Vr G alaxies contain th in gas layers in th e g alactic plane w hich ex ten d beyond th e
lum inous disc.
T h e ro tatio n al velocity o f th is gas is m easured by doppler-shifts
in th e ato m ic sp ectra. T he m ost com m only used tracers are th e 2 1 cm spin-flip
tra n sitio n in ato m ic hydrogen and an em ission line in carbon m onoxide.
O u r solar system sits a t a galacto cen tric d ista n ce R sun « 8.5 kpc, near th e edge
of th e M ilky W ay’s lum inous disc. T h e m e asu re d ro ta tio n curve for th e M ilky W ay
galaxy is shown in Figure (2.4) [39]. T h ere is no K ep lerian fall-off o u tsid e o f th e solar
circle; v rot rem ains essentially co n stan t as fa r o u t as can be m easured. All m e asu re d
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
29
260
O 240
CO
3
220
-
Solar Radius
-
O
> 200
-
180-1
4
8
12
Radius (kPc)
16
F igure 2.4: Polynomial interpolation of the measured rotation curve of the Milky Way
Galaxy.
g alactic ro ta tio n curves rem ain flat well beyond th e radius o f th e lum inous disc.
A flat ro ta tio n cu rv e suggests a m ass co m p o n en t whose d en sity p scales as A j.
A d d itio n a l evidence from th e sta b ility o f sp iral a rm s points to th is co m p o n en t b ein g
sp h erically d is trib u te d . T h is d a rk co m p o n en t w hich surrounds th e g alactic disc is
called th e halo.
T h e local halo d en sity can be d e te rm in e d by co n stru ctin g m ass m odels o f th e
galax y w hich re p ro d u c e th e ro ta tio n curve. T h e different com ponents can be d e­
scribed by te rm s in th e M iyam oto a n d N agai P o te n tia l in ( R , z ) c o o rd in a tes[40].
(2.34)
W here M t- is th e m ass o f th e f-th c o m p o n en t, a,- is th e scale radius an d 6 t- is th e scale
thickness. T h e ro ta tio n velocity an d d e n sity a re given by
4 ttG
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
(2.35)
30
A m odel which m a tc h es th e M ilky W ay ro ta tio n cu rv e from ~ 1 0 pc to 20 kpc is
shown in F ig u re (2.5). T h is m odel co n tain s four co m p o n en ts, a com pact nucleus, a
c en tral bulge, a disc, a n d a halo. T h e m ass and scale facto rs for these com ponents
are su m m a riz ed in th e ta b le in F ig u re (2.5). T h e in fe rred local halo d en sity from
th is m odel is ~ 3 .4 x 10 -2 5 g /c m 3.
250 h
total
_ 200
^ disc
o
<u
3
halo
150
100 - !
> “
•'
50-!,
bulge
nucleus
04
0
10
20
30
40
50
Radius (kPc)
C o m p o n en t
N ucleus
B ulge
Disc
H alo
M (10 11 M 0 )
0.06
a (k p c)
b (kpc)
0 .0 0
0 .1 0
0 .0 0
1.60
3.92
6.13
15.70
0.18
O.SO
5.14
15.05
F igure 2.5: Calculated rotation curve for a four-component mass model of a galaxy.
T h e re a re m any variatio n s th a t also rep ro d u ce th e ro ta tio n curve, giving a va­
rie ty of e stim a te s for th e local halo density. G ates, G y u k , an d T urner a tte m p te d
to q u a n tify th e local halo d en sity by co n stru ctin g m illions o f m odels, co m p arin g
th e m to observations, a n d com piling th e d is trib u tio n for th e halo densities in vi­
able m odels[41]. W hile n o t a tru e likelihood analysis, th e resu ltin g d istrib u tio n of
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
31
halo densities resem bled one. T h e ir q u a n tita tiv e e stim a te s for th e local halo d en sity
are 5.3i|;o><10 -2 5 g /c m 3 ( ~ 300 M eV /cm 3) for a sp h erical halo, an d 9 .2 t3;f x 10 -2 5
g /c m 3 ( « 550 M e V /c m 3) for th e m ore realistic m odels w here th e halo is fla tte n e d
by ro ta tio n . T h e u n c ertain ties reflect th e full-w idth a t half-m axim um of th e d is tri­
butions.
Self-gravitating system s w hich are d issip ativ e will collapse in to discs. T h is is
th e case for baryonic m a tte r. All o f th e non-baryonic can d id ates are dissipationless,
so th ey would re m a in spherically d istrib u te d , w ith only m inor p e rtu rb a tio n s from
th e ir extrem ely w eak g ra v ita tio n a l in teractio n s. T h is is why th e y are reasonable
candidates to m ake up g alactic halos.
P ierre Sikivie has suggested th e possibility of s tru c tu re in halos co m prised of
dissipationless d a rk m atter[42].
It is ty p ically assu m ed th a t th e halo is ap p ro x i­
m ately isotherm al as a result o f a process called violent relax atio n . W hen th e d ark
m a tte r first falls in to th e p o te n tia l well o f th e galaxy, it has a very narrow velocity
dispersion. O ver tim e , g ra v ita tio n a l in teractio n s w ith inhom ogeneities, e.g. globu­
lar clusters etc., cause th e m a tte r to th erm alize, th a t is assum e th e c h arac te ristic
velocity dispersion o f th e halo. Sikivie argues th a t th e re should be flows in th e halo
associated w ith m a tte r th a t has only recen tly fallen in to o ur p o ten tial well. T h is
late-infall m a tte r w ould not have th erm alized a n d w ill m ain tain its in itia l velocity
dispersion.
N on-virialized flows produce stru c tu re s in th e halo, in p a rtic u la r caustics. C au s­
tics are locations in space w here th e d a rk m a tte r d en sity diverges. T h e d en sity
rem ains finite because th e tin y velocity dispersion becom es significant w hen th e
density is sufficiently high.
Tw o kinds of caustics a re created , inner a n d o u te r.
O u te r caustics are spherical an d occur a t th e tu rn a ro u n d radius o f a p a rtic u la r flow
of d ark m a tte r. In n e r caustics are rings lo cated a t th e radius of closest ap p ro ach ,
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d e term in e d by th e an g u lar m o m e n tu m o f th e particles in th e flow. T h e ch aracter­
istic scale for th e caustics sh o u ld b e th e p ro d u ct of th e age o f th e g alax y a n d th e
velocity of ro tatio n .
T h e caustic rings should a p p e a r as b um ps in th e ro ta tio n curve. Sikivie has
used a self-sim ilar infall m o d el to p re d ic t th e radii, an d claim s to see evidence for
s tru c tu re in m easured ro ta tio n curves[43]. So far, th e resu lts a re suggestive, bu t
inconclusive. T h e im p o rta n c e o f th is suggestion is th e possible en h an c e m en t o f th e
halo density a t o u r location.
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33
2.2.5
The Axion as a CDM candidate
H aving e sta b lish e d th e existen ce o f d a rk m a tte r, in p a rtic u la r non-baryonic d ark
m a tte r, th is sectio n describes axions in th e c o n tex t o f a c a n d id a te for th e cold d ark
m a tte r in o u r g alac tic halo. Scenarios for axion p ro d u c tio n in th e early u n iv erse are
described, follow ed by a discussion o f th e ir energy d is trib u tio n .
W hen th e te m p e ra tu re o f th e universe T was g re a te r th a n th e te m p e ra tu re of
th e P eccei-Q u in n sy m m e try b reak in g T p q , th e P Q sy m m e try was u n b ro k en an d
there w ere no axions. W h en th e te m p e ra tu re cools below T pq, P Q sy m m e try is
sp o ntaneously broken, a n d axions a re form ed as m assless G oldstone bosons. T h e 0
p a ra m e te r o f th e Q C D vacuum assum es a ran d o m value betw een 0 and 2 t t . Topo­
logical d efects a re c re ate d in th e form o f axionic strin g s. E ventually, th e universe
cools below th e Q u ark -H ad ro n tra n sitio n a t T = T q c d , w here th e P Q sy m m e try
is ex p licitly b ro k e n by in sta n to n effects, giving axions a sm all m ass. 0 begins to
oscillate a b o u t 0, solving th e stro n g -C P problem .
T he o scillations o f th e axion field ab o u t 0 = 0 is a co h eren t source of axions. T h e
abu n d an ce d e p en d s on th e in itia l value o f 0 , so th is scen ario o f axion p ro d u c tio n is
called vacu u m -m isalig n m en t p ro d u c tio n . T h e e x p e c te d size o f th e in itia l m isalign­
m ent dep en d s on th e tim in g o f in flatio n , b u t it is ty p ic a lly assum ed to b e th e rm s
average of a u n ifo rm d istrib u tio n o f all values fro m -tt to tt, 0 rTns =
W ith this
assu m p tio n th e a b u n d an c e o f axions Qa from v a cu u m -m isalig n m en t is[35]
n.A * = 0.85 x
f 1 2 ^ ') }
V m a J
\
AqcD
J
*
(2.36)
w here A q c d is th e Q C D scale facto r. Axions p ro d u c e d th is way w ith m a ~ 10 fj,eV
would d o m in a te th e C D M in th e universe.
T h e re a re tw o o th e r scenarios for axion p ro d u c tio n in th e early un iverse, b u t
this e x p e rim e n t is n o t sensitive to th e resu ltin g a x io n fro m e ith e r one. T h e first is
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34
th e rm a l p ro d u c tio n th ro u g h th e processes 7 + Q — Q + a and iV -f- k —>- N
a w here
Q is a q u a rk and N is a nucleon. T h e axion ab u n d an ce from th e rm a l p ro d u c tio n
is [35]
^
(2-37)
In o rd e r to be cosm ologically significant, th e m ass would have to be 0 (1 0 0 eV ).
T h e se axions have been ruled out on a stro p h y sic a l grounds, an d w ould have d ecayed
a lre ad y (ra~n <C t 0). T h e decay p ro d u c ts, specifically m onochrom atic p h o to n s, have
b een searched for unsuccessfully[44].
T h e th ird , and perhaps m ost co n tro v ersial scenario of axion p ro d u ctio n is th e
ra d ia tio n from axionic strings form ed by th e SSB o f
U(1)pq.
As
long as in flatio n
occurs a fte r th e SSB, a netw ork of axionic strin g s will fill th e universe. T h e n etw o rk
of strin g s will dissip ate energy by ra d ia tin g axions. A lot of th eo retical u n c e rta in ty
rem ain s (see e.g. [45, 46, 47, 48, 49]), b u t m o st predictions for m a are on th e o rd e r
o f 100 /.ieV, o u t of reach for th e c u rre n t m icrow ave-cavity detectors.
T h e e x p ected signal from axion to p h o to n conversion depends on th e en erg y
d istrib u tio n o f galactic halo axions. T h e ir to ta l energy is th e su m of th e ir rest m ass
m ac 2 a n d k in etic energy | m au2, w here v is th e ir velocity dispersion. If th e d a rk
m a tte r halo is therm alized, T u rn er calcu lates th e kinetic energy d istrib u tio n to b e
M axw ellian d istrib u te d , w ith a boost from th e m o tio n of th e e a rth w ith re sp ec t to
th e g a lac tic rest frame[50]. This b o o sted M axw ellian shape is show n in F ig u re (2.6)
for axions w ith m a = 4 /zeV an d v = 270 k m /se c , th e m easured value for o u r halo.
Qa = ( f ) 2 ~ 1 0 - 6.
If th e axion is discovered, careful m e asu re m e n ts o f this d istrib u tio n sh o u ld reveal
a n n u a l m o d u latio n s due to th e e a r th ’s m o tio n a ro u n d th e sun. T h e w id th w ould be
a m e asu re o f th e galactic halo v elocity dispersion.
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35
3
cs
S—
<L>
£
o
A E /E * 10
CL
E.'a
E-E.
F ig u re 2.6: Expected thermalized axion lineshape. E a = m ac2 is the rest mass of the
axion.
If Sikivie is co rrect, an d th ere is a significant p o rtio n of th e halo which is
no n -th erm alized , th e ir d istrib u tio n should be a series of very n arrow peaks w ith
Q a ~ 1 0 -IS [52]. T h is is im p o rta n t since peaks th is narrow could be searched for
m uch m ore sensitively th a n th e relatively b ro ad sp e c tru m from th erm alized axions.
T h e sp e c tru m o f these peaks an d th e ir an n u al m o d u latio n s would provide a w ealth
of in fo rm atio n on g alactic dynam ics.
T h e coherence len g th o f th e axion field is d e te rm in e d by th e de Broglie wave­
len g th Xq = h f p = h / ( m av). A ssum ing v is th e v elo city dispersion of th e halo an d
m a ~ 1 p e V , Ad = 0 ( 103 m ). Finally, if pa m pkaio-, th e n u m b er d en sity of axions
n a is a p p ro x im a te ly 1 0 13/c m 3.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
36
2.2.6
Allowed Axion Mass Range and Two-Plioton Cou­
pling
Since all values of th e P Q sy m m e try b re a k in g scale / pq solve th e stro n g -C P problem ,
m a is co m p letely a rb itra ry . It is e n tire ly possible for axions to solve th e stro n g -C P
p ro b lem w ith o u t being cosm ologically sig n ifican t. O n th e o th e r h a n d , th e a b u n d an c e
o f axions m u st be consisten t w ith cosm ological observations. If m a <§; 1 /ueV, th e n
th e ir a b u n d an c e from (2.36) is Clah 2
1.
If th is is to be co n sisten t w ith th e
ob serv atio n th a t o u r universe is flat, a n d fu rth erm o re th a t Qm ~ 0.3, th e n th e
H u b b le c o n sta n t w ould have to be m u ch larg er th a n 100 k m /s e c /M p c .
T h is is
stro n g ly c o n tra d ic te d by observational ev id en ce, so a lower b o u n d on axions from
th e vacuum m isalig n m en t p ro d u ctio n scen ario is roughly 1 /ueV.
A strophysics plays a n im p o rta n t ro le in placing an u p p e r lim it on th e axion
m ass. T h e rm a lly p ro d u ced axions could escap e from th e in terio r o f stars w ith o u t
in te ra c tin g .
T h is process would very effectively cool th e sta r, sh o rte n in g its life
considerably. T h e a m o u n t of energy ax io n s could carry away goes like th e ir m ass,
a n d th is fact is used to place an u p p e r lim it on m a. T h e m ost strin g e n t o f th ese
b o u n d s com es from Supernova 1987a[53]. N in eteen n eu trin o events w ere d e te c te d
fro m th e ev en t, a n u m b e r consistent w ith th e o re tic a l m odels w here n eu trin o s w ere
th e d o m in a n t source of cooling. A xions w ith m a £ 10- 3 eV w ould have c o m p eted
w ith n e u trin o s, sh o rten in g th e observed n e u trin o b u rst. T his is effectively an u p p e r
b o u n d on m a.
C osm ological and astrop h y sical o b serv atio n s leave a relatively sm all region for
th e axion m ass called th e axion w indow .
l f i e V < m a < 10m e V
(2.38)
T h e re is p le n ty o f th e o re tic a l an d e x p e rim e n ta l u n c e rta in ty in th ese c o n stra in ts, a n d
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
37
th e n u m b e rs q u o te d are so m ew h at co n serv ativ e. For d etails o f th e m a n y cosmolog­
ical, a stro p h y sic a l, an d p a rtic le physics c o n stra in ts on th e axion m ass see [54].
T h e p re d ic te d values o f th e tw o -p h o to n coupling c o n stan t ga^ f are relativ ely
in sen sitiv e to th e p a ra m e te rs o f th e v ariety o f m odels d escrib ed earlier. T his allows
th e p o ssib ility o f a definitive search in se n sitiv ity over m uch of th e axion window.
F ig u re (2.7) sum m arizes th e allow ed reg io n o f axion m ass a n d th e variety o f
m o d e l p re d ic tio n s for ga-n - T h e solid h o riz o n ta l lines a re p red ictio n s of som e of
th e ty p ic a l K SV Z an d D FSZ m o d els; th e tw o v e rtic al lines in d ic a te th e region o f
m ass c u rre n tly accessible to m icrow ave-cavity d e tec to rs. T h e c u rre n t sen sitiv ity is
show n b y th e d a sh e d line. T h e sh ad e d v ertic al lines in d ic a te th e lower m ass lim its
for th e different p ro d u ctio n scenarios. I f th e re is significant strin g -p ro d u ctio n , th e
re su ltin g axion w ould be o u t o f reach for c u rre n t ex p erim en ts. T h is is cu rren tly a
h o tly d e b a te d topic, but in C h a p te r F ive I w ill discuss ways to e x te n d th e reach of
m icrow ave-cavity d etecto rs to cover a significant fractio n o f th e m ass ran g e preferred
by s trin g p ro d u c tio n scenarios, as well as im p ro v e th e sen sitiv ity to cover all o f th e
m odels.
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38
-1 6
DFSZ
models
;vac
1
><o
00
KSVZ
models
-1 8
current
sensitivity
s
|
R.A. Battye et al.
M. Yamaguchi et al.
C. Hagmann et al.
\S N 1 9 8 7 a
-2 0
-3
ma (eV)
F ig u re 2.7: Sum m ary of the allowed axion mass window and predicted two-photon cou­
pling constants. The band between the KSVZ and DFSZ model predictions is the region
of mass currently accessible to microwave-cavity detectors. The references for the three
string constraints are [46, 47, 48]
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39
2.3
Cavity Detection of Axions
A variety o f techniques have been proposed for search in g for very light axions.
p a rtic u la rly th ro u g h th e axio n -p h o to n coupling; for a co m p reh en siv e review see [51].
As m o tiv ated in [51], only th e m icrow ave-cavity tech n iq u e o f Sikivie. p resum ing th a t
light axions d o m in a te th e g alactic halo d ark m a tte r has a realistic chance of finding
th e m . T his is th e ap p ro ach chosen in th is thesis.
T h e idea o f using m icrow ave cavities as axion d e te c to rs was first proposed by
P ie rre Sikivie in 1983[1]. T h e basic idea is to use th e m icrow ave cavity to d e tec t
th e photons from axion-to-p h o to n conversion th ro u g h th e tw o-photon in teractio n
described earlier. T h e coupling c o n stan t ga~n is in c re d ib ly sm all, b u t it is possible
to tak e ad v an tag e o f th e n a tu re of th e axion back g ro u n d .
As shown earlier, th e background axion n u m b e r d e n sity is very high. M oreover,
since axions w ere b o rn as a B ose-E instein condensate, th e y all have th e sam e energy
an d are coherent.
If it is possible to couple th e c o h eren t axion background to
a coherent p h o to n b ackgrou n d , th en th e re should be a sm all n u m b e r of photons
produced by ax io n -to -p h o to n conversion. A s ta tic m a g n e tic field was th e source of
coherent photo n s proposed by Sikivie. T h e process o f a x io n -to -p h o to n conversion
in a background m a g n etic field is shown in F igure ( 2 . 8 ).
By conservation of energy, th e photons pro d u ced in ax io n -p h o to n conversion will
have energy eq u al to th e re st m ass o f th e axion plus th e sm all corrections resu ltin g
from th e ir k in e tic energy discussed in th e last sectio n . Since th e axions all have
th e sam e energy, th e p h o to n s will be m o n o ch ro m atic w ith freq u en cy / given by
h f = m ac2( l + 0(1C T 6)).
A t th e lower en d of th e axion m ass window (1 - 100 /m V ), th e converted photons
are in th e m icrow ave region (1 GHz = 4.14 fieV ). For th is reaso n , m icrow ave cavities
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
40
9 a rr
a
►-
F igure 2.S: Feynman diagram for axion production through th e Prim akoff process.
a re used as th e photon, d e te c to r. Because th e axion b a ck g ro u n d is coherent, th e
converted p hotons a re co h eren t as well, and act as a classical field ex citin g m odes in
a resonant cavity. T h e axion m ass is unknow n, so th e freq u e n cy o f th e cavity m ust
b e tu n e d to m a tc h th e re so n a n t frequency w ith th e ax io n re st m ass.
In a Sikivie-type a x io n d e te c to r, a tu n a b le m icrow ave c av ity sits inside a large
solenoid m ag n et. T h e noise sp e c tru m from th e cav ity is m e a su re d a n d searched for a
p eak in d icativ e o f a x io n -to -p h o to n conversion. A ny c a n d id a te p e ak th a t is found in
th e background noise s p e c tru m m u st pass a very s trin g e n t te st; it m u st only ap p ear
w hen th e m ag n etic field is on, a n d its pow er m u st be field d e p e n d e n t.
2.3.1
Cylindrical Resonant Cavities
T h is section p resen ts th e im p o rta n t electro m ag n etic fo rm u las for th e m icrow ave
cavities used as axion d e te c to rs. T hey are ty p ically rig h t-c irc u la r cylinders w ith th e
g eo m etry show n in F ig u re (2.9).
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41
x
F ig u re 2.9: Right-circular Cavity Geometry.
T h e electric an d m ag n etic fields are fo u n d by solving th e H elm holtz eq u atio n
V 2$ + fc2$ = 0
(2.39)
w here th e vvavenumber k is given by
k 2 = fie J 2 - P 2
(2.40)
P is th e eigenvalue for th e tran sv erse (x ,y ) com ponents.
S olutions to (2.39) a re called m odes o f th e cavity. For a n e m p ty cavity, th e re
a re tw o ty p es of m odes w hich differ by th e b o u n d ary conditions satisfied by th e
fields. T ransverse M agnetic (T M ) m odes have B z = 0 an d satisfy th e b o u n d ary
co n d itio n E z (r = R ) = 0. T ransverse E lec tric (T E ) m odes have E z = 0 a n d satisfy
th e b o u n d a ry condition
|r_ fl = 0 .
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42
T h e field $ in (2.39) corresponds to E z for T M m odes an d to H z for T E m odes.
Since th e tw o -photon coupling is p ro p o rtio n al to E - B , an d th e solenoidal m agnetic
field only has a z-com ponent, axions will only couple to T M m odes. I will ignore
T E m odes for th e rest of th e section.
T h e lo n g itu d in al com ponent of th e electric field E z for T M m odes in a rightc irc u lar cavity found from th e solution o f (2.39) is[55]
E s (r, 6 , 2 , t) = E aJ m
e±im* cos
e *'"4
(2.41)
m ,n , and p are m ode indices (T M mnp m ode), a mn is th e n th root of J m(x). m is th e
n u m b e r of a z im u th a l nodes (m = 0 , 1 ,2 . . . ) , n is th e n u m b e r o f rad ial nodes (n =
1,2,3 . . . ) , an d p is th e n u m b er of longitudinal nodes (p = 0,1,2 . . . ) .
T h e frequency o f th e T M mnp m ode is
/-
= 2^ p V ^ + (t )2
<2-42>
B ecause th e axion couples m ost strongly to th e T M 0lo m ode, it is th e only one
used in th is ex p erim en t. T h e expressions for E z a n d /oio are
E z (r, 6 , 2 , t) = E 0Jo (2 .4 0 5 -^ ) eiu,t
(2.43)
fM M H A = i t i E
( , . 44)
W h en a m e ta l tu n in g rod is placed inside th e cavity, a th ird kind o f m ode w ith
E z = B z = 0 is possible.
T hese are called T E M m odes an d do n o t couple to
axions. T h e y are solutions to th e tw o-dim ensional e le c tro s ta tic eq u atio n s, and th e ir
frequency depends only on th e length o f th e cav ity cl.
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43
T h e q u a lity fa c to r o r Q o f a cav ity is a m e a su re o f th e pow er loss in th e walls
as well as th e sharpness of th e response to e x te rn a l ex citatio n s. It is defined as th e
ra tio of th e stored energy to th e pow er loss p e r cycle.
<3 = u j j r
(2.45)
w here cjq is th e resonant frequency. B ecause th e w alls are lossy, th e electric field
sjJtyt
.
oscillations are d a m p ed E z (t) = E z (0)e 2£? e tuJ° . T h e frequency s p e c tru m o f th e
energy is found from th e sq u are o f th e m a g n itu d e o f th e Fourier tra n sfo rm o f E z(t).
T h e result is th e com m on L o ren tzian reso n an t line sh ap e
I ^ M 2 = ------------ P° , - t t
(w - UJo)2 +
(2-46)
w here Pq is th e pow er on resonance. T h e b a n d w id th A / o f a cav ity m o d e is defined
as th e full-w idth a t h alf-m ax im u m of (2.46) w hich is equal to
Q is p ro p o rtio n al
to th e ra tio o f th e cav ity volum e V to th e volum e occupied by th e fields in th e walls
S 8 w here S is th e surface a re a a n d S is th e sk in -d e p th given by
4= v^7
( 2 -4 7 )
w here p is th e resistiv ity o f th e walls. Q for th e TM oio m ode in a rig h t-circu lar
cav ity is
<?= r r F
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( 2 ' 4S)
44
2.3.2
Expected Signal Power from Axion Conversion
T h e s ta rtin g p o in t for c a lc u la tin g th e signal pow er from a x io n -to -p h o to n conversion
is th e in te ra c tio n L agrang ian fo r th e tw o-photon in te ra c tio n w hich is eq u al to th e
in te g ra tio n o f th e L agrang ian d e n sity over th e cav ity volum e.
L a~f-y = ( C d V = f £fa r ,^ a(<)E • B dV
JV
(2.49)
JV
E x p a n d in g th e electric field o f th e cavity E into o rth o n o rm a l m odes
E =
£-\e.\
(2.50)
A
. y J ^ e ( x ) e x (x ) ■e .v (x ) d V = 5\.v
(2.51)
Ideally, th e m a g n etic field B is solenoiclal, so it can b e ex p ressed a s B = B 0b: (x )z .
w here B 0 is chosen such th a t
J v B - B d V = B 20 V = B l J b l{x) d V
(2.52)
So th e n o rm a liz a tio n for b~ is
± J v ftx)JV= 1
(2.53)
In s e rtin g th ese in to th e in te ra c tio n Lagrangian gives
— ( J d 1a( 0 B a ^ ^ E \a .\
A
(2.o4)
w here
O*a
C\ = J bz( x ) e \ ( x ) ■z d V
(2.55)
Focussing on a single m o d e A
L \ — ga^<pa( t) B 0E .\a x
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(2.56)
F ig u r e 2.10: Equivalent circuit for one mode of a sin gle-port cavity.
A single m ode of a one-p o rt cav ity can be m odeled u sing th e eq u iv alen t circuit
shown in F ig u re (2.10). T h e voltage source va(t) rep resen ts th e e x c ita tio n of th e
cavity by axion-to-photon conversion. T he transform er re p re se n ts th e variable cou­
pling of th e cavity to th e ex tern al am plifier represented by R
4.
T h e L agrangian for
th e circuit is given by[56]
1
q2
Ldr = ^ L q 2 - — -{- v aq
(2.57)
T h e resistor is represented by a dissipation function F r
Fr =
(2.58)
T h e E uler-L agrange e q u atio n for th e circuit is
d_ ( 8 L c ir \
dt ( dq J
dLdr
dFR _
dq + dq
T h e source te rm for v a{t) is qva(t) which is eq u al to L \. T h e
^
^
charge q isnorm alized
by e q u atin g th e sto red energy in th e cavity m ode U\ to th e sto re d energy
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in th e
46
circu it Uc-
Ux = \ ! v E a ' T > xdV = \ l v e(x)EA ' E a d V = \ E x V
U
°
2
(2.61)
C
(2.62)
q2 = E f C V => q = E x V C V
Q arn
/*\
“ *( t) = T
(2.60)
=
(2.63)
......
#
T aking th e F ourier T ransform
/
\
@a~cy B 0 CX\(pa ( ^ )
v - (uj) = —
7 W
(2.64)
~
T h e m easured pow er P ( uj) is given by
V2
Ra
(2.65)
T h e relation betw een VQ a n d va is found from th e eq u iv alen t circuit. T h e sim plified
resistance R is re la ted to th e cavity resistance R c by R = R c( 1 + /?) w here (3 is th e
coupling p a ra m e te r defined by
(3 =
n
2r a
( 2 .6 6 )
Rc
T h e coupling p a ra m e te r describes th e stre n g th of coupling betw een th e e x tern al
circu it and th e cavity. C ritical coupling is defined by th e condition th a t th e load
resistan ce equals th e cavity resistance, m eaning th a t ( 3 = 1 . (3 > 1 is referred to as
overcoupling a n d (3 < 1 is referred to as undercoupling, or w eak coupling.
T h e tran sfer function for th e cav ity m ode from th e sim plified circuit m odel is
Va
V l+ 8 2
1 + (3 u)
- 1
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(2.67)
47
Q u is th e u n lo a d ed Q o f th e circu it. V0 is sim p ly re la te d to Vi in th e sim plified
circuit
v- = T T f v '
(2 '6 8 )
So th e tran sfer fu n c tio n o f th e original c irc u it o n th e left o f F ig u re (2.10) is
V0
ua
0
1
1 + (3 \ / l
S2
(2.69)
T he m easured pow er is
pi , _
1
9 l ^ B 20 a \ ]±
^
( l + / 3 ) 2 l + 82 R a C V
T he to ta l pow er fro m axion-to-photon conversion Pa-*~t is found by in te g ra tin g E q u a ­
tion (2.70)
P ^ = J p (u ) ^
A ssum ing th a t Q a
(2.71)
Q u , 8 is essentially c o n sta n t over th e sm all frequency b a n d '
where <pa{w) is n on-zero and can be b ro u g h t o u tsid e o f th e in teg ral
=
B-Hr
I 2
1
ti-ryB Z al
(1+{3)2 1 + S 2 R aC V
j , w2 d u
I2 ^
(2-72)
T he axion d e n sity pa is related to th e ax io n field <ba by
/
°°
roc
\ M t ) \ 2 dt = m l
-O C
rim
|<paM | 2 ^
J
— CO
(2.73)
w /t
T h e second e q u a lity follows from P arsev al’s th e o re m . P lugging th is into E q u a tio n
(2.72) gives
p
=
P2
1
9a-ryB0a l pa
{ l + P ) 2 l + 52 R a C V m 2a
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^
}
48
a2
In tro d u cin g th e fo rm facto r C \ = y%, a n d rep lacin g th e circuit p aram eters R a
an d C w ith th e m e asu re d q u an tities j3 a n d Q u using R a C =
gives th e axion-
to -p h o to n conversion pow er in th e n o n -tra n sp a re n t e V /se c u nits.
1 c/
P .-w =
a~ ^
a~n(! a B 02 V u 0Q uC x
(1 +( 3 ) 2 l + 5 2 m 2a
reV
[se d
(2.75)
A ssum ing th a t p a = phaio ~ 0.45 G e V /c m 3, a n d K SV Z axions, th e dim ensionless
q u a n tity
7Tla
= 4.78 x 10 43. T h e q u a n tity B 2V is an energy, so th e conversion
from n a tu ra l u n its to cgs is sim ple by co m p arin g th e expressions
u [eV] = - B 2V
L
U [erg] = — B 2V
'bn
(2.76)
C o n v ertin g to w a tts, a n d norm alizing to ty p ic a l e x p e rim e n ta l p aram eters gives th e
e x p ec te d signal pow er for th e conversion o f K SV Z axions to photons in a m icrow ave
c av ity placed inside a solenoidal m agnetic field
/> _
=
3.32*10
(l^5o) (§)
(2 '77)
T h e form fa c to r C \ is used q u ite often a n d is given ex p licitly by
A
1 ( f v E ,\ • B d V ) 2
B l V f v e(x)\E>\* d V
T h e expression for C \ shows why th e TM oio is u sed alm o st exclusively in cav ity
d e te c to rs. T h e in te g ra l o f E • B is zero for all T E a n d T E M m odes, an d falls rap id ly
to zero for all T M m o d es except th e TM oio m ode.
T h e ex p ected pow er is 0 (1O -22 W ), a tin y pow er indeed. B eing able to d e tec t
th e se signals requires th e w orld’s m ost sen sitiv e m icrow ave receiver.
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49
2.3.3
Power Combining Multiple Cavities
From E q u a tio n (2.44), th e frequency of th e TMoio m o d e scales as R ~ l . T his m eans
th a t exploring h igher frequencies requires sm aller ra d ii cavities. T h e m agnet volum e,
m eanw hile, rem ain s th e sam e, and placing a single, sm aller cav ity into th e m agnet
w ould be inefficient.
O ne obvious solution is to p lace m u ltip le cavities into th e
m ag n et.
O p e ra tin g N in d ep en d e n t cavities would req u ire N sets o f cryogenic am plifiers
an d receiver electronics. O n to p of th a t, th e noise backgrounds a d d in q u a d ra tu re .
T his m eans th a t o p e ra tin g /V ind ep en d en t cavities re su lts in only a y /W im provem ent
in signal-to-noise ra tio (S N R ).
A n a lte rn a tiv e is to tak e advantage o f th e coherence of th e axion signal and
com bine th e o u tp u ts o f th e N cavities which are tu n e d together. T h is gives a factor
o f iV im provem ent in th e SN R ratio over th e single sm all cavity. This is possible
as long as th e axion signal is coherent over th e v o lu m e of all th e cavities. T h e de
B roglie w avelength is 10-1000 m , so this is easily th e case.
A sch em atic of a n N -p o rt power c o m b in e r/d iv id e r is shown in Figure (2.11).
T h e b o tto m p ictu re shows how N voltages w ith freq u en cy ui an d phase differences
4>i com bine. Unless th e phases are th e sam e, th e re will be interference term s an d
th e pow er o u tp u t will n o t b e th e sum of th e in p u t pow ers.
To u n d e rsta n d how th e signals from m u ltip le cav ities will com bine, it is necessary
to know th e a m p litu d e an d phase of th e cav ity tra n s fe r fu n ctio n for a p a rtic u la r
cav ity m ode.
T h is can be found from th e tw o -p o rt cavity equivalent circuit in
F ig u re (2.12). T his is id en tical to th e one-port m o d el o f th e previous section w ith
a n a d d itio n a l p o rt for coupling power into th e cavity.
T h e a m p litu d e o f th e tra n sfe r function |FT(u;)| fo r th e equivalent circuit o f Fig-
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50
V
-o —=eJmt
Vn
y ejtot o-o
v
Vn
V ej(“t+<>i) o
eiwt
o —
— (Vje-i't’i -i
VN
(-VNe->+<)
V Nei(“t+^ ) q
F ig u re 2.11: Schematic of an N -Port Power Com biner/D ivider.
ure (2.12) is th e sam e one fro m th e previous section w ith th e inclusion o f two
coupling c o n sta n ts th is tim e 0 i an d 0 2 0102
1
m “ )l = TTa = i + h + f o V T T P
(2.79)
w here 5 is given by
5=
Q u
1+01+02 w
£)'
i
(2.80)
T he p h a se o f th e tra n sfe r fu n ctio n ©(a;) is sim ply t a n -1 (^)- A p lo t of th e tra n sfe r
function for a m ode w ith f Q = 1 G H z a n d Q u = 100000 is show n in F ig u re (2.13).
T h e tra n s fe r fu n c tio n for pow er-com bining N cavities is given by
Veout
Vi,
0ju>t N
N
J<t>i
0102___________
(2.81)
i= 1 1 + 0 1 + 0 2 y f l + S ?
T he pow er is found by m u ltip ly in g (2.S I) by its com plex c o n ju g ate. T h e e x p erim en ­
tal v erification of th is fo rm u la is discussed in C h a p te r Four.
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51
I
•Rc
• O-
1:N ,
Figure 2.12: Equivalent circuit for one mode of a two-port cavity.
1 .0 -1
Amplitude
...... Phase
0.8 -
- 0.5
0.0
-
X
0.4-
--0.5
0 .2 -
-
-
1.0
—1.5
Frequency
F ig u re 2.13: Amplitude and phase of the cavity transfer function for one mode.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
2.3.4
Numerical Resonant Frequency and Form Factor Cal­
culation
T h e re so n an t frequency a n d form fa c to r for th e cylindrical cavity w ith different
tu n in g ro d configurations is calcu lated using a G auss-Seidel relax atio n m eth o d based
on a finite-difference a p p ro x im atio n to th e wave e q u atio n . T h e TMoio frequency is
in d e p e n d e n t of cavity len g th so th e calcu latio n s can be perform ed in two dim ensions.
M ost calcu latio n s a re perform ed on a 100 x 100 g rid , th e e rro r in th e calcu lated
frequency is less th a n 0.5% for th e e m p ty cavity case.
F ig u re (2.14) shows th e calcu lated E z for th e cav ity w ith a single dielectric
tu n in g ro d in th e c en te r of th e cavity, a n d slightly offset. T he d ielectric increases
th e field in its vicinity, a n d lowers th e reso n an t frequency below th a t of th e e m p ty
cavity. F ig u re (2.15) shows th e calcu lated E~ w ith a single m etal rod in two different
positions.
T h e field inside th e ro d is zero, a n d th e cav ity is effectively sm aller,
hence th e reso n an t frequency is raised from th e e m p ty cavity value. U nless two
rods are used, th e re is a range of frequencies aro u n d th e em p ty cav ity value th a t is
inaccessible.
F ig u re 2.14: Calculated longitudinal electric field for the TM oio mode of a circular cavity
with a single alumina tuning rod (er = 9.23).
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53
F ig u re 2.15: Calculated longitudinal electric field for the TM 010 mode of a circular cavity
with a single metal rod.
T h e calculated E z values are used to calcu late th e form fa c to r from E q u atio n
(2.78). T his is a volum e in teg ral, b u t E z is c o n sta n t along th e z direction. B z is
found from th e B iot-S avart law for a sim ple solenoid.
As m en tio n ed in C h a p te r
T h ree , th e a ctu a l m ag n et c u rre n t configuration is slightly m ore
com plicated, b u t
th e field is w ell-approxim ated by a sim ple solenoid.
F ig u re (2.16) shows th e calc u la te d form facto r versus freq u en cy for a d ielectric
rod, assum ed to be a lu m in a (er = 9.23). It is q u ite low because th e field is con­
c e n tra te d inside th e dielectric, w h ere C is reduced by er .
F ig u re (2.17) shows th e
c alcu lated form facto r for a m e ta l ro d . T h e form fa c to r
w ith a single m e ta l ro d
is h igher th a n w ith a n alu m in a ro d , b u t it is still d egraded from th e e m p ty cav ity
value of 0.69.
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54
0 .3 5
C = -338 + 13.21f -16.5ir + 6.92f3
0.3-
2
u
CC
U-H 0.25
s-i
o
&
0. 2 -
0.15
0.750 0.800 0.850 0.900 0.950 1.000 1.050 1.100
F requency (G H z)
F ig u re 2.16: Calculated form factor of the cavity with a single alum ina rod with r / R =
0.17.
s °-45l
■4-1
■*
U
cC
PP
J-i
o
pp
0.35
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
F r e q u e n c y (G H z)
F ig u re 2.17: Calculated form factor of th e cavity with a single metal rod with r / R = 0.26.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
2.3.5
Expected Scan Rate
In th is section, I e stim ate th e freq u en cy scan ra te using th e pow er derived earlier.
T h e s ta rtin g p o in t is th e definition o f th e signal-to-noise ra tio
SN R = —
0~N
(2.82)
w here Ps is th e signal power and crjv is th e size of th e noise flu ctu atio n s.
G au ssian noise, a n =
For
w here P n is th e average noise pow er an d N is th e
n u m b e r of sam ples. If th e sam ple tim e is r , th e b an d w id th B is r -1 , an d th e to ta l
m e a su re m e n t tim e t = N r . R ew ritin g (2.82) in term s o f P s ,P n ,B , a n d t
S iV R =
(2.83)
Pn
For th e rm a l noise, Pn = k g T sB w here k g is B o ltz m an n ’s c o n sta n t an d Ts is th e
sy ste m noise te m p e ra tu re , so th e S N R becom es
SM R = T“ ^ r \ / r r
fx/ Qj -s
(2.84)
y -D(i
B a is th e axion b an d w id th which is a p p ro x im a te ly 10"6/ a for th erm alized axions.
T h e ideal scan ra te for a desired S N R is one axion b a n d w id th in th e tim e t a
fou n d fro m (2.84)
df_B.
s - i :
^ _ (SNR)*k%T?B.
t' ~ —
—
(2 '8o>
Pa-y-/ in te rm s of th e non-fixed q u a n titie s, neglecting th e cav ity response (S = 0 ),
fro m (2.77) is
/ > _ = 1-32 x 1 0 -
W^
( ^
)
(£ )
(2.86)
P lu g g in g (2.S6) in to (2.85) gives th e sca n r a te in term s o f th e e x p erim en tal p a ra m ­
e te rs
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56
dt
-
28 2
^
02
( - ± - ) 2 ( ± U i) 2( ^ - ) 2
V T. J VI G H z )
sec (1 + / 3 )4 K S N R j
(_ 3 -_ y
V160000/ \ Q . o /
(2.87)
'
’
T h e a c tu a l scan ra te is b e tte r since th e lo ad ed cav ity b an d w id th B c > B a, each
step covers
J 2- = (1 + 0)^0- bins. T h u s, th e e stim a te d scan ra te for th e microwave-
cav ity axion d e tec to r is
d£
dt
_
~
MHz
02
f 4 \ 2 /4 .5 A V /
f0 \ 2
° day (1 + 0 ) 3 \ S N R J V T s ) \ l G H z )
( l^ o ) (& ) (§ )*
<«>
Tw o im p o rta n t special cases are c ritic a l coupling (,8 = 1 )
MHz
d
L
dt
'
day
4 \ 2 M .5 A' \ 2 f
f0 V
K S N R J V Ts J \ 1 G H z ) V160000 J
f
2
( 9 ± ) ( Q .X
VlO6/ VO.57
(2.89)
an d tw ice overcoupling (0 = 2) w hich gives th e th eo retical m ax im u m scan rate
ld
~
dt
~
9 3
Mil ( 4 Y(4-5 KY( fo Y( Q
*]
day
\SNRJ
V Ts
J
\lG H z)
U 60000/
(&) ( £ ) 2
T h e re are m an y com plications th a t com e from overcoupling th e cavities, so th e
e x p erim en t is o p erated w ith th e c av ities critic a lly coupled. T h e im p o rta n t result is
th e q u a lita tiv e behavior of th e scan ra te
if
it
flo p p y *
Ti
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1 '
57
T h is expression shows w hy so m uch effort is p u t in to lowering th e sy stem n o ise
te m p e ra tu re .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
58
Chapter 3
Apparatus
T his c h a p te r describes th e h a rd w are re a liza tio n of th e S ikivie-type axion d e te c to r
detailed in th e last ch ap ter. F ig u re (3.1) is a d iag ram of th e m ag n et c ry o sta t an d
d e tec to r in se rt. In this search , four co p p er-p la ted stainless steel cav ities sit in a
s ta tic 8.0 T m a g n etic field su p p lied by a su p erco n d u ctin g solenoid. T h e reso n an t
frequencies o f th e cavities a re tu n e d by m ov in g ceram ic or m e ta l posts. Pow er from
th e ind iv id u al cavities is co upled by sm all electric field probes to a pow er co m b in er
w here it is com bined in p hase a n d fed to th e first cryogenic am plifier. T h is h a rd w are
is described in th e following fo u r sectio n s.
A fter fu rth e r room te m p e ra tu re am p lificatio n , th e signal from th e cavities is fed
into a su p er-h etero d y n e receiver. In th is receiver, th e pow er from a sm all b a n d
centered a ro u n d th e cavity freq u en cy is m ixed down to audio freq u en cy a n d its
pow er s p e c tru m is m easured using a F a st F o u rier T ransform (F F T ) an aly zer. T h ese
pow er s p e c tra a re analyzed for signals o f ax io n -to -p h o to n conversion. D etails o f th e
room te m p e ra tu re electronics a re in S ections 5 an d 6 .
T h e rest of th e c h ap te r describes th e rem ain in g in s tru m e n ta tio n o f th e d e te c to r
and th e c o m p u te r system w hich ties e v ery th in g together.
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To big
Helium
vacuum
pump
Cryostat vessel
Liquid Helium reservoir for
microwave cavities
Liquid Helium reservoir for
magnet
Amplifiers, cooled to 1. 3K
Power Combiner
Frequency tuning mechanism
Coupling mechanism
Copper plated Stainless Steel
Cavity
Superconducting magnet coil,
60cm ID, 110cm long
7.5 Tesla field at 4.2 Kelvin
Weight: 6 tons.
Figure 3.1: The U.S. Axion search detector.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
60
3.1
The Magnet
T h is section describes the co n stru ctio n , o p eratio n an d p erfo rm an ce o f th e m agnet
used in th is experim ent.
3.1.1
Construction
T h e m a g n et used in this axion search was b u ilt by W ang N M R of L iv erm o re, CA[57].
It is a su p erco n d u ctin g solenoid, w ound w ith copper-clad n io b iu m -tita n iu m (N bT i)
w ire. T h e len g th of th e coil is 112 cm w ith a bore radius o f 26.5 cm . For stab ility
considerations, it is a high-inductance design with L m = 533.5677. T h e six-ton coil
is com pletely im m ersed in liquid h elium (LHe) which m ain tain s its te m p e ra tu re at
4.2 K.
T h e coil was designed to m axim ize th e sto red m agnetic energy ( B 2 V ) w here V is
th e cavity volum e. T he prim ary con sid eratio n in m ost m agnets is field hom ogeneity
w hich leads to an increased n u m b e r o f w indings a t the ends o f th e coil.
T hese
w indings c re a te a m ore uniform field along th e axis of th e coil, b u t m u ch o f th e ir
m ag n etic energy is outside th e solenoid. T h is m agnet has a u n ifo rm d istrib u tio n
o f c u rre n t along th e coil. A lth o u g h th e a c tu a l w ire configuration is com plex, the
field of th e coil is well ap p ro x im ated by th a t of a sim ple solenoid w ith a uniform
c u rre n t d istrib u tio n . Figure (3.2) shows th e m ag n etic field o f a solenoid calcu lated
by n u m erical in tegration of th e B io t-S av art law, norm alized to B 0, th e field on axis
in th e m id d le o f th e solenoid.
T h e m a g n et dew ar contains a n in su la tin g vacuum , and th re e layers o f rad iativ e
shielding betw een th e coil and th e bore, inclu d in g a th erm al shield fixed a t 77 K by
liquid n itro g en (LN2). This ’w arm -b o re’ design allows th e in sert to b e p laced into
th e m a g n et w arm w ithout w arm ing th e coil or boiling excess h eliu m . K eep in g th e
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
61
F igure 3.2: Longitudinal component of the magnetic field of a solenoidal magnet calcu­
lated using the Biot-Savart Law.
coil cold typically requires 60 liters of LHe an d 10 lite rs o f LN2 p e r day.
T h e final im p o rta n t asp ect o f th e m agnet design is th e fact th a t it can n o t o p erate
in p e rsiste n t m ode.
A p e rsiste n t m ode su p erco n d u c tin g m a g n et has a com plete
su p erco n d u ctin g p a th once it is fully charged, lead in g to very stab le o peration.
In ste ad , th is m ag n et requires a p erm an en tly co n n ected pow er supply, an d th e leads
betw een th e coil a n d th is su p p ly are resistive. T h e lead s from th e coil to th e top of
th e m a g n e t c ry o stat a re m ad e from several layers o f c o p p e r foil, an d o p tim ized for
vapor cooling w hen placed in th e exhaust stre a m o f cold h eliu m gas . C ooling th e
copper in th is way lowers its resistance. T he leads fro m th e pow er supply to th e top
of th e c ry o sta t a re m a d e from s tra n d e d copper w elding cable. T h e a ctu a l values of
these le ad resistances w ill b e discussed la te r in th e sec tio n on perform ance.
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62
Software
PID
Loop
Voltmeter
F igure 3.3: C ircuit diagram for regulating the m agnetic field.
3.1.2
Operation
F igure (3.3) shows a sc h e m a tic o f th e m agnet circuit in clu d in g th e im p lem en tatio n
of c o m p u ter control a n d feed b ack stab ilizatio n . A P ro p o rtio n al-In teg ral-D eriv ativ e
(P ID ) feedback loop is im p le m e n te d in softw are to co n tro l ra m p in g th e m agnetic
field up and dow n an d to sta b iliz e th e field against d rifts in le ad resistance. T h e
c u rre n t in th e coil is d e te rm in e d by m easuring th e vo ltag e across a high-precision
sh u n t resistor ( R s in F ig u re (3.3)) using an HP3458A Volt m e te r [58]. R s = 200fiQ so
I m = 5000 x Vs. T h e c o m p u te r reads this voltage an d sets th e H P59501 D igital-toA nalog C onverter (D A C ) v o lta g e to th e ap p ro p riate value. T h e D A C voltage d irectly
controls th e o u tp u t vo ltag e o f th e m ag n et power supply. In th is fashion, th e voltage
Vps is continuously a d ju ste d to e ith e r m ain tain a fixed ch arg in g o r discharging ra te ,
o r to hold th e field s te a d y a t th e desired set-p o in t. T h e n o rm a l o p eratin g p o in t
o f th e m agnet d u ring sin g le-cav ity o p eratio n was I m = 224 A corresponding to
B 0 = 7.612 T . For fo u r-ca v ity o p eratio n , this was in creased to I m = 235 A w ith
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
63
B 0 = S.O T .
F rom F igure (3.3), w hen c u rre n t is flowing th ro u g h th e circ u it, th e re is a voltage
drop given by :
V d ro p
=
I m
R s
+
2 I m R c
+
2 I
t t iR
( 3 •1 )
l
w here R c is th e resistan ce o f th e cables b etw een th e pow er su p p ly a n d th e m ag n et
leads, a n d
in
R
is th e resistan ce of th e leads. T h e feedback co m p en sates for changes
l
an d also adds a p ositiv e (negative) voltage for ch arg in g (discharging).
V d ro p
D uring charging Vps = Vdrop + Vcharge w here Vcharge is th e desired charging volt­
age. T h e feedback com pensates for th e change in
V d ro p
as
I m
increases m ain tain in g
Vm = Vcharge ■ T h e ra te of change o f I m is given by:
d im .
j
V n
J7~ —
= ~r
a t
V'-*—')
L m
Vcharge is typically 1 - 2 volts, so I m ~ 10 — 15A / h r . It tak es one day to ram p
th e m ag n et to full field, th is is no p roblem since th e m a g n et is u sually kept a t full
field for m o n th s a t a tim e .
F or stab le o p e ra tio n V ps = Vdr0p ± A F w here A V is a sm all co rrectio n voltage
for sm all errors betw een th e m easu red c u rre n t an d th e s e tp o in t. T h e m in im u m size
o f A V is d e term in e d by th e resolution of th e D AC, w hich will be discussed in th e
n e x t section.
Finally, for discharging th e m ag n et Vps = Vdr0p — Vuscharge■ Feedback again
m a in ta in s
V m
—
— V u s c h a r g e
b u t th e free-w heel diode show n in F ig u re (3.3) lim its
Vdischarge < 0.7V . T his gives th e m ax im u m discharging r a te —I m < 5A / h r . R am p ­
ing dow n th e m a g n et from full field requires alm o st tw o days.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
64
3.1.3
Performance
All of th e voltages show n in Figure (3.3) a re logged a n d can be used to assess th e
perform ance of th is m a g n e t. O u r p rim ary concern is field s ta b ility to m in im ize eddy
current h e atin g in th e cavity. A changing m ag n etic field will in d u ce an e m f in th e
copper wall o f a c a v ity causing o hm ic h eatin g . T h is is a n ex tre m ely im p o rta n t issue
when considering going to lower te m p e ra tu re s.
As s ta te d in th e previous section, th e sm allest c o rrectio n voltage AV" w hich can
be applied to th e coil is d eterm in ed by th e reso lu tio n o f th e D AC. T h e HP59501
DAC used p re sen tly has a resolution o f 10 m V a n d a ra n g e o f 10 V . T h e re is a 1/5
voltage div id er b etw een th e o u tp u t of th e DAC a n d th e in p u t to th e pow er supply
control, so th e reso lu tio n an d range becom e 2 m V a n d 2V respectively. A control
voltage of 0 - 5 V corresponds to an o u tp u t voltage V ps of 0 - 10 V . T h is m eans
th a t th e sm allest AV" is 4 m V an d th e range of V ps is 0 - 4 V.
A conservative e s tim a te is th a t th e reg u latio n o f th e field takes place every th irty
seconds. T h e change in cu rren t in th is tim e is
4772V"
A Im = ^ 3 ^ ( 3 0 s e c ) = 2 3 x 10
A
(3-3)
For norm al o p e ra tio n ( I m = 224A)
A Im
. -_ 6
= 10 “ 6
(3.4)
T his m eans th a t th e field should be reg u lated to 1 p p m . From d a ta ta k e n over a
random ly chosen five-day period d u rin g single-cavity o p e ra tio n , B Q = 7.616 ± ( 1 .8 x
10- 5 ) T , an d Do = 2.4 x 10- 6 . T h is im plies th a t th e a c tu a l reg u latio n is o n ly good
to 2.4 p p m , a n d th a t th e DAC reso lu tio n is n o t th e lim itin g factor. T h e reason for
th e discrepancy is show n in Figure (3.4), a plot o f B 0 over th e 5 d ay p e rio d which
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
65
cleaxly shows a dally m o d u la tio n . T h e source o f th is m o d u la tio n is th e v ariatio n o f
R c w ith te m p e ra tu re as show n in F igure (3.5) a p lo t o f th e cab le resistance versus
th e te m p e ra tu re in th e e x p erim en t hall. T h e v alu e o f Rj_, fro m th e d a ta is lOO^zfi
w ith no significant v ariatio n s, except for s h o rt p e rio d s d u rin g LHe fills because of
increased gas flow p a st th e leads.
1 . 0 0 10'5
5.00 10-6
‘■••V
"-1
* * .* f ! ** *
o
® 0.0010°
o
CQ
-
1
: A v>
■j
_ 1
•
-
-
i;
m i '
_.
-5.00 10"6
-
1.00 10-5
0.00
1.00
2.00
3.00
4.00
5.00
Time (days)
F igure 3.4: Daily modulation o f th e magnetic field.
T his d a ily m o d u la tio n is very slow a n d so it does n o t cause a problem as fa r
as e d d y c u rre n t h e a tin g is concerned. P la n s fo r o p e ra tin g a t lower te m p e ra tu re ,
how ever, re q u ire m a g n etic field regulation g o o d to 0.1 p p m o r b e tte r.
T h is can
easily b e achieved w ith a h ig h e r resolution D A C , b u t th is v a ria tio n in R c increases
th e re q u ire d d y n a m ic range, w ith a co rresp o n d in g loss o f reso lu tio n .
T h e b e st so lu tio n to th is problem is tw o-fold. F ir s t, p lace th e cables in a m o re
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
66
2.98- £
JZ
O
cn
O
-
2.96
2.94
t h
0£l
Al
2
ffi
2.92
2.9-.
U
R [10'3Ohm] = 2.811 + 0.006 T[C]
2.S6-
2.84--5.00
10.00
15.00
20.00
25.00
30.00
Shed T (C)
F ig u re 3.5: Resistance of the magnet supply cables versus tem perature in the experiment
hall.
te m p e ra tu re stab le en v iro n m en t to lower th e resistan ce variations. Second, em ploy
tw o D A Cs in th e regulatio n . O ne DAC has low er resolution and higher dy n am ic
ran g e to tra c k large, slow variations. T h e second DAC should have high resolution
an d a lim ite d dynam ic range an d be used to fine tu n e th e field. This co m b in atio n
should easily give an o rd e r of m a g n itu d e im provem ent in field stability.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
67
3.2
Insert Cryogenics
O ne o f th e tw o significant sources o f b ack g ro u n d noise in th is ex p erim en t is th e
Jo h n so n , o r th e rm a l noise from th e cavity. T h is c o n trib u tio n is reduced by cooling
th e c av ities to 1.4 I< using superfluid 4 He.
Vacuum Pump
LHe @ 4.2 K
Heat Exchanger
Valve
Cavity Space
LHe @ 1.4 K
F ig u r e 3.6: B asic co m p o n en ts o f a 1.4 K L am b da-R efrigerator.
A sch e m a tic of th e cooling system is g iven in F ig u re (3.6). LHe flows from a
reservoir a t 4.2 K , through a sm all capillary, co n tro lled by a valve on to p o f th e
cavity. T h e LHe is precooled by th e e x h au st as it passes th ro u g h a h eat exchanger.
O nce in sid e th e cavity space, it form s a p o o l a t th e b o tto m of th e vacuum can
w hich is p u m p e d on by a large R oots p u m p . T h e p ressure inside th e cav ity space
is ty p ic a lly 1 to rr which corresponds to a te m p e ra tu re of 1.4 K for th e LHe. Below
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
68
th e A-point (2 .IS K ), superfluid 4H e is fo rm ed w hich is very effective a t eq u alizin g
th e te m p e ra tu re everyw here inside th e c av ity volum e. At ab o u t 1.2 Iv, th e eq u ilib ­
riu m vapor p re ssu re of 4He d ro p s to zero, so no am o u n t of p u m p in g w ill low er th e
te m p e ra tu re fu rth e r. T he co n su m p tio n o f LHe for th e insert cryogenics is a b o u t 10
lite rs /d a y .
C ooling th e cav ity fu rth er w ould re q u ire e ith e r a 3He sy stem o r a d ilu tio n re­
frig erato r, b o th expensive item s. T h is is o n ly w orthw hile w hen th e c a v ity is th e
d o m in a n t source o f noise. As w ill b e discussed la te r, th e co n trib u tio n o f th e cryo­
genic am plifiers is roughly 3 K , so cooling th e c av ity to 1.4 Iv is a d e q u a te .
3.3
Cavities and Tuning Rods
T h e rig h t-c irc u la r m icrow ave cav ities w ith d ia m e te r ~ 10 cm and le n g th 1 m w ere
c o n stru c te d fro m 304 stainless ste e l a n d su b seq u en tly copper-plated.
T h is offers
tw o ad v an tag es over th e tra d itio n a l m e th o d o f m ak in g high-Q cavities o u t o f solid
h ig h -p u rity copper. F irst, stain less steel is m ore rugged and easier to m a ch in e to
tig h t to leran ces th a n copper. M ore im p o rta n tly , th e microwave p ro p e rtie s (i.e. Q)
d e p en d only on th e surface p ro p erties, w hile th e edd y -cu rren t h e atin g fro m a v ary in g
m a g n e tic field d ep en d s on p ro p erties o f th e b u lk . T h e low resistiv ity of c o p p e r w hich
gives rise to th e high Q, also c o n trib u te s to h ig h er eddy cu rren t h e a tin g th a n in
h ig h er re sistiv ity m aterials such as stain less steel. In th e unlikely e vent o f a m a g n e t
q u en ch , th e com pressive force on a solid co p p er c av ity is m uch g re a ter th a n th a t for
a stain less ste e l cavity. P la tin g several sk in -d e p th s o f copper on th e in sid e o f th e
stain less-steel cavities gives th e m h ig h Q w ith o u t excessive ed d y -cu rren t h e a tin g .
To facilitate plating and allow the interchange o f tuning rods, the cavities are
m ade in three pieces, the cylindrical wall and two end-caps. In the T M 0io m ode,
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
69
c u rre n t flows across th e jo in t betw een th e e n d ca p s an d th e cylinder, so th is m u st
be a very tig h t seal. F igure (3.7) shows a cro ss-sectio n of th e knife-edge seal used
to secure th e end-caps. A to ta l of 40 b o lts closely sp aced aro u n d th e circum ference
press th e e n d -c ap in to a knife-edge m a ch in ed in to th e cavity wall.
T h ese seals
have w orked very well, th e re has been no n o tic e a b le d eg rad atio n in Q a fte r several
openings o f th e cavities.
Cavity Cylinder
Pressure Plate
(Not Threaded)
Compression Ring
(Threaded)
Cavity Top Plate
Knife Edge Seal
F ig u re 3.7: Cross-section of the knife-edge seals used for the cavity end-plates.
W ith in a g e o m etric al factor, th e Q o f a c a v ity is th e ratio of th e volum e V to
th e p ro d u c t o f th e surface area S tim es th e sk in d e p th S.
Q ocJ s
F rom E q u a tio n (2.47), 6 oc y/p, w here p is th e re sistiv ity of th e walls.
(3 '5)
T h is is
im p o rta n t, b e ca u se m a n y copper alloys w ith slig h tly higher resistiv ity th a n p u re
copper h a v e s m o o th e r surface finishes w hen p la te d . T h e sm aller surface a re a m ore
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
70
th a n com pensates for th e higher resistance.
For th is reason, we use an organic
co p p er alloy called U B A C copper for plating[59]. A fte r th e cavities were plated,
th e y were vacuum baked a t 400° C for four hours an d slow ly cooled. This process
n o t only rem oves a lot of im p u rities from th e copper, it also enlarges th e average
dom ain size, decreasing th e resistivity.
T h e surface resistan ce o f th e co p p er an d th e p la tin g thickness are determ in ed
w ith a transm ission m easu rem en t o f th e T M 0io m ode a t room te m p e ra tu re . T h e
setu p for transm ission m easu rem en ts is described in d e ta il la te r in th is ch ap ter, th e
resu lts are shown here in F ig u re (3.S). T h e value o f th e u n lo ad ed Q from th e weakly
coupled transm ission m easu rem en t was 44400.
-10 H
W eak C oupling
-12
-14
i0
Critical Coupling
-16
£
ZJ -18
1
-20
C
C3
-24
1159.8
1159.9
1160.0
Frequency (MHz)
1160.1
1160.2
Figure 3.S: Power transmission measurement for an empty cavity.
To m easure th e u n lo ad ed Q of a cav ity it m u st b e v ery w eakly coupled. This
can lead to weak signals, w hich m ay drop below th e noise floor, especially a t room
te m p e ra tu re . W hen th e cav ity is critically coupled, th e m easu red Q is one-half th e
unloaded Q and th e signal is m uch stro n g er. C ritical coupling can b e d eterm in ed by
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
71
0-1
03
3
0
1
-10
*
-20
CU
■a
CJ
U
o -15
c:
o
>
■3 -25
GS
-30 —1
1159.90
1160.00
Frequency (MHz)
1160.10
F ig u re 3.9: Power reflection measurement for the m ajor port on an em pty cavity.
perform ing a reflection te st o n th e m a jo r p o rt. At c ritic a l coupling, th e re is m in im a l
pow er reflected from th e cav ity on resonance. T h e re su lt of th is te s t on th e e m p ty
cav ity is show n in F ig u re (3-9), an d th e tran sm issio n curve for c ritic a l coupling is
also in F igure (3.8). T h e critically coupled Q was 22S00, which im p lies a n u n loaded
Q of 45600, in reasonable agreem ent w ith th e d irect m e asu re m e n t. U sing E q u a tio n s
(2.48 and 2.47), th e re sistiv ity of th e co p p er is 1.72 x lO ~ 8 flm a t 300 K.
E q u a tio n s (2.48 a n d 2.47) b o th assum e th a t betw een collisions, th e electro n s
in th e m e ta l tra v e l in sp a tia lly c o n sta n t electric fields. T h is is d e sc rib e d b y th e
co n dition 8
/, w here I is th e electro n m ean -free-p ath . In no b le m e ta ls such as
copper, I becom es very large a t low te m p e ra tu re s an d th e classical th e o ry breaks
down.
W h en £
/, th e local re la tio n betw een th e c u rre n t d en sity a n d e lec tric field
J = crE is not valid.
T h is is called th e anom alous skin effect.
In th is regim e,
th e effective skin d e p th o f th e m e ta l is in d ep en d en t o f th e re sistiv ity a n d varies as
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72
5 oc / - ^[60].
B ecause th e re sistiv ity o f co p p er decreases w ith te m p e ra tu re , th e classical th eo ry
p re d ic ts t h a t th e Q o f th e cavities sh o u ld in crease as well.
Below a b o u t 50 Iv,
c o p p er is in th e anom alous skin effect reg im e a n d th e Q becom es in d ep en d en t of
te m p e ra tu re . T h e anom alous Q is ro u g h ly a fa c to r o f five g re a te r th a n th e ro o m
te m p e ra tu re value.
T h e p la tin g thickness is sim p ly d e te rm in e d fro m th e reso n an t freq u en cy o f th e
w eakly co u p led cavity. T h e m e asu re d freq u en cy o f th e TMoio m ode was 1.16 G H z,
im p ly in g a c a v ity radius of 9.900 cm u sin g E q u a tio n (2.44).
T h e rad iu s on th e
m achine draw ings was specified as 9.925 cm w ith a to leran ce o f a p p ro x im a te ly 1
m il. T h e difference betw een th e tw o is th e th ick n ess o f th e co p p er p la tin g , roughly
10
m il as desired.
F ig u re 3.10: Geometry of a cavity w ith a single tuning rod.
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T w o ty p es of tu n in g rods are used to a d ju st th e reso n an t frequency o f th e cavities.
D ielectric rods m a d e from solid alu m in a lower th e frequency from th e e m p ty cavity
value. M eta l rods, m ad e from cap p ed hollow co p p er tu b e s which are p lated and
a n n ealed w ith th e sam e process as th e cavities, raise th e cavity frequency. T he
cylin d rical rods, w hich ru n th e full len g th of th e cavity, are m oved fro m near th e
wall to n e a r th e c en te r by tu rn in g a sh aft a t the en d o f sm all horizontal b ars as shown
in F ig u re (3.10). T h e shafts are tu rn e d using piezoelectric a c tu a to rs described in
th e n e x t section.
T h e re a re two p o rts on each c av ity for inserting and e x tra c tin g pow er. Linear
a n ten n a s form ed by rem oving th e o u te r copper-cladding from RG-402 sem i-rigid
coaxial cable a re used as electric-field probes. T h e coupling s tre n g th (j3) for each
p o rt is controlled by changing th e in sertio n d e p th of th e an ten n as.
T h e m inor
p o rt is fixed w ith a very sh o rt a n te n n a (/3 <C 1 ) an d is used for in sertin g power
for tra n sm issio n m easu rem en ts. T h e coupling of th e m a jo r p o rt is v ariable from
stro n g overcoupling (j3
1)
to weak coupling and is connected to th e com biner and
cryogenic am plifier. T h e insertio n d e p th o f th e m a jo r p o rt a n te n n a is also controlled
by piezoelectric a c tu a to rs , an d in n o rm a l o p eratio n is a d ju sted to m a in ta in critical
coupling (/3 = 1 ).
F ig u re (3.11) shows a m easu rem en t o f the m ode stru c tu re for a cav ity w ith a
single a lu m in a ro d as th e ro d was m oved from n e ar th e wall to th e c en te r of th e
cavity. T M m odes all shift to g e th e r as th e tu n in g ro d is m oved. T h e m odes which
rem ain flat a re th e T E m odes w hose frequency is in d ep en d en t of th e tu n in g rod
position.
T h e dielectric co n stan t o f th e alu m in a was d eterm in ed by fittin g th e
tu n in g curve of th e TMoio m ode w ith calculations o f th e relax atio n code using th e
know n g e o m etry of th e cavity. T h e only free p a ra m e te r was th e d ielectric constant
of th e ro d . T h e resu lts of th e best-fit (er = 9.23) is shown in Figure (3.12).
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74
l-2 i
1.1
0.8
TM
0 .7
Tt
0
0 (rad)
Figure 3.11: Mode m ap of the cavity with a single alumina rod with r / R = 0.17.
A m ore com p licated m o d e m a p for a single m etal rod is shown in F igure (3.13).
T here are two reasons for th e co m p licated stru c tu re , a higher d en sity o f T E m odes
a n d the ap p earan ce o f T E M m odes. T h e high d ensity o f T E m odes is a consequence
o f th e higher frequency, as well as th e asp ect ratio o f th e cavity. T E M m odes ap p ear
w ith th e in tro d u ctio n o f co n d u cto r inside th e cavity a n d are spaced every 150 MHz
in a 1 m long cavity.
Figures (3.11) an d (3.13) b o th show exam ples o f m ode crossings, regions w here
th e frequency of a T M m o d e is d eg en erate w ith e ith e r a T E o r T E M m ode. A
typical m ode crossing region is shown in Figure (3.14).
If th e re were cylindrical
sym m etry, th e tw o m odes w ould cross, b u t th e sy m m e try is broken by th e tu n in g
rod. Sim ilar p h en o m en a a re com m on in deg en erate p e rtu rb a tio n theory, including
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75
° C a lc u la te d
— M ea su r e d
— 0.95-
N
X
S-
0.9-.
^
0.85-■
>v
o
c
cr
o
£
0 .8 0 .7 5 -J—
1.0
1.6
2.1
2.7
3.2
T u n in g R o d A ngle (rad )
F ig u r e 3 .1 2 : M ea su red freq uency o f th e T M oio m od e v ersu s tu n in g rod an g le. T h e
p o in ts sh ow n are ca lcu la ted u sin g th e relaxation cod e w ith th e know n ca v ity g e o m e tr y for
a d ielectric c o n sta n t o f 9 .2 3 .
th e ap p ea ra n ce o f b a n d gaps in sem iconductors. O rd in arily , th e stra ig h t a n te n n a
w ould n o t couple to e ith e r a T E or T E M m ode, b u t in th e m o d e crossing region
(/2
> f > f i in F ig u re (3.14)), th e tw o m odes m ix an d each has a T M co m p o n en t.
T h e inaccessible regions are rescanned w ith th e cavities filled w ith LHe, sh iftin g th e
m ode crossing freq u en cy dow n roughly 3%.
F igure (3.15) shows th e m easu red un lo ad ed Q o f a cav ity w ith a single alu m in a
ro d a t a te m p e ra tu re o f 4.2 K . Regions w here Q uni rises ra p id ly are m ode crossings
w ith th e in d ic a te d T E m odes. T h ere a re two different values a t each freq u en cy
w hich d ep en d on w h e th e r th e tu n in g ro d passes close by th e m a jo r p o rt or n o t.
T h e fields are e n h an c e d n e a r th e rod, so w hen th e tu n in g ro d is close to th e m a jo r
p o rt a n ten n a , th e coupling is m uch stro n g e r th a n w hen it is on th e o th e r side o f th e
tu n in g curve. T h e d ip n e a r 800 M Hz is a resu lt o f th e tim in g ro d b ein g in close
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
76
0 (ra d )
F ig u re 3.13: Mode map of th e cavity w ith a single metal rod with r / R = 0.26.
p ro x im ity to th e m a jo r p o rt.
A t 4.2 K , Quni is essentially in d e p e n d e n t o f frequency, in d icativ e th a t th e losses
are d o m in a te d b y th e dielectric. If th e w alls w ere th e d o m in an t source o f losses,
Quni w ould n o t im prove significantly w ith fu rth e r cooling of th e cavity sin ce th e
skin d e p th o f th e copper walls is a lre ad y well in th e anom alous regim e. As th e
cav ity was cooled to 1.4 K, Q uni im p ro v ed u n til th e te m p e ra tu re reached a b o u t 2 .5
K , and th e n re m a in e d essentially c o n sta n t, in d icativ e th a t th e loss ta n g en t o f th e
d ielectric fin ally b ecam e negligible. T h e im p ro v em en t in Q uni fro m 4.2 to 1.4 Iv was
a b o u t a fa c to r o f two.
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77
O
c
(D
f2 - -
13
O’
CD
W
TE or TEM
Li-
Figure 3.14: Frequency versus tuning rod angle showing a typical mode crossing region
involving a TM mode and a stationary T E or TE M mode.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
7S
1 60x10
TE,
140
120
100
Tuning Rod Close to Minor Port
Tuning Rod Close to Major Port
800
850
950
900
Frequency (MHz)
1000
1050
F igure 3.15: Measured Qun/ of the TMoio mode versus frequency. Crossings with known
T E modes are indicated. The dip around 800 MHz is caused by the tuning rod coming in
close proximity to the strong port.
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79
3.4
Piezoelectric Tuning and Coupling Mechanism
A chieving sufficient control o f th e c av ity re so n an t frequencies a n d couplings requires
rep ro d u cib le n a n o m eter scale m o tio n s a t 1.4 K in a n 8 T m ag n etic field. B ecause o f
lim ite d space in m u ltip le-cav ity d e te c to rs, th e drivers m u st be c o m p act. In a d d itio n ,
th e y should cause m in im al h e a tin g in th e c av ity space.
T h e tu n in g an d coupling m ech an ism s on th e single cav ity d e te c to r relied o n
200
s te p /re v o lu tio n s te p p e r m o to rs c o n n e c te d to g e a r re d u c tio n sy stem s o n to p
o f th e cavity. T h e tu n in g m ech an ism u se d a 42000:1 re d u ctio n w hile th e coupling
m ech an ism h a d an 800:1 re d u c tio n . T h e p h y sical reso lu tio n was a d e q u a te , b u t th e re
w ere several short-com ings th a t m ad e th e g e a r sy stem u n d esirab le for th e fo u r-cav ity
array.
W ith th e m echanical gears, th e s c a tte r in th e ste p size a t 1.4 K was 500 Hz,
a p p ro x im a te ly th e sam e as th e sm allest re p ro d u c ib le freq u en cy step . T h e m ovem ent
was very slow, a c o m p lete rev o lu tio n o f th e tu n in g ro d took several hours.
The
vacuum ro ta ry feedthroughs w ere s u b je c t to w ear, a n d th e h e a t co n d u ctio n th ro u g h
th e m ech an ical coupling to th e cav ity w ould b e p ro b le m a tic for fu tu re designs. M ost
im p o rta n tly , th e g e ar sy stem was n o t v e ry co m p act.
For th e four-cavity array, a n e n tire ly new ap p ro ach was used, b ased on piezoelec­
tric a c tu a to rs. P iezoelectric sy stem s h a d b een p reviously used in e ith e r cryogenic
e n v iro n m en ts or high m ag n etic fields, b u t n o t b o th .
3.4.1
The Piezoelectric Effect
P iezo electric m a te ria ls undergo d im e n sio n al changes in response to ap p lied e lec tric
fields.
C onversely, m echan ical p re ssu re re su lts in a voltage p ro p o rtio n al to th e
ap p lied pressure. T h is p ro p e rty m ak es th e m very useful as elec tric to m ech an ical
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
80
tra n sd u ce rs.
M odern piezoelectric m a te ria ls axe ceram ics w hich have b een poled, th a t is cooled
fro m above th e ir C urie te m p e ra tu re in a stro n g electric field. C eram ics are p a rtic u ­
la rly useful because th e y are v ery h a rd , m achinable, chem ically in e rt, an d re sista n t
to atm o sp h eric conditions. U nlike n a tu ra l piezoelectric cry stals, th e m ech an ical and
electrical axes in ceram ics can b e o rie n te d relativ e to th e ir shape.
T h e s tre n g th of th e piezoelectric effect is described by a te n so r relatio n Si =
d ijE j. Si is th e stra in in th e zth d irectio n , E j is th e j t h co m p o n en t of th e ap p lied
electric field, and
is th e s tra in coefficient.
B y convention, d irectio n 3 is th e
poling direction. T h e piezoelectric a c tu a to rs used in this e x p erim en t are arran g ed
as e x p an d ers, th e stra in a n d ap p lied field a re b o th parallel to th e poling axis. T h e
relevant s tra in coefficient is ^33 w ith ty p ical values <r/ 33 r j 450-650 x 10 ~ 12 m /V [61].
T h e efficiency of th e piezo electric effect is expressed as th e m echanical coupling
k2
f
2
_ e n erg y c o n v e rte d f r o m electrical to m e ch an ica l
in p u t electrica l e n e rg y
^
T h e response of th e m a te ria ls is very rap id , allowing d y n am ic o p e ra tio n . W h en
th e frequency of th e d riving vo ltag e is eq u al to th a t of a m ech an ical resonance, th e
response is enhanced by th e m ech an ical Q o f th e m ass-spring sy stem form ed b y th e
a c tu a to r an d its m ass load. T h e Q can b e ~ 1000, so th e effect is q u ite stro n g . T h e
frequencies of th e m echanical resonances are com plicated fu n ctio n s o f th e m a te ria l,
geom etry, loading, te m p e ra tu re , a n d d riv in g voltage. For th is reason, th e o p tim a l
d riv in g frequency m u st be found em pirically. T h e equivalent circu it m odel d escrib ed
la te r helps in th is process.
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81
Motion
Alumina W edge
Positive Electrode
Piezoelectric Element
Negative Electrode
Alumina Body
Figure 3.16: Micropulse L-104 Cryogenic Piezoelectric A ctuator.
3.4.2
Micro Pulse Cryogenic Actuators
T h e piezoelectric a c tu a to rs used are a m odified version o f th e L-104 a c tu a to r sold
by M icro P u lse S ystem s, Inc[62].
As seen in Figure (3.16), two pieces o f Lead-
T ita n a te -Z irc o n a te (P Z T ) are m o u n te d betw een th e alu m in a wedge a n d th e body
o f th e a c tu a to r. In th e stan d a rd version, th e bod y is m ad e from nickel an d serves
as th e neg ativ e electro d e for th e p iezoelectrics. In this ap p licatio n , how ever, we
m odified th e body to be nickel p la te d a lu m in a to m atch th e th e rm a l e x p an sio n of
th e d riv e r an d alu m in a wedge. T h e P Z T pieces are m o u n ted to th e electro d es using
a conductive epoxy.
T h e poling d irectio n of th e P Z T is in d ic a ted by th e arrows in F ig u re (3.16).
W h en a voltage is ap p lied to one o f th e p o sitiv e electrodes, th a t piezo e x p an d s
causing th e wedge to m ove the driv in g piece.
T hese a ctu a to rs a re designed for re so n a n t o p eratio n . T h e re so n an t freq u en cy is
ty p ic a lly in th e range o f 130-160 kH z d e p en d in g on te m p e ra tu re . H ow th ese values
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
F ig u r e 3.1 7 : Equivalent circuit o f a piezoelectric actu ator.
a re d e term in e d is th e su b je c t o f th e next section.
3.4.3
Piezoelectric Actuator Impedance Measurements
T h e m echanical reso n an ce freq u en cy of th e L-104 a c tu a to rs varies w ith applied volt­
age, te m p e ra tu re , lo ad in g a n d several o th e r facto rs. A t room te m p e ra tu re , it is well
known to be a ro u n d 130 kH z, b u t th ese devices h a d n ev er been used below 77 K so
it was necessary to c h a ra c te riz e th e m a t cryogenic te m p e ra tu re s to d e te rm in e the
resonant freq u en cy a n d s tre n g th o f th e piezoelectric effect.
T he various p iezo electric p ro p e rtie s are d e term in e d by m easu rin g th e electrical
im pedance u sing th e equ iv alen t circu it of F ig u re (3.17). Co is th e electrical capaci­
tance, N is th e tra n s d u c e r ra tio o r v o lta g e /u n it force, C m is th e elastic com pliance,
M p is th e m ass, a n d Ftp is th e d am p in g . T h is circu it can be sim plified as show n on
th e right of F ig u re (3.17). T h ese m odels h ave lim ited q u a n tita tiv e usefulness be­
cause th e c irc u it p a ra m e te rs vary w ith voltage, loading, te m p e ra tu re an d frequency.
T h ey are still useful, how ever, in e stim a tin g th e o p tim a l d riv in g frequency a n d the
relative s tre n g th o f th e p iezo electric effect a t different te m p e ra tu re s.
The basic b e h a v io r o f th e c irc u it can b e seen by neg lectin g th e d am p in g resistor
( R = 0). T h is is th e eq u iv alen t circu it in th e lim it o f infinite m echanical Q . T he
im pedance Z ( uj) is
( }
oj(C q +
C t - c^T x C iC o )
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
(
}
83
T h e re are two special frequencies, cor a n d u a. A t
Z(ujr ) = 0, this is th e
re so n an t frequency. A t th e a n ti-re so n an t freq u en cy
= <
jJ* ( ^ t Cl ) , Z (u,y) = oo.
T h e o p tim a l d riv in g frequency coopt lies in th e ran g e u r < uiopt < u:a, so m easu rin g
ujr an d u)a co n strain s ujopt.
A dding th e d am p in g resisto r keeps Z(u>a) finite, a n d slig h tly m odifies u>r a n d u>a.
A c alcu lated im p ed an ce curve using th e eq u iv alen t c ircu it is show n in Figure (3.18),
displaying th e resonance an d a n ti-reso n an ce. T h e m e asu re d curve in Figure (3.19)
d e m o n stra te s th e q u a lita tiv e ag reem en t b etw een th e c irc u it m o d el an d th e real actu a to r.
70x10'
-1 0 -2 0 -
^0
o
Q
-3 0 -4 0 -
so
40
-5 0 -6 0 -
i 1 1 1 1 I 1 ■~r~‘ I 1 1 1 ■ i ' ■ ■ 1
140
150
170
160
Frequency (kH z)
180
140
150
160
170
Frequency (kHz)
180
F ig u re 3.18: Typical impedance curve a s ca lcu la te d from th e eq u ivalent circu it.
0
12-
o
Q
a
es
-
-
20-
-3 0 -4 0 -
2150
160
170
F requency (kH z)
180
160
150
170
Frequency (kHz)
F igure 3.19: M easured im p ed a n ce cu rve a t 4 .2 K.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
180
84
T h e m echanical energy is rep resen ted in th e circuit by Ci an d th e in p u t electrical
energy is rep resen ted by C q. T h e stren g th o f th e p iezo electric effect, given by k 2
is sim ply
T his can be re w ritte n in term s o f th e resonance a n d an ti-reso n an ce
frequency §£• = y | — 1 .
T h e electrical c ap a c itan c e is m easured using a c ap acitan ce m e te r. T h e a c tu a to r
is sim p ly a p arallel p la te c ap acito r, so a m easu rem en t o f Co is easily co n v erted to
d ielectric c o n sta n t er . T h e m e asu re d value o f th e relativ e d ielectric c o n sta n t of th e
P Z T a t 300 K , 77 Iv, and 4.2 K is shown in Figure (3.20). T h e rapid drop in er as
T —> 0 is com m on in ferroelectrics and is p rim arily responsible for th e te m p e ra tu re
dependence of th e resonant frequencies.
C3
C
1000
1
90 0
^
800
u
600-
0
50
100
150
200
Temperature (K)
250
F igure 3.20: Measured relative dielectric constant of the L-104 actuator determined by
a DC capacitance measurement as a function of tem perature.
T h e im p ed an ce of th e a c tu a to rs was m easured a t 300 K , 77 K , and 4.2 K using a
sinew ave g e n erato r a n d an oscilloscope in th e a rran g em en t show n in F ig u re (3.21).
R s represents th e in p u t resistan ce of th e oscilloscope w hich was 1 M fi. T h e piezo­
electric im p ed an ce was d e te rm in e d by th e relativ e a m p litu d e a n d p hase o f V\ and
V2 as th e frequency of th e sinew ave was varied. T h e resu lts for th e th re e m easu re-
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
S5
F igure 3.21: Circuit used to determ ine the electrical impedance of the L-104 actuators
as a function of frequency.
m erits are shown in F igures (3.22 - 3.24). T h e curve a t 77 K w as ta k en w ith th e
a c tu a to r subm erged in LN 2, th e acoustic im p ed an ce o f th e LN2 excessively dam p ed
th e resonances, m aking th e m easu rem en t essentially useless. F o rtu n ately , 77 K is
no t an operating p o in t for th e a c tu a to rs so a re p e at m e asu re m e n t was n o t necessary.
Subm erging th e a c tu a to rs in LHe did not p resen t sim ilar p ro b lem s.
As seen in Figures (3.22 a n d 3.24), th e fu n d am en tal reso n an ce sh ifts from 130
kH z to 160 kHz as th e te m p e ra tu re is lowered from room te m p e ra tu re to 4.2 Iv. T he
resonance also becom es m u ch sh arp er, show ing th e need for p recise selection of th e
driving frequency for cryogenic o p eratio n . T h e stre n g th of th e p iezo electric effect in
P Z T a t 4.2 Iv is found to be ro u g h ly 20% o f its room te m p e ra tu re value, consistent
w ith available d ata.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
8G
<H>
oo
eo
o
6-
a
2
M
N
4-
'So
Im
C3
2-
020-1
-40-i
-60-|
-80-|
-100
-
^ nr
§8
I
120
140
160
180
Frequency (kH z)
200
r | I' I . | I I I I
120
140
160
180
Frequency (kH z)
200
F ig u re 3.22: M easured impedance curve a t 300 Iv.
2
1.8
N
1-6-
r
I
140
160
180
Frequency (kH z)
120
i
i
i
I
i
i
i
I
i
i
i
I
i
i
140
160
180
Frequency (kHz)
i
200
F ig u re 3.23: M easured impedance curve a t 77 Iv.
o-i
4 .0 -
-1 0 ao
t>
Q
“
__s
N
pu
C3
2. 0 -
-2 0 -3 0 -
cP
o
-4 0 -
o
-5 0 -
1. 0 120
140
160
180
Frequency (kH z)
200
SP
o°
120
o
%
<$
8
140
160
180
Frequency (kHz)
F ig u re 3.24: M easured impedance curve a t 4.2 K.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
200
87
3.4.4
Piezoelectric Tuning Mechanism
Alumina Shaft
Clamp
Brass Ball
Alumina Disc
L-104 Piezo Element
■SST Mounting Plate
Top Plate
Guide Bearing
Tuning Rod
F ig u re 3.25: Cross-section of the piezoelectric tuning mechanism.
F ig u re (3.25) shows a cross-section of th e p iezo electric based tu n in g m echanism .
T h re e of th e L-104 a ctu a to rs are positioned a t th e vertices of an eq u ilateral tria n g le
c e n te re d a b o u t th e tu n in g rod shaft. T h e a c tu a to rs are oriented so th e ir d riv in g
m o tio n is in th e ta n g e n tia l direction. T h e th re e a c tu a to rs are o p e ra ted in p a ra llel
to spin a ceram ic disc w hich sits on to p o f th e m .
T h e tu n in g rod is suspended from a brass b all w hich clam ps to th e sh aft. T h e
b ra ss b all serves as a self-aligning m echanism , m a in ta in in g equal weight d is trib u tio n
on th e th re e a c tu a to rs . T h e ro tatio n o f th e disc is coupled to th e tu n in g ro d sh aft
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
88
by th e friction betw een th e brass ball a n d a cone m achined into th e center o f th e
disc. G uide b earin g s in th e top an d b o tto m p la te s o f th e cavity allow th e shaft a n d
rod to tu rn freely.
F igures (3.26) a n d (3.27) show th e resu lts o f te sts o f th e revolution speed o f th e
tim ing m ech an ism versus driving frequency a n d ap p lied DC voltage. R esonances o f
th e in d iv id u al piezoelectric actu a to rs show u p as dips in th e revolution tim e versus
frequency plo ts.
O nly two dips a p p e a r in th e counter-clockw ise test because th e
resonance of tw o o f th e individual a c tu a to rs overlap in th a t case, explaining w hy
th e frequency a t w hich th ey overlap corresponds to th e fastest ro tatio n speed. B y
way of c o m parison, th e revolution tim e for th e m echanical gears on th e single c av ity
was several h o u rs.
In th e p re se n t configuration, th e a c tu a to rs w ere all selected random ly, w ith th e
ex p ectatio n th a t th e individual resonances w ould b e very close to g eth er, an d cer­
tainly fairly b ro a d . As th e im pedance te sts d e m o n stra te d , th is is no longer th e case
a t cryogenic te m p e ra tu re s. In fu tu re designs, th e im p ed an ce tests described e arlier
should be used to m a tc h actu ato rs w hich h ave sim ilar resonant frequencies.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
89
M o­
90-
£
u
t>
vi
100-
60-
£
H
50-
c>
40-
f= 134 kHz
n o -:
VDC= 16 V
,_v 8 0 o
<
L> 7 0 J=
I i i i |
130
i i i |
132
i i i |
134
r i i |
136
i i i |
138
80-
60“
40
20-
-i—r-i—p-i—n —|14
140
16
V oltage (V )
F req u en cy (kH z)
F ig u re 3.26: M easured clockwise rotation tim e o f th e piezoelectric tuning mechanism at
room tem perature as a function of driving frequency and applied dc voltage.
70-
80 H
f = 134 kHz
60£
60-
o
£
H 40£
U
U 20-
50
St 40u
o
301|
I
130
I
I
|
I
132
I
I
|
I
134
I
!
|
I I
136
F req u en cy (k H z)
I |
138
i i i
10
I i i i I i i i l i i
12
14
16
i I i i i l
18
20
V o lta g e (V )
F ig u re 3.27: M easured counter-clockwise rotation tim e of th e piezoelectric tuning mech­
anism a t room tem perature as a function of driving frequency and applied dc voltage.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
90
3.4.5
Piezoelectric Coupling Mechanism
Cable Guide (Limits Travel)
RG402 Coax
Ceramic Cable Sleeve
Sleeve Guide
Cover Plate
Set Screws
Set Screw
Leaf Spring
■Piezo Mounting Bracket
Spring Plate
Bail Bearing
Piezo Actuator
Cover Plate
Top Plate Clamp
F ig u re 3.28: Piezoelectric-based linear coupling mechanism.
F ig u re (3.28) shows th e p iezo electric-b ased lin ear coupling m echanism .
Tw o
op p o sin g L-104 a c tu a to rs are used to d riv e a ceram ic sleeve u p an d dow n. S im ilar
to th e tu n in g m echanism , th e a c tu a to rs a re o p e ra te d in p arallel, w ith e ith e r b o th
u p p e r o r b o th lower piezoelectrics en erg ized to g e th e r. T h e ceram ic sleeve is epoxied
a ro u n d th e m a jo r p o rt a n ten n a . T h e ra n g e o f tra v e l is lim ite d by th e sleeve h ittin g
th e to p p la te o f th e cavity or a cab le g u id e on th e to p . T h e c u rre n t design allows a
to ta l tra v e l o f 0.75 inches, sufficient to v ary th e coupling fro m to ta l w ith d ra w a l of
th e a n te n n a to stro n g overcoupling.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
91
F igure (3.29) shows th e m easured Q o f a cavity versus a n te n n a in sertio n d ep th
a t 4.2 K. As th e a n ten n a is w ithdraw n from th e cavity, j3 —> 0 an d Q l — Qunh th is
is seen by th e fla tte n in g of th e curve in F ig u re (3.29). Q uni = 110000, so a t critical
coupling, Q l — 55000, well w ith in th e range o f travel o f th e linear drive.
100x10
80-
O'
60
40-
20•Full Insertion
o-i
Insertion Depth
Figure 3.29: M easured Q of the TMoio mode versus strong port insertion depth. The
flattening of the curve as the probe is withdrawn indicates weak coupling (/3
1)-
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
92
3.4.6
Piezoelectric Driving Electronics
T he block d ia g ra m of th e piezoelectric d riv in g electronics is show n in F igure (3.30).
T heir p u rp o se is to provide high-voltage ac p o w er to th e different sets of piezoelectric
a ctu a to rs a t th e p ro p er frequency for a co n tro lled len g th o f tim e. T h e voltages,
frequencies, a n d d u ratio n s m ust all b e c o m p u te r controllable. D irection control is
provided by a sw itch system which d irects th e pow er to p ro p e r sets o f a ctu a to rs.
T h e s te p size is easily controlled by v ary in g th e tim e th e various a c tu a to rs are
energized. How ever, since it is a reso n an t process, it is highly non-linear. Linear
behavior is achieved by breaking large step s in to several sm all ones.
T h e fu n ctio n g e n erato r is an H P33120A A rb itra ry W aveform G e n era to r w hich
is used to o u tp u t a low-voltage square wave w ith th e p ro p er d riv in g frequency and
num ber o f cycles. T h e frequency can be v aried for each set o f a c tu a to rs, and th e
num ber o f cycles corresponds to th e sm allest reliab le step size.
T h e n u m b e r o f sm all step s taken is co n tro lled by th e BN C 500A P u lse G en erato r.
This o u tp u ts a series of T T L pulses, each one trig g erin g th e function g e n erato r to
o u tp u t a sq u are wave b u rs t.
A d ju stin g th e n u m b e r of pulses, an d th e spacing
betw een th e m controls th e overall ste p size a n d th e d u ty cycle of th e sm all steps.
T he d u ty cycle is kept aro u n d 50% so th a t th e sm all steps are in d ep en d en t.
S quare w ave pulses are fed into th e pow er am p lifier shown in F igure (3.31). T h e
F E T s sim p ly a c t as sw itches so th e o u tp u t is sw itch ed betw een 0 and th e ap p lied dc
voltage Vd C i depen d in g on w hether th e in p u t is positive o r negative. T h e b ipolar
push-pull p a ir am plifies th e in p u t voltage to sp eed th e sw itching of th e F E T s. Vqc
is su p p lied by a p a ir of H P6033A pow er su p p lies co nnected in series an d is ty p ically
in th e ra n g e 12 - 24 V.
F ig u re (3.32) shows th e ta n k circu it, c o n tain in g n o t only th e fixed cap acitan ce in
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
93
th e driv in g c irc u it C t, b u t also th e c a p a c ita n c e o f th e leads Cc an d th e p iezo electric
cap a c itan c e Cp. T h is is called a step -u p c irc u it because on resonance V0f Vi = j Q t ,
t h a t is, th e o u tp u t voltage is step p e d -u p by th e Q o f th e ta n k circu it Q t . In te rm s
o f th e circ u it p a ra m e te rs, Q t = u jq L /R . T h e values o f L an d C are chosen to m a tc h
th e electrical resonance of th e ta n k c irc u it w ith th e m echanical reso n an ce o f th e
a c tu a to rs .
E ach cav ity has its ow n ta n k c irc u it, w hich c an drive one of four sets o f a c tu a to rs
co rresp o n d in g to th e four d irectio n s u p , dow n, clockwise, an d counter-clockw ise.
T h is allows all four cavities to be tu n e d sim ultaneously, w hile it is n ev er n ecessary
to m ove two directio n s on th e sam e cav ity a t once. T he d irectio n is c o n tro lle d by
o p tic a l relays, a c tu a te d by a 9 V dc signal. T h e 9 V signal is ro u te d to th e relays
w ith a n H P34970A m u ltip lex o r. T h e softw are prevents m ore th a n one d ire c tio n of
a n y cav ity from being energized.
T h e m in im u m rep ro d u cib le step size a t 4.2 K was found to b e roughly 10 cycles.
T h e corresponding cavity frequency reso lu tio n was b e tte r th a n 100 H z, a b o u t 1
p a r t in 107, im p ly in g physical m otions o f th e tu n in g rods on th e o rd e r o f 50 n m .
F ig u re (3.33) shows a h isto g ra m of th e cav ity frequency shifts for a series o f 500
ste p s ta k e n a t 4.2 Iv. T h e sm allest rep ro d u cib le frequency shift w ith th e m ech a n ic al
g ears on th e single cavity was 500 Hz, w ith a s ta n d a rd d ev iatio n of a b o u t 500 Hz.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
94
# S te p s -
Pulse
Generator
Period -
Trigger
'r
# Cycles --------- ■*
Frequency -
Function
Generator
*
Power
Amplifier
21
1r
Tank
Circuit
'
Direction
CW Piezos
Switch
System
CCW Piezos
Up Piezos
Down P iezos
F igure 3.30: Block diagram of the piezo driving electronics.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
95
10 kQ
1kQ
0.001 \iF
2.0V
0.001 jiF
-2.0V
0V
F ig u re 3.31: Circuit diagram for the piezoelectric power amplifier.
-°— WM/ WV-v,-i — 1i------— i1------------------------*
—
<=----- ’ UOOUU L JV
WWWV r -C , r ~ c c Z ~ c p
n
: :
VQ
^
^
••
:___________ i
F ig ure 3.32: Tank circuit including the capacitance of the leads and piezoelectric actua­
tors.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
96
p = 9 4 .4 Hz
a = 13.6 Hz
N = 500
■r~i
40
60
80
100
120
140
160
180
Frequency Shift (H z)
F ig u re 3.33: Histogram of frequency shifts showing th e minimum repeatable step size of
less th an 100 Hz.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout perm ission.
97
3.4.7
Heating from Piezoelectric Actuators
K now ledge of th e ty p ic a l step size a t cryogenic te m p e ra tu re s allows th e e stim a tio n
o f th e h e a tin g cau sed by th e piezoelectrics.
M easurable te m p e ra tu re rises o ccu r
w hen th e p iezoelectrics are o p e ra te d co n stan tly .
T h e piezoelectric q u a lity fa c to r Q p is re la te d to th e loss ta n g e n t ta n 8 a n d th e
stored en erg y U an d pow er loss P by
=
sb 3
<3-s>
Solving for P
P = 2 ir f0U ta n 8
(3-9)
T h e sto red en erg y is sim ply th e energy in th e cap acitan ce
U = ic if
(3.10)
So th e pow er loss c an be ex p ressed by
P = ir f0C V * ta .n S
(3.11)
T h e loss angle, or Q v , is found fro m m easu rem en ts of th e voltage ste p -u p o f th e ta n k
circu it. T h e u n lo a d ed Q o f th e ta n k c irc u it Q u is found by m e asu rin g th e step -u p
w ith th e piezos disco n n ected . T h e loaded Q , Qi is fo u n d by m e asu rin g th e step -u p
w ith th e piezos reco n n ected . Q p is found from th e relatio n
Q~i = Q~u + O'?
(3‘12)
Q u was m e asu re d to b e axound 90 w hile Qi was ap p ro x im a te ly 50. T h e value o f Q p
from (3.12) is a p p ro x im a te ly 100.
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9S
T h e h e atin g is eq u al to th e p ro d u ct o f th e pow er loss an d th e d u ty cycle o f th e
piezos. T h e typ ical ste p tim e for a 2 kHz cavity s te p is roughly 1 m sec, and a step is
ta k en every 100 sec, m ean in g th e d u ty cycle is a p p ro x im a te ly 10~°. For th e tu n in g
m echanism , w ith th re e piezos in parallel, th e c ap a c itan c e C is ab o u t 75 p F , a n d th e
ty p ical driving voltage Vp is 600 V. Plugging th ese values into E q u atio n (3.11) gives
a value for th e h e atin g o f 1 fiW , tru ly negligible.
3.5
Cryogenic Electronics
T his section describes th e tw o com ponents w hich com e betw een th e cavity a n d th e
room te m p e ra tu re electro n ics, th e power com biner a n d th e first am plification stag e.
3.5.1
Wilkinson Power Combiner
T h e m a jo r p o rt a n te n n a s are connected to th e four p o rts of th e power co m b in er
th ro u g h iden tical pieces of 0.047” d iam eter sem i-rigid coax. This cable is coiled to
provide enough flexibility for th e an ten n a range o f trav el.
T h e pow er com biner is a Wilkinson[63] ty p e pow er co m b in er/d iv id er m an u fac­
tu re d by N arda[64]. It has a n insertion loss of less th a n 1 dB and a usable b a n d w id th
of 0.5 - 2.0 G H z. T h e design of a tw o-port W ilkinson com biner is shown in Fig­
u re (3.34). T h e q u arter-w av e section is for im p ed an ce m atching, and in th e usual
case of equal in p u t a n d o u tp u t im pedances (Z,-n = Z out = Z 0) has an im p ed an ce of
y /W Z Q w here N is th e n u m b e r of p o rts. T h e resisto rs are for isolation, otherw ise,
th e o u tp u t p o rt of th e com b in er would be sh o rte d w hen one of th e in p u t p o rts is
sh o rted . T h e value of th e isolation resistor is N Z Q.
W hen all of th e in p u ts are in phase, no power flows th ro u g h th e isolation resistors
a n d th e o u tp u t pow er is th e sum o f th e in p u t pow ers.
W hen th e re is a p h ase
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
99
F ig u r e 3 .3 4 : T w o-w ay W ilk in son P ow er D iv id e r/C o m b in er. For a g en era l N -w ay d evice,
th e im p ed a n ce o f the q u arter-w avelen gth lines is \ f N Z 0 and th e valu e o f th e isolation
resistors is N Z 0 .
m ism a tch , power is d issip ated in th e resistors and th e o u tp u t pow er is reduced from
th e su m of th e inputs. For th is reason, a lot o f care was given to m ak e th e m a jo r
p o rt an ten n a s and connecting cables identical. However, as seen in F ig u re (2.13),
th e p h a se is also a function o f th e cavity frequency an d Q. T h e re w ill alw ays be
som e m ism a tch betw een th e cavities so it is im p o rta n t to u n d e rs ta n d its effects,
especially in term s of th e tra d e -o ff betw een o p tim a l com bining efficiency and tim e
sp en t perfectly m atch in g th e cavities. T h e first resu lt discussed in th e n ex t c h ap te r
is a m easu rem en t v alidating th e pow er-com bining form ula deriv ed in C h a p te r Tw o.
3.5.2
HFET Amplifiers
A long w ith th e cavity Jo h n so n noise, th e d o m in an t b ackground in th e d e tec to r
is th e noise in th e first am p lifier stage. T his noise is c h arac te riz ed by th e noise
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
100
te m p e ra tu re of th e am plifier. T h e th e rm a l noise P/v in a b a n d w id th B is given by
P iV = k s T B w here T is th e te m p e ra tu re . If th is pow er is am plified b y by a noisy
am plifier w ith gain G th e pow er o u tp u t P0 is g re a te r th a n (S'Pv = G k B T B . T h e
e x tra pow er Pn ,a is from th e am p lifier noise, an d is ch aracterized by th e a p p a re n t
increase in te m p e ra tu re of th e in p u t source w hen am plified
P0 = G kBT B + PNtA = G k B ( T + T n ) B
(3.13)
w here T v is defined as th e am p lifie r noise te m p e ra tu re . For m u ltip le stag es, th e
ex tension is straightforw ard. C o n sid er tw o cascaded am plifiers wfith g a in s G \ and
Go an d noise te m p e ra tu re s T v i a n d T v 2 , th e pow er o u tp u t is
Po = G 2G l k B ( T + T Ni) B + k B G 2T N2B
(3.14)
T h e com bined noise te m p e ra tu re TA is found by e q u atin g E q u atio n 3.14 w ith th e
to ta l o u tp u t power w ith P0 = G i G 2 ^ b ( T + T A) B , giving
Ta = T v 1 + 5 ^
(3-15)
T h is is a n im p o rta n t result w hich s ta te s th a t th e noise c o n trib u tio n of a n in d iv id u al
stage o f am plification is eq u al to its noise te m p e ra tu re divided by th e to ta l gain
preceding it in th e chain. T h is is w hy th e first stage of am plification is th e m ost
im p o rta n t, its noise te m p e ra tu re d o m in a te s th e overall system noise.
T h e am plifiers used in th e a x io n d e te c to r em ploy G aAs H e te ro ju n c tio n F ield
Effect T ransistors (H F E T s) a n d w ere m a d e by th e N ational R adio A stro n o m y O b­
serv ato ry (N RA O )[65j. Each am p lifie r stag e em ploys two single-ended H F E T am ­
plifiers in a double-balanced d esig n show n in Figure (3.35). T h e d o u b le-b alan ced
design g re a tly im proves th e in p u t m a tc h o f th e am plifier w ith only a m o d e st increase
in n o ise-tem p eratu re.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
101
90°
90 °
■oOutput
F ig u r e 3 .3 5 : B alanced A m p lifier D esig n .
A single-ended am plifier has an o p tim al im p e d a n c e for m inim um noise te m p e ra ­
tu re w hich differs fro m .th e in p u t im pedance (50 Q in o u r case). A m atch in g n etw o rk
tran sfo rm s th e in p u t im p ed an ce to th e o p tim a l im p ed an ce, b u t this can only b e do n e
over a sm all b a n d w id th . T h e in p u t m a tc h o f th e am plifier would be very good in
th e sam e sm all b a n d w id th , o u tsid e of it, th e re w ould be significant reflection.
In th e double-balanced design, th e sin g le-en d ed am plifiers are m ade as id en ­
tical as possible, a n d o p tim ized for m in im u m noise te m p e ra tu re in th e c e n te r o f
th e desired freq u en cy b an d . T h e 7t / 2 h y b rid sp lits th e in p u t signal equally to th e
tw o am plifiers e x cep t for a tt / 2 phase difference. O u tsid e o f the b an d w id th o f th e
m a tc h in g netw orks, pow er is still reflected b a c k to th e in p u t. T h e reflected signals
recom bine in th e in p u t h y b rid , b u t th e y now h av e a tt phase difference so th e y in­
terfere d estru ctiv ely . T h e reflected signals a rriv e in phase a t th e fo u rth p o rt o f th e
in p u t h y b rid w here th e y are te rm in a te d in a 50 O resisto r. T h e am plified signals axe
recom bined in th e o u tp u t 7?/2 hybrid. O n t h e o u tp u t, th e original 7t / 2 p h ase sh ift
is rem oved a n d th e tw o signals com bine. T h e fo u rth p o rt, w here th e signals in te r­
fere d estru ctiv ely , is te rm in a te d . T h e re su lt is a n am plifier w ith nearly o p tim iz e d
R ep ro d u ced with p erm ission of the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
102
3S
21
20
3
19
IS
18
17
2
16
IS
15
14
360.0
440.0
520.0
600-0
680.0
760.0
840.0
I
360i) 440.0
Frequency (MHz)
520.0
600.0
680.0
760.0
840.0
Frequency (MHz)
F ig u re 3.36: Gain and noise tem perature of the first cryogenic amplifier. D ata supplied
by NRAO.
noise te m p e ra tu re a n d good in p u t m a tc h over th e e n tire frequency b an d . A ll o f th e
am plifiers from N R A O have h ad b e tte r th a n -15 dB in p u t m atch over th e ir e n tire
frequency band.
T h e first am plification stag e consists of tw o N R A O am plifiers cascaded a t th e
o u tp u t o f th e pow er com biner. T h e in d iv id u al gains and noise te m p e ra tu re s are
show n in F igures
(3.36 and 3.37) w hile th e co m b in ed gain a n d noise te m p e ra tu re
is show n in F ig u re (3.38).
T h e noise te m p e ra tu re o f these am plifiers h as been m easured in situ to verify
th e N R A O re su lts a n d as a calib ratio n of th e se n sitiv ity of th e d etecto r. T h e m ea­
su re m e n t is p erfo rm ed by m easu rin g th e noise pow er as th e physical te m p e ra tu re o f
th e c a v ity Tc is varied. T he cav ity is critically coupled, so on resonance it a c ts like
a 50 Q re sisto r a n d em its a noise pow er Pc — k s T cB .
It is know n from previous ex p erien ce th a t th e noise te m p e ra tu re o f th e am plifiers
is in d e p e n d e n t o f physical te m p e ra tu re below 12 K , m ost likely due to inefficient
cooling of th e H F E T channels. T h e te m p e ra tu re d ep en d en ce of th e gain is a c c o u n te d
for b y following th e height of a fixed pow er p e ak in je c ted into th e cavity d u rin g th e
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
103
21
35
20
19
18
25
17
c
Z
16
520.0
600.0
760.0
840.0
Frequency (MHz)
440.0
600.0
680.0
760.0
840.0
Frequency (MHz)
F ig u re 3.37: Gain and noise Tem perature of the second cryogenic amplifier. D ata supplied
by NRAO.
te st.- T h e re su lt is a straig h t-lin e plot o f noise pow er versus cav ity te m p e ra tu re
whose in te rc e p t gives th e noise te m p e ra tu re o f th e am plifier. T h e resu lt of a te st
at 700 M Hz is given in Figure (3.39). T h e m easu red value o f 1.7 K is in agreem ent
w ith th e N R A O value.
E arly in th e com m issioning of th e single-cavity d e te c to r it was discovered th a t
th e noise te m p e ra tu re o f th e am plifiers is sen sitiv e to th e ap p lied m a g n etic field. T h e
field effectively increases th e channel len g th , a n d th e c o n trib u tio n o f reco m b in atio n
noise.
T his is d escribed in Ref.
[6 6 ], a n d th e effect is elim in a te d if th e H F E T
channels a re o rie n te d parallel to th e ap p lied field.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
104
42
40
ei
38
3
es
£O
c
2
32
52D.0
600.0
680.0
760.0
131—
840.0
360.0
440.0
680.0
520.0
Frequency (MHz)
760.0
Frequency (MHz)
F ig u re 3.38: Combined gain and noise tem p eratu re of the cascaded cryogenic amplifiers.
D a ta supplied by NRAO.
5
-D
w.
<c
o
?o
cu
<u
'o
Z
-
2.0
0.0
2.0
4.0
6.0
8.0
10.0
12.0
Cavity Temp (K)
F ig u re 3.39: R esults of an in situ noise tem perature measurement on the NRAO am­
plifiers. The measured noise tem p eratu re of 1.7 K is determined by the negative of the
intercept of the straig h t line of power versus tem perature.
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
105
3.6
Transmission and Reflection Measurement Elec­
tronics
NETWORK
ANALYZER
NETWORK
ANALYZER
SWEEP
GENERATOR
y
SWEEP
GENERATOR
DIRECTIONAL
COUPLER
-3
PORT
f \
n
PORT
-=
son;
MAJOR
PORT
Figure 3.40: Setup for making cavity transmission (left) and reflection (right) measure­
ments. The reflection te st can not be performed in situ because no directional coupler is
permanently installed.
Transmission, an d reflection m easu rem en ts were all p e rfo rm ed w ith a n HPS757D
Scalar N etw ork A nalyzer connected to a n HP83620A S y n th esized Sw eeper. Reflec­
tion te sts also req u ired a directional coupler. T he setu p for b o th te sts is shown in
Figure (3.40).
M icrowave sw itches allow transm ission m easu rem en ts on th e in d iv id u al cavities
to be perform ed in situ. F irst, th e o u tp u t o f th e com biner is sw itched to th e netw ork
analyzer. T he sw eeper pow er is th en sw itched to th e m in o r p o rt of th e cavity to
be m easured. A lth o u g h th e com biner is coupling pow er fro m all four cavities, only
th e response from th e c av ity being ex cited is m easured. T h is w ould not be possible
w ith o u t th e isolation resisto rs in th e pow er com biner. O th erw ise, th e o u tp u t of th e
com biner would be a sh o rt if any of th e cavities were m ism atch ed .
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
106
3.7
Receiver Electronics
SWEEP
NETWORK
A.
U L MIXER
FFT
ATTENUATOR
COM BINER
F ig u re 3.41: Double-Heterodyne Receiver Schematic.
This section describes th e d o u b le-h etero d y n e receiver d o w n stream from th e cryo­
genic am plifiers. T h e sch em atic is shown in Figure (3.41). T h e p u rp o se is to m ix
a sm all frequency region cen tered aro u n d th e cavity resonance dow n to audio fre­
quency w here its pow er s p e c tru m is m easu red using an F F T s p e c tru m analyzer.
A room te m p e ra tu re p o st-am p lifier m an u factu red by M ITEQ [67] sits ato p th e
insert. This raises th e signal to a level sufficient to ren d er d o w n stream noise and
interference negligible a n d overcom e th e a tte n u a tio n o f th e long cables to th e room
containing th e rest o f th e electronics. T h e gain of th is am plifier is show n in Fig­
u re (3.42).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
107
38-
S
■3
37-
35-
200
400
600
800
F requency (M H z)
1000
1200
F ig u re 3.42: Measured Gain of the M ITEQ post-amplifier.
3.7.1
Image-Reject Mixer
T h e radio frequency (R F ) o u tp u t o f th e p o st-am p lifier is fed in to th e first m ixing
stag e of th e receiver. A m ix e r is sim ply a signal m u ltip lie r, th e R F in p u t signal is
m u ltip lied by a fixed frequ en cy signal from a local o scillato r. In frequency space,
th is m u ltip lica tio n becom es a convolution o f th e in p u t s p e c tru m and th e Fourier
tra n sfo rm of th e local o scillato r signal, d e lta fu n ctio n s a t ± / l o w here f i o is th e
local o scillator frequency. T h e resu lt o f th e co n v o lu tio n is very sim ple, an ex act
copy of th e in p u t freq u en cy sp e c tru m shifted dow nw ard by / l o an d an ex act copy
sh ifted u pw ard by J l o - T h e upw ard shifted sp e c tru m is ty p ic a lly filtered o u t, o r is
sufficiently o u t of b a n d to b e safely ignored, leaving as th e o u tp u t a signal w hose
frequency sp e c tru m is shifted dow nw ard by / l o A p o te n tia l p ro b lem w ith sim ple m ixers is aliasing. F requencies which are m ixed
below D C are aliased to p o sitiv e frequency. Hence, freq u en cy com p o n en ts a t / ± /
lo
a re b o th m ixed to th e sam e in te rm e d ia te frequency (IF ). T h e aliased frequency com ­
p o n en ts a re referred to as th e im age. In th e case of th e ax io n d e te c to r th e com ponent
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
108
90°
90°
F ig u re 3.43: Design of an image-reject mixer.
m ixed d ire c tly m ay c o n tain signal, b u t th e im age frequency only c o n trib u tes noise,
so aliasing degrades th e signal-to-noise ra tio (S N R ).
O ne so lu tio n to th is problem is to use an im ag e-reject m ix er (IR M ). A schem atic
o f one is show n in F ig u re (3.43). T h e concept is sim ilar to th e double-balanced
am plifier described earlier, two 7t/2 hybrids are used in such a way th a t in th e
o u tp u t p o rt th e tw o signal com ponents recom bine c o n stru ctiv ely an d th e two im age
co m p o n en ts recom bine destructively. T h e im age signals reco m b in e co n stru ctiv ely in
th e o th e r p o rt o f th e o u tp u t h y b rid which is te rm in a te d .
T h e o riginal IR M was a com m ercial u n it b u ilt b y M IT E Q designed for use a t
an IF o f 10.7 M Hz, a com m only used freq u en cy in FM d em o d u latio n .
T h is IF
m a tc h e d th e p a ssb a n d o f th e c ry stal filter d escrib ed in th e n e x t section. For reasons
d escrib ed la te r, it b ecam e necessary to m easu re th e im ag e-rejectio n for different
in te rm e d ia te frequencies o f th e M IT E Q IR M along w ith tw o IR m ixers b u ilt from
d iscrete co m p o n en ts.
T h e s e tu p for m e asu rin g th e in sertio n loss a n d im ag e-rejectio n of a n IR M is
show n in F ig u re (3.44). For a fixed LO frequency, R F pow er is in jected in to th e
in p u t, a n d th e IF pow er tra n s m itte d m easu red . T h e freq u en cy o f th e R F is sw ept
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
109
RF
Power
Meter
LO
F igure 3.44: Setup for measuring image rejection and insertion loss of an image-reject
mixer.
through, a b a n d cen tered about th e LO frequency. F ig u re (3.45) shows th e resu lts
o f a m easu rem en t on th e M IT E Q IR M w ith f ^ o set to S00 MHz. T h e definitions
o f th e insertion loss (assum ing 0 d B m in p u t pow er) an d im age-rejection are show n.
T h e im age-rejection a t a given IF is sim ply th e ra tio of th e tra n s m itte d pow er at
+ f i F to th e tra n s m itte d power a t —/ i f As seen in F ig u re (3.45), th e im ag e-rejectio n is very good ( « 20 dB ) on ly for a
narrow range o f in te rm e d ia te frequencies. F ig u re (3.46) shows th e im ag e-rejectio n
a n d insertio n loss versus LO freq u en cy for th e M IT E Q a t its design IF o f 10.7 M Hz.
As discussed in th e following sectio n , it is desirab le to change th e IF to 30 M Hz.
F ig u re (3.47) shows th e M IT E Q ’s im ag e-rejectio n an d insertion loss a t th is IF . T h e re
is v irtu a lly no im age-rejection.
For th is reason, a new IR m ix e r was b u ilt from discrete co m p o n en ts.
Fig­
u re (3.4S) shows th e tra n s m itte d pow er versus IF for a LO frequency o f 800 M Hz.
T h e im age-rejection is slightly worse th a n th a t o f th e M IT E Q , b u t it is still ad eq u a te .
T h e new m ix e r is u sab le over a m u c h b ro ad er range of IF . T h e im ag e-rejectio n an d
in sertio n loss for th e new m ixer is show n in F ig u re (3.49).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
no
Oh
I.L .
>«"*N
E
09
T
'w3'
-
10 -
I.R .
E
CA
c
w
H
-3 0
-3 0
-20
-10
0
10
20
30
In term ed ia te F r eq u e n c y (M H z )
Figure 3.45: Transm itted power versus interm ediate frequency for the MITEQ ImageReject mixer. Shown on the graph are the definitions of the insertion loss (IL) and image
rejection (IR) for a given IF, in this case 10.7 MHz. T he LO frequency is fixed a t 800
MHz
T h e R F hybrid lim its th e u sab le R F b a n d w id th o f th e new m ixer to 500 - 1000
M Hz. Figures (3.50 an d 3.51) show th e resu lts for m e asu rem en ts o f th e new m ixer
w ith a different R F h y b rid designed for 1 - 2 G H z o p eratio n .
D uring op eratio n , th e local o scillato r freq u en cy is a d ju ste d so th e cavity fre­
quency is m ixed dow n to th e fixed c ry sta l filter IF , e ith e r 10.7 o r 30.112 M Hz.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
Ill
as
20-
8-
(A
’S I
O
c
-
fs
-1 6 -
-5
12-
2
c
-
200
400
600
80 0
1000
LO Frequency (M H z)
20-
1200
200
40 0
60 0
800
1000
LO Frequency (M Hz)
1200
F ig u re 3.46: Image rejection and insertion loss of the MITEQ IR Mixer measured a t an
interm ediate frequency of 10.7 MHz.
3 -I
-8 -T
m
-
o
12-
-1 4 -
-1 8 200
400
600
800
1000
LO Frequency (M Hz)
1200
20 0
400
600
80 0
1000
LO Frequency (M Hz)
1200
F ig u re 3.47: Image rejection and insertion loss of the M ITEQ IR Mixer measured a t an
interm ediate frequency of 30 MHz. T he image rejection is unacceptably small in this case.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
112
-10 H
eC3
w
H
-3 0
-100
0
50
-5 0
Interm ediate F req u en cy (M H z)
100
F ig u re 3.4S: Transm itted power versus interm ediate frequency for the rebuilt image-reject
mixer designed for 500 - 1000 MHz. The LO frequency is fixed a t 800 MHz
20 H
e
-6 -T
-o
O
co
czj
15-
-7 -
■Ji
.M, iocc
:
I 5J—
-
-9 -
10-
-
12 -
S/i
C
E
0500
8
60 0
700
800
900
LO Frequency (M H z)
1000
500
600
70 0
800
90 0
LO Frequency (M Hz)
1000
F ig u re 3.49: Image rejection and insertion loss of the rebuilt image-reject mixer designed
for 500 - 1000 MHz measured a t an interm ediate frequency of 30 MHz.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
11 3
^
00
-1 5 -
3
■1
o
-
20
-
t)
'ET
OS
O
-c ’
bo -2 5 —
o
-3 0 -
-100
50
-5 0
0
In term ed iate F req u en cy (M H z )
100
F igure 3.50: Transmitted power versus interm ediate frequency for the rebuilt image-reject
mixer designed for 1000 - ‘2000 MHz. T he LO frequency is fixed a t 1200 MHz.
-5H
eo
ea - 1 5 V
20
z/i
-
-
c
0-
-2 5 800
1200
1600
LO Frequency (M Hz)
2000
800
1200
1600
LO Frequency (M Hz)
2000
F ig u re 3.51: Image rejection and insertion loss of the rebuilt image-reject mixer designed
for 1000 - *2000 MHz measured at an interm ediate frequency of 30 MHz.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
114
3
3
tz
M
X
m
\u»
10000.0 20000.0 30000.0 40000.0 50000.0 60000.0
Frequency (Hz)
10000.0 20000.0 30000.0 40000.0 50000.0 60000.0
Frequency (Hz)
F ig u re 3.52: Measured transfer functions of both the old (left) and new (right) crystal
filters.
3.7.2
Crystal Filter and Second Mixing Stage
T h e second m ix in g stage tak es the 10-30 M Hz o u tp u t from th e im age-reject m ix e r
dow n to audio frequencies. A udio frequency w avelengths a re m uch too long to b u ild
a h y b rid , so a second im age-reject m ixer is o u t o f th e q u estio n . Instead, th e signal
is p assed th ro u g h a b an d p ass crystal filter w hich passes th e signal frequency, b u t
not th e im ag e frequency. T h is is followed by a n o rm a l d o u ble-balanced m ixer.
T h e raw d a ta m u st be norm alized by th e tra n sfe r fu n c tio n of th e cry stal filter,
so n o t o n ly m u st it be w ell-m easured, it m u st re m a in stab le. S tab ility is achieved
by housing th e filter in a te m p e ra tu re -re g u la te d h ousing m a in ta in ed a t 40° C.
T h e tra n s fe r fu n ctio n o f th e cry stal filter is m easu red by placing a 32000 K
noise source a t th e in p u t o f th e receiver an d in te g ra tin g for a long tim e w ith th e
F F T s p e c tru m an aly zer described in th e n ex t section.
T h e in teg ratio n tim e for
th e tra n s fe r fu n c tio n m easu rem en t m u st b e m u ch longer th a n th e m e asu re m e n t
tim e since th e flu ctu atio n s a d d in q u a d ra tu re w hen n o rm alizin g th e raw d a ta . T h e
m e asu re d tra n s fe r functions for two c ry stal filters is show n in Figure (3.52).
T h e new c ry sta l filter, show n on th e rig h t o f F ig u re (3.52) has a m uch w id er
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
115
u sab le b a n d w id th a n d less s tru c tu re th an th e o ld er filter. U n fo rtu n ately , its passb a n d is a t 30 M H z, a n d as seen in th e last sectio n , th e re is little im age-reject at
th a t IF for th e M IT E Q im age-reject m ixer. T h e d o u b lin g o f th e noise te m p e ra tu re
fro m th e lack o f im age-rejectio n is com p en sated by th e in creased b an d w id th , so th e
u ltim a te scan ra te w ould b e co m p arab le for th e tw o filters. W ith th is in m in d , th e
new filte r was chosen b ecau se o f th e reduced s tru c tu re in th e tra n sfe r function. In
th e m e a n tim e , th e new im age-reject m ixers d escrib ed in th e previous section were
c o n stru c te d for fu tu re d a ta -ta k in g .
T h e frequency o f th e second local o scillato r rem ain s fixed a t th e value w hich
m ixes th e 10.7 o r 30.112 M Hz IF to th e au d io freq u en cy o f th e F F T sp ec tru m
a n aly z e r d escrib ed n e x t.
3.7.3
Audio Frequency FFT
A fte r th e second m ixing stag e, th e power sp e c tru m is m e asu re d using a S ta n ­
fo rd R esearch S ystem s (SRS) M odel 760 F ast F o u rier T ran sfo rm (F F T ) S p ectru m
A nalyzer[ 6 S]. T h e in p u t is dig itized by a 16-bit an alo g -to -d ig ital co n v erter (A D C )
sa m p lin g a t 256 kH z.
B y th e N yquist S am p lin g T h e o re m , th e m ax im u m signal
freq u e n cy w hich can b e sam p led w ithout aliasing is 128 kH z. For th is reason, th e
in p u t is p assed th ro u g h a n an ti-aliasin g low -pass filter w hich begins rolling off at
100 kH z a n d a tte n u a te s all freq u en cy com ponents above 156 kH z by a t least 90 dB.
A n in d iv id u a l tim e reco rd consists of 1024 sam p led p o in ts w hich are Fourier
T ran sfo rm ed using a n F F T alg o rith m , resu ltin g in a 512 p o in t pow er sp ectru m .
T h e freq u en cy reso lu tio n o f th e pow er sp e c tru m A / is d e te rm in e d by th e inverse of
th e to ta l tim e o f th e tim e record, which in te rm s o f th e sam p lin g frequency f s and
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
116
to ta l n u m b e r of points N is
A/ = £
(3.16)
For a 1024 p o in t tim e reco rd sam pled a t 256 kH z, A / = 250 Hz. T h e frequency
span of th e pow er sp e c tru m is 0 to 128 kH z. B ecause th e an ti-aliasin g filter sta rts
rolling off a t 100 kH z, th e sp ec tru m from 100 - 128 kH z is unusable an d is discarded
by th e analyzer, th e final pow er sp ectru m is 400 p o in ts w ith a sp an from 0 to 100
kHz.
Peaks in th e pow er sp e c tru m narrow er th a n th e b in w id th will be sm eared over
m ultip le bins unless th e frequency is co m m en su rate w ith th e sam p lin g frequency.
T his is d ue to end effects in th e tim e record. T h e s itu a tio n can be im proved by
m u ltiplying th e en tire tim e record by a function w hich rises slowly from zero in th e
beginning a n d falls slowly to zero a t th e end. T h is process is called w indow ing,
and th e fu n ctio n used is th e w indow ing function. Since th e e x p ected axion signal is
several bins wide, this is n o t necessary and th e so-called u niform window is used.
Since th e passband o f th e cry stal filter is only 40 kH z, it is possible to use d ig ital
filtering in th e analyzer to halve th e sp an an d double th e resolution. T h e resolution
could alw ays be im proved by keeping m ore th a n 1024 p o in ts, b u t th is is fixed by
th e h ard w are. In stead , th e in p u t is d ecim ated , an d o n ly every o th e r p o in t is kept.
T his is equivalent to sam plin g a t half th e frequency. T h e in p u t is d ig itally filtered
to a tte n u a te th e frequency com ponents above th e re v ised N yquist ra te . T h e sam e
n u m b er of p o in ts now corresponds to a tim e record w hich is tw ice as long, h alv in g th e
frequency resolution. T h is process can b e re p e ate d for successively sm aller spans,
b u t th e 50 kH z span w ith 125 Hz resolution is well m a tc h e d to th e cavity b a n d w id th
an d e x p ec te d signal w id th .
A d ig ita l h etero d y n in g step was perform ed, allow ing th e 50 kH z sp an to begin
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
117
a t a frequency o th e r th a n D C . It is sim p ly a d ig ital im p le m e n ta tio n o f a m ix e r,
m u ltip ly in g th e tim e record by a sine-w ave. T h is is used to s ta rt th e pow er s p e c tru m
a t 10 kH z, well aw ay from 60 H z in te rfe re n c e a n d th e harm onics.
A n in d iv id u al pow er s p e c tru m ta k es 8 m sec to acq u ire an d F F T , w ith e sse n tially
no d e ad -tim e . To im prove th e SN R , it is necessary to average m an y sp e c tra to g e th e r.
T ypically, th e pow er s p e c tru m is o n ly dow nloaded to th e co m p u te r a fte r a t least
10000 s p e c tra have been averaged, ro u g h ly every SO sec. A t this p o in t, th e c av ities
a re u sually s te p p e d to a new frequency, o r a t least rechecked to verify th a t th e
frequencies have not. d rifted significantly.
In su m m ary , th e raw pow er sp ec tru m .c o n sists o f 400 points covering th e ra n g e
from 10 - 60 kH z w ith a reso lu tio n o f 125 H z, ty p ically rep resen tin g th e lin ear av erag e
o f 10000 in d iv id u al sp ectra. T h e average c av ity frequency step is m uch sm a lle r th a n
th e sp e c tru m b a n d w id th so each freq u en cy is covered by several pow er s p e c tra . T h e
process o f norm alizing, com bining an d searching for peaks in th e d a ta is d e sc rib e d
in C h a p te r Four.
3.7.4
High-Resolution FFT
A n in d e p en d e n t, high-resolution search ch an n el o p erates in parallel to ex p lo re S ik iv ie ’s
id e a of fin e -stru c tu re in th e ax io n signal. T h e 35 kH z signal passes th ro u g h a sixpole c ry sta l filter and th ird m ix in g stag e to shift th e cen ter frequency to 5 kH z.
D u rin g th e 80 seconds th a t th e m e d iu m reso lu tio n channel is averaging s p e c tra , a
P C -b a sed D S P takes a single 50 second sp e c tru m an d perform s a n F F T . T h e re­
su ltin g frequency resolution is 20 m H z, a b o u t th e lim it im posed by th e D o p p ler
sh ift due to th e e a r th ’s ro ta tio n . T h ese d a ta are searched for coincidences b e tw ee n
different scans, as well as coincidences w ith peaks in th e m ed iu m reso lu tio n d a ta .
T h is channel was no t o p e ra te d in th e scan discussed in th is thesis.
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118
3.8
Miscellaneous Instrumentation
T h e rem ain in g in stru m e n ta tio n consists m o stly o f various p ressu re, te m p e ra tu re ,
a n d light level sensors. B a ra tro n c ap a c itan c e gauges m e asu re th e p ressures in th e
in su latio n space betw een th e in sert a n d th e m a g n et b o re, in th e in sert h e liu m reser­
voir, an d in th e cavity volum e. P h o to -d io d es m o n ito r th e light levels w h ich axe
useful to d e tec t a ctiv ity in th e e x p e rim e n t hall. F inally, te m p e ra tu re s in sid e an d
o u tsid e th e hall are m easu red w hich can b e used to d e te c t th e rm a l co rrelatio n s like
th e m a g n et cable resistan ce d escrib ed earlier.
3.9
Computer System
W ith th e exception o f th e cryogen fills a p p ro x im a te ly once p e r week, th e en tire
o p e ra tio n of this e x p e rim e n t. is a u to m a te d .
D uring p ro d u c tio n ru n n in g , th e live
tim e has been well over 90% for periods o f longer th a n one y ear.
T h is perform ance has been achieved u sin g LabV IEW [69] softw are ou a M acin to sh
Q u a d ra 950, a n d recently on a Pow erM ac 7100. O n to p o f th e n o rm al d a ta -ta k in g
o p eratio n s, a n d m agnet reg u latio n d escrib ed earlier, th is c o m p u te r logs all o f th e
housekeeping d a ta (te m p e ra tu re s, p ressu res, voltages, e tc .) to disk. T h is d a ta is
invaluable in th e very ra re in stan ces o f c a ta stro p h ic failure. T h e softw are is designed
to be ru n n in g , regardless o f w h e th er o r n o t th e e x p e rim e n t is ta k in g d a ta .
T h e use of T im b u k tu Pro[70] softw are allows th e e x p e rim e n t to be m o n ito red
a n d co n tro lled from re m o te lo catio n s. It is n o t u n co m m o n for th e e x p e rim e n t hall
to re m a in unoccupied for days a t a tim e u n d e r n o rm al o p e ra tio n s.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
119
Chapter 4
Results
T h is c h ap te r describes resu lts o f th re e tests p erfo rm ed w ith th e e n tire four-cavity
a rra y assem bled. T h e first te st is th e validation o f th e pow er com b in in g form ula
derived in C h ap ter Tw o.
T h e second te st involves a sim u la ted d a ta ta k in g run
w ith o u t th e long in te g ra tio n tim e , th is served to verify th e p erfo rm an ce of th e en tire
m echanical system . Finally, a sm all scan was p erform ed to reach K SV Z sen sitiv ity
for th e first tim e w ith a m u ltip le-ca v ity axion d e te c to r.
T h e scanning ra te for th e fo ur-cavity array is p ro hibitively slow w ith th e alu m in a
rod s p rim arily because o f th e d e g ra d atio n of th e fo rm facto r as seen in F ig u re ( 2 .1 6 ) .
P lugging th e value of C fro m F ig u re (2 .1 6 ) an d th e sy stem noise te m p e ra tu re T cav +
T a m p
= 4 .5 K into th e scan ra te fo rm u la (2 .S 9 ) gives a s c a n n in g r a te o f 160 k H z/d ay ,
com pared to a typical scan ra te o f 1 M H z/d ay for th e single cavity. W ith th e m etal
rods, th e scan ra te w ill be m u ch faster, but H F E T am plifiers in th is region were
n o t available a t th e tim e o f com m issioning. T h e receiver chain is well u n d ersto o d ,
so it was possible to v a lid a te th e m u ltip le-cav ity a rra y concept w ith o u t a long ru n
of p ro d u ctio n d a ta ta k in g . H F E T am plifiers in th e 1-2 G H z range w ill be available
in th e Fall of 2 0 0 0 a t w hich tim e full-scale o p e ra tio n o f th e fo u r-cav ity a rra y will
com m ence.
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120
4.1
Power Combining Test
SWEEP
GENERATOR
POWER
DIVIDER
MINOR
PORT'
MAJOR
PORT
CAVITY &
TUNINGROD
POWER |
COMBINER
AMPLIFIER
NETWORK
ANALYZER
Figure 4.1: Setup for measuring four-cavity power combining.
It is extrem ely im p o rta n t for th e analysis o f th e d a ta to u n d e rsta n d th e pow er
com bining of four in d e p e n d e n t cav ities. E q u ally im p o rta n t, it helps in th e design
a n d ru n plan to u n d e rsta n d th e effects o f cav ity frequency an d Q m ism a tch . If a lot
o f tim e m ust be sp en t m a tc h in g th e cavities, th e dead-tim e increases significantly
so a trade-off is necessary.
A form ula for th e ph ase-sen sitiv e com bined pow er from four cavities was deriv ed
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
121
in C h a p te r Tw o a n d th is se c tio n describes th e te st p erfo rm ed to v a lid a te th a t ex­
pression. T h e se tu p for th is te s t is show n in Figure (4.1). Tw o id e n tic a l W ilkinson
co m b in er/d iv id ers are used. P ow er fro m th e sweep g e n erato r is fed th ro u g h th e di­
v id er and in to th e four m in o r p o rts. T h e m a jo r p o rts are c o n n ec te d to th e com biner
a n d th en to th e netw ork a n a ly z e r th ro u g h one stage o f am p lificatio n .
Before th e com bined m e a su re m e n t is m ade, th e tran sm issio n cu rv e for each cav­
ity was m easu red in d iv id u a lly to d e te rm in e th e resonant frequency, Q , a n d power
on resonance. T h e n , th e y w ere c o n n ected to th e com biner a n d m e asu re d sim u ltan e­
ously. Figures (4.2 - 4.6) a re th e re su lts o f five different te sts p erfo rm ed w ith th e four
cavities in differing sta te s o f m ism a tc h . T h e solid curves a re th e re su lts pred icted
from E q u atio n (2.81) using th e m e asu re m e n ts from th e in d iv id u al cav ities, an d are
n o t th e resu lts of fittin g . T h e d o tte d curves in d icate th e resu lts o f sim p ly adding
th e tra n s m itte d pow er from th e fo u r cav ities w ithout ta k in g th e re la tiv e phases into
account. I define th e co m b in in g efficiency as th e ratio of th e a re a u n d e r th e solid
curve to th e a re a u n d e r th e d o tte d curve, it is a m easure o f how m u c h pow er is lost
in th e isolation resistors o f th e com biner. T h e agreem ent b etw een th e th eo ry and
ex p erim en t is excellent a n d I co n clu d e th a t th e pow er com bining is w ell-understood.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
122
20x10
E
QJ
£o
cu
o
O Measured
■ Combined in Phase
Simple Sum
15-
10 -
£
CO
C3
5-
o-l
809.8
809.9
810.0
Frequency (MHz)
810.1
810.2
F igure 4.2: Power combining measurement w ith all four cavities within 1 kHz (cavity BW
= 80 kHz). T he circles are the measured values, th e solid line is the expected curve from
equation (2.81), and the dotted curve shows th e results of simply adding the individual
cavity transm ission curves.
O Measured
—— Combined in Phase
Simple Sum
E
E
CO
e5
C
Um
809.8
809.9
810.0
Frequency (MHz)
810.1
810.2
F ig u re 4.3: Power combining m easurem ent with one cavity offset 100 kHz.
R ep ro d u ced with p erm ission o f th e copyright ow ner. Further reproduction prohibited w ithout p erm ission .
o Measured
— Combined in Phase
— Simple Sum
8 x 10'
£
E
E
C/3
e
C3
Urn
1 -.
809.8
809.9
810.1
810.0
Frequency (MHz)
810.2
F igure 4.4: Power combining measurement with one cavity offset 100 kHz, and a second
one offset 150 kHz.
O Measured
—— Combined in Phase
- - Simple Sum
£
£
E
V)
0c3
809.8
809.9
810.0
Frequency (MHz)
810.1
810.2
F igure 4.5: Power combining measurement with one cavity offset 100 kHz, and a second
one offset 110 MHz.
R ep ro d u ced with p erm ission o f the copyright ow ner. Further reproduction prohibited w ithout p erm ission.
124
O Measured
——Combined in Phase
- - Simple Sum
£
£
£o
cu
-o
cj
gc/5
e
6
4
2
809.8
809.9
810.2
810.1
810.0
Frequency (MHz)
810.3
810.4
F ig u re 4.6: Power combining measurement with one cavity offset 100 kHz, and a second
one offset "200 kHz.
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125
4.2
Scanning Test
T h e n e x t te st perform ed w as a d ry ru n o f th e e x p erim en t. T h e cavities were a t 1 .4
K, a n d th e m ag n etic field w as a t 8.0 T . T h e o b jectiv es o f th e te st were to verify
th a t th e w hole sy stem w orked, a n d to te s t a feedback m ech an ism for controlling th e
step size.
Degree
A fo
■o # Cycles
Exponential
Filter
Setpoint
F ig u re 4.7: Block diagram of the piezo feedback system.
T h e first step was to m a tc h all o f th e cavities to th e sam e frequency, in th is
case 810.90 M H z. A ro b u st m e th o d o f co n tro llin g th e step size is th e P ro p o rtio n alIn te g ra l (P I) feedback loop show n in F ig u re (4.7). I im p lem en ted th is m eth o d for th e
m ech an ical gears and is has p ro v en to b e q u ite successful. B ecause of th e G aussian
n a tu re o f th e steps (see F ig u re (3.33)), th e feedback is p erfo rm ed on an ex p o n en tial
average o f th e c av ity freq u en cy sh ifts, a n d n o t ju s t on th e la te st m easu rem en t. T h e
expression for an ex p o n en tial filter o f degree D is
N e w A v e ra g e = { 0 U A v e r a 9 e l » ( P ~ 1) +
V*™ )
(4
4)
A s im u la te d scan was th e n s ta rte d , a s te p was ta k e n w ith all fo u r cavities, a n d th e n
all fo u r frequencies an d Q ’s w ere m e asu re d . In a real scan, th e F F T w ould have
in te g ra te d for several m in u te s a t each ste p , b u t th a t tim e was reduced to a few
seconds. T h e step size for each c av ity was a d ju ste d by its in d iv id u al feedback loop
to keep th e average frequen cy sh ift a t 2 kHz.
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1 26
ion
Cavity # I
Cavity #2
Cavity #3
Cavity #4
\
o
-10 -J
810.90
810.95
811.00
811.05
Average Frequency (MHz)
811.10
811.15
F ig u re 4.8: Frequency deviations in scan test.
F ig u re (4.8) shows th e four cavity- frequencies w ith th e ir average su b tra c te d , as
a fu n ctio n of th e ir average freq u en cy over th e first 250 kHz of step p in g . T h e cavity
b an d w id th was 8 kHz, so all fo u r cavities stayed w ithin one b a n d w id th for th e entire
tim e , often m uch closer th a n th a t. F ig u re (4.9) shows th e com bining efficiency versus
frequency w hich rem ained above 90% for th e en tire scan. It w ould have rem ained
closer to 95% except for th e d e v iatio n m a d e by cavity 1.
T h is result was very good new s. A tran sm issio n m easu rem en t ta k es a few seconds
to perform , so it was hoped t h a t th e step p in g would be reliable e n o u g h th a t only one
c a v ity w ould have to b e m e asu re d p e r cycle. This proved to be th e case, th e re were
no large deviations or o th e r p ro b lem s observed over th e course o f several hundred
step s, so certain ly every four step s is a safe tim e interval. O f course, w hen a large
d e v iatio n is m easured, th e th re e previous m easurem ents becom e q u estio n ab le, b u t
th is a p p ears to be a very sm all p e rc en ta g e of the tim e.
A fte r this te st, I m ade a few m odifications to th e step size reg u latio n .
If a
c a v ity is m ore th a n one ty p ic a l ste p size ahead of w here it sh o u ld b e because of an
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
127
1.00 —r
0 .9 8 0 .9 6 0 .9 4 ■2 0 .9 2 S 0.90
0 .8 8 -
0.8 6 810.90
810.95
811.00
811.05
Frequency (MHz)
811.10
811.15
F ig ure 4.9: Power combining efficiency versus average cavity frequency in th e scan test.
unusually large step , a step is skipped. Likewise, if it is two or m ore ste p s behind,
e x tra step s are tak en . A ny tim e th e com bining efficiency drops below a p reset level
(90 - 95% ), all four cavities are m a tc h ed again. M atching all four cav ities can take
a ro u n d a m in u te , as experience is gained in production running, th e th re sh o ld for
com bining efficiency m ay b e ad ju sted .
Before th ese te sts, I perfo rm ed a M onte C arlo sim ulation w here I a ssu m e d th a t
th e four cavities step p e d w ith G aussian d istrib u te d frequency shifts w ith a typical
m e an and s c a tte r found in previous te sts. T h e m easurem ents were w ith in th e error
of th e sim ulation, b u t th e com bining efficiency was always b e tte r th a n p red icted .
T h is led m e to in v estig ate th e independence of th e steps.
A series of fixed sized step s was ta k en w ith one of th e cavities, a n d th e fre­
quency shifts w ere m easured . T h e dev iatio n s from th e m ean freq u en cy shift were
d e term in e d , a n d th e au to co rre latio n was tak en . T h e result is shown in F ig u re (4.10).
If th e steps w ere in d e p en d e n t, th e re w ould b e a peak at zero offset a n d n o th in g else.
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128
T h e dip a t ± 1 b in shows th a t th e steps axe n o t q u ite in d e p en d e n t. T h e dip is neg­
ative, m ean in g th a t th e a d ja c e n t deviations are o f o p p o site sign, i.e. a sm all step
is followed by a large one o r vice versa. If th e ste p s w ere in d ep en d en t, th e cav ity
w ouldn’t try to ” c a tc h -u p ” a fte r too sm all o f a ste p , b u t th e re is som e m em ory in th e
system w hich a p p e a rs to b e beneficial. Figure (4.11) shows th e sam e m easu rem en t
when feedback is in tro d u c e d . T h e dip is sm eared b y th e ex p o n en tial filter, th e re is
less of a ten d en c y to c a tc h u p . For this reason, th e degree o f th e filter is usually k ep t
a t 2 or 3, enough to sm o o th fluctuations w ith o u t c o m p letely losing th e m em ory.
150
100
oc
50.-
cu
_o
3
<
-50
-100
-20.0
-15.0
-10.0
-5.0
0.0
5.0
10.0
15.0
20.0
O ffset
F ig u re 4.10: A utocorrelation of the frequency shifts with fixed step size.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
129
150
100-
c
c
a>
50 ...
<
-50
-20.0
-15.0
- 10.0
-5.0
0.0
5.0
10.0
15.0
20.0
O ffset
F ig u re 4.11: A utocorrelation of the frequency shifts w ith feedback-controlled step size.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
130
4.3
Data Scan
Since it w ould m a rk a very im p o rta n t m ilesto n e, a sm all scan was p erform ed w here
sen sitiv ity to K SV Z model axions was ach iev ed over a sm all frequency b an d . T h e re
was no im age rejection at th e in te rm e d ia te freq u en cy used, so th e sy stem noise
te m p e ra tu re was doubled, reducing th e scan ra te to 40 kH z/day.
T h e first step was to critically couple all fo u r cavities.
a w eakly coupled transm ission m easu rem en t; a t 1.4 Iv
coupling, Q l =
\ Q
Un i
Q
Q
u n i
u n i
was d e term in e d w ith
= 220000. A t c ritic a l
so the m ajo r p o rt co u p lin g o f each cavity was a d ju sted u n til
th e Q was approxim ately 110000.
N ext, th e resonant frequency of each c a v ity was m atched to th e s ta rt freq u en cy
S12.54 M Hz. T h e integration tim e req u ired for th e cavity b an d w id th was several
hours.
T he pow er spectra were dow nloaded to th e co m p u ter every 80 seconds,
corresponding to th e average o f 10000 in d iv id u a l traces. Each tim e th e sp e c tru m was
dow nloaded, th e frequency of one o f th e cav ities was rechecked w ith a tra n sm issio n
m easu rem en t.
T h e cavities v/ere cycled th ro u g h , so each cavity was re-checked
roughly every 5 m inutes. T h e cavities re m a in e d very stable th ro u g h o u t th e e n tire
scan.
A fter sufficient integration tim e a t a p a rtic u la r frequency, a 15 kH z step was
ta k e n an d all four cavities were rem atch ed . S ince th e in teg ratio n tim e was so long,
it was possible to spend the necessary tim e to m a tc h all cavity frequencies w ith in 2 0 0
Hz w ith o u t having a significant d e ad -tim e p ercen tag e. T h e 15 kHz step was eq u al
to tw o cavity bandw idths, th e region was covered in two scans, th e frequencies in
th e second scan w ere offset 7.5 kHz from th e first scan. T h e frequency range covered
was 812.54 to 812.62 MHz.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
131
4.3.1
Data Combining
T h e individual pow er s p e c tra m u st b e norm alized a n d com bined to fo rm th e overall
noise pow er sp e c tru m . Since th e axion pow er in a given frequency b in d ep en d s on its
offset from th e c av ity reso n an t frequency, a w eighted averaging m e th o d is em ployed.
90x10'
8070-
cu
6050-
20
30
40
Intermediate Frequency (kHz)
50
F igure 4.12: Typical raw F F T power spectrum with 10000 averages.
A ty p ic a l raw sp e c tru m is show n in Figure (4.12). T h e s tru c tu re is d o m in a te d
by th e c ry sta l filter tra n sfe r function. T h e cavity noise sp ec tru m is fo u n d by nor­
m alizing th is sp e c tru m w ith th e m e asu re d cry stal filter tran sfer fu n c tio n . T h e filter
b a n d w id th is 40 kH z, a sp an req u irin g roughly one day of in te g ra tio n for th e d a ta
scan.
T h is m eans th a t th e cry stal filter tra n sfe r function m u st b e averaged for
several days. T h is was perform ed d u rin g th e assem bly process.
T y p ical n orm alized s p e c tra are show n in Figures (4.13 an d 4.14), F ig u re (4.14)
contains a peak from a n e x tern al rad io source.
T h e L orentzian sh a p e is d u e to
coupling betw een th e cav ity a n d th e first cryogenic amplifier[73]. N oise from th e
te rm in a tio n in th e in p u t h y b rid o f th e first am plifier travels to th e c a v ity w here it is
reflected a t th e m a jo r p o rt a n d th e n am plified w hen it reenters th e a m p lifier in p u t.
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132
6.4
S 6.3
CL.
6.2
I | I I I 11U I I [ I I I I | I I I I | I I I I | I I I I | I
20
30
40
50
Intermediate Frequency (kHz)
20
30
40
50
Intermediate Frequency (kHz)
F igure 4.13: F F T power spectrum after crystal filter norm alization (left) showing the
five-parameter fit and fit-normalized spectrum (right) .
;
7 . 4 -j
=L 7 -2 -i
% 7 .0 -j
£
6.8-
m
&UMi i,
Mil JuaVHAj
l,.|l
r Tf1
£
3
C.
*
6.6-
C/3
111 i\ if ri r r 111 111 1 11 ii i »11 1 r r p r n -i
20
30
40
50
Intermediate Frequency (kHz)
20
30
40
Intermediate Frequency (kHz)
F ig u re 4.14: F F T power spectrum after crystal filter norm alization (left) showing the
five-parameter fit and fit-normalized spectrum (right). This spectrum contains a peak
later determined to be from an external source.
T h is effectively increases th e noise te m p e ra tu re aw ay fro m th e cav ity resonance.
T h is sp ec tru m is u n d e rsto o d w ith a n equivalent circu it m o d e l an d is norm alized
w ith a fiv e-p aram eter fit. T h e fits a re show n as solid lines on th e left o f F ig u res (4.13
a n d 4.14), and th e fit-n o rm alized tra c e s are show n on th e rig h t.
T h e re is a sm all sy ste m a tic in th e fit-n o rm alized sp ec tra . T h e ex act origin of th e
effect is unknow n, b u t it o rig in ate s in th e IF electro n ics. Since it is in d e p en d e n t of
th e cav ity frequency, th e sy ste m a tic is e lim in a te d b y n o rm alizin g each fit-n o rm alized
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
133
0 .4 -
0. 2 -
o
-0 .4
20
30
40
Interm ediate F req u en cy (k H z)
50
F ig u re 4.15: Residual of the F F T fit function found by averaging all of the fit-normalized
spectra.
s p e c tru m by th e average of all fit-n o rm alized sp e c tra ta k e n d u rin g th e scan. T h e av­
erage o f th e s p e c tra is show n in F igure (4.15), n o rm alized to th e s ta n d a rd dev iatio n
in an in d iv id u a l sp ec tru m .
A fte r th e sy ste m a tic norm alizatio n , th e resu ltin g flat noise sp e c tru m is w eighted
an d co -ad d ed using th e m e th o d developed by Ed Daw[71]. A fter th e n o rm alizatio n
p ro c e d u re , th e flu c tu a tio n in th e j t h b in of th e ith s p e c tru m 5,-j is dim ensionless.
T h e s ta n d a rd d ev iatio n in th e zth sp e c tru m <r,- is given by
<Ti =
1
n\
(4.2)
J= 1
F or G au ssian noise, th e 07 =
w here N is th e n u m b e r of averages, ty p ically 10000.
T h e flu c tu a tio n s a re converted to W atts by m u ltip ly in g b y th e average noise power
Pnj in th e j t h b in . Pnj = fcsTj A / w here A / is th e re so lu tio n of th e F F T s p e c tru m
(125 H z) a n d T j is th e sy stem noise te m p e ra tu re in th e j t h bin. T h e sy stem noise
te m p e ra tu re is given by T j = T cav -f ctjTamp w here ocj is a fa c to r w hich accounts
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
134
1 .0 1 5 -
1. 0 1 0 -
1.005Cavity Freq.
1.000 -
0.995 -L
20
30
40
50
Intermediate Frequency (kHz)
F ig u re 4.16: Noise tem perature correction factor a j for a typical spectrum . The correc­
tion is only about 1 percent for this case.
for th e increase in th e effective am plifier noise te m p e ra tu re o u tsid e of th e cavity
b an d w id th . Figure (4.16) shows a j for a typical sp ectru m . T h e s ta n d a rd deviation
for th e j t h bin of th e fth s p e c tru m becom es cr,j = Pnjcr,-.
T h e fluctuation in th e j t h b in o f th e com bined d a ta Sj is th e w eighted average
o f th e n sp ectra which cover th a t frequency 5j = X3"=i
T h e w eighting factor
W{j w hich m axim izes th e com bined SN R is
h -Paxit> x 7 1
(4.3)
= - * tj r
w here h,j is th e cavity pow er-com bining form ula norm alized so th a t h = 1 on reso­
n a n ce w hen all four cavities a re p erfectly m atch ed . Pa{ is th e th e o re tic a l signal power
in th e zth sp ectru m found fro m E q u atio n (2.77) w ith 5 = 0. W ith th is w eighting
fa c to r, th e fluctuations, s ta n d a rd deviation, an d th eo retical signal pow er in th e j t h
b in o f th e com bined d a ta are
*j =
(4.4)
i)M -
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
135
" = E ,(M W /« S
(4 ' 0)
x M jP M r t
Pi ~ e , ( m w / 4
, lfn
„
(4-6)
T h e signal-to-noise ra tio in th e j t h b in is given by
SNRj = — =
£
h?-P2^
a ij
(4-7)
T h e d a ta stre a m w hich is searched for peaks is th e flu ctu atio n s in u n its of sigm a
d
(4.8)
§L =
T he signal pow er a n d rm s noise level p er 125 H z is show n in Figure (4.17).
T h e signal s p e c tru m w ill be sm o o th er in p ro d u c tio n ru n n in g w hen sm aller step s are
taken, an d SN R fla tte n in g scans a re perfo rm ed . T h e S N R from th is d a ta can be
quickly tu rn e d into a n exclusion lim it on th e tw o-photon coupling co n stan t ga~n-
4.3.2
Single-bin Power Excess Search
T h e com bined d a ta s tre a m can be used to place a lim it on ga^
for axions w hich
deposit all of th e ir pow er in a single 125 Hz bin. T h is d a ta s tre a m is shown in Fig­
ure (4.18). T h e tw o larg e flu ctu atio n s w ere associated w ith e x te rn a l radio sources,
as d e term in ed by a s p e c tru m an aly zer a n d an e x te rn a l a n te n n a in th e ex p erim en t
hall. No o th e r c a n d id a te peaks passed th e cu t a t 3.3 a . From [71], a single-bin
signal w ith SN R o f 7.25 w ould pass th e 3.3 cr cu t w ith 90% confidence. T h e absence
o f peaks m eans th a t th e pow er in th e j t h bin excluded w ith 90% confidence is
P (9 0 % )j
P ( K S V Z )j
SNRj
7.25
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(4.9)
136
.-21
.-22
U+
<L>
£
o
cu
KSVZ
,-23
noise
. 2-
8 1 2 .5 4
812.56
8 1 2 .5 8
81 2 .6 0
Frequency (M H z)
812.62
F ig u re 4.17: Signal and noise power/125 Hz versus frequency.
T his is co n v erted to a lim it o n ga-n using E q u a tio n (2.75). Since P ( K S V Z ) oc g*
g 2a^ ( 9 0 % h
P ( 9 0 % )/
g*{KSVZ)s
P{KSVZ)j
(4.10)
T h e freq u en cy d ep en d en ce o f ga-n is often rem o v ed by norm alizing w ith th e axion
m ass m a. In th e K SV Z m odel, m a is re la te d to g a.n using E quations (2.7) an d
(2. 10).
-2
f 5Wy.V G e V
\m a )
eV 2
a \ 2 (10 7 G e V ) - 2
(0.6 e V ) 2
-» (s )
(4.11)
T h e re su ltin g exclusion p lo t for single-bin ax io n p eak s is show n in Figure (4.19).
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
E x tern a l S ig n a ls
O,
E
3.3 a
o<L>
ex,
0T3
U
’o
C/3
z
CQ
812.54
812.56
812.58
812.60
Frequency (MHz)
812.62
Figure 4.18: Fluctuations for single-bin peak search.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
ma(|J.eV)
3.36392
3.36408
3.36424
1-17
>4>
>
4)
o
3
a
00
,-18
KSVZ
812.56
8I2_58
812.60
812.62
Frequency (MHz)
F igure 4.19: Single-bin axion exclusion region from the four-cavity scan.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
139
4.3.3
Six-bin Power Excess Search
If galactic halo axions a re th erm alized , th e signal in th e d a ta sh o u ld b e approxi­
m a te ly 800 Hz w ide. T h is is roughly six bins in th e co m b in ed d a ta . To search for
th is signal, a six-bin ru n n in g average was a p p lied to th e c o m b in ed d a ta . T his d a ta
stre a m is show n in F ig u re (4.20). T h e s ta n d a rd d e v ia tio n is re d u c e d as expected,
and this search is n o t sen sitiv e to th e narrow radio p eak s seen in th e single-bin d ata.
T h e non-G aussian n a tu re o f th e d a ta is re la te d to th e IF sy ste m a tic d escrib ed earlier
a n d th e fact th a t a d ja c e n t bins are c o rrelated because o f th e ru n n in g average. T h e
problem is th a t th e s ta tis tic s are no longer G aussian, a n d a M o n te C arlo m eth o d
m u st be used to d e te rm in e th e sensitivity. Peaks w ith th e e x p e c te d M axw ellian
lineshape w ere in je c ted in to th e d a ta w ith varying pow er levels to d e te rm in e which
pow er level was d e te c te d w ith 90% confidence w ith th e c u t p laced a t 1.1 a as seen
in Figure (4.20). A fte r th is was d eterm in ed , conversion to a n ex clu sio n lim it on ga^
is identical to th e single-bin case. T h e exclusion plot for th e rm a liz e d g alactic halo
axions is show n in F ig u re (4.21).
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812.54
812.56
812.58
812.60
Frequency (MHz)
812.62
F igure 4.20: Fluctuations for six-bin peak search. The stru ctu re in the d a ta is related to
the IF system atic and the correlations between adjacent bins after the running average.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
maOieV)
336392
3.36408
336424
.-17
><u
>
u
O
.-IS
E
KSVZ
812.56
812.58
812.60
812.62
Frequency (M Hz)
Figure 4.21: Six-bin axion exclusion region from the four-cavity scan.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
142
4.4
Conclusions
T h e engineering ru n of th e fo u r-cav ity a rra y was very successful. T h e p iezoelectric
a c tu a to rs w orked a t 1.4 K a n d 8 T . T h e step reso lu tio n was ex cellent, a n d th e
h e a tin g was m inim al. F ind in g th e o p tim a l d riv in g frequencies was ted io u s, b u t the
re su lts w ere consistent betw een cooldow ns so it only h a d to be done once.
The
te sts also p o in te d o u t several im p ro v em en ts w hich will b e m a d e before th e s ta r t of
p ro d u c tio n ru n ning.
T h e re was som e cross-talk b etw een th e lin ear an d ro ta ry drives caused by ca­
p a citiv e coupling betw een th e leads. A t ro o m te m p e ra tu re an d even a t 77 Iv, this
is n o t a p ro b lem because th e p iezo electric c ap a c itan c e is g re a te r th a n th is p a ra sitic
c ap a c itan c e . B u t, as seen in F ig u re (3.20), a t cryogenic te m p e ra tu re s, th e d ielectric
c o n sta n t o f th e P Z T drops an d th e p iezo electric cap a c itan c e becom es co m p arab le
to th e cable capacitance. S e p a ra tin g th e leads for th e lin ear drives an d th e ro ta ry
drives sho u ld allev iate this p roblem .
T h e sy stem w orked well enough to p ro d u c e th e first lim it w ith a m u ltip le-cav ity
d e te c to r. W ith a few im p ro v em en ts, p ro d u c tio n d a ta ta k in g will com m ence w ith
a scan ra te com parable to th a t o f th e single-cavity d e tec to r. Figure (4.22) shows
th e lim its c u rre n tly set by m icrow ave c av ity searches, in d ic a tin g th e coverage o f our
single a n d four-cavity detecto rs.
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143
Rochester-BN L-FNAL
U. Florida
LLNL/MIT/UF
DFSZ
1-cavity
4-cavity
M (p.eV)
a
F ig u re 4.22: C urrent exclusion region for axion-photon coupling from microwave cavity
searches.
R e p ro d u c e d with permission of the copyright owner. Further reproduction prohibited without perm ission.
Chapter 5
Future Directions
T h e work described in this thesis has d e m o n stra te d th a t pow er-com bining m u ltip le
cavities is a viable m eans of e x ten d in g th e m ass coverage of m icrow ave c av ity axion
detecto rs. T h e piezoelectric a c tu a to rs developed for cryogenic o p e ra tio n in a high
m agnetic field w ill play a v ita l role in fu tu re generations of th e e x p erim en t.
P red ictio n s of m a from strin g m odels are typically 0 (1 0 0 ^ eV ), req u irin g cavities
w ith /oio ~ 25 GH z. T he tech n iq u e of pow er-com bining signals from m a n y sm all
cavities is only p ra ctical for frequencies up to « 3 GHz, because th e n u m b e r of
cavities required scales as /q 10, w here /oio is th e frequency of th e TMoio m o d e. T h e
n u m b e r of cavities N c which fill th e m a g n et volum e scales as N c oc R ~2£~l . Naively,
one w ould ex p ect N c oc /q 10 since fo w oc R ~ l . B u t, in order to keep th e TMoio
m ode d o m in an t, th e aspect ra tio o f th e cavities m u st be m ain tain ed , so £ oc R . As
th e radii of th e cavities decreases, th e len g th s m u st decrease as well a n d as a resu lt
N c oc /oio- A n a lte rn ativ e for reaching higher frequencies is th e stra teg ic p lacem en t
of m e ta l posts inside a single larg er cavity. [6 ] P relim in ary work on th is is alread y
beginning.
F ig u re (5.1) shows a tria n g u la r la ttic e o f 19 posts inside a circu lar cavity. W ith
r / R = 0 . 1 , th is a rran g em en t raises th e T M 0io frequency by a facto r o f 5 co m p ared
R e p ro d u c e d with perm ission of the copyright owner. Furth er reproduction prohibited without permission.
145
T
2R
F ig u re 5.1: Layout of a multi-post cavity with a triangular lattice of 19 posts.
to th e em p ty cavity value. T h e form factor is ss 0.5, a reaso n ab le value for a cav ity
axion detecto r. It is im p o rta n t to note th a t th e u sa b le volum e in th is configuration
is m uch g re a ter th a n th a t o f th e corresponding single c av ity w ith th e sam e resonant
frequency.
T uning these cavities can not be done w ith a sim ple tu n in g rod as before.
T ran slatio n al sy m m e try in th e axial direction m u s t b e m a in ta in e d to avoid m odelocalization, so in sertin g a sm all dielectric rod along th e axis is also unacceptable.
M odeling indicates th a t th e fu n d am en tal freq u en cy scales rou g h ly as a -2 , w here a
is th e n earest-neighbor spacing in th e la ttic e, so th is m u st b e changed to tu n e th e
cavity.
A sim ple m e th o d for changing th e la ttic e sp acin g is show n in Figure (5.2). T h e
cen tered square la ttic e on th e left can be b ro k en dow n in to two square la ttic e s
w ith spacing 2a. O ne of th e la ttic es is tra n s la te d as in d ic a ted , an d th e s tru c tu re
tran sfo rm s into th e re c tan g u la r la ttic e shown on th e rig h t. T h e tu n in g b a n d w id th
A /o io //o io is ~ 1 0 %.
A p ro to ty p e c av ity w ith 85 posts has been c o n stru c te d . T h e posts are arran g ed
as tw o in tertw in ed sq u are la ttic es o f 36 an d 49 p o sts respectively. One la ttic e is
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
146
2a
o
o o
I
© ©I ©1 •
o o o
©i ©I ©i ©
o o o
©I ©I 9 i ©
H
t
i
3
\*~
®o ®o ©o
© o
•
o
e
o
©
O © O © O ®
F ig u re 5.2: Possible tuning scheme for the multi-post cavities. As the offset between
the two square lattices is changed, the arrangem ent changes from a triangular lattice to a
rectangular lattice.
A
2
H
l \
\
v
/
\\
-4 5 -
-50 - <
J
/
17 I 1 I I I I I' I I I I I I I I I I I I I I I I I I I I I IT
7 .8 0
7 .8 4
7 .8 8
7 .9 2
F requency (G H z)
3ffl
IT
£
p
cu
-4 2 :
- At“
AT -
£
-4 6 -
H
I
\
M
/
:
"O
y
CA
/
-3 8 :
~4 0 ~-
00
(£
-a
o
=
1 A\
A /
IV
1 V J/
-4 0 -
4\
\
M
/
/
M
lLl
A
A
/ \
ran
CQ -35“
•o
:
i i i i i"Ti i n i T m i »i i i i t i i i i i i i i r
7 .8 3 5 7 .8 4 0 7 .8 4 5 7 .8 5 0 7 .8 5 5 7 .8 6 0 7 .8 6 5
F req u en cy (G H z )
F ig u re 5.3 : Transmission m easurem ent on a prototype cavity containing 85 posts showing
the lowest TM modes. The graph on th e right is a close-up of the TMoio mode.
fixed, w hile th e o th e r can b e m o ved as a group, tu n in g th e cavity.
F ig u re (5.3)
shows th e lowest th re e T M m o d es for th e p ro to ty p e cavity.
M ore stu d y is required to d e te rm in e if th is is a feasible m e th o d to ex p lo re higher
frequencies. T h e biggest concerns are lim ited tu n in g b a n d w id th , c o n stra in ts on the
ph y sical dim ensions to avoid m o d e-lo calizatio n , possible d e g ra d a tio n o f th e form
fa c to r as th e cavity is tu n e d , a n d th e d e n sity o f in te rfe rin g T E a n d T E M m odes at
h ig h er frequencies. R esolving th e se issues could e x te n d th e m ass ran g e o f m icrow ave
c av ity axion searches by a n o th e r d ecad e.
T h e piezoelectric based tu n in g a n d coupling m ech an ism p ro v id e high s te p resolu-
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
147
tio n in a very co m p act package, m ak in g th e m essen tial for m u ltip le cavity d etecto rs.
A d d itio n ally , th e y g en erate negligible h e a t d u rin g n o rm a l o p eratio n .
T h e u ltim a te goal o f th is e x p erim en t is to scan th e low est tw o decades o f th e
axion w indow (1 - 100 /^eV) a t D FSZ sensitivity. T h is will req u ire a t least an order
o f m a g n itu d e im provem ent in power sen sitiv ity , m o re if we w ant to increase th e scan
ra te a n d re la x th e a ssu m p tio n th a t th e e n tire local halo d en sity is com prised o f ax­
ions. T h e only p ra c tic a l w ay to achieve th is is by low ering th e physical te m p e ra tu re
o f th e cavities an d th e noise te m p e ra tu re o f th e first am plifier.
T h e cavities c an be cooled to m illikelvin te m p e ra tu re s w ith a d ilu tio n refriger­
a to r. T h e cooling pow er is q u ite low, so th e d e te c to r m u st b e th e rm a lly isolated;
th is is n e arly im possible w ith th e m echanical connections req u ired for th e step p erm o to r based tu n in g a n d coupling m echanism s. W ith th e piezoelectric a ctu a to rs,
th is p ro b lem is solved.
A lth o u g h th e physical te m p e ra tu re of th e cavities could b e lowered below 100
m K , th e b e st H F E T am plifiers available to d ay have noise te m p e ra tu re s above 1 K. It
is unlikely th a t th e y will soon get below 500 m K . Soon, th e H F E T s will b e replaced
w ith a new k in d o f m icrow ave am plifier, based on dc S u p erco n d u ctin g Q u an tu m
In terferen ce D evices (dc SQ U ID s) being developed a t UC B erkeley. T hese am plifiers
a re a p p ro ach in g q u a n tu m -lim ite d noise te m p e ra tu re s (35 m K a t 500 M H z). In an
a p p e n d ix , I describ e these am plifiers along w ith m y w ork in Jo h n C larke’s group at
B erkeley.
T h is m u ltip le cavity search m arks th e end of th e so-called ’Second G en eratio n ’
axion search ex p erim en t.
Efforts axe now shifting to dev elo p m en t of th e T h ird
G e n e ra tio n e x p erim en t, u tilizin g dc SQ U ID am plifiers an d a d ilu tio n refrigerator.
A lth o u g h no axion was found, th e ex p erim en t has b een a g re a t success. W ith th e
ex p erien ce a n d technological advances o f th e p a st few yeaxs we a re now in a position
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
to perform th e definitive axion search.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
1*9
Appendix A
dc SQUID Amplifiers
T h is ap p en d ix describes th e theory a n d c o n stru c tio n o f th e dc S u p erco n d u ctin g
Q u a n tu m Interference D evice (dc SQ U ID ) b a se d m icrow ave am plifiers being devel­
oped for th e n e x t generatio n m icrow ave c a v ity axion d e tec to r.
T hese am plifiers
offer an o rd er of m a g n itu d e im p rovem ent in noise te m p e ra tu re over sta te -o f-th e -a rt
H F E T am plifiers, a n d are ap p ro ach in g q u a n tu m -lim ite d p erform ance. A sch e m a tic
of a dc SQ U ID am plifier is show n in F ig u re ( A .l) . A su p erco n d u ctin g loop w ith
tw o Josephson ju n c tio n s in p arallel encloses a to ta l flux 4>. T h is flux is th e su m of
a fixed flux su p p lied by th e bias coil a n d th e signal flux supplied by th e in p u t coil.
T h e phase difference betw een th e two sides o f th e loop is affected by ch an g in g $
resu ltin g in interference effects analogous to th e tw o-slit ex p erim en t in optics. T h e
re su lt is a very sensitive flux-to-voltage tra n s d u c e r[74].
A fter a b rie f overview of su p e rc o n d u c tiv ity a n d th e Josephson relatio n s, th e
design of dc SQ U ID am plifiers is discussed. F in ally , resu lts are presented on som e
devices w hich have been fa b ric a ted a n d te s te d a t U C Berkeley.
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150
Input Coil
+o
Bias
o+
+ a
o-
Flux Bias Coil
F ig u re A .l: Schematic of a dc SQUID configured as an R F amplifier. The X ’s mark the
Josephson junctions interrupting the superconducting loop.
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151
A .l
Superconductivity Overview
S u p erco n d u ctiv ity resu lts fro m th e spontaneous sy m m e try b reaking (SSB) o f th e
U ( l) p h ase invariance of e lectro m ag n etism which is linked to charge conservation
a n d gauge invariance. T h is sy m m e try also explains w hy th e p h o to n is m assless.
T h e G oldstone bosons asso ciated w ith this SSB are C ooper p airs, tw o electrons
w ith opp o site w avenum ber a n d spin com bine to form spin-0 p su ed o p articles. T he
g ro u n d s ta te a fte r SSB is a B ose-E instein condensate consisting o f zero -m o m en tu m
C ooper p airs d escribed by th e ir d en sity p s and phase
6
. T h e w avefunction ib is given
bv
tb
= -4=eJ&.
*■'
‘
\J Ps
Below th e c ritic a l te m p e ra tu re Tc. th e electrons form cooper p airs, all w ith th e
sam e phase. T his selection o f a p a rtic u la r phase is responsible for th e U (l) sym ­
m e try being sp o n tan eo u sly broken.
A fter the SSB, th e p h o to n is m assive inside
th e su p erco n d u c to r a n d since forces m ed ia ted by m assive gauge bosons are finite in
e x te n t, electric a n d m a g n etic fields o n ly have a lim ited e x te n t in su p erco n d u cto rs
d en o ted by th e p e n e tra tio n d e p th A. For typical su p erco n d u cto rs, A ss 50 nm . All
m a g n etic fields a n d electric c u rre n ts in a su p erconductor a re confined to w ith in th e
p e n e tra tio n d e p th of th e surface.
T h e p hase 0 is re la te d to th e g eneralized m om entum p by p = hW9. p is related
to th e c en te r of m ass velocity v an d v ecto r p o ten tial A by p = 2m ev — 2 eA . In
te rm s of th e su p erco n d u c tin g c u rre n t density j = —2 eps < v > , p can also be
expressed as
p = —— j — 2 eA
e-Ps
(A .l)
T h e B ohr-S om m erfeld q u a n tiz a tio n § p • df = n h gives
fp -d e =
j> j ■d£ — 2e
A ■d£ = n h
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(A.2)
152
Perform ing th e in teg ral inside th e su p erco n d u c to r w here j = 0 , an d usin g Stoke’s
T heorem gives
2e [ (V x A )-dS = 2e f B
JA
JA
dS = 2 e $ = n h
(A-3)
E quation (A .3) shows th a t th e flux enclosed by a su p erco n d u ctin g loop is q u an tized
in units of th e flux q u a n tu m 3>0 = A =2.0678 x 10 -1 5 T -m 2. T his is a n im p o rta n t
principle for th e o p e ra tio n o f SQ U ID s.
A.2
Josephson Junctions
'/W y M ^ /Y sy /ss///y //s/sy ///s
^^Superconductor
^ //////////////////<
Insulator
R
C ___
lc sin (5)
,„,777777777777777777m
jgzzSuperconductor^
m y/s'/y//s/yjr/ys//y/s/A
Figure A.2: Josephson junction (left) and the corresponding RCSJ circuit model (right).
A Josephson ju n c tio n is shown sch em atically on th e left of F ig u re (A .2). Tw o
superconductors are s e p a ra te d by an in su la tin g lay er w hich is sufficiently th in to
allow tu n n e lin g o f cooper pairs. T his leads to th e ac a n d dc Josephson effects. In
th e absence o f an y elec tric o r m ag n etic fields, a d c c u rre n t flows across th e ju n c tio n .
This is th e dc Josep h so n effect, and th e c u rre n t d e n sity J is given by
J = J c sin(£)
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(A .4)
153
5=
0 2
— 0i is th e p h a se difference betw een th e tw o su p erco n d u cto rs. J c is th e c ritic a l
density, th e m a x im u m zero-voltage c u rre n t fo r th e ju n c tio n ; it is a fu n ctio n o f th e
g eo m etry a n d p ro p e rtie s o f th e ju n c tio n m a te ria ls.
W hen a dc voltage V is applied to a Jo se p h so n ju n c tio n , ac c u rre n t oscillations
occur across it. T h is is th e ac Josephson effect, w hich occur w ith th e Josep h so n
frequency ujj
ojj
2eV
= 5 ——
( A. 5)
A dc voltage o f 1 fj,V corresponds to a Jo se p h so n freq u en cy of 4S3.6 M Hz.
A.2.1
RCSJ Circuit Model
A circuit m odel for a c u rre n t biased Jo sep h so n ju n c tio n is shown on th e rig h t o f
F igure (A .2). W h en th e in p u t c u rre n t I exceeds / c, th e re are th ree p arallel branches
for th e c u rre n t, th e Josephso n tu n n elin g c u rre n t I j = / c sin(5), th e ju n c tio n capaci­
ta n ce Ic =
an d th e ju n c tio n resistan ce I r =
T h e resistance is th e p arallel
co m b in atio n o f th e single-electron tu n n e lin g re sista n c e Rn and th e e x te rn a l sh u n t
resistance R s. In alm o st all cases R s <C R ^ so R ~ R s. T h is circuit is referred to as
th e R esistively a n d C ap acitiv ely S hu n ted J u n c tio n (R C S J) model.
T h e R C S J m o d el is useful for calcu latin g th e I-V curve for a Josephson ju n c tio n .
I is equal to th e su m o f th e c u rre n t th ro u g h th e th re e branches I = I c + I r + I j U sing (A .5) to convert voltages to phase differences gives
I = - ^ 5 -t- -^j^> + h sin(5)
(A . 6 )
E q u a tio n (A . 6 ) describes a driven d am p ed -o sc illa to r w ith ch aracteristic freq u en cy
u!p called th e p la sm a frequen cy o f th e ju n c tio n given by
=
1/75J
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( A ' 7)
154
1614-
12
>
-
10 -
2-
‘
o4
400
0
800
1200
time (psec)
F ig u re A.3: Voltage across a current-biased Josephson junction found from Equa­
tion (A.6 ) for I > Ic. T he measured dc voltage is shown by the dashed line.
T h e Q o f th e ju n c tio n is given by ujpR C . T h e o u tp u t is a series of pulses w ith
frequency
u j j
.
As long as th e Jo sep h so n frequency is m uch g re a te r th a n th e m ea­
su re m en t frequency, th e m e asu re d voltage will b e th e tim e average o f this signal.
F ig u re (A.3) shows th e v o ltag e across th e ju n c tio n as a function o f tim e; th e d ash ed
line shows th e average v o lta g e w hich is actu a lly m easured.
T h e re are two sp ecial cases w here E q u a tio n (A . 6 ) can be solved analytically,
w hen Q -C 1 and w hen Q
1
. W h en Q >
1
th e ju n c tio n is said to be u n d er-
d a m p ed . U nd erd am p ed ju n c tio n s disp lay hysteresis in th e ir I-V ch aracteristics a n d
a re n o t useful for S Q U ID s.
O verdam ped ju n c tio n s have Q «
1,
a n d th e capacitance can be ignored in
(A . 6 ). E xternal sh u n tin g resisto rs are o ften used to overdam p Josephson ju n c tio n s.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
155
N eg lecting C, eq u atio n (A . 6 ) becom es a first-o rd er differential eq u atio n .
6
'~>eR
= — ( I - I c sin(5))
n
(A.S)
S e p a ra tin g variables a n d in te g ra tin g over one p erio d
r2*
d6
2e R
I
,
r . fJ~ = - r - T
Jo I — I c sin(o)
n
(A .9)
P e rfo rm in g th e in teg ral and solving for th e p erio d T gives
T = - ^ - (
2irIc \
2cR \ J P - I * )
(A .10)
T h e tim e averaged voltage V is found from tim e-averaging (A.S).
K = i 5
2e
= l
2e
l
/
T Jo
r ^
dt
= l | :
2e
T
(a .n )
P lu g g in g (A .10) into ( A .ll ) gives th e I-V relatio n for a n o v erd am p ed Josephson
J u n c tio n w ith / > I c.
V = R y J P - Pc
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(A . 1 2 )
156
A.3
SQUID Basics
I
I
ml
F ig u re A.4: C urrents in a dc SQUID.
F ig u re (A.4) shows th e c u rre n ts in a dc SQUID w ith a c o n sta n t in p u t c u rre n t
I . I s is th e circu latin g s u p e rc u rre n t necessary to m a in ta in flux q u a n tiz a tio n . T h e
c u rre n ts / i and I 2 th ro u g h th e tw o ju n c tio n s are given b y I\ = \ I + I s, I 2 = \ I — IsU sing (A .4), I can be w ritte n as
I = h + h = / c(sin( 6'1) + sin (£ 2)) = 2 /c sin[^(<5'1 -f &2)] cos[^(£i — £2)]
w here
and
8 2
are th e p h a se differences across th e two ju n c tio n s.
Som m erfeld relatio n re w ritte n in te rm o f th e phase
6
(A .13)
T h e B ohr-
(§S/Q = 27rn) c o n stra in s th e
to ta l p hase change aro u n d th e loop to b e an integer m u ltip le of 2tt. T h is lead s to a
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157
relatio n betw een <5i a n d 52
<$1
— S2 = 2ir(n + -^-)
^O
(A .14)
n can be set to zero w ith o u t loss o f g en erality . In s e rtin g (A .14) in to th e eq u atio n
for / (A .13)
/ = 2 I c cos
sin [|(5 x + S2)]
(A.15)
T h e dc SQ U ID acts as a single Josephson J u n c tio n w ith a flux d ep en d en t critical
cu rren t I = / ' sin(V ) T h e critical c u rren t o f th e SQ U ID I'c = 2 I c cos
betw een 0 a n d 2 I c as th e flux changes from 0 to <&0.
8
varies
' = |( 5 i + S2) is sim ply th e
average o f th e tw o p h a se differences. F igure (A .S) show s th e V-<1> ch aracteristics of
a dc SQ U ID for different values of th e bias c u rre n t.
T h e I-V c h a ra c te ristic s o f a dc SQ U ID c a lc u la te d fro m (A . 12) for th re e different
values o f $ are show n in F ig u re (A . 6 ). For a given b ias c u rre n t I > 2I c, th e o u tp u t
voltage of th e S Q U ID changes as th e flux varies from a n in te g e r m u ltip le o f
<& 0
zo
a h alf-integer m u ltip le. T h e to ta l change in v o ltag e is th e m o d u la tio n voltage AV'
in d ic a ted on th e I-V c u rv e for th e case [ = 2[c. T h e p lo t on th e rig h t o f F ig u re (A . 6 )
shows th e v a ria tio n o f A V w ith bias c u rre n t. T h e m a x im u m m o d u la tio n occurs for
/ = 2 Ic.
T h e to ta l flux in th e loop $ is com prised o f tw o co m p o n en ts, th e e x tern ally
applied flux
an d th e flux from th e self-in d u ctan ce o f th e loop
= L I S- I s is
th e circ u latin g s u p e r-c u rre n t given by
Is =
— I 2 ) = 7c sin
cos (S')
(A. 16)
In th e case o f no b ias c u rre n t / = S’ = 0, th e re la tio n betw een th e e x tern al flux an d
th e flux th ro u g h th e loop is
^
-$ = - + An5m
(/7_T $)\
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( ?)
158
2.0
! = 4IC
0.0
0
I
2
3
4
F igure A.5: V - <2> Characteristics for a dc SQUID calculated from the RCSJ model for
different bias currents. The self-inductance of the SQUID loop has been ignored.
w here f3m =
is called th e m agnetic h y steresis p a ra m ete r. If th e inductance of
th e loop is to o high, th e flux induced by th e circu latin g su p er-cu rren t is enough
to cause th e flux in th e loop to increase by an in te g e r m u ltip le of $ 0, causing
un w anted hysteresis in th e relatio n betw een
an d $ x. T he condition for avoiding
th is hysteresis is
> 0
(A -18)
<&x=0
N e ar $ x = 0 , 3> ~ 0, so sin ( ^ ) ~ | r a n d th e co n d itio n (A .IS) becom es
*x=0
1 - TT&n
> 0
(A .19)
T h e condition for avoiding m agnetic hysteresis is seen to be (3m < K Figure (A .7)
shows calculations of $ versus $ x in th ree cases: h y ste re tic (j3m > - ) , non-hysteretic
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1 59
3.0
1.0
Z5
o.s
2.0
os
0.6
>
<
<t>= (n +■0.25)<t>t
0.4
t.O
0.2
0.5
0.0 J
0.0
1.0
1.5
0.0
2.0
I/(2IC)
t.o
1.5
2.0
3.0
35
4.0
I/(2IC)
F ig u re A . 6 : IV C haracteristics (left) and Modulation Voltage A V for a dc SQUID calculated from the RCSJ model.
(/?m < 7 ), an d th e c ritic a l p o in t (/3m '= -7T) . W hen th e bias c u rre n t is not zero,’ the
•
m axim um allow ed value of (3m is found from n u m erical sim u la tio n .
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16 0
4 -,
0
2
1
0
3
4
>X/ O o
Figure A .7: Total flux versus applied flux for a dc SQUID, showing the hysteresis for
0m > h-
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161
A.4
dc SQUIDs as RF Amplifiers
n+1
F ig u re A.S: Flux biasing of a dc SQUID.
T h e V -$ cu rv e in F ig u re (A.S) show s how th e dc SQ U ID can be flux biased to
act as a se n sitiv e flux-to-voltage tra n sd u c e r. A flux is ap p lied by a n external coil
as show n in F ig u re (A .l) so th a t th e flux in th e loop is eq u al to (n + ^)$o- A tin y
in p u t flux a p p lie d th ro u g h a sep a ra te in p u t coil causes th e o u tp u t voltage to change
significantly. T h e sen sitiv ity is given by th e slope of th e V -$ curve a t th e bias p o in t
** = §£■
T h e p a ra m e te rs o f th e SQ U ID are u su ally chosen to o p tim ize th e energy resolu­
tio n e ( / ) , defined as th e noise energy p e r u n it b a n d w id th a n d found from num erical
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162
sim ulations to be[75]
£ (/) = K>kBT j ! £
(A .20)
/?c is called th e S tew art-M cC u m b er p a ra m e te r. /3C < 0.7 is th e condition for th e
ju n ctio n s to be n o n -h y ste re tic, it is equivalent to Q 2 w here Q is th e q u ality facto r of
th e junctions defined e arlier. T h e energy resolution is im p ro v ed by lowering L an d
C.
T h e in d u ctan ce L sho u ld be sm all, b u t still allow flux to couple to th e SQ U ID ,
th e practical lim it is ro u g h ly 10 pH . W hen L is specified, I c is fixed by th e condition
,8 m ;$ 1 which is found to be o p tim al from num erical sim u latio n s.
T h e critical
cu rren t density should b e as large as possible, so th e ju n c tio n area is kept to a
m inim um . E ventually, lith o g rap h y lim its how sm all C can b e m ade, usually aro u n d
1
pF . Finally, th e co n d itio n (8 C < 0.7 fixes th e value o f th e sh u n t resistance R , w hich
is typically 10 - 2 0
M ost dc SQ U ID am plifiers are c o n stru cted using th e design of K etchen an d
Jaycox shown in F ig u re (A .9)[76]. T he SQ U ID loop is a sq u are w asher m ad e from
niobium (N b). T h e loop is closed by a se p a ra te N b c o u n terelectro d e connected to
th e w asher by th e tw o sm all ju n ctio n s a n d th e e x te rn a l sh u n t resistors. A spiral
in p u t coil is form ed on to p o f th e w asher, sep a ra te d by a n insulating layer. An
ex tern al coil, n o t shown, supplies th e bias flux.
Originally, th e in p u t signal was coupled betw een th e en d s of th e coil, form ing
a sim ple tran sfo rm er w ith th e in p u t coil as th e p rim a ry a n d SQUID loop as th e
secondary. T his w orked for frequencies up to 200 M Hz, a t which frequency th e
p arasitic capacitance betw een th e coil a n d th e SQ U ID w ash er becom es too g reat.
R ecently, this p ro b lem was solved by coupling th e in p u t signal betw een one end
of th e coil and th e SQ U ID w asher, w hich a cts as a g ro u n d plane for th e coil. A
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
F ig u re A .9: L ayout o f a typical square w asher dc SQ U ID amplifier.
m icro strip resonator is th u s form ed b y th e o p en -en d ed s trip lin e whose im p ed an ce
is d e te rm in e d by th e in d u c ta n c e o f th e in p u t coil an d its g ro u n d plane, and th e
c ap a c itan c e betw een th e m . N ear th e fu n d a m e n ta l freq u en cy o f th e strip lin e, th e
gain o f th e am plifier is e n h an ced by th e Q o f th e re so n ato r
o=i f
(A-21)
w here Z{n is th e source im p ed an ce a n d Z Q is th e strip lin e im pedance[77].
T h e square-w asher SQ U ID S b u ilt a t B erkeley h a d in n e r a n d o u te r dim ensions
of 0.2 m m x 0.2 m m an d 1 m m x 1m m , an d th e in p u t coils had a w id th of 5 ^ m
and len g th s ranging fro m 6 -71 m m .
A .5
dc SQUID Measurements
T h e g ain o f th e dc S Q U ID am plifiers is m easu red using th e setu p shown in Fig­
ure (A . 10).
T h e b a re SQ U ID is v e ry b ro a d b a n d so a cold 20 dB a tte n u a to r is
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164
p laced a t th e in p u t to p re v e n t noise fro m th e R F g e n erato r from s a tu ra tin g th e
am plifier. T h e o u tp u t 4 dB a tte n u a to r is for m a tc h in g th e SQ U ID o u tp u t to th e
e x te rn a l electronics. T h e m e a su re d g ain curve for a dc SQ U ID am plifier w ith a
s ix -tu m in p u t coil is shown in F ig u re (A . 1 1 ).
220 £2
Input
-r
56 £2
180 £2
(jEEb O u tp u t
180 £2
oooB T S
F ig u re A. 10: Schematic of th e setup used to make SQUID gain measurements.
C
*3
O
4 -i
1000
1050
1100
Frequency (MHz)
1150
F ig u re A .11: Measured gain of a six-turn dc SQUID. The outerm ost turn of the input
coil was shorted to ground.
T h e g ain b a n d w id th is d e te rm in e d by th e fu n d a m e n ta l frequency an d Q o f th e
in p u t coil. T h e frequency is in v ersely p ro p o rtio n a l to th e in p u t coil le n g th t. Fig-
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
165
u re (A . 12) shows th e m easured resonant freq u en cy versus coil length for four different
coil len g th s, d e m o n stratin g th e ex p ected £- 1 b eh av io r. T h ere is less coupling to th e
SQ U ID w asher as th e coil length is d ecreased , b u t th is design should be fu n ctio n al
up to ~ 5 -7 GHz.
5000-r
10300
I[mm] - 0.6
4000--
N
X
2 3000-c0u)
§•
2000 - -
1000 - -
2.000
10500
19.000
Coil Length (mm)
F ig u re A. 12: M icrostrip resonant frequency versus coil length.
expected l ~ 1 behavior with a 0 .6 mm correction.
36.000
The solid line is the
T h e b a n d w id th of these am plifiers is lim ite d , a n d it would be inconvenient to
be replacing th e m often on th e real d e te c to r. F o rtu n ately , th e reso n an t freq u en cy
of th e coils can be varied in situ by placing a variab le reactance a t th e op en en d of
th e strip lin e . T his changes th e phase sh ift o f th e reflected wave a n d hence changes
th e re so n an t frequency. In principle, th e tu n in g ran g e is 0 .5 /o to / c, w here f 0 is th e
re so n an t frequency of th e open-ended line, as th e lo ad is varied from a sh o rt to an
open.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
166
T uning o f these am plifiers has b een accom plished by connecting a p a ir o f G aAs
v a ra cto r diodes across th e previously open end o f th e m icro strip , show n in Fig­
u re (A.13)[78]. W hen th ese diodes a re reverse biased, th e w id th o f th e ir dep letio n
layers varies w ith th e applied voltage, m ak in g th e m voltage co n tro lled v ariab le ca­
p acito rs. Figure (A .14) shows n in e g ain curves for th e sam e am plifier as th e v aracto r
voltage was varied from -1 V (sm all forw ard bias) to + 2 2 V (stro n g reverse bias).
T h e resonant frequency is progressively lowered from 195 to 117 M H z as th e ca­
p acitan ce is increased; th e gain is c o n sta n t to w ith in 1 dB . M easu rem en ts o f the
noise te m p e ra tu re a t 4.2 K w ith an d w ith o u t th e varactors revealed no discernible
difference.
'- p ' tioo
Input +=+56 £2
220 £2
—~
. T
ood v«-|
Varactor
I 1 j —|
,
22 £2
1---------------
56 £2
jr-m
180 £2
i l lB
^ d o o d b “j
—|-0 i ■■O utput
iso a
I
F ig u re A. 13: Same configuration as th e gain measurement with the addition o f the
varactor and its DC bias.
T h e noise te m p e ra tu re of th e SQ U ID am plifiers was m easu red using a h eated
resistor, sim ilar to th e m eth o d d escrib ed in C h a p te r T h ree for th e se n sitiv ity cali­
b ra tio n . T h e do m in an t source o f noise in these devices is th e Jo h n so n noise from
th e resistive shunts across th e Jo sep h so n ju n c tio n s. T h is noise scales lin early w ith
te m p e ra tu re , so th e noise te m p e ra tu re o f dc SQ U ID s is ex p ected to b e p ro p o rtio n al
to th e ir physical te m p e ra tu re u n til e ith e r th e q u a n tu m lim it (Ta = hu/2kB )[75] is
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
167
PQ
3
2 0 -=
c
:
22V
-1 V
30-
15-
10
-
5 -1
100
120
140
160
180
200
220
Frequency (MHz)
F ig u re A . 14: Measured gain curves of a ‘2 1-turn dc SQUID as the voltage across a varactor
connected between the outerm ost turn and ground is varied from -1 to 22 V.
reached o r ho t electro n effects in th e sh u n ts becom e d o m in a n t.
F or a n early device w ith an 1 1 - tu rn in p u t coil a t 4.2 K , th e g ain was 22 dB on
resonance, a n d th e noise te m p e ra tu re T a was 0.9 ± 0 .3 Iv, in c lu d in g an e stim a te d 0.4
K c o n trib u tio n from th e ro o m -te m p e ra tu re postam plifier. S u b seq u en tly , to reduce
this noise co n trib u tio n , a cooled, single-stage H F E T am plifier w as used as a p o sta m ­
plifier. F ig u re (A .15) shows th e m easu red noise te m p e ra tu re , w hich has a m in im u m
of 0.25 ± 0.1 K a t 365 M Hz for a b a th te m p e ra tu re of l.S K . In th is m easu rem en t,
th e low -noise b an d w id th is lim ite d by th e H F E T p o stam p lifier a n d n o t th e SQ U ID .
In a fu rth e r a tte m p t to red u ce th e noise te m p e ra tu re , SQ U ID s w ere cooled to 0.4
- 0.5 K in a ch arcoal-pum p ed , single-shot 3H e cry o stat. T h e H F E T p o stam p lifier
was m o u n te d on th e 1 K p o t.
W ith in th e e rro r bars, th e noise te m p e ra tu re a t
each freq u en cy scales w ith th e b a th te m p e ra tu re . T h e s y ste m noise te m p e ra tu re
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
168
a t 438 M Hz was 0.50 ± 0.07 K , o f w hich 0.38 ± 0.07 Iv was c o n trib u te d b y th e
p o st am plifier[7 9].
A t 500 m K , th e noise te m p e ra tu re s a re alre ad y w ithin a facto r of fo u r o f th e q u an ­
tu m lim it, w hich for a 500 M Hz am p lifie r is ap p ro x im ately 35 m K . D e m o n stra tin g
a q u a n tu m lim ited am plifier will re q u ire a m u ch q u ieter p o stam p lifier. T ow ard this
e n d , a second SQUID has been u se d as a postam plifier to th e in p u t S Q U ID . To
p re v e n t th e tw o S QUIDs from in te ra c tin g , it was necessary to in sert a n 8 dB a t­
te n u a to r. T h e resonant freq u en cy o f th e second SQUID was tim e d w ith a v a ra cto r
dio d e to m a tc h th a t of th e first S Q U ID . T h e m ax im u m pow er gain a t 386 M H z was
33.5 ± 1 dB , as shown on th e rig h t o f F ig u re (A .15). Noise te m p e ra tu re s below 100
m K have been m easured using c asc a d e d SQ U ID s cooled in a d ilu tio n refrig erato r.
W ork is continuing to d e m o n stra te q u a n tu m -lim ite d noise p erform ance.
35-
30-
.£ 25
0 .2 0.0 362
364
Frequency (MHz)
366
368
340
360
38 0
400
Frequency (MHz)
420
F ig u re A .15: Measured noise tem p eratu re (left) and gain for a dc SQUID amplifier with
a 21-turn input coil at a physical tem p eratu re of 2 K.
A .6
Conclusion
dc SQ U ID based R F am plifiers w h ich co u ld g reatly im prove th e s e n s itiv ity o f m i­
crow ave c av ity axion d etecto rs h a v e a lre a d y b een fab ricated and te ste d a t U C B erke­
R e p ro d u c e d with perm ission of th e copyright owner. Further reproduction prohibited without permission.
169
ley.
Progress is being m a d e to w ard d e m o n stratin g an am p lifier w ith q u an tu m -
lim ited noise te m p e ra tu re , as well as e x ten d in g the frequency ran g e. T h e highest
frequency am plifier te s te d so fa r is slightly over 3 GHz. T h e o n ly rem ain in g prac­
tic a l issues w ith th e am plifiers involve th e fin al packaging so th e y can o p e ra te in a
real axion d etec to r as o pposed to th e shielded confines o f th e la b o ra to ry . T h e m ost
im p o rta n t step is th e design a n d co n stru ctio n of a c o m p en sa tio n coil to provide
a m agnetic field-free en v iro n m e n t for th e am plifiers close to th e high-field region
containing th e cavities.
R e p ro d u c e d with perm ission of the copyright owner. Further reproduction prohibited without permission.
170
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