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 Material Information Title: A search for cosmic axions Physical Description: Book Language: English Creator: Hagmann, Christian A Copyright Date: 1990
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A SEARCH FOR COSMIC AXIONS

By

CHRISTIAN A. HAGMANN

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

1990
UNIVERSITY OF FLORIDA LIBRARIES

ACKNOWLEDGEMENTS

It is my great pleasure to thank my advisors Profs. N. Sullivan and D.

Tanner for their guidance, valuable advice and general support during the time

of my research. I am grateful to Prof. P. Sikivie for his continued interest in

my research project and many stimulating and helpful comments. It was a great

honor for me to participate in the Florida Axion Project and become a member

of a new and exciting branch of physics.

My thanks go also to Profs. G. Bosman, H. Campins, J. Ipser, and M. Meisel

for their interest and patience while serving on my committee. The experiment

would not have been possible without the support by the cryogenic group under A.

Hingerty and G. Labbe, the machine shop under B. Fowler and the electronics shop

under L. Phelps. I am grateful to C. Porter for providing the 3-D plotting program.

I would also like to mention my fellow graduate students and the staff members

of the Physics Department for creating a pleasurable working environment.

Page

ACKNOWLEDGEMENTS .. ................ . ii

ABSTRACT . . . . . . . . . . . . . .

CHAPTERS

1 INTRODUCTION .... ....... .. .. ... ... ... 1

2 ORIGIN AND PROPERTIES OF THE AXION . . ... .. 5

The Axion and the Strong CP Problem . . . . . . . 5
Axions in Astrophysics and Cosmology . . . ... ..... . 14
Axions as a Dark Matter Candidate . . . . . .... 25
Detection of Galactic Axions. ... . . . . . . 29

3 DESCRIPTION OF EXPERIMENTAL APPARATUS ...... .37

Magnet and Cryogenic System .... . . . . ..... 37
The Cavities . . . . . . . . . . . . 40
The Microwave Amplifiers ................... 57
Room Temperature Data Acquisition Hardware . . . ... .66

4 DATA ACQUISITION AND ANALYSIS . . . . .... 73

The Search Process .............. ....... . 73
The Data . . . . . . . . .. . . . . 75
Limit on the Electromagnetic Coupling of the Axion ...... .79

5 SUMMARY AND CONCLUSION . . . . . . .... .83

APPENDICES . . . . . . . . . . . . .. 86

A CAVITY DEVELOPMENT . . . . . . . .... 86

111

Page

B SOFTWARE LISTINGS .................. 93

REFERENCES ..... ..... . . . . . . 120

BIOGRAPHICAL SKETCH ................... 125

Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

A SEARCH FOR COSMIC AXIONS

By

Christian A. Hagmann

August 1990

Chairman: Neil S. Sullivan
Cochairman: David B. Tanner
Major Department: Physics

A search experiment has been carried out to look for relic cosmic axions

trapped in the halo of our Galaxy and limits on their abundance are presented

for the axion mass range of 5.48 x 10-6 eV < ma < 6.77 x 10-6 eV and 7.47 x

10-6 eV < ma < 7.60 x 106 eV.

The detector consisted of a high-Q microwave cavity immersed in liquid

helium and permeated by a strong magnetic field. A low noise microwave receiver

with a cryogenic HEMT amplifier as its first stage was coupled to the cavity. The

noise power within a cavity bandwidth was spectrum analyzed and searched for

narrow peaks resulting from axion to photon conversions via the Primakoff effect.

The resonance frequency was changed by displacing a dielectric rod inside the

cavity.

CHAPTER 1
INTRODUCTION

The axion is a hypothetical particle which was proposed to explain why the

strong interactions conserve the symmetries of P and CP. The Lagrangian of

QCD contains a symmetry violating term

Sg2
L= Tr (G, G"V ) (1.1)
16 xr2

where the Gp,'s are the gluon fields, g is the QCD coupling constant and 0 E

(0,27r), is a free parameter of the theory. To be consistent with experiments, 8

has to be very small (9 10-9), which seems unnatural.

Peccei and Quinn [1] presented an elegant solution to this problem by in-

troducing a special spontaneously broken global U (1) symmetry into the QCD

Lagrangian. This broken symmetry has the effect of making 8 a dynamical vari-

able, which adjusts itself to the CP conserving value 0 = 0. As pointed out by

Weinberg [2] and Wilczek (3], there must exist a Goldstone boson associated with

the spontaneously broken U (1)pQ symmetry; it was named the axion.

Massless at the classical level, the axion acquires a small mass due to quan-

tum effects. The mass was originally thought to lie in the keV range, making

it accessible to accelerator searches and beam dump experiments. However, this
1

particular search was unsuccessful, and a new kind of 'invisible' axion was pro-

posed [4, 5]. This new axion has a much smaller mass and is very weakly coupled

to ordinary matter. Mass and couplings of the axion are suppressed by the PQ

symmetry breaking scale vpO, with the mass given by

a 10-eV(102 Ge) (1.2)
VPQ

and with all couplings proportional to ma.

Invisible in most terrestrial experiments, axions can have important impli-

cations for astrophysics and cosmology. Thermally produced axions can freely

stream out from the interior of stars and contribute to their radiative energy loss.

In order not to cool stars excessively, the axion couplings need to be sufficiently

small, and this constraint places an upper limit on the axion mass. An even better

limit comes from the recent observation of supernova SN1987a. Axion emission

can dominate neutrino emission for a small axion mass. Yet the observed neutrino

pulse can account for all of the energy released during the collapse. This can be

used to place an upper bound on the axion mass of ma 103 eV.

Cosmological considerations provide a lower limit on ma. When the temper-

ature in the early universe falls below the PQ scale, U (1)pQ gets broken sponta-

neously and topological defects in form of cosmic strings will appear. This string

network will oscillate, causing the emission of axions, which contribute to the

present energy density of the universe and would close it for ma ~ 10-41 eV.

If an inflationary period occurs after the PQ phase transition, the strings will

become diluted and no axions are produced from strings. However in that case,

axions are produced due to the initial misalignment of the vacuum angle 0.

For ma ~ 10-5 eV, cosmic axions from string radiation or initial vacuum

misalignment constitute a significant fraction of the energy density and would be

all or part of the dark matter which is known to exist in our universe. In particular

our own galaxy has a dark halo of unknown composition which could conceivably

The experiment described herein has made an attempt to detect these galac-

tic axions for an axion mass near 10-5 eV. As first proposed by Sikivie [6], the

detection scheme was based upon the electromagnetic coupling of the axion field

a, given by

Lary = ga- a E B (1.3)

with gao- being proportional to ma. In a static magnetic background field B0, this

coupling stimulates the conversion of axions into photons. B0 provides a virtual

photon and a real photon carries off all the axion energy. Since galactic axions

are nonrelativistic, the photon energy, w = ma (1 + 0 (10-6)), will have only a

small spread. The frequency of these photons lies in the microwave region, and the

decay rate of axions is resonantly enhanced in a high-Q microwave cavity tuned

to the axion energy.

The cavity is kept at liquid helium temperature to reduce its thermal emis-

sion and is coupled to a very sensitive microwave receiver. The signature of an

axion conversion signal would be an increase of the power inside of the cavity

above the noise floor. The cavity has to be tunable in order to search a wide range

of possible axion frequencies. Tuning is accomplished by moving dielectric and

(or) metal rods inside of the cavity, yielding tuning ranges of about 10-20 %. To

date, two cavities have been constructed, which enabled us to cover the frequency

range

1.32 GHz < f < 1.63 GHz and 1.80 GHz < f 1.83 GHz (1.4)

corresponding to axion mass ranges of

5.48 x 10l-eV < ma < 6.77 x 10-6eV and
(1.5)
7.47 x 10-6eV < ma < 7.60 x 10-6eV

No positive signal coming from axion conversion has been detected so far,

and this result enables us to place an upper limit on the axion coupling for the

above mass range, assuming that the galactic halo is axionic.

CHAPTER 2
ORIGIN AND PROPERTIES OF THE AXION

The Axion and the Strong CP Problem

Quantum chromodynamics (QCD) the gauge theory of strong interactions,

successfully describes many aspects of quark physics such as scaling and asymptotic

freedom. The Lagrangian is usually written as

1 "/
L QCD = Tr GuGP" + C k (iPD, nk) qk (2.1)
k

where
Gp, = &,A, OA, ig [A,, A,]

Duqk = (a, ig Ap) qk (2.2)
8
A = Aa a/2
a=l

and where the Guv's are the gluon fields, the Aa's are the Gell-Mann matrices

and k runs over all quark flavors : u,d, s, c,t, b The classical QCD Lagrangian

possesses a high degree of symmetry and in particular conserves P, CP and T.

Due to the non-Abelian nature of the gluon fields, there exist vacuum states

of topologically distinct finite energy field configurations [7] labeled by a winding

number n. Upon quantization, tunneling solutions (instantons) appear, and the

true vacuum state becomes a linear combination of the In) states. The ground
5

state has the form of a Bloch wave and is given by

|9) = -ino I (2.3)
n

The '0-vacuum' modifies the Lagrangian that enters the path integral formulation

of QCD by adding a new term :

Og2
L ef = L CD + 1r2Tr (GyvGPv) (2.4)

where G/u = e'"7P Gp and is the dual of GP The symmetry properties of

GG are the same as of FF in electrodynamics and are odd under P, CP and T.

The coefficient 0 is an arbitrary constant with modulus 2r, and the effective

Lagrangian violates P, CP and T, provided 0 5 0, 7r. The best known physical

manifestation of the 6-vacuum would be a nonvanishing electric dipole moment of

the neutron, dn. The current upper bound of dn is [8]

Idn|l 10-25 e cm (2.5)

which constrains 0 to be very small, namely

IO\ <' 10-9 (2.6)

The strong CP problem is a question of how to make 0 naturally small.

The situation becomes even more mysterious if the electroweak interactions

are included. The spontaneous symmetry breaking of the electroweak SU(2)

U(1) gauge group is responsible for giving mass to the quarks. The mass matrix

in the Lagrangian is in general off-diagonal and non-hermitian, i.e.

L mass = -4Ri Mij qLj qLi (Mt)ij qRj (2.7)

One can diagonalize the mass matrix Mij with two special unitary transformations

acting on the left and right quark fields respectively,

qLi ULij qLj
(2.8)
qRi URij Rj ,

leaving the mass matrix in the form Mij = ImijI 6ij eia. The common phase

factor a can be removed by a chiral U(1)A transformation with

qLi e-ia/2 qLi
(2.9)
qRi ei/2 qRi ,

bringing the mass term into its final form :

L mass = mk qk qk (2.10)
k

The latter transformation affects the QCD angle 0 through the well known

Adler-Bell-Jackiw (ABJ) anomaly [9] of the U(1)A axial current:
ni
Jg = Eqk 7 5 qk
k (2.11)
2
DJ5 = 2nf ---Tr (GG ") + mass term contributions

The effect of performing a chiral rotation a/2 is to change 0 according to

0 -+ + nf a = 0 + arg det M- 0 (2.12)

Sis the effective CP violating parameter in the QCD Lagrangian, and it replaces

8 in Eqs. (2.4) and (2.6).

The most convincing solution of the strong CP problem was put forward

by Peccei and Quinn [1]. They postulated the existence of a new spontaneously

broken global U(1)pQ symmetry, thereby making 0 a dynamical variable which

adjusts itself naturally to zero. The U(1)pQ chiral symmetry is spontaneously

broken at a scale vpQ, and there exists a Goldstone boson called the axion [2].

The axion field transforms as the phase of a complex scalar field:

a -* a + vpQ 6a (2.13)

where 6a is the infinitesimal parameter of the U(1)pQ transformation. Exact

Goldstone bosons have only derivative couplings, but here it is postulated that

the U(l)pQ symmetry suffers from the same ABJ anomaly as the ordinary QCD

U(1)A symmetry. This anomaly results in a potential for the axion and gives it a

mass, making it a pseudo-Nambu-Goldstone boson. The modified Lagrangian is

2 1
Leffl = LQCD +(+ ) 1 Tr (GapvG^') + "da 'a + L (8~a, ) (2.14)

where ( is a model dependent constant and L (ma, 0) specifies the couplings to

fermions. The axion potential is minimized for [1]

(a) = -0 vpQ/f (2.15)

which is the vacuum expectation value of a. The physical axion field corresponds

physical = a (a) (2.16)

The first axion model embedded the U(1)pQ quasi-symmetry in the standard

electroweak theory. The standard Higgs double 4 was replaced by two separate

doublets [10]

I = ~1 e 2 = v2 e (2.17)

where

x= 2 v= + v = (/GF)- 250 GeV = vp (2.18)
V1

and where GF is the Fermi constant. The vacuum expectation values v1 and v2

break down the symmetry SU(2)W 0 U(1)y 0 U(1)pQ into U(1)EM giving rise to

four Goldstone bosons. Three of them are absorbed through their couplings to the

gauge fields generating mass for the vector bosons W+, W-, Zo. The remaining

Goldstone boson is the axion.

Current algebra techniques are used to calculate the mass and the lifetime

of the axion. The result is [2, 10, 11]

f nf 1 f_
ma = m -(x + -)
vpQ 2 1+z (2.19)
1
S12nf (x + -)keV

where m, and f, are the pion mass and decay constant and z mu/md 2 0.56.

For ma < 2me the dominant decay channel is a -+ 27 with the lifetime

(105eV 5
5a V) sec. (2.20)
\Ma m

This kind of axion has been searched for unsuccessfully in accelerator experi-

ments. One important parameter is the branching ratio for kaon decay into axions

which is calculated to be [12]

BR(K+ -* r + a) > 1.3 x 10-5

(2.21)

while the experimental upper limit is 3.8 x 10-8 [13]. Another negative result

comes from heavy quarkonium decays where the qq -- a + process has not been

observed.

Not long after these negative results, new axion models were proposed mak-

ing the axion very weakly interacting and impossible to see in accelerator exper-

iments. Two models of the 'invisible' axion are usually considered by particle

theorists. The KSVZ or hadronic axion [4] leaves the standard electroweak model

intact and introduces a heavy exotic quark Q and a complex scalar field a, both

of which carry Peccei-Quinn charges and are singlets under SU(2)W 0 U(1)y.

Under the PQ symmetry, they change according to
Q e5SCeQ
(2.22)
a -- e-2ia

while leptons and ordinary quarks are singlets. The Yukawa coupling between Q

and a is

Ly = -f QL QR f*QR* QL (2.23)

and the modified Higgs potential has the form

V(,, or) = -4 $j ,a*a + A(t(t)$)2 + Aoj2 + AtPt\$ a*u (2.24)

where 4 is the Higgs doublet from the standard model. Ly and V((, a) are left

invariant under U(1)pQ.

R: QL -QL QR -- +QR, a - (2.25)

is required to avoid a bare mass term mQQ The a field develops an arbitrary

large vacuum expectation value (a) = vpQ above the electroweak scale, and the

phase of a becomes the axion field a. The axion couples to ordinary quarks

and photons indirectly through Q and gluon loops. There is also a higher order

coupling to electrons via radiative corrections. The axion mass is given by

KSVZ J- f7 6 x 10-6 eV 1012 GeV (2.26)
1+ z VpQ VPQ

while the photon coupling is

La, = 9- aFPv~Pv= gaaE B, (2.27)

with

KSVZ a E 2 24+ z
S27r pQ+ 3 Z 1 + (2.28)

=1.93 x 10-15 GeV-1 (E 2 4+ z ma
(N 3X3 1+ez 10l+ l-5 eV)

where a f :- is the fine structure constant and E/N is the ratio of electromag-

netic and color anomaly of U(1)pQ.

The DFSZ axion [5] model has two Higgs doublets as in the original model.

In addition there exists a scalar field a analogous to the one in the KSVZ model

with VEV vpQ > v. Both a and ordinary fermions carry PQ charges. The a

field couples indirectly to quarks and leptons via a quartic interaction between a

and i, P2, with the latter fields having Yukawa interactions with fermions. The

relevant interaction between the Higgs fields is given by

V) = A (T2a2) + h.c. (2.29)

which fixes the PQ transformation properties of a relative to i1, 2 and the

fermions. In this scenario, the mass and couplings are similar to the KSVZ axion

with
DFSZ f(2.30)
ma = nf m, (2.30)
1+2z VPQ

and
DFSZ Q= 8n8 2 4+z
ga77 vpQ 3 1 + z (2.31)

=1.38 x 10-15 GeV- ( 105eV)

The PQ symmetry breaking scale vpQ for both KSVZ and DFSZ axions is con-

strained from above only by the GUT scale mplanck 1019 GeV, corresponding

to a mass ma 10-12 eV.

Axions in Astrophysics and Cosmology

Astrophysical considerations provide powerful constraints on invisible axion

parameters. Axion emission contributes to the energy loss of stars and affects

their evolution. The energy transport out of stars in the form of axions is maxi-

mized for axion couplings yielding a free mean path of roughly one stellar radius.

Increasing the coupling strength leads to axion trapping inside an axion sphere

whereas decreasing the coupling allows free streaming but reduces the number of

axions produced. In order not to exceedingly shorten observed stellar lifetimes,

cooling through axions must be a sufficiently weak process and thus yields limits

on axion couplings.

In main sequence stars and red giants the dominant processes for axion

cooling are Compton-like scattering [e- + 7 e- + a] and bremsstrahlung [e- +

Z (e-) -* e-+ Z (e-) +a ], whereas for KSVZ axions with no tree level coupling to

leptons, the most important loss mechanism is the Primakoff effect [7 + Z (e-) -

a + Z (e-) ]. The associated Feynman diagrams are shown in Fig. 2.1. Applied

to red giants, these processes lead to an upper bound on the axion mass in the

KSVZ model of [14]

mKSVZ< 0.7eV 0E .92 (2.32)
a EIN 1.92

7 a

7

Z,e

e e

b)

/

SN

/

e

K '
ir

Z,e

Z,e

d)

Fig. 2.1: Feynman diagrams of axion production mechanisms in stars with a)
Primakoff effect, b) Compton-like scattering, c) nucleon-nucleon bremsstrahlung
and d) electron-nucleon and electron-electron bremsstrahlung.

Ze

a)

N

c)

In the DFSZ model, axion cooling will prevent red giants from entering the helium

burning phase unless [15]

mDFSZ < 10-2eV. (2.33)

For an axion mass ofma 2 0 (10-300keV), the previous arguments are no longer

valid because thermal axion production is suppressed by the Boltzmann factor

e-a. However, this mass region is already excluded by laboratory experiments

[13].

The most recent astrophysical limit comes from supernova SN 1987a. Dur-

ing core collapse, the enormous gravitational binding energy of ~ 1053 erg was

released and radiated away. According to the standard picture, this energy was

carried away in the form of a neutrino pulse. Due to the high density and temper-

ature of the core, neutrinos are trapped inside a neutrino sphere and are only ra-

diated from the surface. Axions with sufficiently small mass and couplings stream

freely and could greatly accelerate the cooling process if produced abundantly. The

dominant production process for axions in supernovae is nucleon bremsstrahlung

[N + N -* N + N + a] (see Fig. 2.1) with a cross section proportional to ma.

For increasing axion mass, trapping occurs and the cooling efficiency decreases.

Therefore one ends up with an axion mass range for which axion emission domi-

nates or is comparable with neutrino emission. The neutrino pulse from SN 1987A

[16] was consistent with a purely neutrino-cooled core, which yields the excluded

mass region [17]

10-3eV < ma < 2eV, (2.34)

which is valid for the KSVZ and DFSZ model. The combination of astrophysical

bounds allows a mass range for DFSZ axions of

mDFSZ 10-3eV (2.35)

and two windows for the KSVZ axion of

nKSVZ <5 10-3eV and 2eV mKSVZ 5eV (2.36)

Another important bound on the axion mass comes from cosmological considera-

tions. In the early universe, axions may have been in thermal equilibrium with the

primordial plasma. A rough criterion for a particle to be in equilibrium is F > H

where I is the interaction rate with other particle species and H is the universal

expansion rate or Hubble constant. Because the interaction rates between axions

and other particles scale like the axion mass, only axions heavier than 10-3eV

will be in thermal equilibrium at some time. These thermally produced axions will

eventually fall out of equilibrium as the the universe expands and their number per

comoving volume will remain constant except for the decay channel [ a -- 27 ]. The

contribution that these axions add to the present energy density of the universe

is given by [18]

Thermal h2 10-2 (ma/eV,) (2.37)

where Q = p/pc and 100 h km sec-1 Mpc-1 is the present value of the Hubble

constant. From Eq. (2.37) it is clear that thermal axions could make up a signifi-

cant fraction of the present energy density only for mass ranges which are already

excluded by stellar evolution.

A possibly much more important contribution [19, 20] to the energy density

comes from the misalignment of the initial 6 value with the CP conserving value

of 0 = 0. When the temperature of the early universe falls below the PQ breaking

scale vpQ, the Higgs field a takes on the form

a = vpQ eia/vPg (2.38)

where the phase assumes a random value in the range 0 < a < 27r VpQ and the

axion field will in general be a function of position a = a (x) However if a period

of inflation occurs during or after the PQ phase transition, a small patch of space

will grow exponentially during that epoch and render a constant in the observable

part of universe. As the temperature of the universe reaches T c 1 GeV, instanton

effects become important and the axion acquires a small mass given by [20]

ma(T) 0.1 ma(T = 0) (37 T > AQCD (2.39)

where AQCD 200 MeV and is the QCD scale, below which quarks are confined

in hadrons. Below AQCD the axion mass will assume its value given by Eqs. (2.26)

and (2.30). A finite axion mass provides a potential for a of the form (see Fig. 2.2)

V(a) = 1ma(T)2 a2 + O(a4) (2.40)
2

which drives the axion field to the CP conserving value a = 0. Actually, in some

axion models, V(a) can have N degenerate minima (N = N in Eq. (2.14)) which

can be modeled as

V(a) = m2 (T)( cos)) (2.41)

and can lead to a domain wall problem [21]. In the DFSZ model, N = 6.

When the axion mass ma(T) becomes larger than the Hubble factor H(T),

the axion field begins to oscillate at a frequency determined by ma(T). The

Lagrangian governing the axion field dynamics is

L = (a, a)2 V(a), (2.42)

yielding the equation of motion

d2 d
d2 (a) + 3H(t) d (a) + m2(t) (a) = 0 (2.43)
dt2 dt a

T << 1GeV

T )) 1GeV

Fig. 2.2: Temperature dependent axion potential V(a, T).

where H(t) = 1/2t is the expansion factor during the radiation dominated phase

of the universe. For rha/ma < ma, Eq. (2.43) has the approximate solution

(a) = A(t) cos (ma t) (2.44)

where A(t) satisfies

d(m A2) = -3H (ma A2) (2.45)

Equation (2.45) can be integrated to yield

(ma A2)f/(ma A2)i = (Rf/Ri-3 (2.46)

The coherent oscillations of (a) correspond to a zero momentum Bose condensate

of axions which is diluted by the expansion of the universe. The energy density

(in comoving coordinates) stored in those oscillations is given by

1 ,2 (d(a)2 1 22 1 2
Pa = m2(a)2+ m av Q (2.47)

where 0i, the initial misalignment angle of the axion field in our universe, has

uniform probability of falling within the interval [-7r, 7r]. In addition, there will

be dissipation due to axion decay [a -+ 2 7] and [4a -- 2a] which turns out to be

negligible in comparison with Hubble expansion. The contribution to the present

energy density has been calculated to be [19, 20]

f2a h2 = 0.26 x 100 0.7 A 10saeV -18 (2.48)
20 10 V 0i

where A200 is the QCD scale in units of 200 MeV. In an inflationary universe,

2 = 1 and constrains the RHS of Eq. (2.48) but does not provide a definite limit

on ma.

Without inflation or if the reheating temperature after inflation was above

the PQ scale, a different scenario [22, 23] takes place. When PQ breaks, the

Higgs field a = vpQeia/vPQ will acquire random angles in causally unconnected

regions of spacetime. Then the present observable universe has not evolved from

a domain of spatially uniform (a), but originated from many regions with initially

uncorrelated (a). The Lagrangian of this system is given by

1 A2
L = 12 *a (a )2 T > 1GeV. (2.49)
2 4

Since a is single valued, the total change in phase around any closed path must be

A0 = 2rn (2.50)

where n is an integer, called the winding number. For n 0, it is not possible to

shrink the loop radius to zero without encountering a singularity where the Higgs

field vanishes. The resulting tubes of false vacuum are called strings. The core

1
6 (2.51)
V\ VpQ

and the field far away from the core has the asymptotic form

a = vpQein (2.52)

where 9 is the polar coordinate about the string axis. The energy per unit length

of a straight global string is

2 2 Q 10 12R
p -n2 vR2 + r 27rr dr 27rn2 pQ (2.53)

where the first term is the contribution from the core and R is the cutoff length

which is typically the distance between neighboring strings. Strings with n 1

are unstable and decay into elementary ones. Unlike gauge strings, global strings

have most of their energy outside their cores.

As the universe expands kinks will come within the horizon, and the string

network starts to oscillate, causing the emission of axions. Furthermore, strings

will intercommute and exchange partners leading to the production of loops which

collapse and disappear. Simulations [24] of cosmic string evolution suggest that

a scaling solution will be quickly reached with 0(1) strings per horizon volume.

This implies that the ratio Pstring/Prad ln(R/6), and strings will not dominate

the energy density of the universe. In order to satisfy the scaling solution, axions

have to be radiated at a rate

d 1 d
na = -3 H(t) na + Prad (2.54)
dt w(t)d

where
(2.55)
w(t) d Prad/d t d d dw w

Here d2prad/dt dw is the radiation spectrum at time t, na is the number density

be integrated and one obtains

1 [t d p(t')
na(t) t32 t3/2w(t')(2.56)

where tpQ is the time of PQ breaking. The all important ingredient of this

equation is the radiation spectrum. One group of authors [22] argues that axion

radiation from strings is peaked at the longest possible wavelength, which is equal

to the string size. Thus w(t) 2 1/t and integration of Eq. (2.56) yields

na(tQCD) vQ ma(tQCD)ln w(t) (2.57)

where tQCD is the time of quark confinement and the axion mass turns on. Another

group [23] claims that the spectrum is 1/k with cutoffs at 1/t and 1/6. The axion

number density in this case is

na(tQCD) 2p ma(tQCD) w(t) In (2.58)

which is different from the previous estimate by a logarithmic factor of order 100.

After tQCD, domain walls bounded by strings will form which decay predominantly

into gravitational radiation. Therefore the axion energy density at present is

ma 1
spring M13aeV if w(t) 2 1 and
(2.59)
'O"a if w(t)" In
.string 10-5eV t 6 .

In summary, cosmological considerations cannot provide a strict bound on

the axion mass. If there was no inflation or if inflation occurred before PQ sym-

metry breaking, then a lower limit on ma of (10-5 10-3)eV exists, whereas

in case of inflation after PQ, no conclusive statement can be made.

Axions as a Dark Matter Candidate

Cosmologically produced axions can, for certain values of mr and 8i, make up

a sizeable fraction of the present energy density of our universe. Axions originating

from vacuum misalignment or string oscillations are born cold and never come into

thermal equilibrium. Thus cosmological axions are an ideal candidate for cold dark

matter which is currently popular in theories of galaxy formation.

Evidence for dark matter has existed for a long time, starting with the

measurement of the rotational velocity of the Coma cluster. These measure-

ments yielded a much too large value if one considers only the masses in form

of stars. Zwicky [25] applied the virial theorem and estimated a mass-to-light

ratio of M/L 50h in solar units which is about 20 times larger than in the

solar neighbourhood. He concluded that most of the mass of the Coma cluster

is invisible or dark. More recent measurements [26] on clusters yield an average

value of M/L ~ 400h.

Another indication for dark matter comes from rotation curves of spiral

galaxies. This type of galaxy makes up about 65% of all observed galaxies, among

them the Milky Way. Doppler shift measurements of the rotational velocity of

tracer stars versus distance from the center yield surprisingly flat rotation curves.

By applying Kepler's law

m Vo (r) GmM(r)
= (2.60)
r r2

one expects Vrot to fall off as r-1/2 outside of the luminous component, if there is

no dark halo. Here m is the mass of the tracer star and M(r) is the mass within

radius r from the center of the galaxy.

Figure 2.3 shows a rotation curve of NGC 3198 fitted by a disk and a halo com-

ponent. The halo model used was

Phalo() = Phalo(0)(1 + (2.61)

where a is linked to the core radius and -y 2. The inferred M/L of spiral galaxies

is typically ~ 30 40 or even bigger because the extent of the halos is not known.

200 . .
NGC 3198

150

S' : halo
E
- 100
.-

50 disk

0

0 10 20 30 40 50

Fig. 2.3: Fit to observed rotation curve with exponential disk and spherical
halo for NGC 3198. The parameters in Eq. (2.61) are a = 1.3kpc, 7 = 2.05,
Phalo(O) = 0.0063 Mpc-3. (After [27])

Several attempts were made or are underway to measure the density of the

universe over very large scales. The infrared survey [28] done by the IRAS satellite

has measured the density to lie within 0.3 12 < 1.2, while another test comes

from the redshift-number count by Loh and Spillar [29] yielding 0.7 < 0 < 1.3.

This is in agreement with the theoretically preferred value Q = 1 (M/L ~ 1600)

as predicted by inflationary models.

Therefore a great deal of dark matter of unknown composition is likely to

exist. An important constraint on its composition comes from primordial nucle-

osynthesis. From the relative abundances of elements produced in the Big Bang,

mainly 2H, 3He, 4He and 7Li, one can estimate [30] the density of baryonic matter

( protons and neutrons ) to be 0.03 < Obaryon 5 0.12. For this reason, most

of the dark matter has to be of some exotic form, if one assumes 0 = 1 as in

inflationary models. Many candidates have been put forth, among them massive

neutrinos, supersymmetric particles and the axion. They can be classified into

'cold dark matter' and 'hot dark matter'. The former type (e.g. axions) is slow

moving at the time of structure formation and can clump on all scales causing

small structures to appear first ('bottom-up'). The second group (e.g. light neu-

trinos) is freely streaming on galactic scales and produces large structures first

('top-down').

It might be the case that neither of the two groups can by itself explain

the structure of the universe without some further ingredient. Axions however

remain on the top of the list as potential candidates for dark matter because their

existence is needed in particle physics.

Detection of Galactic Axions

Turner [20] has calculated the local halo density, using a similar model as in

Eq. (2.61), to be

Phalo z 5 x 10-25 g cm-3 0.3 GeV cm-3 (2.62)

The halo is assumed to form an isothermal sphere with a Maxwellian velocity

distribution

p(v)d3v = Ne v2 d3v (2.63)

where 3 = 3/(v ), (v2) ~, 280 kmsec-1 and N is a normalization constant. This

distribution is broadened by the earth's rotation around the center of our galaxy.

The kinetic energy of an axion is

E = ma (v Ve)2 (2.64)
2

where ve a 220 km sec-1. The probability distribution of E becomes

p(E) = N' e- E/ma sinh(0 ve f2E/ma)

(2.65)

which is plotted in Fig. 2.4. The figure shows that galactic axions have a fractional

width 1/Qa 0 O(10-6).

At present, the cavity detector is the most promising detector for galactic

axions as proposed by Sikivie [6]. It uses the electromagnetic coupling of the axion,

which can be written as

La = -" ga a F F = g. a E B (2.66)

This coupling allows for axion-photon transitions in a background field B0 (Eo).

B0 provides a virtual photon (see Fig. 2.5) and the axion energy is carried away

by a single real photon of frequency

S= ma(1 + O(10-6)) (2.67)

For axion masses with ma = 0 (10-5) eV, the photons are in the microwave region,

and as a result, the most suitable detector is a microwave cavity.

The axion field can be described as a classical field because of its high number

density (~ 1013cm-3) and is coherent on laboratory scales with a de Broglie

wavelength of
h 10-5eV
Aa = 10m () (2.68)
ma v m

With BO = i BO, the coupling of the axion field to a given cavity mode is

gaa7Bo J d3x Enp (x, t).- (2.69)

1.0
0.9
0.8
0.7
0.6
n 0.5
0.4
0.3
0.2
0.1
0.0
0.0 5.0E-7 1.0E-6 1.5E-6 2.0E-6
E/m,

Fig. 2.4: Kinetic energy distribution of galactic axions with earth motion
included. The fractional width is 10-6.

Fig. 2.5: Feynman diagram of a -y conversion in background field B0.

where Enlp (x, t) is the electric field of that mode and V is the cavity volume.

Enip (x, t) satisfies

V x V x Enp (, t) 6 (x)WnIp Enlp (x, t) = 0 (2.70)

where we have allowed the dielectric constant of the material within the cavity

to be spatially dependent. From Eq. (2.69), only TMnl0 modes couple in first

approximation. The equation of motion can be derived from

L = (e (x) E2 x,t) 1 B2 (x, t) g a (t) E (x, t)-B (x, t) (2.71)

yielding

V x V x E (x) e (x) w2 E (x) = w2 gayy a (w) BO (2.72)

assuming a is spatially independent. This equation can best be analyzed [31] by

expanding the fields into cavity modes e (x)nlp with the normalization

e (x) e (x)np-.e (X)nlp, d3x = V bnn' 11' 5 pp (2.73)

Thus
E (x)= A np e (x)nlp and
n,l,p (2.74)
BO(x) = e (x) B0 'Inlp e (X)nlp
n,l,p
where

r7nlp = e (X)nlp dx (2.75)

For a particular mode one obtains
2
Anlp 2 2 2- ga (w) B 7 nlp (2.76)
Wnlp
and the average energy in the cavity mode becomes

Up = dwV (2.77)

For a cavity of quality Q, Eq. (2.77) becomes

Un2p 2 2V a( )2 4/Q (2.78)
U7"p = a 7nlp W2 w2)2 +,4/Q2/ 2 -

The power from axion to photon conversions P = w U/Q is [6, 31]

Pnlp g2 Cp V Bo Min (Qnlp, Qa) (2.79)
Ma

provided a (w) peaks at Wn/p, and we have defined the dimensionless form factor

Cnlp
(fv z -e(x)nlpd3x)2
nl = r2 = 3 (2.80)
Cra -nip V fv e(x) e (x) d3lp

For a cylindrical cavity one has
4
Cnlp ( 2 0 6p0 (2.81)
(Xon)

where XOn is the nth zero of the Bessel function Jo(x) and e (x) = 1. It is preferable

to use the TM010 mode (C010 = 0.69 for e (x) = 1) because the higher modes have

a rapidly decreasing form factor.

In the DFSZ model, P from Eq. (2.79) becomes
S Pa \ ma (Qnlp
Pn 4 x 10-23 W np- ) (27r(3GHz)
S 4 10 8 T 5 x 10-25 g/cm3 2(3GHz) 105
(2.82)

where the quantity Qnlp is the loaded quality factor of the cavity given by

1 1 1
L + -- (2.83)
QL Qw Qh

Here 1/Qh is the contribution from the coupling hole and 1/Qw is the contribution

due to absorption into the cavity walls. The power transferred to the receiver is

Pr QL P (2.84)
Qh

where we have dropped the mode indices. Since usually QL < Qa, one can look

at Qa/QL frequency bins simultaneously. Because of the low signal power level, it

is necessary to average over some time. The sensitivity of a microwave receiver is

expressed in terms of its system noise temperature Tn. The signal to noise ratio

s/n for an integration time t in a bandwidth B is

n Ts BB Pr /'t-/B (2.85)
n Tn kBTn

where T, is the signal temperature. The time t required to achieve a given signal

to noise value s/n is therefore

t = 2 ( B. (2.86)
\n/ Pr

The search rate df/dt at which frequency bands can be searched is proportional

to (1/QL)/(QLI/Qh) which is maximized for a Qh value of

1 2
Qh Qw

2 1
3 QL

(2.87)

The search rate df/dt for the optimum coupling configuration (Eq. 2.87) is [31]

2 (Bo)42 () )2(5K)2

( fiHz)

105
(2.88)

df 0.3 MHz 4 2 V
dt year s t/n 10f

CHAPTER 3
DESCRIPTION OF EXPERIMENTAL APPARATUS

Magnet and Cryogenic System

The magnet [32] used in this experiment was a superconducting (NbTi)

solenoid which was 40 cm long and possessed a bore of 17.1 cm. The central field

was 8.6 T at a current of 88 A. This provided an average field of 7.5 T in a volume

of about 8 liters. The magnet was charged by a IPS-100 power supply [32] and

typically run in persistent mode at full field. Each of the two vapor cooled current

leads consisted of 48 strands of 10 mil tungsten wire.

The magnet, cavity and cryoelectronics were housed in a superinsulated

dewar [33], which could hold up to 70 liters of liquid helium. A lambda fridge

[34] lowered the bath temperature in the lower part of the dewar to 2.2 K (see

Fig. 3.1). The fridge (see Fig. 3.2) consisted of a stainless steel tube (1.3 cm dia.),

which contained a heat exchanger and a flow impedance.

The heat exchanger was made out of a 1 m long piece of 1 mm o.d. Cu-Ni

tubing, while the impedance was made out of a 3 cm long section of a 0.1 mm

i.d. Cu-Ni tube. Liquid helium entered the exchanger through a hole in the

tube at T = 4 K, was precooled and exited through the impedance. A vacuum

pump [35] was connected to the steel tube and caused the helium which collected
37

He vapor

X- ridge

cavity

magnet
(persistent mode)

. . . . . . . . .

..

. .n . .

- IHe @ T=4K

-.- amplifier

--He @ T=2K

Fig. 3.1: Sketch (not to scale) of dewar containing magnet, cavity and am-
plifier.

to pump

heat exchanger-

impedance

superfluid helium -

copper plug

copper

brush

Fig. 3.2: Sketch (not to scale) of A-fridge.

inlet

at the bottom to become superfluid. A copper plug with brush acted as a heat

exchanger with the surrounding helium bath. The cooling power of the fridge was

approximately 1 W. Cooling down the magnet and cavity from T = 4K to T = 2K

took about a day after which the temperature stabilized at the A-transition where

the heat capacity and thermal conductivity of 4He becomes large. At the top of

the bath, the temperature was T = 4 K at atmospheric pressure. Due to the poor

thermal conductivity of normal fluid 4He, a transition region with a temperature

gradient formed between the top and the position of the fridge.

The thermal conduction losses into the dewar were minimized by using cryo-

genic coaxial cables [36] for all microwave links and manganine wires for the tem-

perature sensors. The helium consumption was about 12 liters/day when running

at T = 4 K and 20 liters/day when using the A-fridge. In the latter case, trans-

fers were made every two days. During transfers, the bath temperature increased

slightly and data acquisition was stopped briefly. Shortly afterwards the bath

stabilized at the original temperature and data taking resumed.

The Cavities

Two cylindrical cavities were built out of oxygen free copper [37], each con-

sisting of a tube and two end plates. After rough machining, the surfaces were

smoothed with # 600 sandpaper and finally electropolished [38] to reduce further

their roughness. The pieces were then clamped tightly together in order to make

good electrical contact. No vacuum seal was required since during detector oper-

ation, the cavity was filled with liquid helium which entered through the coupling

and tuning rod holes. Figure 3.3 shows cavity I, which was used in our first scan.

Cavity II was identical to cavity I except that the radius was z 10 % smaller than

in cavity I.

Several holes were made in the plates for coupling and tuning purposes.

The larger one of the two coupling ports was overcoupled and connected to the

receiver. It consisted of an inductive loop (shown in Fig. 3.4) made out of a piece

of 50 Q coax cable. The coupling strength was adjustable by varying the insertion

depth of the loop. The smaller one consisted of a weakly coupled probe (shown in

Fig. 3.4), which was used to measure the resonance frequency and quality factor

in transmission.

The cavities were operated in the TM010 mode, which yields the maximum

form factor C. The electric and magnetic fields (shown in Fig. 3.5) of that mode

for a cavity filled with a medium with dielectric constant E are [39]

E = EO ( 2.405 p) e-it
(3.1)
(2.405 p)
B = -iVeEoJl1 R

minor port

Hl

major port

fine
tuning
rod

coarse
tuning
rod

h-- 15,2 cm

Fig. 3.3: Cavity I with 2 dielectric tuning rods. The larger rod is moved
laterally while the smaller one moves vertically.

38.1

cm

I I
LL~LLL~-L

SMA connector

BeCu fingers

-- Teflon dielectric

Fig. 3.4: Coupling loop and coupling probe.

N
C

where Jn is the Bessel function of order n, R is the cavity radius and

2.405 c
= R(3.2)

is the resonance frequency.

Furthermore, the quality factor of that mode is [39]

LR
Q= R+L (3.3)
R+L 6

where L is the length of the cavity and 6 is the skin depth. At liquid helium

temperatures, copper is in the extreme anomalous skin depth regime such that

[40]

( c2 me F 1/3 (3.4)
8 7-2wne-2

Here, me = 9.11 x 10-28 g is the electron mass, e = 4.80 x 10-10 esu the electron

charge, vF the Fermi velocity and n the conduction electron density. For Cu [41],

vF = 1.57 x 108 cm sec-1, n = 8.50 x 1022 cm-3 and the anomalous skin depth is

S=2.8 x 10- cm G )1 /3 (3.5)

For cavity I, the theoretical Q is

(3.6)

Qwall = 2.7 x 105

1.0
0.9
0.8
0.7
0.6
0.5
0.4-
0.3
0.2
0.1
0.0 /
0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
p/R

Fig. 3.5: Electric and magnetic field profiles for the TMo10 mode.

while the measured Q was

Qwall (measured) = 1.6 x 105 (3.7)

which is in reasonable agreement with the theoretical value.

Tuning of the cavities was accomplished by moving dielectric or metal rods

inside of the cavity. Dielectric rods, in general, decrease the resonance frequency

whereas metal rods increase it. Our dielectric was the low loss ceramic compound

Mg2Ti 03/Mg A1204 (SMAT-9.5) [42] with a dielectric constant of e 9.5 and

a loss tangent tan 6 < 10-5. The loss tangent was measured by placing a disk

shaped sample inside a lead plated superconducting cavity. The microwave losses,

1/Q, are

1/Q = 1/Qwall + 1/Qdielectric + 1/Qcoupling (3.8)

For a weakly coupled cavity possessing negligible wall losses, 1/Q is then deter-

mined by the losses in the dielectric and tan 6 = (/Q, where ( is a geometrical

filling factor of order unity.

Each cavity contained a large dielectric or copper rod which was moved side-

ways for coarse tuning and a small dielectric rod for fine tuning. This configuration

was chosen in order to avoid longitudinal mode localization [43], which leads to

degradation of the form factor C.

The larger rod was moved through an arc shaped slot cut in both end caps.

The position of the rod was adjusted manually by rotating a stainless steel rod

on top of the dewar. The movement was translated by gears into the motion of

two arms supporting the rod through teflon holders which were heat shrunk to

the dielectric (copper) rod. The rod was further secured against vibrations by

The fine tuning rod was moved vertically through a hole in the top plate

of the cavity. The position of the rod was adjusted by a dove tail slider on top

of the cavity and was driven by a computer controlled stepper motor [45] on top

of the cryostat. The total rod travel distance of the fine tuning rod was 16 cm

corresponding to roughly 18000 steps of the motor. The tuning range of this rod

was approximately 20 MHz.

Table 3.1 summarizes the parameters of the cavities used in this experiment.

Figures 3.6, 3.7 and 3.8 show tuning curves calculated for the case of a single

dielectric or metal rod moving radially inside a cylindrical cavity. The modes were

obtained by solving the 2D wave equation using a relaxation method. Figure 3.9

shows the calculated form factor C. In calculating resonance frequencies and form

factors, we neglected the presence of the fine tuning rod in the cavity. Figures

3.10 and 3.11 are 3D-plots of the electric field of the TMo10 mode for a cylindrical

2.0
Rod

3.0 4.0 5.0
Displacement (cm)

6.0

Fig. 3.6: Cavity frequency versus radial displacement of a dielectric rod for
R = 7.62 cm, r = 0.64 cm, L = 38.1 cm and E = 9.5. The TMO10 mode used in
the search is shown along with the TE modes in its vicinity.

1.7

1.6

0.0

E113

TM01 0

TE ,

ITE 12

1.8 .. . .

1.7 TE

1.6 TM010
CD
1.5

Q 1.4

_: 1.3 TE 1

1.2

1 .1 ..... '
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Rod Displacement (cm)

Fig. 3.7: Cavity frequency versus radial displacement of a dielectric rod for
R = 6.83 cm, r = 0.64 cm, L = 38.1 cm and e = 9.5. The TM010 mode used in
the search is shown along with the TE modes in its vicinity.

2.1

2.0 2 1

1.7 1TE
L-1.9
CD

U 1.8

L 1.7 j TE 11
LL

1.6

1 .5 . . . . . . . . . . . . .. . .
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
Rod Displacement (cm)

Fig. 3.8: Cavity frequency versus radial displacement of a metal rod for
R = 7.62 cm, r = 0.62 cm and L = 38.1 cm. The TMO10 mode used in the search
is shown along with the TE modes in its vicinity. To avoid cluttering, the higher
TM modes as well as the TEM modes are not drawn.

1.0
0.9
0.8
0.7
0.6
o 0.5
0.4
0.3
0.2
0.1
0.0
0.

0

1.0

2.0 3.0 4.0 5.0
Rod Displacement (cm)

6.0

7.0

Fig. 3.9: Form factor C of the TM010 modes used in the search. Curve 1
is calculated for a radially moving dielectric of radius r = 0.64 cm and dielectric
constant E = 9.5. Cavity radius R = 6.83 cm. Curve 2 shows the same for
R = 7.62 cm. Curve 3 is calculated for a metal rod of radius r = 0.62 cm and
cavity radius R = 7.62 cm.

3

- 1

IIIIIII 1

Fig. 3.10: 3D-plot of the electric field for the TM010 mode of a cylindrical
cavity with a radially displaced dielectric rod. R = 6.83 cm, r = 0.64 cm, E = 9.5
and rod displacement from center is 3.2 cm.

Fig. 3.11: 3D-plot of the electric field for the TM01O mode of a cylindrical
cavity with a radially displaced metal rod. R = 7.62 cm, r = 0.62 cm and rod
displacement from center is 2.1 cm.

2.0 .. . .

2 -- ^ -^-^--- ------ -- -- -- -- - -iiz -:** * *
I TM

TE 1.0 TE2

0-
I

TTM
c 0.5

L.
LL
0.0 ........
0 2 4 6 8 10 12
Rod Travel (mm)

Fig. 3.12: Measured mode crossing between the TE112 mode and the TM010
mode.

-35

-40

-45

-50

-55
0.0

0.1 0.2 0.3 0.4

Frequency (MHz) 1.51455 GHz

Fig. 3.13: Transmitted power through cavity II near the TM010 resonance.
The cavity was overcoupled with QL = Qwall/3 40000.

-50 L-
0.0

0.1 0.2 0.3 0.4

Frequency (MHz) 1.51465 GHz

Fig. 3.14: Reflected power off major port of cavity II near the TMo10 reso-
nance. The return loss is 10.5 dB.

Table 3.1 Summary of cavity parameters. L = 38.1 cm in all cases.

R [cm] tuning rod tuning range [GHz] Qwall C
7.62 dielectric 1.32-1.44 1.5 x 105 0.50
6.83 dielectric 1.44-1.60 1.2 x 105 0.45
7.62 metal 1.56-1.85 0.5 x 105 0.60

cavity with a displaced dielectric and metal rod respectively. As seen from the

tuning curves in Figs. 3.6, 3.7 and 3.8, a number of crossings between the TM010

mode and TE modes occur. In the neighbourhood of the crossings, the modes will

mix and repel each other leading to holes in the spectrum. Figure 3.12 displays

such a crossing as measured in cavity I. The holes are typically a few hundred

kHz wide. Figure 3.13 shows a transmission spectrum of the overcoupled cavity II

near the TMo10 resonance, with Qwall = 2 Qhole, while Fig. 3.14 shows the power

reflected off the major port for the same resonance.

The Microwave Amplifiers

The cryogenic microwave amplifiers were purchased from Berkshire Tech-

nologies [46]. The first of the three stages consisted of a high-electron-mobility

transistor (HEMT), while the remaining stages utilized ordinary GaAs microwave

transistors. Upon cooling, the noise temperature of these devices drops dramati-

cally. A photograph of one of the amplifiers is shown in Fig. 3.15, whereas Fig. 3.16

shows an amplifier located in the cryostat.

The noise temperature of the amplifiers was carefully measured using a vari-

able temperature technique [47]. A 50 Q termination acted as the noise source

and was connected to the amplifier input. The amplifier excess noise is obtained

by varying the temperature of the load and measuring the output noise of the

amplifier. Figure 3.17 shows the test setup including load and amplifier.

The termination consisted of a metal film rod resistor [48] on a BeO substrate

with one end soldered to the center conductor of a section of cryogenic coax cable.

The other end was soldered to a copper plate, which had a thermometer and heater

attached to it. The time constant of a few seconds was determined by a thermal

link consisting of two 10 cm long strands of AWG-18 copper wire between the

copper plate and the helium bath. The microwave feedthrough of the vacuum can

was a commercial hermetic connector [49] with the rubber o-ring replaced by an

indium o-ring. The emitted noise power of the termination in a bandwidth B is

P = kB TB (1 Ipl2) (3.9)

where kB is Boltzmann's constant, T is the load temperature and p is the reflection

coefficient. The power return loss of the unit was measured to be greater than

20 dB, thus giving only a small correction to the perfectly matched case. The

Fig. 3.15: Photograph of L-band HEMT amplifier with top lid removed. The
input is to the left and three amplification stages are visible.

Fig. 3.16: Photograph of cryostat showing magnet and cavity. The fine
tuning drive is in the center, and the adjustable coupling is to the right. The
amplifier and circulator are located near the top.

S-- to spectrum analyzer

amptlfler under test

Helum

stainless coax

vaCUUM

hermometer

termination

Fig. 3.17: Setup of noise temperature measurement.

heater

Cu late

electrical loss of the line between load and amplifier was less than 0.1 dB and was

neglected.

The noise power at the output of the amplifier is

P = kB (Ta + T) BG (3.10)

where Ta nad G are the amplifier noise temperature and gain respectively. This

power was further amplified by a room temperature amplifier [50] and finally de-

tected by a HP 8569B spectrum analyzer [51]. For a given frequency, the noise

temperature is obtained by measuring P (Eq. 3.10) for different T's and fitting

these data with a straight line to extract Ta. The results for two cryogenic ampli-

fiers are given in Figs. 3.18 and 3.19.

For these ultra low noise amplifiers it is necessary to take into account the

noise contributions from the other stages in the chain. The formula for cascaded

amplifiers is

T = Ti + T2/G1 + T3/G2 + "'" (3.11)

where the Tn's and Gn's are the noise temperatures and gains respectively. The

post amplifier had a gain of 50 dB and a noise temperature of 200 K, making

all following contributions negligible. The two cryogenic amplifiers had gains of

34 dB and 35 dB with minimum noise temperatures of order To 3 K. Thus the

0 1I . I . . .
1.2 1.3 1.4 1.5 1.6 1.7 1.8
Frequency (GHz)

Fig. 3.18: Noise temperature of Berkshire L-1.5-30H at a physical tempera-
ture of T= 4 K. The power gain is 34 dB over the shown band.

2-

1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3
Frequency (GHz)

Fig. 3.19: Noise temperature of Berkshire S-2.3-30HR at a physical temper-
ature of T= 4 K. The power gain is 35 dB over the shown band.

noise from the post amplifier adds a small amount of extra noise to the system

( 0.1 K) which is included in Figs. 3.18 and 3.19.

In the experiment, magnetically shielded cryogenic circulators [52] were in-

serted between cavity and amplifier for impedance matching. This procedure was

necessary because the cavity impedance changed rapidly across the resonance, and

the amplifier was optimized for a constant input impedance of 50 f. The circu-

lator uses its third arm to absorb any reflected power coming from the amplifier

input. The third port of the circulator was also used to bring microwave power to

the major port for measurement of its reflection coefficient. In order to keep room

temperature noise from entering the amplifier or cavity, a 50 dB attenuator [53]

was added. Assuming the coax cable is properly terminated at room temperature,

the noise power per unit bandwidth emitted by the attenuator is [54]

kBT = kBToa + kBTatt(1 a) (3.12)

where a is the power loss and To ~ 300 K. Therefore, the attenuation is sufficient

to reduce the incoming noise to a level equivalent to the bath temperature.

Room Temperature Data Acquisition Hardware

Figure 3.20 shows the complete experimental setup and data acquisition

system. The output of the cryogenic amplifier was fed into a post-amplifier [50]

and subsequently into the first mixing stage.

The image reject mixer (see Fig. 3.21) was assembled from commercially available

components. The principle of an image reject mixer is to suppress the unwanted

sideband through destructive interference. The image reject ratio of the mixer

refers to the power ratio at the mixer output of the two sideband frequencies.

For a given intermediate frequency wIF, the mixer is sensitive to two side-

bands represented by the voltages V = cos [(w wIF)t ], where w is the LO fre-

quency. The RF hybrid [55] splits this signal into two components with a 900

phase shift relative to each other:

\< [ ( 1 )cs [( O i(F)] 3
cos [(w wF)-t I` \cos [(w WiF)t + 7/21) (3.13)

It is followed by two identical doubly balanced mixers [56] which down converted

the signal to an IF frequency, i.e.

1 cos[(wiwlF)t] 1 (cos[wIFt (3.14)
V\2cos[(w w )t + r/2] J) 2\cos[wF t /2] '

where the mixer losses have been neglected. The LO [57] output power level was

set to 10 dBm and was divided in phase by a power splitter [58]. The IF frequency

for tiagnostics Image reject
I I Post Amp mixer IF Amps band pass filter

Mixer

superconducting magnet
< B > = 7,5 T

Fig. 3.20: Block diagram of experimental setup. The room temperature part
consists of a two-stage superheterodyne receiver followed by a real time spectrum
analyzer.

Mixer

1-^

RF In -- IF out

900 o Power 90
Lk V y Splitter Hb
Hybrid Hybrld
50 R0 50 0

Mixer

Fig. 3.21: Block diagram of image reject mixer. The image reject ratio was
S20 dB in the 1-2 GHz band.

of 10.7 MHz was determined by the center frequency of the crystal filter following

the mixer. Finally the IF signals were combined in an IF hybrid [59] where the

the two sidebands appeared at two different ports. The effect of the hybrid can

be represented by
l c cos [wF t )

S(s[wt 0 2])' (3.15)
cos [w IF t 7 /2j) 0
4 cos [wIF t + 7r/2

where the upper and lower entries in the bracketed expressions on the RHS give

the voltages at the two output ports. As a result, the two sidebands appear at

different ports and the undesired one is terminated. The RF loss of the unit was

S8 dB and the image reject ratio was measured to be > 20 dB (see Fig. 3.22).

The signal was further amplified by two IF amplifiers [60], with a combined

gain of 60 dB. The signal then passed through a 8 pole crystal filter [61] before

entering the second mixing stage. The filter had a center frequency of 10.7 MHz

and a 3 dB pass band of 30 kHz. The attenuation was ? 3 dB over the useful band

(see Fig. 3.23) and was slightly temperature dependent. To stabilize the response,

the filter was enclosed in a box and its temperature held at T = (29 .1) C by a

temperature controller [62].

The second mixing stage comprised a single double balanced mixer [63], and

a local oscillator [64]. The audio signal covering DC to ? 30 kHz was finally passed

40

30

o
*r-4

rC
cr

LJ
o

E
CD

i-

1.0 1.2 1.4 1.6 1.8 2.0

Frequency (GHz)

Fig. 3.22: Image reject ratio versus frequency of first stage mixer for an IF
of 10.7 MHz. The solid curve is for fLO > signal, while the dotted curve is for the
reverse.

0

-5

-10
rn

-15 -

0

ro
-2 -20 -

C -25

4-')
.a,

-30

-35

-40
10.68

Fig. 3.23: Response of crystal filter.

10.69 10.70 10.71

Frequency (MHz)

10.72

to an AC amplifier and sampled by an A/D converter with a resolution of 16 bit.

The sampling rate was set to 70 kHz in order to oversample slightly. The signal

was then spectrum analyzed and averaged in real time. For each spectrum, 64

samples were taken yielding a 32 point power spectrum. The samples were stored

in a buffer memory, and while one set of samples was taken, the previous array

was fourier analyzed and averaged. The power spectra were computed by a TI

TMS320C25 digital signal processing chip which is a fixed point device with an

instruction cycle of 100 ns. All digital processing components including a RAM

of 64 kbytes for data and program storage were on a single board [65] which was

integrated into a Z-386 workstation [64].

CHAPTER 4
DATA ACQUISITION AND ANALYSIS

The Search Process

Data taking in this experiment was a time consuming and repetitive task.

Therefore as much as possible of the process was automated and runs were left

unattended for many hours.

The main data acquisition program was called FNDAX2 and is listed in

the Appendix. It controls various instruments with GPIB commands or by RS232

serial line. Before the program is run, the coupling strength of the major port is

set manually to the desired value. The program starts by measuring the resonance

frequency, Q and reflection coefficent of the TM010 mode. For this purpose, a set

of three HP33311B microwave switches [51] were installed and controlled by a

HP59306A relay actuator [51]. First CW power from the LO is directed to the

weakly coupled minor port and the transmitted power is measured for several

frequencies around the resonance by a HP438A power meter [51]. These data are

then fitted by a Lorentzian to obtain resonance frequency and Q. The LO is then

switched to the cryogenic circulator and reflected power is measured on and off

resonance which yields the coupling strength of the major port.

At this stage, the LO enters the data acquisition mode and is switched to the

73

image reject mixer. The host PC then starts the spectrum analysis program which

was downloaded into the DSP board prior to the run. The A/D converter samples

in interrupt mode the audio signal coming from the amplifier chain and stores

the data in one of two buffer memories. The TMS320C25 chip in the meantime

works on the other buffer and computes a 64 complex point FFT resulting in a

32 point power spectrum. The time for one spectrum is 0.9 ms and a total of

105 spectra are taken and added together in approximately 90 seconds. After the

spectrum is completed, the host uploads the data and normalizes it. In order to

get a flat baseline, the frequency response of the crystal filter has to be divided

out. Moreover, the noise output of the cryogenic HEMT amplifier can deviate

from a flat spectrum if temperature imbalances in the circulator-amplifier-cavity

system exist. To account for all those effects, a running average is calculated from

the last 30-40 scans and used for normalization.

For each bin, the deviation from the mean is calculated according to

P-P
#sigma= (4.1)
P

where N is the number of averages and the spectrum is displayed on the screen.

The program searches the spectrum for peaks above 2 a in single bins and com-

binations of two neighboring bins. The bin width of 1 kHz roughly matches the

expected width of the axion line. If a peak appeared, another spectrum is taken

and compared with the first. In the first cavity scans, the spectra were added to-

gether, but in the later scans the two spectra were checked for coincidence peaks.

This process was repeated up to four times and the peak was flagged for later

rechecks if it proved to be statistically significant. Each spectrum is written into

a file along with the values of the cavity parameters and the temperature of the

bath. The computer then resets the cavity frequency by sending a number of

pulses to the stepping motor and the cycle repeats itself. A typical run produces

about 400 spectra corresponding to a tuning range of about 8 MHz. The data are

later backed up on floppy disks for permanent storage.

Occasionally, a test signal was introduced into the system. The CW signal

was generated by a HP8350A [51] sweeper phase locked to an EIP578 [67] mi-

crowave counter for frequency stability. The power was attenuated to ~ 10-20 W

and fed into the minor port of the cavity. This procedure served mainly for the

frequency calibration of the detector.

The Data

To this date (July 90), five scans have been completed. Cavity I with a

dielectric tuning rod was run twice at T = 2 K and T = 4 K, while cavity II and

cavity I with a metal rod were run once at T = 2 K.

A typical spectrum obtained during a scan is shown in Fig. 4.1 whereas

Fig. 4.2 is an example of a spectrum with a candidate peak. A number of similar

peaks were found and rechecked after the scan with the magnetic field turned off.

None of the candidate peaks survived this test, i.e., all persisted when the field

was off. Most of the peaks were narrow (< 200 Hz) and are assumed to be pickup

from computer clocks in the vicinity of the experiment. The number of peaks were

quite high and could have been reduced by putting the apparatus into a screened

room. The scans have some holes in the covered frequency range due to crossings

with TE modes. The data from the four scans is summarized in Table 4.1.

Table 4.1

Scan Frequency Coverage [GHz] Holes [GHz] Peaks

I,II 1.32-1.44 1.346096-1.346382 32

1.444693-1.445210
III 1.44-1.60 1.449171-1.450566 62

IV 1.60-1.63 1.602404-1.602991 18

V 1.80-1.83 2

5 10

Frequency

15 20 2!

(kHz) 1.603021 GHz

Fig. 4.1: Sample spectrum of cavity output. la corresponds to a power
P 3 x 10-22 W in the cavity.

0

-2

-4

-5

-8

-10
0

I

0 5 10 15 20 25
Frequency (kHz) 1.533251 GHz

Fig. 4.2: Spectrum with candidate peak.

Limit on the Electromagnetic Coupling of the Axion

Since no peak with the signature of an axion signal was found, it is possible

to put an upper limit on the electromagnetic coupling of the axion, provided the

galactic halo is axionic. The noise fluctuations in the detector have a Gaussian dis-

tribution, and one can calculate the probability for a given signal to be detectable.

Our detector is looking for 2a peaks above the noise floor. The probability for

a negative fluctuation of > 2o is 0.025 or in other words, a > 4a signal will be

seen at a 97.5% confidence level. In terms of the system noise temperature Tn,

bin bandwidth B and # of averages N, the minimum detectable signal power is

(at 97.5% C.L.)

Pmin = 4 kB Tn B /N (4.2)

where kB is Boltzmann's constant. Tn is given by

Tn = Tbath + Tamp (4.3)

where Tamp is almost solely determined by the contribution from the cryogenic

amplifier. Table 4.2 summarizes the parameters entering the calculation of Pmin

for the different runs.

Scans I and II are special in that the same frequency range was covered

twice. By calculating the combined probability for finding a peak, one obtains a

Table 4.2

power somewhat smaller than in Eq. 4.2. The values for Pmin in Table 4.2 are

to be compared with the power from axion to photon conversion going into the

detector. With the cavity overcoupled (Qwall = 2Qhole), only a fraction 2/3 of

the power in Eq. (2.82) is leaving the cavity. Moreover, the cavity response to

axion decay is falling off away from resonance, and the power is further reduced

by a factor a with 0.5 < a < 1.0.

Table 4.3 lists the average power from a -+ y conversion for the individual

scans. Also shown is the axion mass and the upper limit on ga2 (for Pa = Phalo),

which is plotted in Fig. 4.3. This limit is valid assuming the power Pd is falling

in a single bin and is worse by a factor '2 if it falls into two bins.

Scan N B [kHz] Tbath [K] Pmin [W]

I,II 105, 105 1.1 4.2,2.2 0.7 x 10-21

III 105 1.1 2.2 1.0 x 10-21

IV 2 x 105 1.1 2.2 1.3 x 10-21

V 105 1.1 2.2 1.1 x 10-21

81

Table 4.3

1.0E-24

1.0E-25

,1.0E-26

S1.0E-27 -

N1l.OE-28
0
S1.OE-29

1.0E-30

1.0E-31
4.0E-6

1.0E-5 2.0E-5
ma (eV)

Fig. 4.3: Experimental limit on the electromagnetic coupling gayY of the
axion. Also shown is the limit obtained by the RBF collaboration [68]. The solid
and dashed straight lines are the theoretical values of ga77 for the DFSZ axion
and the KSVZ axion (E/N = 0).

CHAPTER 5
SUMMARY AND CONCLUSION

The experiment described herein represents one of the first attempts to de-

tect dark matter in our galaxy. The precise nature of the dark matter is not

known, but axions are a serious candidate for it. The first cosmic axion search

was carried out in a pioneering experiment by the RBF collaboration [68]. This

experiment is the second of its kind.

In the frequency range covered, no axion signal was found. Our detector

was probably not sensitive enough to detect axions, even if they existed in our

galactic halo and with a mass within the searched range. To have a good chance

of detecting galactic axions, the sensitivity of the apparatus needs to be improved

by roughly three orders of magnitude in case of the DFSZ axion (see Fig. 4.3).

Nonetheless, the present data constrain the coupling gay7 of any type of axion if

they are the dark matter in the halo. The limits shown in Fig. 4.3 are expressed

in terms of the experimental sensitivity, which goes like g2 7. The limit on g2

obtained by the RBF collaboration was improved by roughly an order of magnitude

over the frequency band covered so far. This improvement was mainly a result of

a lower system noise temperature resulting from the use of HEMT amplifiers and

of lower bath temperatures.

Besides setting a limit on the axion coupling, these two prototype exper-

iments have also helped to gain know-how for the construction of an improved

detector system. Our microwave receivers have a demonstrated sensitivity to

signals as small as 10-21 W. Since there is not much hope to greatly improve

the amplifier technology in the immediate future and since steady-state magnetic

fields are at present limited to about 20 T, the only possible way to reach the

DFSZ limit in the near future is by increasing the cavity volume.

If the present experimental parameters Tn, C, Q, B0 and d(lnf)/dt are not

appreciably changed, achieving the required sensitivity means increasing the vol-

ume to several thousand liters. The fundamental resonant frequency of such a

cavity is around 100 MHz. Searching for the axion at those frequencies would

be as interesting as at 1 GHz or 10 GHz given the uncertainties in cosmological

calculations of the axion density as a function of the axion mass ma.

A realistic search should of course cover a very large frequency band while

maintaining high sensitivity and large volume. The difficulties associated with

realizing these constraints increase rapidly with increasing frequency [41]. The

most promising approach at present seems to be dividing the available volume into

smaller and smaller cells with growing resonance frequencies. The cavities have

to be ganged together to the same frequency and their output combined in phase.

85

The complexity of such a system will limit the number of cells to a few hundred

and allow the detector to work up to a few GHz. The higher frequency (mass)

ranges could then be explored by a detector with a much higher magnetic field

over a smaller volume. Finally, the chances for a successful axion experiment could

be greatly improved, if the allowed window of axion masses could be narrowed.

APPENDIX A
CAVITY DEVELOPMENT

One of the challenges in the design of a full scale axion cavity detector is

the development of large volume / high frequency microwave cavities [43]. The

ideal cavity should be operated in the lowest TM mode for maximizing the form

factor C. One way of raising the fundamental frequency is to insert metal posts

into the cavity. This would mostly affect the TM modes, because the boundary

conditions for these modes require Eta to vanish on all metal surfaces. On the

other hand, the posts would not, in a first approximation, change the TE modes,

whose boundary conditions require the normal derivative of Htan to be zero on the

surface. Since the mode density of the TE modes increases as f2, the frequency

lifted TM mode would have to cross more and more TE modes during tuning. This

mode crossing problem will eventually become so serious that the holes arising

from the crossings would consume a large portion of the searched band. Another

danger associated with using many posts is transverse mode localization. This

effect is caused by asymmetries in the post configuration leading to unpredictable

localizations of the fields.

One alternative to using posts is to employ identical cavities and combine

their output signals. The previously described problems of localization and reso-

nance crowding would disappear but at the cost of larger complexity. The cavities

86

would have to be tuned individually to the same frequency and their power com-

bined in phase and brought to the front end of a single amplifier.

We tested this method at room temperature for the simplest case of two

cavities. The test setup is shown in Fig. A.1. The empty cavities had TM010

resonance frequencies of 1.67 GHz with Qwuall 3 x 104. The frequencies differed

initially from each other by 0.7%, but could be made to match by inserting a small

teflon tuning rod (0.4 cm in diameter) into one of the cavities.

Two identical Wilkinson powerdividers/combiners [58] were used to split and

combine the signals in phase. A schematic circuit diagram is displayed in Fig. A.2.

A signal of voltage V ~ e-iwt entering on the left of Fig. A.2 is split into equal

amplitude signals V ~ e-iwt// on the right. When used as a combiner with

amplitudes a e-iwt and be-iwt-id, the output voltage is (a + be-i~) e-iwt/V,

and power is dissipated in the internal resistors unless the two inputs are exactly

balanced. In the experiment, one of the ports of each cavity was critically coupled,

while the other was weakly coupled.

The cavity can be modeled by a series RLC-circuit with resonance frequency

2 = 1/LC and Qw = woL/R. The input impedance of a single port cavity near

resonance is given by

Zc + iwL + O (2 Q w Wo+ 1) (A.1)
OR sWC WO

LC

circulator

Teflon rc

- -I-

weakly c.

circulator

Spectrum [
analyzer

Fig. A.1: Test setup of two frequency-matched cavities.

50 n

-2 50 0
V/4 /

50 Q

A/4 \
-2 50 n

50 0

50

50 Q

Fig. A.2: Circuit diagram of Wilkinson powerdivider/combiner.

where 3 = Qw/Qh is the coupling parameter and Zo is the characteristic line

impedance, usually 50 Q. The input impedance is equal to the line impedance for

a critically coupled cavity (/ = 1) on resonance, making the cavity reflectionless.

For a two port cavity, Eq. (A.1) still applies but R is modified to include the

additional losses from the second port. A signal transmitted through a critically

coupled cavity (while the second port is weakly coupled) will suffer a phase shift

V ~- ei with

tana=( wL /2R 2 QLw x (A.2)
wC /w0

and the amplitude is decreased by a factor 1/v1- relative to the value on

resonance.

The output voltage of the power combiner for the LO frequency at the reso-

nance frequency of the first cavity and variable resonance frequency of the second

cavity is then

V 1 1 A.3)
V -V(1 + -- 1 eia (A.3)

and the power

VI =4 1 + 32) (A.4)
Vo 4 1 + X 2

This relationship is plotted in Fig. A.3. Also shown are the measured values

which are in good agreement with the theoretical curve. Operation of a multi-

cavity detector would require keeping the cavity resonance frequencies within a

91

-2
c \

-3

-4

5 . . I .. I I I , *
-3 -2 -1 0 1 2 3

X

Fig. A.3: Transmitted power for the configuration shown in Fig. A.1. The
LO frequency is fixed at the resonance frequency of one cavity while the second
cavity is tuned with a teflon rod.

92

few percent of the cavity width in order to avoid mismatch losses. In summary,

this experiment has demonstrated the feasibility of using phased multicell cavities

as axion cavity detectors.

APPENDIX B
SOFTWARE LISTINGS

program fndax2

c data acquisition program for axion search
c
character frec*20,filnm*20,filcal*20,card*2,bell*l,dot*l,ampl*10
character dir*l
integer*4 spein(32),speina(32),nav,np,npar,ncal
parameter (np=8,ncal=40)
external funct
real*8 spel(32),spe2(31),devl(32),dev2(31),omega,devs(5,32)
real*8 cal(32),f,df,aa(np),bb(np),parmt(3),del,fold,dw
real*8 chisq,chipr,alamda,parmr(3),pl,pr,pin,fl,fr,avv,avc,pref
real*8 stdev,call(0:ncal-l,32),skk(32),dell,fstart,fstop
logical peakl(32),peak2(31),peaklp(32),peak2p(31),peak,peakh
data frec(l:2) /'fr'/, frec(13:14) /'gz'/, fsam /70000./,
1 stdev /1.0d-7/, nav /100000/, bell(l:l) /7/, tc /4.3/,
2 del /2.0d-5/, npar /3/, dot /'.'/,nmax /4/,
3 ampl(l:2) /'ap'/, ampl(8:9) /'db'/ ,amp /-10.0/
c instrument initialization :
open (1,file-'ieeeout',status-'old',access-'sequential')
open (2,file-'ieeein',status='old',access='sequential',form=
1 'binary')
write (1,*) 'reset'
write (1,*) 'remote 01,02,11,13'
write (1,*) 'clear 01,02,11,13'
write (1,*) 'output 01;%A1B2A3'
write (1,*) 'output 02;am0pc'
write (1,*) 'output 11; f4 r4'
write (1,*) 'output 13;kb96en'
rewind 1
write (*,10)
write (*,12)
write (*,9) amp
write (ampl(3:7),8) amp
write (*,713)
if (dir.eq.'I'.or.dir.eq.'i') then

mult- -1
else
mult- 1
endif
write (*,13)
write (*,14)
ilen- 0
15 ilen- ilen + 1
if (filnm(ilen+l:ilen+l).ne.' ') goto 15
write (*,16)
write (*,716) nmax
write (*,17) tc
write (*,18) fsam
write (*,19) nav
nrep- nav/100000
ncount- ifix (1.0e7/fsam) 1
write (*,20)
open (3,file-filcal,status-'old')
if (card.ne.'/*') goto 21
avc O.OdO
do 25 k-1,32
25 avc avc + cal(k)
close (3)
avc avc/32.
do 27 k-1,32
do 28 1-ncal-ncals,ncal-l
28 call(l,k)- cal(k)/ncals
do 27 l-0,ncal-ncals-1
27 call(l,k) O.OdO
c
c start big loop :
c
nstept- 0
nsp- -1
write (filnm(ilen+l:ilen+4),'(a)') '.log'
open (4,file-filnm,status-'new')
write (4,123)
do 300 1l-l,ns
nsd- 0
peak .false.
peakh .false.
do 301 k-1,32

peakl(k) .false.
301 peaklp(k) .true.
do 302 k=l,31
peak2(k) .false.
302 peak2p(k) .true.
write (filnm(ilen+l:ilen+4),23) 11+1000,dot
write (*,35) film
open (3,file-filnm,status-'new')
write (3,*) '# of averages : ',nav
c measure f and q in transmission :
write (1,*) 'output 01;A1B2A3'
write (1,7) ampl(l:9)
rewind 1
ff f
df- 1.5*f/(parmt(2)*np)
do 40 k-l,np/2
if (ll.eq.l) then
ff ff df
else
ff ff + mult*df
endif
bb(k) ff
40 call getp (bb(k),frec,aa(k))
41 if (ll.eq.l) then
ff ff df
else
ff ff + mult*df
endif
bb(np/2+l) ff
call getp (bb(np/2+l),frec,aa(np/2+l))
if (aa(np/2+l).le.aa(np/2).and.aa(np/2).gt.3.e-7)
1 goto 45
do 42 k-l,np/2
bb(k)-bb(k+l)
42 aa(k)-aa(k+l)
goto 41
45 do 48 k-np/2+2,np
if (ll.eq.l) then
ff ff df
else
ff ff + mult*df
endif
bb(k) ff
48 call getp (bb(k),frec,aa(k))
256 parmt (1) 1.0d-30
do 50 k-l,np
parmt (1) dmaxl (parmt(1),aa(k))
50 if (parmt(l) .eq. aa(k)) parmt(3) bb(k)
alamda -1.
55 chipr chisq
call fit (bb,aa,stdev,np,parmt,npar,chisq,alamda,funct)
if (abs(chisq-chipr)/chisq .gt. 0.0001) goto 55
dell abs(fold parmt(3))
if(ll.gt.l) nstep-idnint ((del/dell)*nstep)