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 3D 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.
TABLE OF CONTENTS
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 106 eV < ma < 6.77 x 106 eV and 7.47 x
106 eV < ma < 7.60 x 106 eV.
The detector consisted of a highQ 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 (9 109), 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 10eV(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 ~ 1041 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 ~ 105 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
be made out of axions.
The experiment described herein has made an attempt to detect these galac
tic axions for an axion mass near 105 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 (106)), 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 highQ 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 1020 %. 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 10leV < ma < 6.77 x 106eV and
(1.5)
7.47 x 106eV < ma < 7.60 x 106eV
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 GellMann 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 nonAbelian 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 '0vacuum' 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 6vacuum would be a nonvanishing electric dipole moment of
the neutron, dn. The current upper bound of dn is [8]
Idnl 1025 e cm (2.5)
which constrains 0 to be very small, namely
IO\ <' 109 (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 offdiagonal and nonhermitian, 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 eia/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
AdlerBellJackiw (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 pseudoNambuGoldstone 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
to excitations about the minimum:
physical = a (a) (2.16)
The first axion model embedded the U(1)pQ quasisymmetry 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 105
(2.21)
while the experimental upper limit is 3.8 x 108 [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 PecceiQuinn 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  e2ia
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.
In addition a discrete symmetry
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 106 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 1015 GeV1 (E 2 4+ z ma
(N 3X3 1+ez 10l+ l5 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 1015 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 1012 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 Comptonlike 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) Comptonlike scattering, c) nucleonnucleon bremsstrahlung
and d) electronnucleon and electronelectron bremsstrahlung.
Ze
a)
N
c)
In the DFSZ model, axion cooling will prevent red giants from entering the helium
burning phase unless [15]
mDFSZ < 102eV. (2.33)
For an axion mass ofma 2 0 (10300keV), the previous arguments are no longer
valid because thermal axion production is suppressed by the Boltzmann factor
ea. 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 neutrinocooled core, which yields the excluded
mass region [17]
103eV < 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 103eV (2.35)
and two windows for the KSVZ axion of
nKSVZ <5 103eV 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 103eV
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 102 (ma/eV,) (2.37)
where Q = p/pc and 100 h km sec1 Mpc1 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/Ri3 (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
radius is given by
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
1 1 d2rad (2.55)
(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
of radiated axions and Prad is the energy density of radiation. Equation (2.54) can
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 105eV 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 (105 103)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 masstolight
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 r1/2 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
Radius (kpc)
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 Mpc3. (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 redshiftnumber 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 ('bottomup'). The second group (e.g. light neu
trinos) is freely streaming on galactic scales and produces large structures first
('topdown').
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 1025 g cm3 0.3 GeV cm3 (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 kmsec1 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 sec1. 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(106).
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 axionphoton 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(106)) (2.67)
For axion masses with ma = 0 (105) 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 (~ 1013cm3) and is coherent on laboratory scales with a de Broglie
wavelength of
h 105eV
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 5.0E7 1.0E6 1.5E6 2.0E6
E/m,
Fig. 2.4: Kinetic energy distribution of galactic axions with earth motion
included. The fractional width is 106.
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 1023 W np ) (27r(3GHz)
S 4 10 8 T 5 x 1025 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 IPS100 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. CuNi
tubing, while the impedance was made out of a 3 cm long section of a 0.1 mm
i.d. CuNi 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 Afridge.
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 Atransition 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 Afridge. 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) eit
(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 72wne2
Here, me = 9.11 x 1028 g is the electron mass, e = 4.80 x 1010 esu the electron
charge, vF the Fermi velocity and n the conduction electron density. For Cu [41],
vF = 1.57 x 108 cm sec1, n = 8.50 x 1022 cm3 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 (SMAT9.5) [42] with a dielectric constant of e 9.5 and
a loss tangent tan 6 < 105. 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
springloading the supports with BeCu contact strips [44].
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 3Dplots 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
L1.9
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: 3Dplot 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: 3Dplot 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 1.321.44 1.5 x 105 0.50
6.83 dielectric 1.441.60 1.2 x 105 0.45
7.62 metal 1.561.85 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 highelectronmobility
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 AWG18 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 oring replaced by an
indium oring. 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 Lband 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 L1.530H 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 S2.330HR 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 postamplifier [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 twostage 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
Load  Load
Mixer
Fig. 3.21: Block diagram of image reject mixer. The image reject ratio was
S20 dB in the 12 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
*r4
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 Z386 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 circulatoramplifiercavity
system exist. To account for all those effects, a running average is calculated from
the last 3040 scans and used for normalization.
For each bin, the deviation from the mean is calculated according to
PP
#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 ~ 1020 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 1.321.44 1.3460961.346382 32
1.4446931.445210
III 1.441.60 1.4491711.450566 62
IV 1.601.63 1.6024041.602991 18
V 1.801.83 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 1022 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 1021
III 105 1.1 2.2 1.0 x 1021
IV 2 x 105 1.1 2.2 1.3 x 1021
V 105 1.1 2.2 1.1 x 1021
81
Table 4.3
1.0E24
1.0E25
,1.0E26
S1.0E27 
N1l.OE28
0
S1.OE29
1.0E30
1.0E31
4.0E6
1.0E5 2.0E5
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 knowhow for the construction of an improved
detector system. Our microwave receivers have a demonstrated sensitivity to
signals as small as 1021 W. Since there is not much hope to greatly improve
the amplifier technology in the immediate future and since steadystate 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 ~ eiwt entering on the left of Fig. A.2 is split into equal
amplitude signals V ~ eiwt// on the right. When used as a combiner with
amplitudes a eiwt and beiwtid, the output voltage is (a + bei~) eiwt/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 RLCcircuit 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 frequencymatched 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:ncall,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 /1.0d7/, nav /100000/, bell(l:l) /7/, tc /4.3/,
2 del /2.0d5/, 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)
read (*,*) f
write (*,12)
read (*,*) parmt(2)
write (*,9) amp
read (*,*) amp
write (ampl(3:7),8) amp
write (*,713)
read (*,'(a)') dir
if (dir.eq.'I'.or.dir.eq.'i') then
mult 1
else
mult 1
endif
write (*,13)
read (*,*) nstep
write (*,14)
read (*,'(a)') film
ilen 0
15 ilen ilen + 1
if (filnm(ilen+l:ilen+l).ne.' ') goto 15
write (*,16)
read (*,*) ns
write (*,716) nmax
read (*,*) nmax
write (*,17) tc
read (*,*) tc
write (*,18) fsam
read (*,*) fsam
write (*,19) nav
read (*,*) nav
nrep nav/100000
ncount ifix (1.0e7/fsam) 1
call adset (ncount)
write (*,20)
read (*,'(a)') filcal
open (3,filefilcal,status'old')
read (3,*) ncals
21 read (3,'(a)') card
if (card.ne.'/*') goto 21
avc O.OdO
do 25 k1,32
read (3,*) n,cal(k)
25 avc avc + cal(k)
close (3)
avc avc/32.
do 27 k1,32
do 28 1ncalncals,ncall
28 call(l,k) cal(k)/ncals
do 27 l0,ncalncals1
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,filefilnm,status'new')
write (4,123)
do 300 1ll,ns
nsd 0
peak .false.
peakh .false.
do 301 k1,32
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,filefilnm,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 kl,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.e7)
1 goto 45
do 42 kl,np/2
bb(k)bb(k+l)
42 aa(k)aa(k+l)
goto 41
45 do 48 knp/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) 1.0d30
do 50 kl,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(chisqchipr)/chisq .gt. 0.0001) goto 55
dell abs(fold parmt(3))
if(ll.gt.l) nstepidnint ((del/dell)*nstep)
