This figure "fig1-1.png" is available in "png" format from:


http://arXiv.org/ps/


This figure "fig1-2.png" is available in "png" format from:


http://arXiv.org/ps/


This figure "fig1-3.png" is available in "png" format from:


http://arXiv.org/ps/


THEORY FOR THE DIRECT DETECTION OF SOLAR AXIONS BY COHERENT
PRIMAKOFF CONVERSION IN GERMANIUM DETECTORS



R. J. Creswick, F. T. Avignone III, and H. A. Farach
University of South Carolina
Columbia, South Carolina 29208 USA


J. I. Collar
CERN, CH-1211
Geveva, 23 Switzerland


A. O. Gattone
Department of Physics
TANDAR Laboratory, C. N. E. A.
Buenos Aires, Argentina


S. Nussinov
Tel Aviv University
Ramat Aviv, Tel Aviv, Israel


K. Zioutas
University of Thessaloniki, GR-54006
Thessaloniki, Greece


Abstract
It is assumed that axions exist and are created in the sun by Primakoff conversion
of photons in the Coulomb fields of nuclei. Detection rates are calculated in germanium
detectors due to the coherent conversion of axions to photons in the lattice when the
incident angle fulfills the Bragg condition for a given crystalline plane. The rates are
 1 Aug 1997 correlated with the relative positions of the sun and detector yielding a characteristic
arXiv: v1 1 Aug 1997 recognizable sub-diurnal temporal pattern. A major experiment is proposed based on a
large detector array.


Introduction

The putative pseudo-goldstone axion [1,2] had its inception almost twenty years

ago following the suggested solution to the strong CP problem by Peccei and Quinn [3].

This prompted many theoretical investigations and experimental searches.

An extensive review of axion phenomenology, and their effects on stellar

processes, and bounds, was given by Raffelt [4]. A detailed treatment of solar axions, and

of a proposed method of detecting them, was given by van Bibber et al. [5]. In this letter

we analyze a completely different direct-detection technique based on coherent Primakoff

conversion of axions to photons in the lattice of a germanium detector when the incident


angle satisfies the Bragg condition. Some of the basic concepts have been presented

earlier in different contexts [6,7].
The axion-photon interaction Lagrangian is Lint = (1/4 M) a FF , where
"a" is the pseudoscalar axion field, F is the electromagnetic field, and 1/M ga is the
axion-photon coupling. In the Born approximation the differential cross section for

elastic axion-to-photon conversion off an atom of charge Z (see Fig. 1) is:


2
d ga
= F2 2
2 sin 2 , (1)
2 ( )
a


d 16


where 2 is the scattering angle and F(2) is the atomic form factor. We have

independently verified this cross section given previously [6,7]. Note that the axion-to-

photon cross section is twice as large as that for photon-to-axion conversion because the

photon has two helicities.

The form factor, Fa(2), is that used in earlier work [7] and can be written in two

forms:

Zek2 Zek2
F (2) = = F , , (2)
- - (q k)
a a
r 2 + 2k2 1
( -cos2) = r 2 +q2

0 0


where r0 is the screening length, and q=2k sin is the momentum exchange. The

differential cross section can then be rewritten as follows:


2 2 2
2 2
h
q 4k - q
d Z
= . (3)
d M
c
2 2 - 2
16 2 2
r0 + q






Averaging equation (3) over all solid angle yields:


2 2
h 2
( 1 d Z 2
) = d = +1
ln 1 + 42
1 , (4)
4 d 8 2 2
M c 42 -





where r0k, and Z2
2/8M2c2 = 1.15  10-44 cm2
0 for Z = 32, where
= (ga  108 GeV)2 is the (unknown) relative strength of the axion-photon coupling.


The cross section for = 1 integrated over the energy range 0 to 12 keV is

1.46  10-43 cm2.

The expected flux of axions from the sun has been calculated by van Bibber et al.

[5]. The spectrum is continuous with a peak near 4 keV and falls off exponentially

beyond about 10 keV. The results of these detailed numerical calculations are well

approximated by the empirical form:
3
d (E/E
0 )
= 0 , (5)
/
dE E
0 eE E0 - 1
where 0 = 5.95  1014 cm-2 sec-1. We have used this same expression but have adjusted

the parameter E0 = 1.103 keV to take into account a small change in the solar model core
temperature when helium and metal diffusion are included [8].

For very light solar axions the Primakoff process in a periodic lattice is coherent,

similar to Bragg reflection of x-rays, which leads to the Bragg condition that the
2
momentum transferred to the crystal must be a reciprocal lattice vector, G = (h,k,l)

a0
where a0 is the size of the conventional cubic cell and h, k, and l are integers [9]. If we
assume the charge distribution in the crystal is simply a superposition of atomic charge

distributions, the rate of conversion per unit flux of axions of energy Ea and momentum

k a to photons of energy E and momentum k is:

.
dN V 1 d
; E, k ) = (
c)3 |S(G)|2 (G) (Ea - E) ( ka - k - G) (6)
d (Ea, ka 2 2
v G E d
c a



where V is the volume of the crystal, vc is the volume of the unit cell, and S(G) is the
structure function for the crystal, which for germanium is:


i
(h+k+l) i
2 (h+k) i (h+l) i (k+l
)
S(G) = 1+ e 1 + e + e + e
. (7)




Table 1 gives the parameters of the most relevant reciprocal lattice vectors of germanium.

If we integrate over all final photon states we find the total conversion rate of

axions with momenta ka and energy Ea to be


.
dN V d 1

c |G|2
|S(G)|2 E
a -
d = 2
c 2 . (8)
v G d (G)
c |G|2 2ka . G


Note that in equation (8) the conversion rate is essentially zero unless the energy and

momentum of the axion exactly satisfies the Bragg condition. In a real crystal, the

outgoing x-ray very probably produces a photo-electron after traveling a distance of order

microns which tends to smear out the delta function. However, so long as the mean free

path of the x-ray is long compared with the size of the unit cell (as it is in germanium), the

broadening of the delta function will always be much less that the actual energy resolution

of the detector and can be neglected.

If we now multiply this by the flux of solar axions, (5), the total rate of

conversion of axions is,
.
dN

^ V d 1 c |G|2
(k, E |S(G)|2 d E
dE a) = 2
c 2 a - , (9)
v G d (G) dE ^
c |G|2 2k . G
^
where k is the unit vector from the sun, and d/dE is evaluated at the axion energy

^
Ea =
c |G|2/ 2k . G . This can be written in the compact form:


.
dN . 2 3
^ 42 -g
(k, ) = M |S(g)|2 ( - ( g)) (10)
dE d N0
g g2 + 2 e - 1

2
c
where Md is the mass of the detector in kg, g = (h, k, l), = a0/r0, = ,
a0E0
. 2
(
0
g) = g2 0
, and N0 = 0 61
. / kg d for germanium.
2k^ . g ( 3
2 a )

0


^
For a given direction of incident axions, k, the Bragg condition for each reciprocal

^
lattice vector such that k . G > 0 determines a narrow range of energies. A germanium

detector has a characteristic low-energy resolution, E ~ 400 eV. We take this into

account by replacing the delta function in (10) by a gaussian with the same fwhm as the

detector. Finally we calculate the total counting rate in a range of energies of width ,


+ 2 .
^ dN ^
R (k,) = d

(k, ). (11)
dE
- 2


^
In figure 2 we show the expected counting rate, R (k(t),) R(t), as a function of

time over a single day. The position of the sun is calculated using the U. S. Naval

Observatory Vector Astronomy Subroutines (NOVAS) [10].

The pronounced variation in the axion counting rate as a function of time suggests

that events in the detector be analyzed in terms of the correlation function:


T
= R(t)- , (12)
0 [ R]n(t)dt

where R(t) is the theoretical instantaneous axion counting rate in a given energy interval, T

is the time during which the detector is on, R is the average of R(t) over this period and
n(t) is the number of counts at time t in a short time interval t. Note that if n(t) is

uncorrelated with the position of the sun, then ~ 0 within statistical fluctuations,

whereas if n(t) contains an axion component, will increase proportionally to T.

We have carried out Monte Carlo simulations of a germanium detector with

realistic background rates taking the range of energies from 2 to 10 keV in 0.5 keV

intervals. We find that an array of 400, oriented single crystal n-type detectors, each of

0.25 kg operating for 10 years can set an upper limit on the scale factor ,


-
10 4 . (13)

This corresponds to a 95% CL bound on the axion-photon coupling constant

ga = 1/Mc2 5  10-10 GeV-1 for axion masses between 0 and about 1 keV.
Similar results could be obtained using ~160, single crystals of p-type detectors,

each of 0.65 kg. A negative result would exclude a significant portion of the remaining

ga - ma phase space. A pilot experiment has been done which has accumulated data for
approximately 1.92 kg-years and is being analyzed by the methods discussed above [11].

This result was derived using the correlation function of equation (11). It is not clear that

more sensitive analysis techniques employing pattern recognition, for example, might not

be more effective. these are under investigation at present.

This large array of low energy threshold, low background detectors with excellent

energy resolution would also be a sensitive detector for weakly interacting massive

particles hypothesized as cosmic cold dark matter.


The present technique could also be applied to low temperature sapphire

detectors planned for dark matter searches, and to tellurium oxide detector arrays being

constructed for double-beta and dark matter searches. In these searches, nuclear recoil

events are of interest, whereas for an axion search, only events associated with photon

signals are relevant. The same is true for large arrays of Ge detectors which also employ

cryogenic bolemetric techniques. For these to be more effective than an ordinary low-

background Ge detectors, the raw background due to photons would have to be low.


Acknowledgments


This work was supported by the US National Science Foundation Grand

INT930INT1522, and the University of South Carolina Summer Institute for the

Foundations of Physics. S. Nussinov would like to acknowledge a BSFG (Israel USA

Foundation Grant). The authors would like to thank J. A. Bangert of the U. S. Naval

Observatory for supplying their vector astronomy subroutines, and Y. Aharonvov and E.

A. Paschos for helpful discussions.


References




[1] S. Weinberg, Phys. Rev. Lett. 40 (1978) 223.

[2] F. Wilczek, Phys. Rev. Lett. 40 (1978) 279.

[3] R. D. Peccei and H. Quinn, Phys. Rev. Lett. 38 (1977); Phys. Rev. D 1 6

(1977) 1791.

[4] G. G. Raffelt, Phys. Repts. 198 (1990) 1.

[5] K. van Bibber, P. M. McIntyre, D. E. Morris, G. G. Rafelt, Phys. Rev. D39

(1989) 2089.

[6] W. Buchmller and F. Hoogeveen, Phys. Lett. B237 (1990) 278.

[7] E. A. Paschos and K. Zioutas, Phys. Lett. B323 (1994) 367.

[8] John Bahcall (Private Communication).

[9] Charles Kittel, "Introduction to Solid State Physics", Second Edition, John Wiley

& Sons Inc. London, pp 50.

[10] Kaplan, et al. Astronomical Journal 97 (1989) 1197.

[11] F. T. Avignone et al. (The SOLAX collaboration) to be published (1997).


Captions for the Figures


Figure 1. Axion conversion to a photon in the Coulomb field of the nucleus by the

Primakoff effect.


Figure 2. Theoretical prediction of the count rate of photons converted from axions

incident at a Bragg angle, for a detector located at Sierra Grande, Argentina

(41o 41 S, 65o 22 W). The rate was calculated for 1/Mc2 = ga = 10-8
GeV-1. The location that was chosen is where the pilot experiment is being

performed.



