Detection rate of weakly interacting massive particles

Physics Letters B 308 (1993) 411-417 North-Holland

PHYSICS LETTERS B

Detection rate of weakly interacting massive particles A. G a b u t t i a n d K. S c h m i e m a n n Laboratory for High Energy Physics, University of Bern, CH-3012 Bern, Switzerland Received 3 March 1993 Editor: L. Montanet

An analytic expression allowing to evaluate the spin-independent detection rate of weakly interacting massive particles as a function of the detector energy threshold for any dark matter mass and absorber material, is presented. The detection rate is calculated taking into account the dynamics of the dark matter particles and the loss of coherence in the neutral-current interactions due to the nuclear form factor. The amplitude of the seasonal modulation in the rate is discussed, showing that, in the case of detectors made of heavy nuclei, the sign of the modulation in the counting rate can be different from the amplitude of the modulation in the dark matter flux.

After the proposal o f Drukier and Stodolsky [ 1 ] and the work o f G o o d m a n and Witten [ 2 ] attention has been devoted to the possibility of detecting weakly interacting massive particles ( W I M P s ) via elastic neutral-current scatterings with nuclei. Constraints on the strength o f the weak coupling depending on the dark matter candidates have been already given by experiments with germanium [ 3,4 ] and silicon [ 5 ] diode detectors and with NaI crystals [ 6 ]. Dirac and Majorana neutrinos and sneutrinos with masses less than 40 G e V / c 2 have been excluded by LEP [ 7 ]. Recently, cryogenic detectors with potentially low energy thresholds have been proposed like germanium [8] and silicon [9] bolometers or tin and zinc Superheated Superconducting Granule (SSG) detectors [10]. The modulation in the dark matter flux due to the Earth's motion around the Sun was proposed as a possible signature to distinguish the W l M P signal from other sources of background particularly for SSG detectors [ 11 ]. The theoretical bases relevant to the dark matter detection have been discussed by several authors, see ref. [ 12 ] for a review, but there is a lack in the literature o f a formula allowing to evaluate the detection rate of W I M P s for any dark matter mass in heavy and light absorber materials. In this paper we derive an analytic expression to estimate the detection rate of weakly interacting massive particles as a function of Elsevier Science Publishers B.V.

the detector energy threshold. The calculations are valid for spin-independent interactions and the rate is evaluated taking into account the dynamics o f the dark matter particle halo and the loss of coherence in the W l M P - n u c l e u s interactions due to the nuclear form factor. The choice of an appropriate nuclear form factor and the amplitude of the seasonal modulation in the delection rate are discussed. The strength o f the weak coupling, the number density o f particles and the characteristic velocities of the halo appear explicitly in the final formulas for the detection rate leaving open the choice of the dark matter candidate and o f the details in the halo model. The presented analytic expression allows to evaluate the detection rate for any dark matter mass in heavy and light absorber materials. The calculated detection rates and the amplitudes o f the seasonal modulation for ~'SSn, 74Ge and 28Si absorbers are compared at the end o f the paper, showing that the sign of the modulation depends on the absorber material and on the dark matter mass. The proposed [2 ] interaction of W I M P s with matter is via coherent elastic neutral-current scattering. The elastic scattering cross section for particles with coherent weak interactions is assumed to be isotropic and velocity independent and has the form

41 1

Volume 308, number 3,4 j~zr2 v 2 ~ 2 G2 O'el . . . . r e d ' ' " ~

PHYSICS LETTERSB

(m 2) ,

(1)

where Mred ( G e V / c 2) is the reduced mass of the WIMP-nucleus system, ~ 7 = N - ( 1 - 4 sin2O)Z is a linear function of the number of protons (Z) and neutrons (N) in the target nucleus and Y= ( YL+ YR) is the hypercharge mean value of the dark matter candidate. Since the value of the weak mixing angle is sin20= 0.2325, the elastic cross section is essentially proportional to N 2. For WIMPs in the nonrelativistic limit, the coupling constant G is generally assumed to be less or equal to the Fermi constant GF depending on the dark matter candidate [ 3-6 ]. The recoil energy transferred to the nucleus in the center of mass system in an elastic dark matter scattering is given by 2 mred

E=-M-~W2(1-cos0)

(GeV),

(2)

with 0 the scattering angle, Mn ( G e V / c 2 ) the nuclear mass and w ( m / s ) the dark matter velocity with respect to the Earth's frame. When the de Broglie wavelength of the m o m e n t u m transfer is small compared to the nuclear radius, the WIMP does not interact with the total weak charge of the nucleus but can resolve the nuclear structure. The loss of coherence can be introduced by multiplying the coherent differential cross section with a nuclear form factor. The use of an exponential form factor has been proposed by several authors [ 3,11,13 ]. It is assumed that WIMPs have vector coupling to Z bosons and scatter from nuclei by Z exchange in analogy to the process considered in ref. [ 1 ] for neutrinos. For spin-independent interactions, the expression of the nuclear form factor can be derived from the analogy between neutrino-nucleus scattering and electron-nucleus elastic scattering [ 14 ]. The nuclear form factor is determined by the distribution of the weak charge and by the nuclear radius. For small m o m e n t u m transfer to the nucleus q = 2x/2x/2x/2x/2x/2x/2MnnEthe , form factor for spin-independent WIMP interactions has the form f ( k 2 ) = exp ( - bk 2) with k = q/h the inverse de Broglie wavelength of the m o m e n t u m transfer. Assuming a gaussian density distribution for the number A of nucleons, the nuclear radius is R = 1.2A 1/3 (fm) and b = ~R 2. The dark matter differential cross section can be evaluated from 412

1 July 1993

the isotropic elastic total cross section 0"elof eq. ( 1 ) and from the recoil energy E and has the form (do-)do-el ~7 c = ~-

If(k2)

O-o,Mn

,2= dae, dQ d0 ,2 dl2 dO dE [f(k2)

(E)

- 2w2M~ed exp - ~

(m2/GeV),

(3)

where the nuclear coherence energy is Ecoh=

3h2 (_.~),/3 2MnR 2 - - 3 0 0 × 10 -6

(GeV).

Typical values for Eeoh are between 10 keV and 40 keV in heavy materials and around 200 keV in light absorbers. It is important to note that the differential cross section defined in eq. (3) is proportional to N 2 and goes to zero at high recoil energies due to the exponential form factor. Eq. (3) is obviously not appropriate for pure incoherent interactions occurring at high momentum transfer with a finite cross section proportional to N [ 15 ]. To account for the incoherent contributions in the final detection rate for spinindependent interactions, we propose to use an approximation for the nuclear form factor given by 1

I f ' ( k 2 ) 1 2 = If(k2)12+ ~ [ 1 - I f ( k 2 ) l 2] , leading to the differential cross section do)

do'~l F= ~ I f ' ( k 2 ) 12

=(1_

l~(de~ N]\~']c

+ I dGl m dE

(m2/GeV) '

(4)

where ]f(k2) ] 2 is the probability for having coherence with the differential cross section do-el/dE and [ 1 -- ] f ( k 2) 12] is the probability for having an incoherent interaction of the WIMP with a single nucleon with a cross section proportional to N. The contribution of the incoherent term is not important at low dark matter masses or at small recoil energies where the amplitude of the exponential form factor is always bigger than 1/N. When the form factor is very small and the interaction is strongly incoherent, the differential cross section of eq. (4) converges to the incoherent cross section. In the evaluation of the detection rate both differential cross sections will be used.

Volume 308, number 3,4

PHYSICS LETTERS B

The differential cross sections, defined above, depend on the velocity of the WIMPs and on the dynamics of the dark matter halo. In the currently accepted model, the halo is assumed to be gravitationally trapped in the galaxy with a MaxwellBoltzmann velocity distribution in the galactic rest frame [ 11,12 ]. The average halo velocity depends on the details used in the dark matter halo model and is estimated to be Vav= 270 + 25 k m / s [ 3,11-13 ]. The net velocity of the Earth through the dark matter halo is affected by seasonal variations because of the circulation of the Earth around the Sun. Introducing the dimensionless dark matter speed with respect to Earth ,r.= x/~2 w~ Vav and the time dependent dimensionless Earth speed with respect to the halo t/(t), the Maxwell-Boltzmann velocity distribution can be written in the Earth's frame as

F(w) d w =

~--6~Z exp(--Z2--t/2) sinh(2zt/) dw

~/~ va~ V,vt/

× { e x p [ - (Z-t/)2] - e x p [ -

()~+q)2]} dw.

(5)

The seasonal variation of the velocity distribution of WIMPs is introduced in eq. (5) with the modulation in time of t / ( t ) = t / ( 0 ) + A t / c o s ( ~ o t ) where t/(O) =x/~ V~ V,v, ~o=2~/yr and At/accounts for the amplitude of the modulation. The orbital velocity of the Earth around the Sun V0= 30 k m / s is tilted by ~0=60.2 ° with respect to the orbital velocity of the Sun around the center of the galaxy. Considering a net speed of the Sun with respect to the galactic rest frame of V=232 k m / s the Earth's velocity has a maximum of 247 k m / s on June 2nd and a minimum of 217 k m / s on December 4th [ 11,12 ]. The dimensionless amplitude of the modulation is then At/= ~ ( Vo/V,v ) cos ( ~0). In June, F (w) is shifted to higher velocities and dark matter particles will deposit in average more recoil energy than in December. Considering an average halo velocity of V~v= 270 km/s, the most probable dark matter velocities with respect to the Earth's frame are w~p = 321 k m / s and Wmp=298 k m / s in June and December respectively (see fig. 1 ). The velocity distribution of WIMPs in the Earth's frame has to smoothly converge to zero at the escape

l.O

1 July 1993

,

i

,

i

,

~

i

,

i

,

i

,

i

'

i

,S

0.0 0

' ' ' ' . . . . 1130 200 300 400 500

600

700

800

Dark Matter Velocity [kin/s] Fig. 1. Velocity distributions of the dark matter particles in the Earth's frame on June 2nd and on December 4th. To evaluate the detector rate, the distributions are truncated at the escape velocity we~= 805 k m / s .

velocity west. The value of the local galactic escape velocity Vestcan be derived from the circular velocity of the galaxy. Currently used values for Ves~are approximately 575 + 50 k m / s [ 11,12 ] leading to w~sc= 805+ 50 km/s. The Maxwell-Boltzmann distribution F(w) is highly suppressed for W~We~ and, to simplify the calculations, the dark matter interaction rate will be calculated using the distribution of eq. (5) truncated at the escape velocity We~. For we~¢=805 k m / s the distribution F(w) has a finite value at the escape velocity which is three orders of magnitude less than the value associated to the most probable velocity Wrap. The recoil energy spectrum is not significantly affected by the choice of the cut off value because the differential cross section is proportional to 1/w 2 and the flux of particles with velocities close to We~cis very small. Furthermore, WIMPs with velocities w ~ w ~ have mostly scattering interactions with high recoil energies and the already small differential cross section is suppressed by the nuclear form factor. The flux at Earth of dark matter particles is given by the integral over the velocity distribution of (p/MxC2)wF(w) where p/Mx c2 (m -3) is the halo number density. In the truncated distribution, the total flux is 99.86% of the value associated to the non truncated distribution. As a result, the error introduced in the evaluation of the detector rate using the velocity distribution of eq. (5), truncated at West, is negligible. For a given dark matter velocity the differential 413

Volume 308, number 3,4

PHYSICS LETTERS B

detection rate in a detector with energy threshold Eth is given by

1 July 1993

~ 2 3we$c

X//•thm n

Zth-- 2 VavMred '

Emax d Rate p NA wF(w) f da dw - M x c2 A d -~dE

(g-lm-l),

or=

Eth

(6) where Emax= 2MZ~aw2/M, (GeV) is the maximum value of the kinematical recoil energy transferred to the nucleus ( 0 = n) and NA/A (g- 1) is the number of nuclei in 1 gram of detector material with NA the Avogadro number. The detection rate is calculated by integrating the differential rate over the dimensionless dark matter velocity X with the integration limits Z~, the dimensionless escape velocity, and Zth, the minimum value needed to have a scattering with the recoil energy threshold Eth. Considering the differential cross section of eq. (3) derived from the elastic cross section multiplied by the exponential form factor, the detection rate (kg- 1 d - ~) is given by

3EcohMn ~ 4Vav ' fl= 4 a ' + ' - ~ '

S Y=r/2S+ot'

22=M~o. V.v, with the parameters ot and 22 expressed in (GeV 2 s3/ m3).

From the differential cross section ofeq. (4), where the correction term accounts for the incoherent interactions, the rate (kg- 1 d - ~) can be evaluated in (Rate)F = (1 -- 1 ) ( R a t e ) c +0.24

2-~z p NA (__G_G~2 _S Y N Mx c2 A \GF] tl

× (8 [ eft(Bt ) + eft(B2 ) - eft(B3 ) - eft(B4) ]

Xese (Rate)c=

r dRate dw J dw d--zZdZ

- ~

~h

=o.24Y2 2 ±

Z

1

[B, e x p ( - B ~ ) - B 2 exp( - B 2)

- B 3 exp( - B 2) +B4 exp( - B 2) ] )

M~C z A \ Gv ] tl

(kg-' d - ~ ) ,

X[fl e x p ( - y )

(8)

where X [erf(A1 ) - e f t ( A 2 ) - erf(A3) + erf(A4) ]

g= (q2+ ½_X2h) (GeV / sa/m 3) .

-exp(-Eth)

In eqs. (7) and (8) the velocities are in m/s, the energies in GeV and the masses in GeV/c z. The derived formulas for the detection rate, eqs. ( 7 ) and (8), allow to evaluate the spin-independent counting rate as a function of the detector energy threshold for any absorber and dark matter mass. As an example, the detection rates and the amplitudes of the seasonal modulation for three different absorbers of the proposed dark matter detectors will be discussed in the following section. In table 1, the expected values on June 2nd of the detection rates (Rate) c and (Rate) V for WIMPs of masses from 1 GeV/c 2 to 1000 GeV/c 2 interacting in i tSSn' 74Ge and 28Si absorbers are compared. The rates are evaluated for a detector threshold Eth= 0 considering a dark matter halo of energy density p = 0 . 4 × 106 G e V / m 3 [ 11-13] with the velocities

Ecoh / × [eft(Bin) - eft(Bz ) - eft(B3 ) + eft(B4 ) ]1

(kg-' d - ' ) ,

(7)

where eft(x) is the error function and

B~ = (Zest +r/),

B2 = (Zth + r/) ,

B3 = (Xesc - q ),

84 = (Xth --t/) ,

414

Volume 308, n u m b e r 3,4

PHYSICS LETTERS B

Table 1 Expected detection rates (kg-~ d-~ ) on June 2nd for l laSn, 74Ge and 2aSi detectors with energy threshold Eth=0. ( R a t e ) c is derived considering only the exponential nuclear form factor while (Rate) F takes into account also incoherent interactions. The rates are calculated with coupling constant G = GF, mean hypercharge value Y= l, energy density of WIMPs at Earth p = 0.4 × 106 G e V / m 3, Vesc= 575 k m / s and Va~= 270 k m / s .

1 July 1993

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1000

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. . _ (Rate)F _._

100

I

i

118Sn

(Rate)c

10 1

a)

WIMP llSSn ( k g - l d - l ) 74Ge ( k g - l d - l ) 285i (kg-I d - l ) mass ( G e V / c : ) (Rate)v (Rate)c (Rate)F (Rate)c (Rate)e (Rate)c 1 10 30 50 70 100 300 500 1000

207 1624 2417 2146 1807 1424 575 361 188

207 1621 2394 2103 1751 1359 515 315 160

124 919 1381 1283 1119 911 383 241 125

124 918 1369 1262 1092 881 360 224 115

34 185 218 191 164 133 57 36 19

34 185 217 189 162 131 56 35 18

0.1 1000

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]

x'x. I

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_ _ . ll8Sn

&--. 100

..~

~..

28Si

- - - -

"7, 10

%-

1 0.1

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b) I

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~" '""~"" ~ - ' ~ " " " ' " ' "" ' " ' " """' " "" '"" " I

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"~'~,

10 20 30 40 50 60 70 80 90 100110120 Detector Energy Threshold Eth [keV]

V ~ = 575 km/s and Vav= 270 km/s. To maximize the interaction rate the coupling constant was chosen to be G=GF and Y= 1. (Rate)c is the low limit of the detection rate because at high recoil energies the differential cross section of eq. (3) is reduced below the incoherent value by the exponential form factor. (Rate) Vhas to be considered a more realistic prediction, because the correction introduced in the differential cross section ofeq. (4) takes into account also incoherent scatterings. The two rates evaluated at zero detector threshold do not differ significantly in light materials or at low dark matter masses ( < 20 GeV/ ¢2) where the amplitude of the exponential form factor is generally bigger than 1IN. The use of the correction factor is relevant only in heavy materials and at high recoil energies where the differential cross section is more affected by the loss of coherence and (Rate)c is smaller than (Rate)F. This is shown in fig. 2a for 50 GeV/c 2 WIMPs interacting in 11aSh. At energy thresholds of 80 keV, (Rate) F is about 10 times higher than (Rate) c, while at zero threshold the two rates are almost identical. In fig. 2b the detection rates (Rate)F in June for 50 GeV/c 2 dark matter particles interacting in ll8Sn, 74Ge and 2ssi detectors are plotted versus the parameter Eth. The loss of coherence in the WIMP-nucleus interactions is stronger in l tSSn (E Cob= 15.7 keV) than in the lighter 28Si (E~oh= 171.2 keV), where the

Fig. 2. Detection rates on June 2nd of 50 G e V / c 2 dark matter particles. (a) i lgSn detector, comparison between (Rate)c calculated using only the exponential nuclear form factor and (Rate) Vwhere the contribution of the incoherent interactions is also accounted, (b) (Rate)F for nSSn, 74Ge and 28Si detectors. The coupling strength and the details of the dark matter halo are the same of table 1. The detector energy thresholds Eth are assumed to be sharp.

exponential form factor is close to unity in the range of energy thresholds considered. As a result, for detector thresholds bigger than 50 keV and dark matter particles with a mass of 50 G e V / c 2, the rate in 28Si exceeds the rate in ~'SSn, although at zero threshold the rate in silicon is ten times less than in tin. The seasonal modulation in the detection rate depends on the WIMP mass and on the absorber material and is shown in table 2 for llSSn, 74Ge and 2 s s i detectors with energy threshold Eth = 0. The values of the coupling strength and of the dark matter velocities Vav and Ves¢are the same of table 1. In silicon, as expected, the rate in June is always higher than in December where the dark matter flux is at the minimum value. In heavier absorbers, on the contrary, at high dark matter masses ( > 50 GeV/c 2 in 11SSn) the December rate exceeds the expectations for June. Such behavior can be understood considering the suppression of the differential cross section due to the 415

Volume 308, number 3,4

PHYSICS LETTERS B

Table 2 Seasonal modulation in the dark matter detection rates for t tSSn, 74Ge ad 2sSi detectors with energy thresholds Et~ = 0. The rate modulation A is defined as the difference between the rate on June 2nd and on December 4th normalized to the rate on December 4th. The indices C and F, the coupling strength and the dark matter velocities are as in table 1. WIMP

llSSn ( % )

74Ge ( % )

1

10 30 50 70 100 300 500 1000

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//

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_,_ ___

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llgsn 74Ge

0.6

(A)F

(A)c

(A)F

(A)c

(A)F

(A)c

5.9 5.1 2.0 -0.2 -1.5 -2.5 -3.7 -3.8 -3.9

5.9 5.1 1.9 -0.4 -1.8 -3.0 -4.8 -5.2 -5.4

5.9 5.4 3.3 1.8 0.7 -0.2 -1.8 -2.2 -2.4

5.9 5.4 3.2 1.6 0.5 -0.5 -2.5 -3.0 -3.3

5.9 5.7 5.2 4.9 4.7 4.5 4.1 4.0 3.9

5.9 5.7 5.2 4.8 4.6 4.4 4.0 3.9 3.7

~.

0.3

50GeV/c 2 W I M P mass

o.o 1.5

100GeV/c 2 W I M P mass

(9)

where Ratej and RateD are the June and D e c e m b e r values o f the detection rate, Mass is the d e t e c t o r mass and T i m e is the effective m e a s u r e m e n t time. In this rough evaluation the error in the detector counting rate is a s s u m e d to be only due to statistical fluctuations a n d not affected by r a d i o a c t i v e background. This is a convenient way to estimate the o r d e r o f m a g n i t u d e for the detector mass a n d the measurement time needed to resolve the seasonal modulation. In fig. 3 the d e p e n d a n c e o f M versus the detector threshold is shown for 50 G e V / c 2 a n d 100 G e V / c 2 W I M P s interacting in l lSSn, 74Ge a n d 28Si detectors with (Time. M a s s ) = 1 kg d. The rates are calculated from eq. ( 8 ) using the coupling strength and the details o f the d a r k m a t t e r halo shown in table 1. It is

.

1.2 .~

.

_._

ll8Sn

- -

74Ge

....

28Si

0.9 0.6 ~

~

-

-

=

~

0.3 0.0

form factor. In December, the recoil energy spectrum is shifted t o w a r d lower values due to the seasonal m o d u l a t i o n in the d a r k m a t t e r velocity a n d the amplitude o f the nuclear form factor is, in average, bigger than in June. This effect is stronger at high d a r k m a t t e r masses a n d in heavy absorbers, p r o d u c i n g an inversion in the a m p l i t u d e o f the rate m o d u l a t i o n . The ability to distinguish the seasonal m o d u l a t i o n in a d a r k m a t t e r e x p e r i m e n t free o f background, can be expressed in terms o f the ratio between the amplitude o f the m o d u l a t i o n and the statistical uncertainty:

416

'

0.9

+

IRatej - R a t e d I x / T i m e ' M a s s , M= x/Ratej +RateD

I

1.5

2ssi (%)

mass ( G e V / c 2)

1 July 1993

\

0

,

10

,

J

,

t

~ ,

i

,

,

20 30 40 50 60 70 80 90 100 Detector Energy Threshold Eth [keV]

Fig. 3. Ratio between the amplitude of the modulation and the statistical uncertainty for detectors with (Time.Mass)= 1 kgd without radioactive background. The coupling strength and the details of the dark matter halo are the same of table 1. i m p o r t a n t to notice that the a m p l i t u d e o f the m o d u lation a n d consequently the quantity M d e p e n d on the detector threshold. In l laSh for instance, the amplitude o f the m o d u l a t i o n at E t a = 0 is negative for 100 G e V / c 2 W I M P s . W h e n the detector threshold is raised, only dark m a t t e r particles with velocity Z > Xth are detected and the rate becomes less sensitive to the D e c e m b e r shift t o w a r d low Z o f the Maxwell-Boltzm a n n velocity distribution. The a m p l i t u d e o f the m o d u l a t i o n decreases with increasing detector thresholds until the fluxes o f detectable W I M P s are equal in June and in D e c e m b e r and M = 0. At higher energy thresholds only the tails o f the velocity distributions o f fig. 1 are i m p o r t a n t and the rate in June exceeds the value o f D e c e m b e r ( M # 0 ) . The optim u m detector energy threshold, to resolve the seasonal m o d u l a t i o n in the detection rate o f 50 G e V / c 2 W I M P s , should be a r o u n d 20 keV in '~SSn, 74Ge and 2sSi detectors. In a realistic experiment the radioactive b a c k g r o u n d has to be considered a n d the curves o f fig. 3 have to be properly corrected. In conclusion, we derived an analytic expression allowing to evaluate the s p i n - i n d e p e n d e n t detection

Volume 308, number 3,4

PHYSICS LETTERS B

rate of weakly interacting massive particles as a function of the detector energy threshold for any dark matter mass and absorber material. We have shown that the amplitude of the seasonal m o d u l a t i o n in the detector rate depends on the dark matter particle mass, on the absorber material a n d on the detector energy threshold. In the case of detectors made of heavy nuclei, the amplitude of the seasonal modulation in the counting rate is d o m i n a t e d by the loss of coherence in the W I M P - n u c l e u s interaction and it is different from the amplitude of the modulation in the dark matter flux. The m e a s u r e m e n t of a m o d u l a t i o n in the detector counting rate c o m b i n e d with its dependance on the detector threshold could be of great use in the background d i s c r i m i n a t i o n in dark matter search experiments. The product between the measurement time and the detector mass for devices made of ~SSn a n d 74Ge has to be of the order of 4 kg d a n d 16 kg d to resolve the seasonal m o d u l a t i o n ( M = 2 ) of 50 G e V / c 2 a n d 100 G e V / c 2 dark matter particles. For the lighter 2sSi, bigger masses or longer measurem e n t times are needed, typically 12 kg d to have the amplitude of the seasonal m o d u l a t i o n two times higher than the statistical error ( M = 2) in the counting rate of 50 G e V / c 2 WIMPs. The above values are an approximate estimation for low radioactive background experiments considering dark matter particles with m a x i m u m coupling strength G = GF a n d a dark matter halo of energy density p = 0.4 × 106 G e V / m 3.

I July 1993

We would like to thank P. Minkowski and K. Pretzl from the University of Bern for helpful discussions. This work was supported by the Schweizerischer Nationalfonds zur F6rderung der wissenschaftlichen Forschung.

References [ 1] A. Drukier and L. Stodolsky, Phys. Rev. D 30 (1984) 2295. [2] M.W. Goodman and E. Witten, Phys. Rev. D 31 (1985) 3059. [3] S.P. Ahlen et al., Phys. Lett. B 195 (1987) 603. [4] D. Reusser et al., Phys. Lett. B 225 (1991 ) 143. [ 5 ] D.O. Caldwellet al., Phys. Rev. Lett. 65 (1990) 1305. [6] C. Bacci et al., Phys. Lett. B 293 (1992) 460. [7] K. Enqvist and K. Kainulainen, Phys. Lett. B 264 (1991) 367. [8 ] D.O. Caldwell,in: Low temperature detectors for neutrinos and dark matter IV, eds. N.E. Booth and G.L. Salmon (Editions Frontieres, Gif-sur-Yvette, 1991 ) p. 387; B. Sadoulet,in: Low temperature detectors for neutrinos and dark matter IV, eds., N.E. Booth and G.L. Salmon (Editions Frontieres, Gif-sur-Yvette, 1991 ) p. 147; T. Shutt et al., Phys. Rev. Lett. 69 (1992) 3425. [9] P.F. Smith et al., Phys. Lett. B 245 (1990) 265; N.J.C. Spooner, Phys. Lett. B 273 ( 1991 ) 333. [ 10 ] m. Drukier et al., Phys. Rev. D 33 ( 1986) 3495; M. Frank et al., Nucl. Instrum. Methods A 287 (1990) 583; K. Pretzl, Particle World 1 (1990) 153. [ 11 ] K. Freese et al., Phys. Rev. D 37 (1988) 3388. [ 12] J.R. Primack et al., Annu. Rev. Nucl. Part. Sci. 38 (1988) 751. [ 13 ] J. Rich, in: Dark matter, eds. J. Audouze and J. Tran Than Van (Editions Frontieres, Gif-sur-Yvette, 1988) p. 43. [ 14] D.Z. Freedman, Phys. Rev. D 9 (1974) 1389. [ 15] I. Wasserman, Phys. Rev. D 33 (1986) 2071.

417