Leroy C., Rancoita P.-G. Principles Of Radiation Interaction In Matter And Detection

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730 Principles of Radiation Interaction in Matter and Detection

and isotropic and consequently, the recoil energy distribution dn

/dE

= 1. Equa-

tion (10.8) becomes:

, T ) =

R,max

R,min

P (E

, E

R,th

(T )) dE

. (10.9)

Using E

R,max

= f

and E

R,min

= f

, we can rewrite Eq. (10.9) as:

, T ) = 1 −

(T )

−

1 − exp

−α

− E

(T )

¶¸

(T )

αE

. (10.10)

A detailed description of the neutron data has to include the energy dependence of

the neutron cross section on

C and

F, which contains many resonances in the

energy region relevant to our example. One can determine the parameters α and

(T ) in Eq. (10.10) and, therefore, the eﬃciency ²

, T ), from the measured

count rate (per second), R(E

, T ), of liquid-to-vapor transitions for monoenergetic

neutrons of energy E

at temperature T :

R(E

, T ) = Ψ(E

) V

, T ) N

), (10.11)

where Ψ(E

) is the ﬂux of monoenergetic neutrons of energy E

, V

is the volume

of the superheated liquid, N

and σ

) are the atomic number density of species

i in the liquid and the corresponding neutron cross section, respectively. The ﬁt of

Eq. (10.11) to the data, giving the count rate as a function of the neutron energy, is

shown in Fig. 10.5 for diﬀerent temperatures. It gives an exponential temperature

dependence for E

(T ). The eﬃciency ²

, T ) is obtained from the ﬁt, as well as

α = 1.0 ± 0.1, found to be temperature independent.

In general, the cross section, σ , of neutron–nucleus interaction is the sum of two

terms:

σ = σ

+ σ

, (10.12)

where σ

and σ

are the spin-dependent and coherent cross sections, respecti-

vely. The coherent cross section goes like A

(A being the atomic mass number of

the nucleus, see Sect. 3.1). In the case of the CDM search experiment PICASSO, the

data are analyzed by exploiting the feature that the interaction of neutrons, with

the superheated carbo-ﬂuorates, is dominated by the spin dependent cross section

F, due to a dominant magnetic term in

F (σ

>> σ

). One should note

that the interaction of neutrons with superheated active gas containing bromine

(A ∼ 80) would be dominated by the coherent cross section. The possibility to

fabricate superheated droplet detectors with an active gas containing bromine has

to be investigated. Returning to our present case, the minimum detectable recoil

energy for

F nuclei is extracted from E

(T ) and is given by [Boukhira et al.

(2000)]

R,min

(T ) = 0.19E

= 9 (keV) × exp

−

T − 20

6.45

. (10.13)

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Superheated Droplet (Bubble) Detectors and CDM Search 731

Combining Eqs. (10.7, 10.13), one ﬁnds the droplet phase transition probability as

a function of the recoil energy deposited by a

F nucleus when it is struck by a

neutron. For instance, at T = 20

◦

C, E

R,min

= 9 keV and

P (E

, E

R,min

) = 1 − exp

−1.0 (E

− 9 keV)

9 keV

For E

= 30 keV and 15 keV, P (30 keV, 9 keV)= 0.9 and P (15 keV, 9 keV)=

0.49. The sensitivity curve shows that the detectors are 80% eﬃcient at 30

◦

C for

≥ 5keV recoils and at 20

◦

C for E

≥ 25 keV recoils. This sensitivity is ex-

ploited in cold dark matter search [Boukhira (2002)]. Knowing the

F nuclei recoil

spectrum expected from neutralino interactions and knowing the detector response

as function of temp erature [Eqs. (10.7, 10.13)], it is possible to determine the detec-

tor eﬃciency for a given neutralino mass and a given operating temp erature. The

temperature dependence of the detection eﬃciency provides a way to discriminate

against background contributions.

In the case of a polyenergetic beam, such as a radioactive source, one has to

integrate Eq. (10.11) over the neutron energy spectrum:

R(T ) =

n,max

ψ(E

) ²

, T ) σ

) dE

, (10.14)

where ψ(E

) is the neutron spectral-ﬂux (Sect. 4.1.3); σ

) is known from tables

of neutron cross section on

C and

F and ²

, T ) has been obtained above. For

a given neutron energy spectrum at low temperature only the high energy neutrons

take part in the process of liquid-to-vapor transition. Since the threshold energy

is decreasing as the temperature increases (Fig. 10.4), low energy neutrons, in ad-

dition to high energy neutrons, are also detected. So, for a polyenergetic neutron

source, ²

, T ) should increase with temperature. At high enough temperature,

all the neutrons in the spectrum are contributing to phase transitions and ²

, T )

becomes constant with temperature since no other neutrons will add their con-

tribution to phase transitions. For a monoenergetic neutron source, there is only

one strong increase of ²

, T ) at the operation temperature corresponding to the

energy of phase transition and ²

, T ) is constant for higher temperatures. The

use of droplet detectors as neutron spectrometers is based on the possibility to

make the detector sensitive to diﬀerent ranges of neutron energies by varying the

temperature of operation.

In order to unify the presentation of count rates and minimal neutron energy

for detectors with diﬀerent neutron threshold energies, it is convenient to use the

reduced superheat parameter, s, deﬁned in Eq. (10.2). As can be seen in Fig. 10.6,

the minimal neutron energy, or threshold energy, is an exponential function of s:

= E

−α

(s−s

)

, (10.15)

where α

and E

are two parameters to be ﬁtted from the data. E

is the threshold

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732 Principles of Radiation Interaction in Matter and Detection

Fig. 10.8 Response of a 1.5 l detector (loaded with 100% C

) to α-particles emitted from the

gel, doped with 20 Bq

232

U. Above 38

◦

C, the detector becomes sensitive to background gamma-

rays [Barnab´e-Heider et al. (2002)].

energy at a value of reference s = s

. With parameter s, the response

∗

for diﬀerent

gases is computed with T

c,eﬀ

= 0.9 T

Figure 10.7 shows the response of two detectors (1 and 2 have diﬀerent active

gases and hence have diﬀerent neutron energy thresholds) to neutrons from an AcBe

source as a function of reduced superheat.

10.1.2 Alpha-Particle Response Measurement

The α-particles produced outside the superheated droplet detectors cannot be de-

tected. Due to their short range in matter, α-particles are stopped in the detec-

tor wall. Only α-particles produced within the volume of the detector can be de-

tected. The heavy salt used to equalize densities of droplets and solution and other

ingredients mixed in the gel at the present stage of detector fabrication contains

α-emitters, such as U/Th and daughters. This α-background is the ultimate back-

ground at normal temperatures of superheated droplet detectors operation in CDM

search experiments since other backgrounds, such as γ-rays and mip’s, contribute

only to the detector signal at higher temperature. The α-response is studied with

detectors doped with a source (

232

U or

241

Am) of a known α-activity, introduced as

a soluble salt, and uniformly distributed in the polymerized gel, but not present in

∗

Vaporization for organic liquids takes place at an absolute temperature which is about 90% of

the critical temperature at atmospheric pressure [Eberhart, Kremsner and Blander (1975)].

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Superheated Droplet (Bubble) Detectors and CDM Search 733

Fig. 10.9 Detector response (count rate) of a 1 l detector loaded with a 100% C

gas (loading

of 0.7%), whose gel is doped with 20 Bqs of

241

Am as a function of temperature. A critical length

of L = 18 R

is necessary to ﬁt the data [Barnab´e-Heider et al. (2004)].

the droplets themselves, since the source compounds used in these spiked detectors

is hydrophilic and the freon droplets are hydrophobic.

Figure 10.8 shows the measured response of a 0.7% droplet loading detector to

α-particles emitted from the gel doped with 20 Bq

232

U. Above 38

◦

C, the detector

becomes sensitive to background γ-rays. Previous studies were performed with a

detector of the same loading [Boukhira et al. (2000)] doped with 10 Bq

241

Am. It

was observed that the count rate increases above 0

◦

C and stays nearly constant

from 20 to 40

◦

C at a level of about 3 counts per minute. From the droplet di-

stribution, the range of α-particles and the activity per unit volume, 8 counts per

minutes is expected from geometrical considerations. Thus, 0.5% of the α-particles

are detected in a plateau region. Above 40

◦

C, the count rate increases fastly. Monte-

Carlo studies [Barnab´e-Heider et al. (2004)] have been performed, where α-particles

were randomly generated in the gel with an energy spectrum corresponding to the

241

Am decay. These Monte-Carlo studies indicate that the experimental eﬃciency

is too high for the phase transition to be caused by elastic collisions between α-

particles and nuclei in the droplets. This leads to the suggestion that the phase

transition is triggered by the α-particles ionization loss in the droplets. The fabri-

cation process leads to no diﬀusion of

241

Am in the droplets and the experimental

eﬃciency is low enough, to discard surfactant eﬀect as described in [Pan and Wang

(1999)]. Under those assumptions, the contribution of the recoil short-range

237

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734 Principles of Radiation Interaction in Matter and Detection

Fig. 10.10 SBD-1000 response (count rate) to 200 keV neutrons as a function of temperature. The

volume of the detector is 8 ml. The simulated response gives a loading of 0.61 ± 0.06% [Barnab´e-

Heider et al. (2004)].

can be neglected. Furthermore, under the assumption that the recoil nucleus trig-

gering vaporization at neutron threshold is ﬂuorine, the dE/dx required to trigger a

phase transition is too high to explain the eﬃciencies seen in the α case. This is not

completely understood. However, it suggests that the minimal energy deposited at

neutron threshold must be deﬁned by the carbon recoil, as assumed in calculating

the eﬃciency η (see above). Taking the probability function, Eq. (10.7), and this

minimal energy requirement, the critical length as a function of temperature and

the value of α were deduced from the ﬁt to the data (Fig. 10.9).

A critical length of L = 18 R

and α = 1 are necessary to ﬁt these data. This

value of the critical length obtained is not inconsistent with the value found in [Bell,

Oberle, Rohsenow, Todreas and Tso (1973)]. The values L = 18 R

and α = 1

are validated from the simulation of the neutron response, using the same energy

deposition and critical length requirements. As an example, the result of the ﬁt to

the data for 200 keV neutrons is shown in Fig. 10.10 for 200 keV neutrons, where the

data at 1 atmosphere are from Fig. 10.2. The simulated response gives a loading

0.61 ± 0.06%, in agreement with the 0.7% loading of the detector used for the

α-particle measurement (Fig. 10.9).

Neutrons, with energy above threshold, produced from radioactivity in the envi-

ronment material or spallation induced by cosmic rays muons, are a possible source

of background for α-particles data. In particular, neutrons are produced by the

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Superheated Droplet (Bubble) Detectors and CDM Search 735

Fig. 10.11 Background counting rate of 1.5 l detector (loaded with 100% C

) as a function of

time. The background count rate depends on the quality-purity of nitrogen used for ﬂushing. Fluc-

tuation stabilized and minimized with liquid nitrogen boil oﬀ [Barnab´e-Heider et al. (2002)].

interaction of cosmic-ray muons with the detector environment, such as shielding,

rocks or concrete walls. In principle, shielding against these background neutrons

can be achieved by their moderation with paraﬃn or water, bringing their energies

below threshold (< 50 keV at room temperature).

10.1.3 Radon Detection

The superheated droplet detectors can serve for the detection of radon,

222

Rn, from

238

U-decay chain. Radon is a radioactive gas created by the decay of

226

Ra. It

decays further with T

1/2

= 3.85 days, E

= 5.49 MeV, to isotopes of

218

Po (T

1/2

3.05 min, E

= 6.00 MeV),

214

Pb (T

1/2

= 26.8 min, β emitter),

214

Bi (T

1/2

19.7 min, β emitter) and

214

Po (T

1/2

= 0.164 ms, E

= 7.69 MeV). Thus, radon

is an α-emitter. When present in the detector environment, it can diﬀuse into the

detector, where it induces an α-signal which can be measured. In the case of an

experiment such as CDM search, radon is a source of background. Flushing the

detector container with pure nitrogen largely reduces radon, achieving a typical

radon count rate of 1–2 events per hour, in the example shown in Fig. 10.11 at

room temperature and at a pressure of 1 atm. Isolation from radon must be secured

during detector fabrication, storage, and operation. Hermetic wall and lid for the

detector container and use of pure compression gas should achieve that goal.

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736 Principles of Radiation Interaction in Matter and Detection

Fig. 10.12 Example of an electrical signal produced by the passage of droplet-to-bubble transition

sound wave through a piezo electric sensor.

10.1.4 Spontaneous Nucleation

In CDM search experiments, a possible source of background events is spontaneous

phase transition due to homogeneous nucleation in the superheated liquid. The

metastability limit of a superheated liquid is described in [Eberhart, Kremsner and

Blander (1975)] and references therein. Vaporization for organic liquids takes place

at an absolute temperature which is about 90% of the critical temperature (T

), at

atmospheric pressure. A very rapid decrease of the spontaneous nucleation ﬂux, P

(bubbles s

−1

−3

) with decreasing temperature is expected [Eberhart, Kremsner

and Blander (1975)] and follows an exponential form [Boukhira et al. (2000)]:

P (bubbles s

−1

−3

) ∝ exp

−

(T )

, (10.16)

where E

is the critical energy, and k

is the Boltzmann constant (see Ap-

pendix A.2). Detectors have been tested in a shielded environment [Boukhira et

al. (2000)]. The temperature dependence of the signal was measured and shown to

decrease by several order of magnitude over a temperature interval of 1

◦

C near the

temperature region T ∼ 0.9 T

, where such nucleation should dominate. Therefore,

spontaneous nucleation is not playing any signiﬁcant role, when the detector is used

at normal operation temperature (often at room temperature).

10.1.5 Signal Measurement with Piezoelectric Sensors

The explosive droplet-to-bubble transition generates an acoustic signal which can be

detected by piezoelectric sensors adapted to the acoustic emission spectrum. These

sensors, two or more, are glued to the surface of the detector container (Fig. 10.1)

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Superheated Droplet (Bubble) Detectors and CDM Search 737

and coupled to high gain, low noise preampliﬁers whose frequency response is opti-

mized to suppress lower frequency acoustic noise. An example of an acoustic signal

transformed into an electronic signal is shown in Fig. 10.12.

The signal produced by the explosive droplet-to-bubble transition is transmitted

through the gel as a pressure front, then through the container wall to the piezo-

sensor. The sound velocity in the gel has been measured to be 1600±100 m/s, close to

the sound velocity in the plastic materials used in the container fabrication [Gornea

(2002)]. The signal shapes and frequency responses are dependent on i) the energy

released in the liquid-to-vapor phase transition, ii) the distance traveled by the

sound in the gel leading to signal attenuation as a function of event-sensor distance,

iii) the temperature and pressure of operation and iv) the recording history of the

detector, mainly the number of events that have occurred before the measured signal

since the last compression of the detector. The dependence of the signal amplitude

as a function of the number of events counted after a recompression and pressure re-

lease cycle, for various temperatures of operation, shows a decrease in the counting

and mean maximum amplitude. This reﬂects the detector depletion, starting with

the largest droplets, fewer in numb er but containing a larger fraction of the active

volume. The amplitude attenuation for various sensor-event distances indicates that

signals can be obtained with adequate eﬃciency up to 20 cm from the source. The

signal amplitude increases with increasing temperature and decreasing pressure of

operation. This observation follows the expectation that the energy released in a

droplet explosion increases with temperature and decreases with pressure. It allows

one to set well-deﬁned limits on the temperature and pressure ranges of opera-

tion. The piezo-sensors are set to discriminate against low frequency noise, while

favoring higher frequencies useful for timing purpose [Gornea (2002)]. Fast Fourier

transform analysis of pulses within sp eciﬁc frequency windows selected by the sensor

response allows acoustic noise rejection, yielding a clean radiation-induced signal at

a cost of a loss (< 10%) of eﬃciency.

10.2 Search of Cold Dark Matter (CDM)

In many models of supersymmetry, the neutralino, χ, is assumed to be the Lightest

Supersymmetric Particle (LSP). Particularly, in models of supersymmetry with R-

parity conserved. The R-parity is deﬁned as

R = (−1)

3B+L+2S

for a particle of spin S, baryon number B and lepton number L: R = 1 for Standard

Model particles and R = −1 for the superpartners. The neutralino is stable and re-

garded by many as the most promising candidate for Cold Dark Matter (CDM). Ex-

periments doing a direct search of cold dark matter, like PICASSO [Barnab´e-Heider

et al. (2005a)], SIMPLE [Girard R et al. (2005)], and others, are based on the tech-

nique of detection of nuclear recoils induced in the elastic scattering of neutralino

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738 Principles of Radiation Interaction in Matter and Detection

on a nucleus N [of atomic mass

∗

and atomic number Z]:

χ + N → χ + N,

as the energy, E

, of neutralinos is very small, for instance E

≈ 40 keV for a neu-

tralino mass of 100 GeV. These experiments measure the neutralino–nucleus cross

section as a function of the neutralino mass (this mass is presently unknown but

expected to be very heavy, 100 GeV or more). The cross sections of these neutralino–

nucleus interaction processes can be divided in two separate types. The coherent

part or spin-independent part, described by an eﬀective scalar coupling between the

neutralino and the nucleus, is proportional to the numb er of nucleons in the nucleus

[i.e., proportional to the square of the mass number (see page 218) of the nucleus

)]. The coherent part receives contribution from the scattering of quark,

χ + q → χ + q,

described by the Lagrangian L∼ χχ¯qq. The second part of the neutralino–nucleus

cross section is the incoherent part or spin-dependent part and results from an axial

current interaction of a neutralino with constituent quarks. This interaction, which

couples the spin of a neutralino to the total spin of the nucleus, depends on the spin

of the nucleus and is described by the Lagrangian L∼ (χγ

χ)(¯qγ

q). Therefore,

depending on the target nucleus atomic mass the experiment can be sensitive either

to spin-dependent or to spin-independent neutralino–nucleus interactions. In parti-

cular, one considers the example of the PICASSO experiment [Barnab´e-Heider et

al. (2005a)], which is measuring the neutralino–

F cross section. This cross section

is largely dominated by the spin-dependent contribution. The measurement of the

spin-dependent neutralino–

F cross section, as a function of the neutralino mass,

provides neutralino–

F and neutralino–nucleon exclusion limits. From the data, one

determines: i) the neutralino–proton and the neutralino–neutron cross sections as

functions of the neutralino mass and, ii) the relation between neutralino–proton and

neutralino–neutron spin-dependent coupling constants. One can also obtain limits

on spin-independent cross section for PICASSO by extracting the neutralino–proton

and neutralino–neutron spin-independent coupling constants from spin-independent

experiments (with large atomic mass m

10.2.1 Calculation of the Neutralino–Nucleon Exclusion Limits

For experiments based on the technique of nuclear recoils detection, one can express

the diﬀerential interaction rate per unit detector mass as (e.g., see [Bottino et al.

(1997); Genest and Leroy (2004); Giuliani (2005)]):

= N

max

)

min

)

f(v)

dσ(v, E

)

dv, (10.17)

∗

The diﬀerence between the nuclear mass and the atomic mass is negligible.

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Superheated Droplet (Bubble) Detectors and CDM Search 739

Table 10.1 value of atomic number Z,

mass number A (see discussion at page

220) and the ratio Z/A for nuclei typi-

cally used in direct searches for dark mat-

ter particle.

Nucleus Z A Z/A

F 9 19 0.47

Na 11 23 0.48

Ca 20 40 0.50

Ge 32 73 0.44

127

I 53 127 0.42

131

Xe 54 131 0.41

where E

is the recoil energy, N

is the number

of target nuclei per unit mass;

is the local density of neutralino(ρ

∼ 0.3 GeV/cm

) of mass m

(in units of

GeV/c

in Sects. 10.2.1-10.2.2), v is the neutralino velocity in the detector rest

frame, with v

min

and v

max

the minimum and maximum velocities, respectively. The

velocities will be converted into E

for the calculations. f(v) is the neutralino

velocity distribution in the detector rest frame and is given by:

f(v) d

v =

exp

−v

3/2

v (10.18)

with the velocity dispersion given by

¯v =



1/2

≈ 270–280 km s

−1

(10.19)

≈ 220–230 km s

−1

). dσ(v, E

)/dE

is the diﬀerential cross section for

neutralino–nucleus scattering and is the sum of two contributions:

dσ

(10.20)

where (dσ/dE

)

is the spin-independent (SI) or coherent part of the cross section

and (dσ/dE

)

is the spin-dependent (SD) part of the cross section.

10.2.1.1 Spin-Independent or Coherent Cross Section

The spin-independent (SI) or coherent cross section is given by ([Bottino et al.

(1997)], for instance)

dσ

πv

[Zg

+ (A − Z)g

]

)]

(10.21)

where: G

is the Fermi coupling constant, G

≈ 1.16610

−5

GeV

−2

(Appendix A.2);

is the nucleus mass, v is the neutralino velocity in the laboratory frame; Z and

A are the nuclear charge and the mass number of the target nucleus, respectively; g

The number of target nuclei per gram of material isN

= N

, where A

is the relative

atomic mass (or atomic weight, i.e., Sect. 1.4.1 and discussion at page 220) of the material and

is the Avogadro constant. For

F, we have A

= 19.