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

Подождите немного. Документ загружается.

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

510 Principles of Radiation Interaction in Matter and Detection

layer-detector interface, in order to reach and penetrate the silicon layer where they

deposit their energy. Charge carriers are then generated in silicon yielding a sig-

nal. The particles trajectories must be contained in the solid angle subtending the

converter layer-detector interface from the interaction location in the converter. The

type of converter used and therefore the mechanism of charged secondary particles

generation exploited depend on the neutron energy. Thermal (very slow) neutrons

∗

have huge nuclear reaction cross sections and, therefore, converters such as

LiF,

113

Cd,

155

Gd,

157

Gd are of standard use for neutrons of these energies. Exam-

ples of nuclear reactions of interest are given in Table 6.3.

One should mention also the possibility of using solid-form or bulk detectors [Mc-

Gregor and Shultis (2004)] for detecting thermal neutrons. These detectors use

semiconductor materials partially composed of a neutron reactive material such as

boron. In these detectors neutron interactions occur inside the bulk detector it-

self. These devices consist of a boron compound upon which conductive contacts

have been aﬃxed on opposite sides. A voltage is applied across the bulk mate-

rial. Then, neutrons can be absorbed directly within the detector volume and the

charged particle reaction products are released within the detector volume. Any

B(n,α)

Li reaction in the bulk produces detectable ionization and, since the ther-

mal neutron boron cross section (Table 6.3) dominates over thermal cross sections of

other elements, one can deﬁne [McGregor and Shultis (2004)] the intrinsic eﬃciency,

eﬀ

th−n

, for detecting thermal neutrons by:

eﬀ

th−n

= 1 − e

−Σ l

, (6.135)

where Σ is the thermal-averaged macroscopic neutron absorption cross section of

boron in the bulk and l is the thickness of the detector. The production of such

bulk detectors and their response to thermal or very slow neutrons are discussed

in [McGregor and Shultis (2004)].

However, the neutron capture reactions, as the ones reported in Table 6.3, have a

cross section dropping signiﬁcantly for higher energies of neutrons (i.e., fast neutrons

such as neutrons of energies from several keV to several MeV and more). Another

type of converter has to be chosen and a diﬀerent mechanism of charged secondary

particles generation has to be exploited. Fast neutrons can be detected via energetic

recoils produced from their elastic scattering on nuclei of a converter. The energy

transferred in elastic scattering from a neutron to a recoil nucleus is the recoil energy

given by:

= E

+ M

)

cos

θ, (6.136)

where E

is the incident neutron energy, M

the neutron mass, M

the recoil

nucleus mass, and θ is the angle in between incident neutron and recoil nucleus

directions (laboratory frame). The energy transfer is maximum for θ = 0, i.e, for

∗

The reader can ﬁnd a classiﬁcation of neutrons in Sect. 4.1.3.2. In addition, the total and elestic

corss sections for netron–silicon scattering are shown in Sect. 4.1.3.1.

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

Solid State Detectors 511

Li(n,T) He

6 4

Converter

Silicon detector

nn n

T α

n n

2828

Si(n,p) Al

Converter

Silicon detector

H(n,n) H

Fig. 6.31 Principle of converter-silicon operation: a)

LiF converter, α and tritium (T) produced by neutron interactions with the converter layer

deposit their energy in silicon. T is dominant (see text); b) CH

converter, proton recoil produced by the elastic scattering of a neutron on the

protons of the CH

converter. A fraction of incident neutrons may traverse the converter ﬁlm and interact directly with Si.

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

512 Principles of Radiation Interaction in Matter and Detection

neutron–nucleus head-on collisions. Then, the energy of the incident neutron is

totally transferred for M

= M

, i.e., E

= E

. Therefore, for fast neutrons,

it is advantageous to select converters with high hydrogen content and restrain

the amount of heavier atoms which reduce E

and absorb the energy of charged

secondaries. Polyethylene (CH

) appears to be an optimal choice as converter for

fast neutrons.

6.7.1.1 Signal in Silicon Detectors for Thermal Neutrons

Standard assumptions are made to estimate the signal generated in a silicon detector

by thermal neutrons [McGregor et al. (2003); Wielunski et al. (2004)]. A silicon

detector, 300 µm thick with no dead layer, is considered with a converter layer

put in front (Fig. 6.31). As reminded above, the eﬃciency of this type of detector

for heavy charged particles is 100%. The ﬂux of thermal neutrons (Sect. 4.1.3.2)

is assumed to be independent of the depth d of the (thin) converter. The heavy

charged particles produced by the thermal neutron interactions in the converter

material have their direction of emission kinematically constrained. Only one of the

charged particle reaction products, which are emitted in opposite directions, may

cross the converter layer-detector interface into silicon. Being heavy, these charged

particles practically travel in straight lines in the converter. The range, R

(E), of

these secondary particles in the converter is calculated from SRIM [Ziegler, J.F.

and M.D. and Biersack (2008b)]. Only charged particles produced in the converter

at a distance from the converter layer-detector interface smaller than their range

(E) in the converter will contribute to the signal in the detector. The probability

of signal generation will then depend on R

(E) and on the number, N

, of incident

neutrons on the converter. It will also depend on the atomic concentration, N

, of

target-isotopes in the converter and on the cross section, σ

, of the neutron-i-target

isotope reaction. Coming back to the example of thermal neutron reactions with

LiF, as the α-particle and triton travel in opposite directions due to the small

kinetic energy of the thermal neutrons, only one of them will be detected from

one neutron interaction (Fig. 6.31). In practice, α-particles which have larger losses

can hardly leave the converter. Therefore, most of the

-particles are stopped in

the converter and do not contribute to the signal (see also [Pospisil (1993)]). In

Fig. 6.31 is also illustrated the case where the α-particle and triton (T) are moving

in opposite direction parallel to the silicon surface and, therefore, never reach the

active volume of the detector and cannot contribute to any signal.

Generally, summing over all particles, i = α, T, produced in the neutron reac-

tion, the probability of signal generation can then be expressed as:

S = Σ

i=α,T

, (6.137)

where

= N

P. (6.138)

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

Solid State Detectors 513

Fig. 6.32 The thermal neutron is produced in A at a distance x from the interface converter-

detector. The opening angle of the cone is α = arccos(x/R).

The factor P in Eq. (6.138) represents the total or integrated geometrical pro-

bability that any charge particle generated in the converter reaches the detector

and produces a signal. This probability is P = 0.25, as calculated below. The point

(vertex), A, is the source of isotropic emission of particles produced by the reaction

of the thermal neutron with an atom of the converter (Fig. 6.32). These particles are

produced in a cone of opening angle α and a particle emitted at angle α stops after

the range R in the converter when its energy has gone down to zero (Fig. 6.32). If

point A is located at a distance x from the cone basis, i.e., the converter layer-silicon

detector interface, one has:

cos α =

(6.139)

α = arccos

. (6.140)

The solid angle, SA, of particle emission is then:

SA =

2π

sin θ dθ dφ = 2π (1 − cos α) (6.141)

compared to the total sphere solid angle, SA

sphere

= 4π.

Then, the probability, P , that a particle emitted at point, A, with a range R in

the converter reaches the converter layer-silicon detector interface at a distance x is

P (x, R) =

sphere

2π (1 − cos α)

4π

1 −

(6.142)

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

514 Principles of Radiation Interaction in Matter and Detection

or, introducing the variable η = x/R,

P (η) = 0.5(1 − η). (6.143)

Integrating from η = 0 (origin of the particle at x = 0) up to η = 1 (origin of the

particle at x = R), one obtains:

P ∼ 0.5

P (η) dη = 0.25. (6.144)

The factor P in Eq. (6.138) representing the total or integrated geometrical

probability that any charge particle generated in the converter goes in a direction

opposite to the converter layer-silicon detector interface shown in Fig. 6.32 can be

expressed as:

P (d − x, R) =

1 −

d − x

. (6.145)

Equation (6.145) is needed when one considers the case of a silicon detector

inserted in between two converter layers.

Using Eqs. (6.137, 6.138) the probability of signal generation per incident neu-

tron can be expressed as:

= Σ

P. (6.146)

As an example of estimate one can calculate the signal expected from thermal

neutrons (neutron energy of 0.025 eV) using a

LiF converter (the Fluorine absorp-

tion cross section of 9.6 mb can be neglected compared to the

LiF cross section of

940 b). One can see in Table 6.3, that from neutron interaction with

LiF, charged

particles possibly contributing to the signal are α (2.05 MeV) and

H (2.71 MeV).

The atomic concentration of target-isotopes in

LiF is

LiF

Nρ

= 6.022 × 10

2.635

25.94

= 6.12 × 10

atoms/cm

where N is the Avogadro constant (see Appendix A.2). The ranges of α (2.05 MeV)

and

H (2.71 MeV) in

LiF, calculated with SRIM [Ziegler, J.F. and M.D. and

Biersack (2008b)], are 6.05 µm and 31.67 µm, respectively. Then, one ﬁnds using

Eq. (6.146)

= 4.56 × 10

−2

+ 8.74 × 10

−3

= 5.3 × 10

−2

. (6.147)

The range of the heavy charged particles is limited (6.05 µm in case of alpha

particles and 31.67 µm for tritons). Therefore, if a neutron is captured further away

from the converter-silicon boundary, the heavy charged particles do not reach the

sensitive volume of the silicon detector. Moreover, as already noted before, some

of the heavy charged particles also travel parallel to the silicon surface and do not

reach the detector active volume. The detection eﬃciency increases initially with

increasing converter thickness because more neutrons are captured in the converter

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

Solid State Detectors 515

Fig. 6.33 Thermal neutron detection eﬃciency is shown as a function of the

LiF converter

thickness [Gutierrez (2007)]. The LiF converter optimal thickness occurs at 8 mg/cm

material. However, the detection eﬃciency starts to decrease at a value, d

opt

, of

LiF thickness. The value of d

opt

is found by using Eq.(6.145), for instance. The

detection eﬃciency, dP

det

, of a neutron by the system of a converter layer and an

adjacent silicon detector is given by

det

= exp(−µx)

1 −

d − x

µ dx, (6.148)

where µ dx is the probability for the interaction of a neutron with a nucleus in

the converter material at depth x; exp(−µx) is the probability that a neutron will

traverse a path of length x without interaction; 1/2 [1 − (d − x)/R] is the proba-

bility that the products of the interaction reaction of the neutron with a nucleus

of the converter material will escape into the silicon detector. It can be deﬁned as

the ratio between the solid angle deﬁned by the cone whose generatrix equals the

path of the secondary particles in the converter material and the total solid angle

[Eq. (6.145)]. Integrating Eq. (6.148) over the converter thickness (d) one ﬁnds

det

−exp(−µd) + 1 −

exp(−µd)

µR

−

µR

. (6.149)

To ﬁnd the converter optimal thickness, d

opt

, one imposes the condition

det

d(d)

= 0, (6.150)

i.e.,

opt

ln(µR + 1). (6.151)

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

516 Principles of Radiation Interaction in Matter and Detection

0 10 20 30 40 50 60 70 80 90

337.5

338

338.5

339

339.5

340

340.5

Recoil angle (in degrees) in the laboratory system

s (q ) [mb/steradian]

Incident neutron

=1MeV

0 10 20 30 40 50 60 70 80 90

53.5

54.5

55.5

56.5

57.5

58.5

Recoil angle (in degrees) in the laboratory system

s (q ) [mb/steradian]

Incident neutron

=14MeV

Fig. 6.34 Diﬀerential elastic (n,p) cross section as a function of the proton recoil angle (in degrees) in the laboratory system [Gutierrez (2007)]: for

an incoming neutron a) of 1 MeV and b) 14 MeV.

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

Solid State Detectors 517

Fig. 6.35 Converter maximal thickness as a function of the proton recoil angle. The incoming

neutrons follow a direction along the normal to the converter area [Gutierrez (2007)].

The tritium signal being about 5 times larger than the α signal (Eq. (6.147)), the

neutron detection eﬃciency will be maximum when using the

LiF optimal thickness

for tritium. Equation (6.151) gives for tritium an optimal

LiF converter thickness

opt

= 30.4 µm = 8.0 mg/cm

. (6.152)

The detection eﬃciency is shown in Fig. 6.33 as a function of the

LiF converter

thickness [Gutierrez (2007)]. A maximum clearly occurs at the thickness of 8 mg/cm

which is the LiF converter optimal thickness. The result is in excellent agreement

with the results of Monte Carlo simulations [Uher et al. (2007a)]. For a boron

converter, there are four contributions to the signal (Table 6.3) and Eq. (6.138)

gives:

Li,I

α,I

Li,II

α,II

. (6.153)

If one uses

N ρ

= 13.03 × 10

atoms/cm

and the values of the cross sections for the neutron nuclear reactions in the

converter. i.e.,

σ = 3571 b for n +

B → α (1.47 MeV) +

Li (0.84 MeV) + γ (0.48 MeV)

and

σ = 269 b for n +

B → α (1.78 MeV) +

Li (1.01 MeV),

one ﬁnds:

= 2.04 ×10

−2

+ 4.23 ×10

−2

+ 1.73 ×10

−3

+ 3.96 ×10

−3

= 6.84 ×10

−2

. (6.154)

The ranges of

Li (0.84 MeV), α (1.47 MeV),

Li (1.01 MeV) and α (1.78 MeV)

B, calculated with SRIM [Ziegler, J.F. and M.D. and Biersack (2008b)], are

1.75 µm, 3.62 µm, 1.97 µm and 4.5 µm, respectively. Higher neutron reaction cross

sections in

B is compensated by smaller ranges and therefore sensitivities achieved

with

B and

LiF converters are comparable.

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

518 Principles of Radiation Interaction in Matter and Detection

0 20 40 60 80 100

0.02

0.04

0.06

0.08

0.1

0.12

0.14

thickness [m m]

Proton detection probability

= 1MeV

Fig. 6.36 Proton detection probability as a function of the polyethylene thickness (in µm) for

incident neutron energy of 1 MeV. This indirectly measures also the fast neutron detection proba-

bility [Gutierrez (2007)].

6.7.1.2 Signals in Silicon Detectors by Fast Neutrons

These signals are produced by the elastic scattering of fast neutrons on nuclei in

the converter. As discussed in Sect. 6.7.1, CH

being hydrogen-rich is an optimal

choice of converter for the detection of fast neutron with silicon detectors. In that

case, the detection eﬃciency, S

eﬀ

, of a Si detector as a function of the converter

thickness d can be expressed as

eﬀ

= ²

1 − e

−Σd

−µd

+ ²

−Σd

, (6.155)

where ²

is the intrinsic eﬃciency for recoil protons (assumed to be 1 as all the

protons reaching the detector generate a signal, 100% charge collection eﬃciency), ²

is the intrinsic detection for direct neutron detection (a fraction of neutron can cross

the converter and produce signals in Si through nuclear reactions, like Si(n,p)Al

and Si(n,α)Mg). In Eq. 6.155, the term ²

1 − e

−Σd

−µd

represents the ﬂux of

recoiling protons produced by neutron interactions with the CH

converter nuclei,

Σ (cm

−1

) is the mean macroscopic neutron-converter cross section. In the case of

polyethylene,

Σ (cm

−1

) =

ρ N

+ n

)

and µ is the proton absorption coeﬃcient in the converter, which can be calcu-

lated from SRIM [Ziegler, J.F. and M.D. and Biersack (2008b)]. Parametrisation of

and σ

as functions of energy may be found in [Mesquita, Filho and Hamada

January 9, 2009 10:21 World Scientiﬁc Book - 9.75in x 6.5in ws-bo ok975x65˙n˙2nd˙Ed

Solid State Detectors 519

(2003)], for instance. The term ²

−Σ d

represents the fraction of neutrons travers-

ing the converter ﬁlm and interacting directly with Si. The coeﬃcient N

is the rate

of neutrons emitted from a source of neutrons (such as an Am-Be source). Equa-

tion (6.155) can be ﬁtted to the data to obtain the detection eﬃciency (count rate)

as a function of the converter thickness. For normal incidence, the number of signals

per incident fast neutron is given by a formula similar [Wielunski et al. (2004)] to

Eqs. (6.137, 6.138) or (6.146)

= Σ

, θ) R

, θ)] P (E

), (6.156)

where E

is the incident neutron energy, N

the number of incident neutrons on

the converter, N

the atomic concentration of the target-i-isotopes in the conver-

ter; σ

, θ) is the neutron-i-target isotope elastic scattering cross section and

, θ)] is the range of the recoil-i produced by the neutron-i-target isotope

elastic scattering. The (n,p) diﬀerential elastic cross section for an incident neutron

of given energy is [Hopkins and Breit (1971)]

σ(θ) = Σ

i=l

1 − 2 cos

, (6.157)

where P

is the ith order Legendre polynomial, θ the scattering angle in the labora-

tory frame. The diﬀerential elastic cross section for proton recoil as a function of the

recoil angle (laboratory frame) is given in Fig. 6.34 for 1 MeV and 14 MeV incident

neutrons. From Fig. 6.34, for b oth energies, there are more protons forward emitted

(θ = 0).

0 2 4 6 8 10 12 14

200

400

600

800

1000

1200

1400

optimal thickness [µm]

Neutron energy [MeV]

Fig. 6.37 CH

optimal thickness (µm) as a function of the incident neutron energy (MeV) [Gutier-

rez (2007)].