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

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750 Principles of Radiation Interaction in Matter and Detection

Table 10.3 Values of hS

i and hS

i for

F calculated from various mod-

els. The corresponding ratios R

≡ C

p(N)

and R

≡ C

n(N)

are also listed.

Reference

0.4751 -0.0087 0.903 0.0003 [Divari et al. (2000)]

0.368 -0.001 0.542 1 × 10

−6

[Leroy (2004b)], g

= 1.25

0.415 -0.047 0.689 0.0088 [Leroy (2004b)], g

= 1.00

0.441 -0.109 0.778 0.0475 [Pacheco and Strottman (1989)]

and, obviously, hL

i = 0.04, hL

i = 0. For instance,

Ge is an odd-neutron nucleus

with J = 9/2, a G

9/2

state, and with µ

exp

= −0.879. Therefore, Eq. (10.91) gives

the value:







i = S

= [−0.879 − 0.0 × (9/2)] /(−3.826 − 0.0) = 0.23,

i = 0

(10.93)

with hL

i = 4.27 and hL

i = 0

In the extreme single-particle model(ESPM), it is assumed that the entire spin

of the nucleus comes from the single last unpaired proton or neutron. Thus, one

ﬁnds for a nucleus with an unpaired proton







i = S

= 1/2 {[J(J + 1) + 3/4 − l(l + 1)]/(J + 1)},

i = 0

(10.94)

and, vice versa, for a nucleus with an unpaired neutron. It gives hS

i = 0.50,

i = 0, and hL

i = hL

i = 0 for

F, and hS

i = 0.50, hS

i = 0, hL

i = 0 and

i = 4.0 for

Ge.

Returning to Eqs. (10.86, 10.87), one considers the eﬀects of meson currents in

the nucleus. It is believed that the dominant one-pion exchange mechanism does

not modify the isoscalar moment

= µ

+ µ

The mean eﬀect of these currents is to introduce additional terms −µ

and +µ

into Eqs. (10.86, 10.87), respectively. Heavy-vector-meson exchange currents con-

tribute a term µ

in Eq. (10.86). Model calculations suggest a form:

= −x (S

− S

), (10.95)

with a factor x to b e determined later. One can rewrite Eqs. (10.86, 10.87) as

= g

J + (G

− g

− x) (S

− S

) − (g

− g

) J

− µ

+(G

− g

+ G

− g

) S

(10.96)

and

= g

J + (G

− g

) (S

− S

) + (g

− g

) J

+µ

+ (G

− g

+ G

− g

) S

(10.97)

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Then, the isoscalar moment is [summing Eqs. (10.96, 10.97)]:

= (g

+ g

) J + (G

− g

+ G

− g

− x) (S

− S

)

+2 (G

− g

+ G

− g

) S

(10.98)

or, using g

= 1, G

= 5.586, g

= 0 and G

= −3.826,

= J + (0.76 − x) (S

− S

) + 1.52S

(10.99)

= 1 + (0.76 − x)

− S

+ 1.52

. (10.100)

The diﬀerence (S

−S

) and µ

are obtained from a global ﬁt to magnetic moments

in each of the two nuclei and the ft-value for the Gamow-Teller β-decay from one

to the other:

− S

)

6170

− 1

J + 1

, (10.101)

where R = g

is the ratio of the axial-vector to vector weak-interaction coupling

coeﬃcients.

For

F, one obtains [Buck and Perez (1983)] from a ﬁt to magnetic moment and

ft-value data: µ

/J = 5.2576, µ

/J = −3.7708 and, therefore, µ

/J = 1.487. The

quantity R(S

− S

)/J = 0.9226 is also found. From the original ﬁt, a quenched

value for R was found:

R = 1.00 ± 0.02.

If one uses this value, one obtains

− S

= 0.9226. (10.102)

The quantity x in Eq. (10.100) can be found from a table [Raman, Houser,

Walkiewicz and Towner (1978)]. For

F, the table gives µ

= −0.018. However, to

adapt the present normalization with that of the table, one needs to multiply the

values found in the table by a factor 2. So, one ﬁnds µ

= −0.036 for

F. The value

of x can be extracted using Eqs. (10.95, 10.102) to ﬁnd:

x =

0.036

0.461

= 0.078. (10.103)

Combining Eqs. (10.100, 10.102, 10.103), one ﬁnds:







= −0.047,

= 0.415,

(10.104)

which are reported in [Engel and Vogel (1989)]. If one uses the unquenched value

R = 1.249 ± 0.006 ≈ 1.25

obtained in neutron decay experiments, one ﬁnds

− S

0.9226

1.25

= 0.738

and x = 0.036/0.369 = 0.098 and therefore, S

= −0.001 and S

= 0.368.

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752 Principles of Radiation Interaction in Matter and Detection

−20 −15 −10 −5 0 5 10 15 20

−10

−8

−6

−4

−2

Tokyo/NaF

SIMPLE

CRESST

NAIAD

DAMA/Xe−2

PICASSO 1pb

Pacheco−Strottman

Fig. 10.14 Exclusion plot a

versus a

for σ

χp

= 1 pb and m

= 50 GeV c

−2

[Genest and

Leroy (2004)], [Leroy (2004b)] using hS

i, hS

i calculated from model [Pacheco and Strottman

(1989)]. The PICASSO result [Barnab´e-Heider et al. (2005b)] is compared to the results of several

experiments: CRESST [Seide et al. (2002)], DAMA-Xe-2 [Bernabei et al. (1998)], SIMPLE [Girard

R et al. (2005)], Tokyo/NaF [Takeda et al. (2003)]. Other updated and new experimental results

can be found in [Genest (2007)].

10.2.1.4 Shell Models Calculation and Validation of hS

i and hS

The nuclear angular moment and the nuclear magnetic moment can be calculated

from the value of hS

i and hS

i, as calculated from shell models, and comparison

with the experimental nuclear magnetic moment can be done (Table 10.2) as one

test of the shell model. Using again notations familiar to shell models, the angular

momentum is:

J = hL

i + hS

i + hL

i + hS

i (10.105)

and the magnetic moment is given by

µ = g

i + G

i + g

i + G

i. (10.106)

The calculated magnetic moment of

Ge with free particle g-spin factors show

signiﬁcant improvement in agreement with measured values if quenched g-spin fac-

tors are used (consistent with isovector quenching found in the sd [Brown and

Wildenthal (1987)] and fp [Richter et al. (1991)] shells

‡‡

when ﬁtting to magnetic

moment data). Using quenched isovector component of the spin, one ﬁnds eﬀective

‡‡

The reader can ﬁnd an introduction to nuclear shells in Sect.3.1.5.

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g-spin factors: G

= 4.80 and G

= −3.04. Then, the calculated magnetic moments

Ge are µ = −1.084 and -0.879 for [Ressell et al. (1993)]-“small” and [Ressell

et al. (1993)]-“large”, respectively.

10.2.2 The PICASSO Experiment, an Example

As said previously the PICASSO Experiment [Barnab´e-Heider et al. (2005a)] is

measuring the neutralino–

F cross section. This cross section is largely dominated

by the spin-dependent contribution. The neutralino induced recoil (R) spectrum of

F is given by [Lewin and Smith (1996)]:

≈ c

0,P

)]

exp

−c

, (10.107)

where F

) is the form factor for spin-dependent intereactions: F

) < 1,

it is due to the ﬁnite size of the nucleus and dependent mainly on nuclear radius

and recoil energy;

0,P

403

[σ

/(1 pb)]

/(0.3 GeV cm

−3

)

¤£

hvχi/(230 km s

−1

)

(10.108)

is the total rate of neutrino–nucleus interaction (assuming zero momentum transfer),

0,P

is expressed in counts per kg and per day; A

is the atomic mass number of

the target atoms (

F in the present case), ρ

in GeV c

−2

the mass density of the

neutralino (the local neutralino mass density at the position of the solar system

is assumed to be 0.3 GeV cm

−3

), hvχi is the relative average neutralino velocity

(230 km s

−1

is the velocity dispersion of the dark matter halo);

= 2

+ m

)



vχ

(10.109)

is the mean recoil energy with m

the nucler mass of the recoil nucleus, F

) ∼ 1

for

F and small momentum transfer; c

= c

= 1 for v

= 0 (v

is the velocity of

the earth relative to the dark matter distribution) [Lewin and Smith (1996)]; and

= 0.75 and c

= 0.56 for v

= 244 km s

−1

) [Lewin and Smith (1996)]. Combining

the

F recoil spectra expected from neutralino interactions [Eq. (10.107)] and the

measured detector threshold for

F recoil energy at a given operating temperature

(T ), one can determine the neutralino detection eﬃciency, ²(m

, T )) as a function

of the neutralino mass and operating temperature. Then, the observable neutralino

count rate, R

obs

, as a function of temperature, neutralino mass and cross section is

given by:

obs

, σ

, T ) =

0,P

, σ

) ²(m

, T ))

= 1.34 R

0,P

, σ

) ²(m

, T )). (10.110)

The cross section, σ

, is given by Eq. (10.62). A combined ﬁt of alpha background

and neutralino response to the PICASSO data brought an upper limit of 1.31 pb on

χp

, 21.5 pb on σ

χn

for a mass m

= 29 GeV c

−2

[Barnab´e-Heider et al. (2005b)].

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754 Principles of Radiation Interaction in Matter and Detection

One can also calculate the a

limits. If one considers the example of a cross

section σ

χp

= σ

SDlim

= 1 pb for m

= 50 GeV c

−2

, one ﬁnds:

24G

χp

= 2.92 pb

−1

. (10.111)

SDlim

being ﬁxed, from the ratio of Eqs. (10.65, 10.66) assuming the same mass

for proton and neutron, one ﬁnds

SDlim

p(N)

n(N)

(10.112)

with

p(N)

(8/π)a

[(J + 1)/J]

(6/π)a

J + 1

= 0.778 (10.113)

and

n(N)

(8/π)a

[(J + 1)/J]

(6/π)a

J + 1

= 0.0475. (10.114)

One has used the values







i = 0.441,

i = −0.109,

(10.115)

given in [Pacheco and Strottman (1989)] (Table 10.3).

One can determine the neutralino mass independent ratio:

SDlim

1/0.778

1/4.75 × 10

−2

= 0.061. (10.116)

Therefore, ﬁxing σ

χp

= σ

SDlim

= 1 pb and choosing m

= 50 GeV c

−2

, one obtains

from Eq. (10.116), the value σ

χn

= σ

SDlim

= 16.37 pb. Then, the value σ

χp

p(N)

SDlim

= 1 pb corresponds to σ

SDlim

χF

= 158.78 pb.

Equation (10.80) for m

= 50 GeV c

−2

reads

2.92

≥

√

1.0 pb

√

16.37 pb

, (10.117)

giving the two exclusion boundary limits [Genest and Leroy (2004)], [Leroy (2004b)]:

= 1.708 + 0.247a

(10.118)

and

= −1.708 + 0.247a

. (10.119)

The lines corresponding to Eqs. (10.118,10.119) represents the PICASSO expe-

riment exclusion limits (Fig. 10.14) for a neutralino of 50 GeV c

−2

mass and

χp

= 1 pb.

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Chapter 11

Medical Physics Applications

The knowledge about the physics governing the interactions of particles with mat-

ter and particle detection ﬁnds applications in the ﬁeld of nuclear medicine imaging

technique. This technique uses the injection into the patient of radionuclides directly

emitting photons, or of radiopharmaceuticals, labeled with a positron emitting iso-

tope. Photons directly produced by radionuclides or produced by the annihilation

of positrons emitted by the radiopharmaceutical with bo dy electrons are detected

by radiation detectors. This allows one to reconstruct three dimensional images

representing the distribution of radioactivity inside the patient’s body and to mea-

sure metabolic, biochemical and functional activities in tissue. Magnetic Resonance

Imaging (MRI) is another imaging technique, which does not require the use of

any radioactive material and uses instead the non-zero nuclear spin, an intrinsic

property found in nuclei (see page 226). MRI uses magnetic ﬁelds

∗

varying from 0.2

to 2 T and radio-frequency (RF) waves to observe the magnetization change of the

non-zero spin nuclei. The hydrogen isotope

H, which has a nuclear spin of

, is a

major component of the human body and is used as the main source of information.

Two techniques exploit the interaction of the produced photons with the ac-

tive material of radiation detectors (imager or scanner): Single Photon Emission

Computed Tomography (SPECT) and Positron Emission Tomography (PET). We

have seen, in Sect. 2.3, that photons interact with matter in several ways. These

are Compton scattering, photoelectric eﬀect, and pair production. The two other

possible interactions are discarded for medical applications: Rayleigh or coherent

scattering (see Sect. 2.3.2.2) and photonuclear absorption (see Sect. 2.3.4). Rayleigh

scattering is a process predominant in the forward direction, i.e., in which photons

scatter from atomic electrons without exciting or ionizing the atom and, therefore,

no energy is absorbed in that process. Also, the incoming photon beam (usually

from a source) is hardly altered by this process. The Rayleigh cross section, σ

coh

(see Sect. 2.3.2.2), may be large for low photon energies (around 1 keV or less) and

rapidly decreases with the photon energy. However, for practical energies faced in

medical applications, σ

coh

is much smaller than the Compton (incoherent) cross

∗

1 T = 10

G. The Earth magnetic ﬁeld is ≈ 0.5 G.

755

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756 Principles of Radiation Interaction in Matter and Detection

section. In particular, at energies where coherent and Compton cross sections com-

pete (the coherent cross section could even dominates the Compton cross section),

they are both largely dominated by the photoelectric cross section.

The photonuclear absorption is a nuclear interaction, in which the photon is

absorbed by the nucleus. It becomes relevant for photons with energy b eyond a few

MeV’s, although smaller than the pair production cross section, but these energies

are not encountered in medical applications. The energies encountered in medical

physics range from a few keV up to a few MeV.

The photoelectric eﬀect is dominant for high-Z materials, while Compton scat-

tering dominates for low-Z materials at the photon energies used in medical physics

(see Fig. 2.75 and consider water as the material closest to tissue). The pair pro-

duction process does not contribute, since the photon energy of the sources used

in SPECT and the photon energy in PET (0.511 MeV) are lower than the thresh-

old energy (2 mc

= 1.022 MeV, where m is the electron rest-mass) for creating an

electron–positron pair.

The detection probability of a photon emitted by a source and experiencing

Compton scattering in the body depends on the amount of energy lost as a result of

that scattering. The initial energy (E

) of a photon and its energy after Compton

scattering (E

) are related through

= E

Á·

1 +

(1 − cosθ)

, (11.1)

where θ is the angle between the initial and ﬁnal direction (after Compton scatte-

ring) of the photon. Basically, the probability of Compton interaction has a weak

dependence on the atomic number and decreases with the photon energy. The pho-

ton does not disappear in the Compton interaction and is available for further

interaction (with a decreased energy) in another detector, giving directional infor-

mation. This will be exploited in Compton camera, as we will see below.

The photoelectric eﬀect (Sect. 2.3.1) is an interaction of a photon with a tight-

bound atomic electron: K-electron, L-electron, M-electron, . . .. The photon of

energy, hν, is completely absorbed and a photoelectron is ejected with a kinetic

energy K

given by

= hν − B

, (11.2)

where B

is the binding energy of the electron. The photoelectric eﬀect is inversely

related to the power of ≈ 3.5 of the photon energy and directly related to ≈ the

ﬁfth power of the atomic number. The photoelectric eﬀect cross section also depends

on the electron shell. The photoelectric cross section as a function of the photon

energy presents several discontinuities at low energy. These discontinuities are called

absorption edges and correspond to energies below which it is impossible to eject

certain electrons from the atom. Below the K-edge, the photon cannot eject a K-

electron but still can eject a L-electron or a M-electron (Sect. 2.3.1). For instance, the

vacancy left in the K-shell after interaction is immediately ﬁlled by the transition

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Medical Physics Applications 757

of one electron of an outer shell accompanied by the emission of X-rays or an

Auger electron. Therefore, contrary to the Compton scattering, the photon is totally

absorbed in the radiation detector and there is no directional information provided

by the gamma or X-rays in the photoelectric process.

11.1 Single Photon Emission Computed Tomography (SPECT)

Single Photon Emission Computed Tomography (SPECT) has become a routine

technique in medical applications [Brooks and DiChiro (1976)]. This gamma-ray

imaging technique proceeds through the injection into the patient of a radioac-

tive substance, which emits photons of well-deﬁned energy. The distribution of ra-

dionuclides, position and concentration inside patient’s body is externally monitored

through the emitted radiation deposited in a photon detector array rotating around

the body. This rotation allows the acquisition of data from multiple angles. This

procedure allows the study of organs behaviors, bringing the possibility to reveal

signs of malfunctioning as early as possible.

Organ imaging requires a radiation of suﬃcient energy to penetrate the body tis-

sues. However, the radiation energy must remain low enough to allow its absorption

in the detecting device. Therefore, photons with an energy ranging between 50 keV

and several hundreds of keV can be used for imaging. Photons in this energy range

are produced by speciﬁc radionuclides. A widely-used radionuclide is

99m

Tc, an iso-

mer of technetium with a half-life of 6.02 hours, which decays emitting 140.5 keV

(89%) photons (see Fig. 11.1).

Other sources like

201

Tl,

178

Ta and

133

Xe, emitting lower energy photons, are

also used. For instance,

201

Tl emits 135 keV (2%) and 167 keV (8%) photons and

(69–83) keV mercury K X-rays (90%).

Photons, produced after injection of

99m

Tc in the patient’s body, will eventually

reach a detector where their energy deposition is measured. The organ structure and

its evolution are then visualized from the resulting photon absorption patterns. The

images are the projection of a three-dimensional distribution onto a two-dimensional

plane. This can be achieved by rotating the detector around the patient. Series of

two-dimensional projections are taken from diﬀerent directions.

To create the two-dimensional projections, Anger cameras are often used. First,

the photon emitted from within the patient crosses a collimator. Then, it reaches a

scintillator. The point of scintillation corresponds exactly to the plane coordinates

of the point of emission. Once the initial photon has reached the scintillator, it

excites all the photomultipliers. Analyzing the intensity of the signal coming from

every photomultiplier allows the determination of the plane coordinates. Finally,

the intensity of every photomultiplier’s signal is added. If it equals the energy of

the photon emitted, the information will be kept and helps the formation of the

image. If the energy is inferior, it means that the photon was scattered. Therefore,

the wrong coordinates were found and the information is rejected.

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758 Principles of Radiation Interaction in Matter and Detection

Fig. 11.1 Photons of 140.5 keV energy emitted by a

99m

Tc source of standard use in SPECT,

decaying to its isomer

99m

Tc, followed by a Compton scattering on a nucleus pro ducing a recoiling

electron. θ is the Compton scattering angle.

These cameras need to be operated with a collimator in front of the detecting

material. The collimator is usually made of a thick high-Z metal plate, drilled with

a huge number of small holes. The collimator holes allow only incident radiation

perp endicular to the detector surface and eliminate the obliquely-incident photons

i.e., the secondary photons produced by the interaction of the primary photons (from

the source) with biological matter (tissue, bones, etc. . .). These obliquely-incident

photons, by activating several detector readout cells, may prevent the image forma-

tion or, at best, degrade the image focusing. Their removal is necessary to obtain a

source image of high quality. Compton scattered photons (see Fig. 11.1) have a lower

energy than primary photons and can be rejected by energy discrimination. For a

given energy window, only Compton photons scattered at suﬃciently small angle

can be detected [Eq. (11.1)]. If one assumes that the detector has perfect energy

resolution and that a rectangular energy window is applied, centered at E

with

a width ∆E (in keV), the maximum allowable scattering angle at the ﬁrst order,

max

, is given by:

max

= cos

−1

1 − mc

∆E [E

(2E

− ∆E)]

−1

. (11.3)

As can be seen from Eq. (11.3), the probability to detect a 140.5 keV photon from

99m

Tc source in a 15% wide rectangular energy window is zero if the scattering

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Medical Physics Applications 759

Table 11.1 Characteristics of semiconductor materials for use in medi-

cal imaging.

Characteristics Si GaAs CdTe HgI

Atomic number (Z) 14 31,33 48,52 80,53

Density (g/cm

) 2.33 5.32 5.80 6.30

Band gap (eV) 1.12 1.42 1.50 2.10

Energy per e-h pairs (eV) 3.62 4.20 4.43 4.15

Electron mobility (cm

−1

) 1350 8500 1000 100

Hole mobility (cm

−1

) 450 400 70 4

angle θ > 45.2

◦

. If the energy window is increased to 20%, the corresponding angle

becomes 53.5

◦

. For a single scattered photon to be detected at a position x of the

collimator, the photon must have been emitted and come out of the b ody along a

path conﬁned to a cone with aperture θ ≤ θ

max

The necessity to use a collimator has the adverse consequence to decrease the

detection eﬃciency and this is a limitation to SPECT. The collimator reduces the

numb er of normal impinging photons to about 10

−4

of their original number, after

passing the collimator. As a consequence, a higher dose has to be given to the

patient in order to provide suﬃcient statistics for elaborating an accurate image.

Semiconductors have been also considered as detecting medium of gamma scan-

ner. The properties of several semiconductors for this purpose are listed in Ta-

ble 11.1. A priori, the low atomic number (Z = 14) of silicon is seen as a handicap

for this type of application, although silicon detectors naturally oﬀer excellent spa-

tial and energy resolutions. However, it is possible to use silicon as active medium

devices of gamma camera by combining silicon with high-Z material (W or Pb,

for instance) to form a sampling calorimeter of high eﬀective atomic number which

will serve as SPECT detector. The Silicon Collimated Photon Imaging Calorimeter,

(SiCPICal) [D’Angelo, Leroy, Pensotti and Rancoita (1995)], consists of a superpo-

sition of 200 active silicon layers interspersed with 120 µm tungsten layers. A silicon

active layer is made of 145 strips, each strip being 400 µm thick. Each silicon layer

is organized in high spatial resolution readout cells (550 ×550 µm

). A detector like

SiCPICal has a high-Z detection volume with high photon conversion, exploiting at

the same time the excellent spatial and energy resolution provided by silicon. The

single readout pixel can b e operated at about 10 MHz, thus no electronic dead time

aﬀects the overall detector performances. This detector can be operated in two dif-

ferent modes. It can select events in which the incoming photons have interacted

either by both Compton and photoelectric eﬀects or by photoelectric eﬀect only. The

operation mode depends on the current discriminator threshold setting.

The formation of a focused image is made possible by the presence of a collimator

with a variable structure, located in front of the detector. This allows one to keep

the minimal image size, of about 0.3 mm

for a point source, independently of the

distance between the collimator and the point source.