Physics and Engineering of Radiation Detection

116 Chapter 2. Interaction of Radiation with Matter

The mass attenuation coefficients for nitrogen and oxygen are given by

[

−

1

ρ

dE

dx

]nitrogen

Bethe−Bloch

=

912. 893

14

[

ln

(

2. 743 × 10 −^3

91 × 10 −^6

)

− 2. 677 × 10 −^3

]

= 221. 91 MeV cm^2 g−^1

[

−

1

ρ

dE

dx

]oxygen

Bethe−Bloch

=

912. 893

16

[

ln

(

2. 743 × 10 −^3

103 × 10 −^6

)

− 2. 677 × 10 −^3

]

= 187. 11 MeV cm^2 g−^1

Now we can employ the Bragg-Kleeman rule 2.4.16 to compute the total stop-

ping power of 5MeV α-particles in air.

[

−

1

ρ

dE

dx

]air

Bethe−Bloch

=(0.8)

[

−

1

ρ

dE

dx

]nitrogen

Bethe−Bloch

+(0.2)

[

−

1

ρ

dE

dx

]oxygen

Bethe−Bloch

=(0.8)(221.91) + (0.2)(187.11)

= 214. 95 MeV cm^2 g−^1

2.4.C BraggCurve

The Bethe-Bloch formulae for stopping power of charged particles we discussed in the
preceding section have an implicit dependence on the energy of the particle through
factors likeβandWmax. As a heavy charged particle moves through matter it
looses energy and consequently its sopping power changes. Since stopping power
is a measures the effectiveness of a particle to cause ionization, therefore as the
particle moves through matter its ionization capability changes. To understand this
dependence, let us plot the Beth-Bloch formula 2.4.12 forα-particles with respect
to theirresidualenergy. By residual energy we mean the instantaneous energy of
the particle retained by it as it travels through the material. For simplicity we will
lump together all the terms that are constant for a particular material. Equation
2.4.12 can then be written as
[
−

dE

dx

]

Bethe−Bloch

=

K

β^2

[

ln

(

Wmax

10 −^4

)

−β^2

]

MeV cm−^1 , (2.4.18)

whereK=0. 30548 ρZq^2 /Ais a constant for a given material. Since we are only
concerned with the shape of the curve and not its numerical value, therefore we
have usedI=10−^4 MeV in the above expression. This value is typical of lowZ
materials. A plot of the above equation is shown in Fig.2.4.5. As one would expected,
the stopping power increases with the residual energy of the particle. Hence as the
particle looses energy it causes more and more ionization in its path until it reaches
the highest point known asBragg peak. After that point the particles have lost
almost all of their energy and get quickly neutralized by attracting electrons from
their surroundings.
The plot shown in Fig.2.4.5 is generally known asBragg curve.Apointtonote
here is that the range of a particle traveling through a material depends on its
instantaneous energy. Hence one could in principle plot the stopping power with