Physics and Engineering of Radiation Detection

2.1. Some Basic Concepts and Terminologies 71

rate of energy loss). It is almost universally represented by the symbolX 0 .Themost
widely used semi-empirical relation to calculate the radiation length of electrons in
any material is given by (54)

1

X 0

=4αr^2 e

NA

A

[

Z^2 {Lrad−f(Z)}+ZL′rad

]

. (2.1.14)

HereNA=6. 022 × 1023 mole−^1 is the Avogadro’s number,α=1/137 is the electron
fine structure constant, andre=2. 8179 × 10 −^13 cmis the classical electron radius.
f(Z) is a function, which for elements up to uranium can be calculated from

f(Z)=a^2

[

(1 +a^2 )−^1 +0. 20206 − 0. 0369 a^2 +0. 0083 a^4 − 0. 002 a^6

]

,

witha=αZ. X 0 in the above relation is measured in units ofg/cm^2 .Itcanbe
divided by the density of the material to determine the length incm. Table 2.1.1
lists values and functions forLradandL′rad.

Table 2.1.1:LradandL′radneeded to compute radiation length from equation 2.1.14
(19)

Element Z Lrad L′rad

H 1 5.31 6.144

He 2 4.79 5.621

Li 3 4.74 5.805

Be 4 4.71 5.924

Others > 4 ln(184. 15 Z−^1 /^3 ) ln(1194Z−^2 /^3 )

The term containingL′radin equation 2.1.14 can be neglected for heavier elements,
in which case the radiation length, up to a good approximation, can be calculated
from
1
X 0

=4αre^2

NA

A

[

Z^2 {Lrad−f(Z)}

]

. (2.1.15)

Fig.2.1.1 shows the values of radiation length calculated from equations 2.1.14 and
2.1.15 for materials withZ=4uptoZ= 92. Also shown are the relative errors
assuming equation 2.1.14 gives the correct values.
Another relation, which requires less computations than equation 2.1.14 or even
2.1.15, is

X 0 =

716. 4 A

Z(Z+ 1) ln(287/

√

Z)

. (2.1.16)

Here also, as before,X 0 is ing/cm^2. This relation gives reasonable results for
elements with low to moderate atomic numbers. This can be seen in Fig.2.1.2,