Tsoulfanidis N. Measurement and detection of radiation

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268

MEASUREMENT

AND DETECTION OF RADIATION

is important for the measurement, it can be eliminated by placing the source

and the detector inside an evacuated chamber. If the use of an evacuated

chamber is precluded by the conditions of the measurement, then appropriate

corrections should be applied to the results.

8.2.2

The Solid Angle-General Definition

To illustrate the concept of solid angle, consider a point isotropic source at a

certain distance from a detector as shown in Fig.

8.3.

Since the particles are

emitted by the source with equal probability in every direction, only some of the

particles have a chance to enter the detector. That portion is equal to the

fractional solid angle subtended by the detector at the location of the source. In

the general case of an extended source, the

solid

angle

is defined by

number of particles per second emitted inside the space defined

by the contours of the source and the detector aperture

number of particles per second emitted by the source

(8.2)

The mathematical expression for

is derived as follows (Fig. 8.4). A plane

source of area

emitting

particles/(m2 s), isotropically, is located a

distance

away from a detector with an aperture equal to

A,.

Applying the

definition given by Eq.

8.2

for the two differential areas

dAs

and

&I,

and

integrating, one obtainst

'~~uation

8.3

applies to isotropic sources: nonisotropic sources, seldom encountered in

practice, need special treatment.

Figure

8.3

The fraction of particles emitted by a point isotropic source and entering the detector is

defined by the solid angle subtended by the detector at the location of the source.

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269

Figure

8.4

Definition of the solid an-

gle for a plane source and a plane

detector parallel to the source.

where

is a unit vector normal to the surface of the detector aperture. Since

r/r

cos

Eq.

8.3

takes the form

COS

"=-/,

dsj

ad7

47~As

Equation

8.4

is valid for any shape of source and detector. In practice, one deals

with plane sources and detectors having regular shapes, examples of which are

given in the following sections.

As stated earlier,

is equal to the fractional solid angle

1).

radiation measurements, it is called either

solid

angle

geometry factor.

In this

text it will be called the solid angle.

8.2.3

The Solid Angle for a Point Isotropic Source and a Detector with a

Circular Aperture

The most frequently encountered case of obtaining a solid angle is that of a

point isotropic source at a certain distance away from a detector with a circular

270

MEASUREMENT

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DETECTION

RADIATION

aperture (Fig. 8.5). In Eq. 8.4, cos

d/r, and the integration gives

From Fig. 8.5,

Therefore, an equation equivalent to Eq.

8.5

i(1

cos 6,)

(8.7)

It is instructive to rederive Eq. 8.7, not by using Eq. 8.4, but by a method

that gives more insight into the relationship between detector size and source-

detector distance.

Consider the point isotropic source of strength

particles per second

located a distance d away from the detector, as shown in Fig. 8.6. If one draws a

sphere centered at the source position and having a radius R, greater than d,

the number of particles/(m2

s) on the surface of the sphere is s0/4.rrR;. The

particles that will hit the detector are those emitted within a cone defined by the

location of the source and the detector aperture. If the lines that define this

cone are extended up to the surface of the sphere, an area

is defined there.

is a nonplanar area on the surface of the sphere. The number of particles per

second entering the detector is AS(S,/4.rr~;) and, using Eq.

8.2,

the solid angle

Figure

8.5

The solid angle between a point

isotropic source and a detector with a

circular aperture.

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271

Source

Figure

8.6

Diagram used for the calculation of the solid angle between a point isotropic source and

a detector with a circular aperture.

becomes

The area

is given

(Fig.

8.7):

/d~,

/(R, d0)(RS

sin

0 d+)

~:/~~d+/'"d0

sin

Therefore, the expression for the solid angle becomes

A, 2~~:(1

cos

00) 1

a=-=

-(1

cos

6,) (8.7~)

47rR: 47rR:

which is, of course, Eq.

8.7.

Eq.

8.5

takes the form [after expanding the square root (Eq.

8.6)

and keeping only the first two terms]

R2 rR2

detector aperture

a=-=-=

4d2 47rd2 47rd2

(8.8)

Equation

8.8

is valid even for a noncylindrical detector

the source-detector

distance is much larger than any of the linear dimensions of the detector aperture.

Example

8.1

typical Geiger-Muller counter is

cylindrical detector with

an aperture

mm in diameter. What is the solid angle if a point isotropic

source is located

0.10

m away from the detector?

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MEASUREMENT AND DETECTION OF RADIATION

Source

(a)

Figure 8.7

(a)

The detector is at distance

from the source.

(b)

The source is assumed to be at the

center of the sphere. The cone defined by the angle

determines the area

(differential area

dA) on the surface of the sphere.

Answer

Using

Eq.

8.5 with

0.10

and

25 mm,

the setup is called a 47r geometry because the detector sees the

full 47~ solid angle around the source.

spherical detector represents such a

case (Fig. 8.8~). If

the setup is called a 27r geometry. Then half of the

particles emitted

the source enter the detector (Fig. 8.8b).

Figure 8.8

(a)

45~

Geometry.

(b)

27r Geometry.

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8.2.4

The Solid Angle for a Disk Source Parallel to a Detector with a

Circular Aperture

Consider a disk source parallel to a detector with a circular aperture (Fig.

8.9).

Starting with

Eq.

8.4,

one may obtain an expression involving elliptic

integral^',^

or the following equation in terms of Bessel function~~,~~:

where

R,/R,,

d/R,,

and J,(x)

Bessell function of the first kind.

R,/d

and

R,/d

are less than

the following algebraic expression is obtained

for the solid angle (see Prob. 8.1):

where

R,/d

Detector

Figure

8.9

disk source and a detector with a circular aperture.

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MEASUREMENT

AND

DETECI'ION OF RADIATION

The accuracy of Eq. 8.10 increases as

\CI

and

decrease. If

IC,

0.2 and

0.5,

the error is less than

percent.

8.2.5

The Solid Angle for a Point Isotropic Source and

Detector with a

Rectangular Aperture

Consider the geometry of Fig. 8.10 with a point isotropic source located a

distance

away from a detector having a rectangular aperture with area equal

ab.

The solid angle is given by4

arctan (8.11)

dda2

If the source is located at an arbitrary point above the detector, the solid

angle is the sum of four terms (Fig. 8.10, each of them similar to

Eq.

8.11. As

Fig. 8.11 shows, the detector is divided into four rectangles

the lines that

determine the coordinates of the point

The solid angle is then

where

for

1,.

. .

is given by

Eq.

8.11 for the corresponding rectangles.

8.2.6

The Solid Angle for a Disk Source and a Detector with a

Rectangular Aperture

Consider the geometry shown in Fig. 8.12. A disk source is located at a distance

above a detector having a rectangular aperture with an area equal to

ab.

It is

assumed that the center of the source is directly above one comer of the

aperture, as shown in Fig. 8.12. The more general case of the arbitrary position

of the source is derived from the present example.

Figure

8.10

The solid angle between a point

isotropic source and a detector with a rectan-

gular aperture. Source

located directly above

one corner of the detector.

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275

Figure

8.11

point isotropic source located at

an arbitrary point above a detector with a

rectangular aperture.

The

solid angle is equal

to four terms, each given by

Eq.

8.11.

The distance

(Fig. 8.12) is equal to

Equation 8.4 is then written as

As in

Sec. 8.2.4, if the ratios

Rs/d, a/d,

and

b/d

are less than 1, the expression

in the braces may be expanded in a series.

only the first four terms are kept,

the result

the integration is

where

a/d

b/d

RS/d

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MEASUREMENT

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RADIATION

- -

Figure

8.12

disk source and a detector

with

rectangular aperture.

If the source is located at an arbitrary point above the detector, the solid

angle is the sum of four terms as shown in Fig. 8.11.

8.2.7

The Use of the Monte Carlo Method for the Calculation of the

Solid Angle

The basic equation defining the geometry factor (Eq. 8.4) can be solved

analytically in very few cases. Approximate solutions can be obtained either by a

series expansion (Eqs.

8.10

and 8.13 are such results) or by a numerical

integration or using other appro~irnations,'~~~~

A general method that can be used with any geometry is based on a Monte

Carlo cal~ulation,~-'~ which simulates, in a computer, the emission and detec-

tion of particles. A computer program is written based on a model of the

source-detector geometry. Using random numbers, the particle position of birth

and the direction of emission are determined. The program then checks whether

the randomly selected direction intersects the detector volume. By definition,

the ratio of particles hitting the detector to those emitted by the source is equal

to the solid angle.

The advantage of a Monte

Carlo calculation is the ability to study compli-

cated geometries. The result has an error associated with it that decreases as the

number of particles studied increases.

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277

8.3

SOURCE EFFECTS

Two source effects are discussed in this section: absorption of particles in the

source, and the effect of the backing material that supports the source. Both

effects are always important in measurements of charged particles. In some

cases, however, they may also be significant in X-ray or thermal-neutron

measurements.

8.3.1

Source Self-Absorption Factor

(fa)

Radioactive substances are deposited on a backing material in thin deposits. But

no matter how thin, the deposit has a finite thickness and may cause absorption

of some particles emitted by the source. Consider the source of thickness

shown in Fig.

8.13.

Particle

traverses the source deposit and enters the

detector. Particle

is absorbed inside the source so that it will not be counted.

Therefore, source self-absorption will produce a decrease of the counting rate

Source self-absorption may be reduced to an insignificant amount but it

cannot be eliminated completely. It is always important for charged particles

and generally more crucial for heavier particles

(p,

heavy ions) than for

electrons.

Source self-absorption, in addition to altering the number of particles

leaving the source, may also change the energy of the particles escaping from it.

Particle

in Fig.

8.13

successfully leaves the deposit, but

loses some energy as

it goes through the deposit. This energy loss is important when the energy of the

particle is measured.

approximate correction for self-absorption can be obtained if the source

emits particles following a known attenuation law. As an example, consider a

source with thickness

(Fig.

8.14)

that has a uniform deposit of a radioisotope

emitting

particles. Assume that the source gives

betas per second in the

direction of the positive

axis. If self-absorption is absent,

betas per second

Figure

8.13

Source self-absorption. Particles may be absorbed in the source deposit.

Source deposit

_---

Detector