Tsoulfanidis N. Measurement and detection of radiation

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408

MEASUREMENT AND DETECTION OF RADIATION

incident

radiation

We"

Diameter

Energy

(keV)

(b)

Figure

1230

(a)

A Ge well-type detector.

(b)

Absolute detector efficiency for a detector with 40-mm

depth and 10-mm-diameter well. Source assumed placed at the bottom of the well.

stronger peaks of higher energy.

PCR

values of

30:l

are common, but higher

values have also been reported.

For the analysis of complex gamma spectra, it is helpful to have an analytic

function that represents the efficiency of the detector as a function of energy.

Many semiempirical equations have been developed to fit the efficiency of

Ge(Li) detectors."-20 Three examples are given here.

PHOTON (GAMMA-RAY AND X-RAY) SPECTROSCOPY

409

Energy,

MeV

Figure

12.31

Photon attenuation coefficients for germanium. The dashed line is the approximate

total linear attenuation coefficient.

The Freeman-Jenkin

qua ti on".

where

photoelectric coefficient

thickness of the detector

Compton coefficient

constants to be determined from measurement

Equation 12.8 was used to determine the

relative

efficiency of a cylindrical and

trapezoidal detector over the range

500-1500

keV, with an accuracy of about

percent.

The Mowatt Equation".

where

lIi

exp

pixi)

product of attenuation factors outside the intrin-

sic region

normalization factor

410

MEASUREMENT

AND

DETECTION

RADIATION

1 OOK

1 OK

loo

Bias 3000 V, compared to spectrum of Nal(TI)

Preamplifier-ORTEC 120-26

Amplifier-ORTEC 450 with 3 ps shaping

Peak-toCompton ratio-32:l

Channel

138 eV

lI1lllllllllllllllllllllllllj

1500 2000 2500 3000 3500

Channel number

Figure

12.32

The 60~o spectrum obtained with a NaI(T1) scintillator and a Ge(Li) detector

(reproduced from

Instruments for Research and Applied Sciences

by permission of EG

ORTEC,

Oak Ridge, Tennessee).

the thickness of the germanium front dead layer

a,,

constants to be determined from measurement

effective detector depth

Equation 12.9 is an improvement over Eq. 12.8 because it takes into account

absorption in the window (through the factor

F')

and in the dead layer of the

detector [through the factor exp(-

p,,a,)].

Mowatt's equation, developed for

planar detectors, gives the efficiency with an accuracy of 1.5 percent over the

range 100-1400 keV.

where

through

are constants to be determined from measured gamma

PHOTON (GAMMA-RAY AND X-RAY) SPECTROSCOPY

411

spectra. Equation 12.10, developed for coaxial detectors, predicts the efficiency

with an accuracy of 0.2 percent over the energy range 160-1333 keV. Further

testing of 12.10 showed2' that the last term involving the constants

and

has a negligible effect on the result.

Equations 12.8-12.10 are given just as examples that indicate the general

energy dependence of the efficiency curve. What is done in practice is to

measure the efficiency as a function of energy using calibrated sources emitting

gammas of known energies and

inten~ities."-'~.~'

table of gamma energies

used for calibration is given in App. C.

12.7.2

Energy Resolution of Ge Detectors

The energy resolution of a Ge detector is given in terms of the FWHM

(r).

The

width

consists of the following two components:

width due to detector effects

width due to effects of electronics

Since these two components are uncorrelated, they are added in quadrature to

give the total width,

r=,/m

(12.11)

As shown in Chap.

the width

is energy dependent and is given by

where

is the Fano factor and

is the average energy needed to produce an

electron-hole pair. For germanium, at the operational temperature of

2.97 eV. Thus,

The width

increases when the detector capacitance increases. The

detector capacitance, in turn, generally increases with detector size and may

change with detector bias. Good Ge detectors have a flat capacitance-bias

relationship over most of the range of bias voltage applied.

The capacitance of the detector has an effect on the energy resolution

because it influences the performance of the charge-sensitive preamplifier that

accepts the detector signal. The contribution of the preamplifier to the value of

increases with the input capacitance. One of the manufacturers, Canberra,

reports a 0.570-eV

with zero input capacitance and a slow increase with

higher values as shown in Fig. 12.32. Clearly, the resolution improves if the

capacitance is kept low. The other component of the input capacitance comes

from items like connectors and cables. Reduction of the length of input cable

and of connectors' capacitance is helpful. For the best resolution with a given

system, the preamplifier should be located as close to the detector as possible.

412

MEASUREMENT

AND

DETECTION

RADIATION

100

Input capacitance, pF

Figure

12.33

The dependence of

on input capacitance for a charge-sensitive preamplifier (from

Ref.

2).

Large Ge detectors commercially available today have capacitance as high

as 30 pF, which results in a value of

1.06 keV (Fig. 12.33). Combining the

two contributions

and

in accordance with

Eq.

12.11, one gets

(keV)

d(0.1283)~~~ (keV)

1.062

(12.14)

typical value of the Fano factor for 0.1

10 MeV is 0.16. Substitution in

Eq. 12.14 gives

(keV)

J(2.63

lop3

)E (keV)

1.06~ (12.15)

Equation 12.15 shows that for low energies, the resolution is determined by

electronic noise. For higher energies, the energy contribution predominates.

Consider two cases:

Case

100 keV

Case 2:

1000 keV

for the

The energy resolution versus gamma energy is shown in Fig. 12.3,

four detectors depicted in Figs. 12.27-12.30. Usually, the resolution is given in

terms of the

FWHM

at 5.9, 122, and 1332 keV.

12.7.3

Analysis

Ge Detector Energy Spectra

Despite the superb energy resolution of Ge detectors compared to that of

NaI(T1) scintillators (see Fig. 12.32), analysis of complex gamma spectra is

PHOTON

(GAMMA-RAY

AND

X-RAY)

SPECTROSCOPY

413

keV

Large LEGe

Energy (keV)

Figure

12.34

Typical energy resolution versus gamma energy for Ge detectors (from Canberra, Ref.

2).

necessary. Figure

12.35

shows a typical Ge energy spectrum. Analysis of the

spectrum entails, first, assignment of energy to the peaks of the spectrum and,

second, the determination of the number of counts

(i.e., the area) for each peak.

The energy assignment to the peaks of the spectrum is accomplished by

calibrating the detector with a source that emits gammas of known energy and

intensity. As explained in Sec.

9.10,

calibration means to determine the con-

Channel

Figure

12.35

gamma-ray energy spectrum recorded with a Ge detector. (This is the spectrum from

a human hair sample irradiated

a reactor.)

414

MEASUREMENT AND

DETECITON

RADIATION

stants of the equation

where

is the channel number (for most systems,

is very small, or zero).

When the energy assignment is performed, the observer should be aware of the

following general features of a gamma spectrum recorded by a Ge detector:

For

2 MeV, the full-energy peak is intense and almost Gaussian in

shape.

2. At higher energies (E,

2 MeV) the double-escape peak (ED,

1.022

MeV) becomes prominent. The single-escape peak is present too (see Fig.

12.7).

3. In spectra taken with thin Ge(Li) detectors, one may see the germanium

"escape" peaks. The escape peaks

(EP)

have energy equal to

where

is the

X-ray energy of germanium. The escape peaks are due to

the loss of the energy carried away by the escaping

X-ray of germanium.

This energy is equal to

9.9

keV for the

and 11.0 keV for the

X-ray of

germanium. Figure 12.36 shows tk,: 13'Ce X-ray spectrum with the escape

peaks marked.

4. When the front surface of the detector is covered by a metal, characteristic

X-rays of that metal are emitted if the incident radiation consists of photons

keV

escape peaks

23.3

escape peak

keV

keV\

Energy,

keV

Figure

12.36

The X-ray spectrum resulting from the decay of '39~e. The Ge escape peaks are

clearly seen (from Ref.

54).

PHOTON

(GAMMA-RAY

AND

X-RAY)

SPECTROSCOPY

415

with energy greater than the

X-ray energy of that metal. Gold, which is

sometimes used, emits a

X-ray with energy 68 keV and five

X-rays

between

and 13 keV.

The determination of the area under a peak-i.e., the absolute intensity of

a particular gamma energy-is not as straightforward as the assignment of

energy, because the area under a peak includes contributions from other

gammas. The methods that have been developed for the determination of the

area can be classified into two groups: methods that treat the data (i.e., counts

per channel) directly, and methods that fit a known function to the data.

Methods that treat the data directly give the area under the peak by adding

the counts from all the channels in the region of the peak and subtracting a

"base background." The methods differ in the way they define the "base7' and

the number of channels that define the peak (a review of the methods is given in

Refs. 22 and 23). Figures 12.37 and 12.38 show graphically three of the methods.

Method

A straight line is used to separate the peak from the ba~e,'~.'~ and

the net area under the peak (NPA) is calculated using the equation (Fig. 12.37~)

R-L+1

NPA

(a,

a,)

where a,

number of counts in channel

channel number at left limit of photopeak

channel number at right limit of photopeak

Method

Here, the background (Fig. 12.373) is defined as the average count in

a region equal to 3r (3 FWHM), extending

1.T

on both sides of the peak. The

gross count under the peak is taken as the sum of all the counts in the channels

corresponding to 3r. The net peak area is then given by

Method

This method, due to ~uittner,'~ fits a polynomial, using the least-

squares technique, to the data from 2kL

and 2kR

channels (Fig. 12.38)

on either side of the peak. The base line is constructed in such a way that at

and XR (shown in Fig. 12.38) it has the same magnitude (p,,p,) and slope

(q,, q,) as the fitted polynomial. The net area under the peak is now given by

the equation

416

MEASUREMENT AND DETEmON OF RADIATION

Channel number

(a)

Channel number

Figure

12.37

Determination of the net area under the peak.

(a)

straight line is used to define the

base.

(b)

average background is subtracted from the gross count.

where the value of

is,

terms of a third-degree polynomial, equal to

PHOTON

(GAMMA-RAY

AND

X-RAY)

SPECTROSCOPY

417

+ +

Measured points

Fitted

polynomial

Base

line

Channel number

Figure

12.38

Determination of the area under the peak using the method of Quittner (from Ref.

25).

Quittner's method is quite accurate if the peaks are separated by about

channels.

Today, the most widely used methods are those that fit an analytic function

to each peak.

After the fit is accomplished, the analytic function is used for the

calculation of quantities of interest such as the area and the centroid (position)

of the peak.

The principle of obtaining the fit is simple and essentially the same for all

the meth~ds.~,

5,26G34

Let yili=

be the experimental point-i.e., the counts in

channels xi(,= N-and f(x, a,, a,,

. .

)

be a function that will represent a single

peak. The parameters

a,,

. .

are determined by minimizing the quantity

y[yi

f(xi, a,, a,,

. .

.)12

i.e., by a least-squares technique, where the weighting factors

usually are the

inverse of the variance of y,. The fitting function consists of a Gaussian plus

modifying functions. Three examples are

Here, a Gaussian describes the peak, and the linear function

Cx de-

scribes the ba~kground.,~

~(x)[l

aI(x

x.)~'

a2(x

x~)~~]

(from Ref.

27)