The calibration of a NaI(Tl) scintillation spectrometer using a photon monochromator

NUCLEAR

INSTRUMENTS

AND

METHODS

31

(I964)

I57-I68;

©

NORTH-HOLLAND

PUBLISHING

CO.

T H E CALIBRATION OF A NaI(Ti) SCINTILLATION SPECTROMETER USING A PHOTON MONOCHROMATOR J.F. H A G U E and R.E. R A N D *

Physics Dept., University College London Received 11 May 1964 The use of a photon m o n o c h r o m a t o r for calibrating a total absorption NaI(TI) spectrometer is described. Response functions have been obtained for photon energies up to 28 MeV and used to construct a response matrix. A particular advantage of this calibration method is that the usual integration of response functions over the energy channels is eliminated. T h e measured matrix, after suitable smoothing and grouping,

was inverted and used in the measurement of a 28 MeV bremsstrahlung spectrum. Agreement of this measurement with theory provides confidence in the calibration. Formulae are given for the errors, due to counting statistics, in the experimentally determined spectrum, special attention being given to the effect of smoothing the matrix and pulse height spectrum and of grouping the channels in the bremsstrahlung spectrum.

1. Introduction Proposed experiments to measure 28 MeV bremsstrahlung spectra using a total absorption NaI(Tl) scintillation spectrometer have necessitated the calibration of this instrument. High energy photons may interact in the NaI(Tl) by three processes : pair production, Compton scattering and the photo-electric effect, each giving rise to fast electrons which lose energy to the crystal. The resulting scintillation is detected by a photo-multiplier whose output is amplified and passed to a pulse height analyser. The response function for mono-energetic photons appears typically as in fig. 7. The peak corresponds to total absorption of the photon energy (with a gaussian spread due mainly to the electronics in the present spectrometer) whilst incomplete absorption of the photon energy gives rise to the low energy taiP-8). Continuous energy spectra are thus smeared out by the response of the spectrometer. The most convenient means of obtaining the true energy spectrum from a measured pulse height spectrum is by the matrix method2,4,5). The notation used in the outline of the theory given by Rand ~) will be used throughout. The pulse height spectrum, P, is related to the energy spectrum, N, by P = mN, where m is the matrix composed of the response functions for mono-energetic photons. Thus N=m-IP, gives the energy spectrum and the calibration reduces to the determination of the elements of m. The statistical errors in N may be calculated once the errors in m and P and the elements of rn -1 have been determined.

1.1. METHODS OF CALIBRATION AVAILABLE

* Now at High Energy Physics Laboratory, Stanford University, Stanford, California.

A useful review of this subject has been given by Koch and WyckoffT). Experimental methods of calibration require mono-energetic ),-ray sources to obtain the spectrometer response functions. Below 4 MeV, radio-active sources have been used for this purpose by most workers in this field, notably Foote and Koch1). At higher energies (up to 20 MeV), (p, 7) reactions may be used provided a suitable proton source is available. Comprehensive lists of suitable reactions have been given by Kockum and Starfelt 5) and Koch and WyckoffT). Koch and Wyckoff s) have also synthesised the response of a NaI(T1) crystal to high energy photons by using the measured response to electrons of suitable energies and assuming that pair production is the dominant process. Results compared very favourably with other methods. Penfold and Leiss 9) have developed a n " activation" method 7) of calibration using the high energy end of the bremsstrahlung spectrum. This method suffers from the requirement for extremely good experimental statistics, involved analysis and the uncertainties in the exact shape of the bremsstrahlung tail. Theoretical calculations of response functions have been attempted. The most successful are the Monte Carlo calculations of Berger and Doggettl0). However, reasonable statistical accuracy above 5 MeV was prohibited by the computer time requiredT). The calibration method described in this paper uses the photon monochromator originally suggested tl) by Koch and Camac, devised by Weil and McDaniePZ) and advocated for this purpose by Starfelt2). The method depends on the fact that, in bremsstrahlung, 157

158

J. F. H A G U E

AND

energy is essentially conserved between the incident electron and the degraded electron and photon. Thus if the incident energy is known, monochromatic photons may be observed in coincidence with selected degraded electrons using the principle illustrated in fig. 1. A successful photon monochromator has been developed at Illinois 11) where it is used in observing photo-nuclear reactions. Moffatt and Stringfellow 13) have used a similar device to calibrate a total absorption (~erenkov detector.

PHOTON ~.MAIN BEAM ( ENERGY //"/--MAGNETIC FIELD k=E°-e) / / ~-DEGRADED /z ..__.~( A/ ELECTRON TARGET, / ~k~ ( ENERGY E ) /DETECTOR

INCIDENT ELECTRON (ENERGY Eo)

ICO,NC,OENCEI 1

Fig. 1. P r i n c i p l e o f p h o t o n m o n o c h r o m a t o r .

The monochromator requires an electron beam of well defined energy which was available from the U.C.L. 29 MeV microtrona4,15). Degraded electrons were analysed by 180 ° uniform field focussing which had the advantage of almost 100% efficiency combined with good resolution. The monochromator was used to obtain NaI(T1) response functions at 6 energies from 4 to 24 MeV while lower energy functions were obtained from the radio-active source measurements. Calibration by means of a monochromator has considerable advantages in that the photon spectrum in each channel follows the 29 MeV bremsstrahlung spectrum over a sharply defined band of energies, so that matrix elements for analysing similar spectra 6) can be obtained directly, without the integration over the energy channels required in all other calibration methods. A further advantage is that the response of the complete spectrometer (scintillator and electronics) is obtained in a single experiment.

2. Design details 2.1. CALIBRATION CHANNELS The 180 ° focussing magnet was provided with baffles to define various bands of degraded electron energies. The initial calibration was performed using

R.E.

RAND

50 channels with an upper energy limit of 28.50 MeV. (Although the final matrix is constructed with far fewer channels, viz. 17, to facilitate inversion, more information for smoothing purposes could be obtained by performing the initial measurements with a larger number of channels. After inversion the channels can again be grouped as required). Photons were only observed when in coincidence with a detected electron in one of the acceptance bands. In this way six channels were calibrated directly, viz. 8, 15, 22, 29, 36 and 43, others being obtained by interpolation. Constant channel width was chosen for convenience in analysis. 2.2.

TARGET

MATERIAL

The m o n o c h r o m a t o r requires that the majority of the bremsstrahlung be produced in the field of a nucleus whose recoil energy is negligible. Electron-electron bremmsstrahlung is thus undesirable and to keep this below 1% of the whole 1~) a target of high atomic number was used. Its thickness was limited by multiple scattering which reduces the efficiency of scattered electron detection. A lead foil of thickness 17 mg was chosen. 2.3. M A G N E T I C FIELD A uniform magnetic field was required for focussing the degraded electrons. The maximum orbit radius, corresponding to 25 MeV electrons in a field of 5 300 gauss, was 6 ". The field had to be constant to at least 0.2% (0.1 of a channel width) over a diameter of 12 tp and not more than 2% down at 1231 t!. To provide a suitable solid angle for electron detection the uniform field region had to be of reasonable height (21~ - ). These requirements were met by using a magnet with pole faces of 14" diameter and a 3 " gap, the necessary uniformity of field being provided by appropriate shims (fig. 2).

4 j

RADIUS (INCHES) 5 6

x~x~,~ P O LE

7

PROFILE

u.

SHIM

z © I ~_

~ 2

.WITH

WITHOUT X

% SHIM

4 Fig. 2. A n a l y s e r m a g n e t p o l e profile a n d field.

THE CALIBRATION OF A NaI(T1) SCINTILLATION SPECTROMETER The uniformity of the field was measured with a proton resonance device, from which a usable signal was obtained in the fall-off region by correcting the gradient of the field with two coils, one each side of the proton sample, carrying equal currents in opposite senses. Direct measurement of the field while the experiment was in progress was complicated by lack of space for a resonance probe. A recycling procedure was therefore devised so that the field could be determined by the magnet current. 2.4. SCATTERING CHAMBER The important practical details of the chamber are shown in fig. 3. The chamber was constructed of aluminium alloy, with " O " ring seals top and bottom. The electron baffles were designed to select electron energies within the required channels with as little spread of the channel edges as possible. The principle of the design is illustrated in fig. 4. Degraded electrons of a given energy E are scattered from the target T at various scattering angles s, mostly in the forward direction. It is desirable to collect as many of these as possible. Suppose an electron channel is defined between horizontal orbit diameters dl and d2 in the uniform magnetic field, where TA1 = d l , and TAe = d2. Then the upper limit to the electron energy may be defined by an infinitely dense baffle on the outside of an arc radius de, centre T. The lower energy limit is defined

by the point A1, assuming that no electron can penetrate TA1. The outer baffle need extend no further than B and C where TAIB = TAlC = 90 °. Then an orbit of diameter slightly greater than d2 which would just miss B, say, would be interrupted by the inner baffle. Orbits of diameter d2 are thus confined to the scattering angles s ~< s0 so that without limiting the solid angle, the inner baffle may be defined by the circles TBA1, TAxC. s0 is much larger than the average electron scattering angle, so that the energy channels are basically rectangular in form. The baffles were constructed of lead in thin brass boxes (to facilitate machining) with dimensions calculated to select the 6 required channels (fig. 3). The baffles were limited by a common line along which the electron detector could slide so as to detect electrons in any desired channel. Dimensions were chosen for a primary electron total energy of 28.10 MeV, with a maximum degraded electron energy of 24.11 MeV. This defined a minimum photon energy in channel 8 of 3.99 MeV. In practice, the limits of the energy channels are not well defined for the following reasons : a. Non-horizontal orbits with energies slightly higher than the upper limit may penetrate the gap. This effect has been calculated, taking into account the angular spread of the electrons due to the bremsstrahlung and multiple scattering and shown to be negligible for all channels.

AR E

\\ TIRON

AND

159

CLIPPER

Fig. 3. Experimental arrangement.

160

J.F.

HAGUE

AND

b. For practical reasons, the outer baffle could not be of the required length BA2C (fig. 4) in all cases. c. The beam position could vary. This effect was minimised by using a thin tape (0.030" wide) as target. d. Penetration of the baffle edges occurs. The energy spread due to such an effect is difficult to estimate but may be minimised by having the electron detector as far back as possible. Electrons striking the baffles and losing energy would then perform orbits of smaller radii and miss the detector. The electron detector consisted of a plastic scintillator coupled to an E M I 6097 S photomultiplier by MA IN

BEAM

Fig. 4. Electron scattering geometry.

means of an aluminium light pipe. A transparent light pipe could not be used since large numbers of degraded electrons would penetrate it and produce 12erenkov radiation. The first part of the light pipe was of narrow rectangular cross section shielded with lead so that electrons passing through the gaps at higher energies could not reach the scintillator. No evidence of interference between channels was found during the experiment. The photomultiplier was shielded from stray magnetic field by an iron cylinder. The beam was injected into the chamber through a Permendur tube (saturation flux density 23.6 kG, supplied by Telcon Metals Ltd.) so that the target could be in the region of uniform field. The tube position could be adjusted by means of a universal vacuum joint17). 2 . 5 . PRIMARY BEAM ENERGY

To assist in definition of the beam energy, brass slits were introduced in the final orbit inside the micro-

R.E.

RAND

tron. The uncertainty in the final energy was then mainly due to the 1 cm hole in the resonator. The magnetic field, which was uniform over the final orbit 15) (diameter 180 cm) was measured with a proton resonance device enabling the absolute beam energy to be calculated to 0.2~. Low energy contamination of the beam due to the last slit was estimated to be negligible. 2.6. ELECTRONICS The total absorption scintillation spectrometer consisted of a 6 " long, 4¼" dia. NaI(T1) crystal (supplied by Harshaw Chemical Co., Ohio) viewed by an E M I 6099 A photomultiplier and enclosed in a solid steel box, 6 " square, to facilitate shielding. The combination of a NaI(T1) crystal (decay constant 0.25 #s) and the venetian blind photomultiplier (transit time spread --~15 ns) produced current pulses of rise time 25 ns. The pulses were passed to earth through a 300 ohm resistor and clipped at 50 ns with a shorted delay line. This produced pulses of total length 75 ns which were fed to the electronics via a White cathode follower. Pulses were restricted to an amplitude of 1.5 V to avoid non-linearity in the photomultiplied3). Measurement of the resolution of the cosmic peak (due to minimum ionizing particles passing right through the crystal) for various E . H . T . values and attenuator settings provided a useful means of checking that no part of the electronics was being overloaded. In particular, the resolution of the cosmic peak was constant to 5% for signals from the photomultiplier of up to 5 V. The use of clipped current pulses, necessitated by the coincidence requirements, produced an additional random spread of the output pulse amplitudes (section 4.5). The pulse from the electron detector was standardised before transmission in order to eliminate spurious noise pulses. A block diagram of the electronics is shown in fig. 5. The Hutchinson and Scarrott 18) pulse height analyser counts only when coincidence occurs between photon, electron and magnetron pulses. Due to the uneven intensity of the microtron beam pulses, a veto pulse, obtained from a scintillation counter activated by radiation near the beam tube, was introduced to prevent counting when the beam intensity was excessive. Special attention was given to the gate and pulse lengthener (fig. 6). It was arranged that the gate should be shut as soon as the first photon pulse had reached its peak, thus simplifying the design of the pulse lengthener and permitting much higher counting rates

THE

CALIBRATION

OF A

NaI(TI)

without risk of pile-up. The gate is opened by the magnetron pulse cutting off V4, so that Vt and V2 conduct equally and a negative pedestal appears on the common cathode. A photon pulse may then pass

PHOTON PULSE ( 75 n sec)

ELECTRON PULSE MAGNETRON (IOO sec) PULSE (3m sec)

!

ATTENUATOR x /2

SCHMITT TRIGGER J

PREAMP

x16

tS

I

GATE HUT

q

LI PULSE ~ I LENGTHENER

OPEN/ SHUT COINCIDENCE~J

t [PULSE HEIGHT

ANALYSER

I COINCIDENCE~J

L •

VETO ~ IPERMISSIONTo ICOUNT [

VETO { SCALER l

Fig. 5. Block diagram of electronics (thick lines indicate fast electronics).

through the White cathode follower V1, Va and reach the first stage of the pulse lengthener. The negative going pulse front appears equally on the cathode, the grid and, consequently, the anode of Va, and is fed to the grid of V2 so that the pulse is not limited. However, the lengthened pulse on the grid of V~, causes this valve to amplify the positive slope of the input pulse so that a large positive pulse on the grid of V2 cuts off V1 and closes the gate. Three further stages of pulse lengthening are included to produce an initially flat pulse from the fast input. The pedestal from the gate is biassed off in the pulse height analyser and provides a convenient means of stabilising the zero pulse height (section 3.3). The gate closes in ~ 60 ns after the peak of a pulse and is linear to ~ 30 V at the cathode of V1 (corresponding to 4 V at the photomultiplier anode). Provision is also made for counting cosmic radiation (section 3.3), by omitting the magnetron gate as shown. The coincidence circuits were of a simple diode type 19) and due to the length of the pulse from the spectrometer, had a resolving time of 150 ns. 2.7. COUNTING RATES The m a x i m u m counting rate permissible was deter-

SPECTROMETER

161

mined by pile-up considerations. The microtron produces 100 p.p.s., each of duration 1.5,us, so that for less than 1% pile-up 25 photon p.p.s, could be permitted. This rate produced about 15 coincidences per minute in the middle channels. Chance coincidences were possible, since not all pulses in the electron detector were associated with coincident photon pulses. The spurious electron pulses were caused partly by background in the detector and partly by associated photons missing the spectrometer (section 3.1). This effect, if large, could further limit the counting rate, the ratio of random to true coincidences being proportional to the counting rate. 2.8.

PILE UP VETO PULSE ~IO/~s~¢ )

AMPLIFIER I x4

SCINTILLATION

NUMBER OF COUNTS REQUIRED

It was difficult to predict the number of counts required in the calibration since the effect of the smoothing technique (described in Appendix 3) on the accuracy was difficult to approximate. However, a pessimistic estimate of the final accuracy could be obtained from Appendix 1 in which it is assumed that all matrix elements are independently measured. Such an estimate will be most accurate for a spectrum analysed and grouped in a small number of channels chosen so that each response function peak is contained mostly in one channel. Calculations of errors in an energy spectrum, due to the matrix statistics were therefore made for a total of n =: 4 channels at ¼ maximum energy (i 3). For a 2% error in this channel due to the matrix statistics, Appendix 1 gives u~ -- 8 500 (with M = 8). Even with this low value of n, it was expected that smoothing would reduce the number of counts required by a factor of at least 2, so that 4 000 counts were obtained in each of the higher energy response functions. If a spectrum is analysed with n = 8 or 16 channels (corresponding to 48 calibration channels) this simple theory predicts errors of 6% and 16% respectively at ¼ m a x i m u m energy. The accurate calculations proved these estimates to be extremely pessimistic as shown in table 2. Analysis demonstrates that the 50 × 50 matrix could not be used directly unless an enormous number of calibration counts could be obtained. At the counting rates estimated in section 2.7, each of the higher energy channels could be calibrated in ~ 10 hours.

3. Experiment 3.1. EXPERIMENTAL ARRANGEMENT The experimental arrangement is illustrated in fig. 3.

162

J.F.

HAGUE AND R.E.

RAND

+200V

33K

,, 0oK

i

__

~

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,1'

ii -

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=

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68K

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-

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"<'~GATEOPEN T© -L~ COINCIDENCE WITHELECTRON

DGLIBLETRIODESE88CC CRYSTALDIODESCV448

I

-~

O,I/uF

150K

7

K

-

47K -400

Fig. 6. Gate and pulse lengthener.

The NaI(TI) crystal " viewed" the target at a distance of 26" through an 8 " × 1l " dia. cylindrical lead collimator. 4" of paraffin wax was inserted into the collimator next to the crystal, to absorb any electrons (due to pair production in the collimator material) and also to prevent the crystal from " c h o k i n g " with low energy photons. The lead shielding around the crystal virtually eliminated background. Lead shielding was also necessary around the electron photomultiplier. Observation of the electron counting rate provided a useful means of monitoring the beam. The ratio of photon pulses to electron pulses was used to line up the spectrometer and to measure the width at half height of the photon distribution at the crystal. It was found that approximately half of the photons did not enter the crystal. Measurements showed that the ratio of random to true coincidences was approximately equal to i~o at 15 coincidences per minute which therefore represented a feasible experimental counting rate. The estimated number of necessary counts was obtained in each energy channel. 3.2. BACKGROUND DETERMINATION The background spectrum was obtained from the random counts on the high energy side of the lowest calibration channel and in tile tail of the highest energy "hannel. No purely experimental method was found of sub-

tracting the random coincidences although the following methods were attempted : a. Random coincidences between photon pulses and delayed electron pulses were recorded separately using various circuit arrangements. However, trigger levels could not be kept sufficiently constant to use methods of this type (see also b). b. The erratic beam from the microtron prevented use of methods depending on the counting rate, which require constant beam current during the run. c. Neither was it possible to fit the measured background to the coincidence spectra well above the peaks of the response functions, since some of the coincidences in this region were due to pile-up produced by the erratic running conditions. None of these methods being satisfactory, the background was removed using the mainly theoretical method described in section 4.2. 3.3.

ENERGY

CALIBRATION

AND

ESTABLISHMENT

OF

REFERENCE PULSE HEIGHTS

Some standard must be available for calibrating the energy scale of the pulse height spectrum each time the crystal is used. The peaked pulse height spectrum due to the cosmic muons was suitable for this purpose. The analyser channel corresponding to zero pulse height was determined by passing pulses from a generator through a calibrated attenuator and into the gate (fig. 5). The zero pulse height was also checked

THE C A L I B R A T I O N

OF A

NaI(T1)

SCINTILLATION

from a calibration curve (fig. 8) of peak pulse height of the response functions against the energy of the photons at the centre of the respective calibration channel. This curve, which was linear up to 20 MeV, was extended to 31 MeV by using a measurement due to Koch and Wyckoff 7) corrected for our slightly shorter NaI(T1) crystal.

4. Data analysis 4.1.

SMOOTHING THE EXPERIMENTAL

RESPONSE FUNC-

TIONS

Each response function Qj (with background) was obtained in the form of a histogram from the analyser. A smooth curve, Q*, was fitted to this distribution by allowing each point on the curve to be influenced by a range of analyser channels through the formula

*2

Qj =

cr Qj+ r where c r is a triangular distribution

such t h a t 2

Cr=

SPECTROMETER

163

b. Energy lost from the crystal by photons escaping after producing low energy Compton electrons. c. Electrons formed near the rear of the crystal escaping with most of the original photon energy. The effects of (b) and (c) were calculated and the tails shown to be horizontal and finite at zero pulse height. The calculated tails were fitted to the distributions at the lowest pulse height available. Some of the response functions obtained are shown in fig. 7. 4.3. EFFECTS OF ERRORS IN THE CALIBRATION SPECTRA

Ideally the spectrum of electrons passing through each pair of baffles should follow the bremsstrahtung spectrum with sharp cut-offs at each boundary. The boundaries were, however, modified by : a. errors involved in assembling the baffles. b. the 180° analyser geometry and scattering in the target (section 2.4). c. variation in microtron beam energy and analyser magnet field (sections 2.5 and 2.3). TABLE

1 and s is chosen to be as large

1

?'=--8

as possible without allowing the smoothing method to influence the results. (Corrections were made for curvature). The smoothing process reduced the statistical errors of the experimental result by a factor ~ ~/s although the errors were no longer uncorrelated. The numerical value of s was determined by the information available concerning the shape of the response functions. For example, it was assumed that no point of inflexion occured except near the peak. Thus s could be chosen so that the statistical error in the second derivative of Q], determined from the coefficients Qj-8, Qj, Qj'+8 was just equal to the magnitude of this derivative estimated from a sketched curve. 4.2.

BACKGROUND SUBTRACTION

Background was subtracted from each experimental distribution by calculating the theoretical height of the tail of the response function. Thus the random coincidence spectrum could be normalised for each distribution. The low energy end of the tail is due to : a. Low energy electrons created by photons in the collimator material and absorber. (Since this effect is small and does not significantly alter the shape of the response functions, detailed calculations were unnecessary).

Percentage errors in matrix elements at peaks of response functions due to Energy channel

8 43

Counting statistics

0.5 0.1

0.09 0.05

0.24 0.03

0.6 1.2

Table 1 compares the errors produced by the above effects with statistical errors in some of the matrix elements near the peaks. Statistical errors are seen to be by far the most important. Statistical fluctuations in gain of the various circuits and similar effects would also be present in all measurements made with the spectrometer and would therefore be automatically compensated by the calibration. 4.4.

CONSTRUCTION OF THE 50 X 50 MATRIX

The m o n o c h r o m a t o r measurements were supplemented by a low energy calibration performed on the same crystal by Stewart 2°) using ),-rays from radioactive sources at 1.28, 2.75 and 4.45 MeV. The pulse height spectra were used to calculate the integrated response functions for channels 3, 5 and 8. The last of these was compared to the m o n o c h r o m a t o r measure-

164

J.F.

HAGUE AND R.E.

RAND

160

300

(b)

140 250 120 Z'~

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(CHANNEL

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_i I,I

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1,2

__

I 1.3

Fig. 7. Some experimental response functions of the NaI(T1) spectrometer obtained w i t h t h e m o n o c h r o m a t o r . T h e curves have been smoothed by the methods described in sections 4.1 and 4.4., and the errors calculated accordingly. (The error bars do not r e p r e s e n t the raw experimental data, but indicate t h e s t a t i s t i c a l uncertainU¢ at various points on the response functions.) a) response functions plus background (solid curve) and background (dashed curve), due to p h o t o n s in channel 15 (average energy 8.3 MeV); b) response functions and background due to p h o t o n s in channel 43 (average energy 24.2 MeV); c) comparison of some normalised response functions.

ment in order to extract the contribution to the resolution from the clipping of the spectrometer pulse in the coincidence experiment, ~ 24%. (Clipping was not used in the low energy measurements). The calculated responses for channels 3 and 5 were smeared appropriately.

All available response functions were normalised to the same peak height and recorded as rows of a 50 × 50 matrix, with the peaks on the leading diagonal up to channel 36. Above this channel, the peak positions were determined by the calibration curve (fig. 8, section 3.3). Since the shapes of the functions changed

THE CALIBRATION

OF A

NaI(T1)

only slowly with energy, graphical interpolation was used to find the missing response functions and to improve the statistical accuracy of the measured functions. Interpolation was performed along lines in the

-31

KOCH& WYCKOFF/~//,

MeV

50

o z.j40 to 20MeV / Z Z

1/

~3o u

~2o z to

IO

I / ~ i IO

36 / 20

i 30

i

I 40

i SO

PULSEHEIGHTCHANNELNo

Fig. 8. Calibration curve : dependence of peak pulse height on p h o t o n energy.

matrix parallel to the leading diagonal, while further smoothing along lines defining a constant fraction of peak pulse height was useful. It was assumed that the functions for channels 44-50 were of the same shape as that for channel 43. The final 50 × 50 response matrix was constructed by normalising the elements in each energy channel to the calculated spectrometer efficiency.

SCINTILLATION

4.6.

CONSTRUCTION AND INVERSION OF THE COARSE MATRIX

To form a statistically useful matrix, m,,~', the channels of the final 50 × 50 matrix, rn,j, were grouped in threes to form a 17 × 17 matrix according to the formula n/n

m~'i'

Ni =

m~l N~

(see appendix 2) where Ng is the calibration bremsstrahlung spectrum given with sufficient accuracy by Schiff21). (The matrix was to be used to analyse similar spectra). This matrix is not presented here as it is peculiar to the electronics associated with the spectrometer. Inversion of the coarse matrix yielded negligible rounding off errors.

165

EFFECT OF STATISTICAL ERRORS IN THE RESPONSE FUNCTIONS

In the 17 channel analysis, the energy channels which were effectively directly calibrated were i = 3, 5, 8, 10, 12 and 15. The number of counts obtained in each is shown in table 2. For analytical purposes, the errors in the coarse matrix elements in these channels were considered to be entirely due to the counting statistics in the measured response functions at the corresponding energies. These errors were calculated from the errors of the smoothed response functions allowing for the correlations introduced by the smoothing (section 4.1). The errors thus obtained were not entirely independent but when treated as such would yield a slightly pessimistic value for the accuracy of the bremsstrahlung spectrum. The final response matrix was smoothed in such a way that the response functions used at a particular energy depended equally on the calibration at that energy and on those at the neighbouring calibration energies. The interpolated elements depended on the neighbouring two smoothed response functions, so that 3 or 4 calibration channels were ultimately involved in each. To calculate the statistical errors (in an energy spectrum) due to the matrix, the theory of Appendix 3, could then be used. The smoothing coefficients were approximated by for --3~< r ~< 3 and i+r= 1, 3, 5, 8, 10, 12, 15; otherwise Note that the normalisation condition is

br.~=-}

br,i=O.

~ ~ br,, = n = 17, (since 2 br,~=l). The r=-s

4.5.

SPECTROMETER

~=1

r=-s

above approximation to b yields a sum 470 less than this value. 5. Measurement of 0 ° bremsstrahlung for lead

In order to test the calibration matrix, the forward bremsstrahlung spectrum from the lead target (due to 27.6 MeV electrons) was measured by omitting the coincidence circuits. Background was negligible. The resulting pulse height spectrum was grouped into the appropriate channels and smoothed using a0 = ½, al = a - 1 = ¼ (see Appendix 2). This ensured that the smoothed distribution (P*), when operated on by the inverse response matrix, was sufficiently consistent with the measured response functions to produce no significant oscillations in the final energy spectrum (N~). It is essential for the smoothing interval to be narrower than the response peaks in order not to influence the experimental results.

166

J.F. TABLE 2.

Counts recorded in calibration channels

i,j

Percentage errors,

Bremsstrahlung spectrum

ei

HAGUE AND R.E. RAND

t~N,/N~, due to counting statistics.t

f}Ni/Ni

I

due to :

4

Grouped spectrum :

,.

N,

~m~

~ej

i

3284 2570 2057 1721 1492 1268 1128 953 855 732 852 634 372 309 97

0.6 0.9 1.2 1.7 2.5 2.8 4.1 5.2 5.4 7.3 7.7 11.0 18.8 23.6 58.8

1.3 1.9 2.3 2.6 2.7 3.3 4.1 5.5 6.3 9.3 8.6 11 14 20 128

1

ONi/Ni due to :

Om0

OPj

2

1.0

1.8

3

1.7

2.3

2.2

2.8

5

1.8

4.1

6

2.2

5.3

7

3.2

7.7

7.9

21.4

ONd Ni

Grouped spectrum

due to :

i

Omo ! ~)Pj

(iooo)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

2629 2082 1718 1470 1280 1111.5 969.2 840.1 734.5 645.9 546.1 404.1 255.8 144.3 61.3 14.2

3592 1030

1260 1140 3870

4720

0

- -

1

0.7

1.3

I 2

1.5

1.6

3

1.6

1.6

4

2.2

3.2

- -

t The 27.6 MeV (i.e. 16.1 channel) bremsstrahlung spectrum (N0 is derived from the smoothed pulse height spectrum (P~)" Errors ~ N~ due to matrix contributions ~rnij, and pulse height spectrum contributions ~Pj, are calculated according to appendices 2 and 3.

The resulting energy spectrum (fig. 9) was compared to the (normalised) theoretical spectrum at 0 ° due to Schiff20, corrected for the solid angle subtended by the collimator, and for the attenuation of photons by the paraffin wax absorber. Experimental errors were such that the spectral shape would not be significantly

z z

2

000

< i U o.

Z

affected by the Born approximation and screening assumptions. The errors in the experimental bremsstrahlung spectrum due to statistical errors in the matrix and pulse height spectrum were calculated using the formulae (2.1) and (3.1). Agreement between theory and experiment is satisfactory, the discrepancy at low energy probably being due to inadequate calculation of the rather poor absorber and collimator geometry. The only other quantity which could produce such an effect is the efficiency of the NaI(T1) crystal. This, however, is well established. Incorrect energy calibration of the pulse height spectrum might be expected to produce a small discrepancy at high energy. To demonstrate the considerable advantages of grouping the smoothed energy spectrum, it was grouped into 8 equal width channels and 4 channels chosen to equalise statistical errors. The errors for the grouped spectra were calculated from formulae (2.2) and (3.2).

I OO0

o u

I

IO PHOTON ENERGY

20

:)7.6 MeV

(McV)

Fig. 9. Measurement of 0% 27.6 MeV bremsstrahlung for lead compared to Schiff theory (normalisation is by area above 7 MeV).

Numerical results are shown in table 2 where the advantages are clearly illustrated. Table 2 may also be used to facilitate the analysis of subsequent bremsstrahlung spectra.

THE

CALIBRATION

OF A

NaI(TI) S C I N T I L L A T I O N

6. Conclusions The particular advantages of the photon monochromator in calibrating a total absorption spectrometer have been demonstrated, although the usefulness of the present apparatus has been restricted by the short duty cycle of the microtron. The reduction in statistical errors obtained by smoothing the experimental results is very significant and is greater than might at first be expected, due to the correlations in the coefficients of the analysed spectra. The advantages of grouping are particularly great where the inverse matrix terms are large and alternate in sign. This occurs in any row or column where the channel width is smaller than the widths of the response function peaks. The measurement of the bremsstrahlung spectral shape provides confidence in the results of the calibration, Schiff's formula having been verified in this energy range by other authors2Z). The authors are extremely grateful to Dr. R.E. Jennings for m a n y useful discussions and encouragment in this project and to Prof. Sir H. Massey for the facilities provided. Thanks are also due to Messrs. R.J. Fisher and R . H . Watson of the Physics Department workshop for construction of the apparatus. One of the authors, (J.F.H.) would like to thank the D.S.I.R. for a maintenance grant.

SPECTROMETER

167

channel k in the calibration experiment. Hence --1 2

(SN0 2---

( m q ) Pj" 1 + ~ j=l

assuming Nkluk is constant when k-~ i. So that one obtains

1)

- - ~ - / - ~ ~,+¥

where M = ~I/R~ is the figure of merit for the spectrometer 6) in which ~ = efficiency and R = resolution. tion.

Appendix 2 S T A T I S T I C A L ERRORS I N V O L V E D I N T H E A N A L Y S I S OF A " SMOOTHED "

PULSE H E I G H T S P E C T R U M

Suppose that the measured pulse height spectrum is represented by P j ( j = 1, 2 ... n). Then the smoothed spectrum may be written 8

PI = S

arPj+r

r=--8

8

where s < nR and ~

ar = 1. The corresponding

~'=--8

Appendix 1

energy spectrum is

APPROXIMATE

FORMULA

FOR

ERRORS

INVOLVED

IN

n

*

-1

N~ = ~ m~j Pj

T H E A N A L Y S I S OF A PULSE H E I G H T SPECTRUM

j=l

The analysis in this section assumes equal channel widths and that all quantities are independently measured. If an energy spectrum is given by

Z-17_ = mij arPj+r n

8

j=l

r=-s

n

Nt = ~ m~lPj

Hence

.f=1

then the statistical error bN~ in Ni is given by 8)

~_

(m~j ) 5=1

(~pj)2 +

(6mej) N~

--1 --

.¢.....4 r=-8

armIj-r

so that the statistical error in N~, due to the measured spectrum is given by

k=1

(The equation is exact when n is the number of channels used in the calibration.) Then assuming that the errors are due entirely to the numbers of counts recorded to find Pj and mu, (~P~)~ = P~

~Pj

and

(~mkj) 2 = m~j/uk

where u~ is the number of incident photons in energy

j=l

{ 5~_ s arms,j-r} -1 2 (Spj)2.

(2.1)

If the spectrum N~ is grouped into n' equal channels so that n/n'

n/n p

n

i

i

t=1

168

J.F.

(w ere

:

It, I n t

ftn/nt

i=(U-1)n/n'+l

HAGUE

AND

)

arl ~ 1=1

mL1_r

(5P¢) 2

RAND

(6N*) 2 "~

br,k-r m*d-r Nk-r ]=1

then the statistical error in N ( , is similarly given by (bN() 2 -

R.E.

(2.2)

=-S

In each of these formulae it is usually convenient and sufficiently accurate to put (6Pj) z -----Pj- P~, assuming that the background errors are negligible.

-I

assuming all the dmkj are independent. In this expression, only certain values of k, for which ~mej is non-zero (the calibration energy channel numbers), need be considered. If, for these values of k, the approximations, br, k - r ~ br and Nk r ~ N~ (for - - s ~< t ~< s), can be made, the expressions can be simplified to : b

(~N,)~ ___ Appendix 3

J=l

STATISTICAL ERRORS INVOLVED IN THE USE OF A " SMOOTHED " MATRIX

Smoothing and interpolating an experimental matrix, m, along lines parallel to the leading diagonal, results in a " smoothed" matrix m*, which may be written (assuming ~ is constant)

(Omen)2

--s

*-1 / 2 ~

=-s

N ~2 (~m~j)~.

(3.1)

k=l

If the spectrum Nl is grouped into n' equal channels, the errors in the resulting spectrum, N~', are given by (5N1')2

t

n

{~

r

n

,-1

mz,j-r

~

2

N~ (6mgj) 2. (3.2)

References m~j =

br,I m i + r d + r 1"~-- 8

where br,~ and s are chosen to be physically reasonable and

~@,

br,i-=- 1

r = --,q

The statistical error in an analysed energy spectrum due to the matrix calibration may then be obtained from the identity n

N~ ~ / .N" m l .] - 1 ~ , mlc*jN1c k=l

3"=i

or ~t

n

8

Ni ~. mij *-'7,{~ br,,~m.+rj+r}N.. j=l

r--~- s

Hence __

(3N~ ,~ _ Omkj

~

*--1 br,~-r mt,i-r Nk-r

i'=--8

so that the statistical error in N, due to the measured matrix is given by

1) R.S. Foote and H.W. Koch, Rev. Sci. Instr. 25 (1954) 746. 2) N. Starfelt, K. Fysiogr. Sallsk. Lund. FSrhandl. 26/5 (1956) 43. 3) A. K a n t z and R. Hofstadter, Phys. Rev. 89 (1953) 607. 4) N. Starfelt and H.W. K o c h , Phys. Rev. 102 (1956) 1598. 5) j. K o c k u m and N. Starfelt, Nucl. Instr. and Meth. 4 (1959) 171. 6) R.E. Rand, Nucl. Instr. and Meth. 17 (1962) 65. 7) H . W . Koch and J . M . Wyckoff, I.R.E. Trans. NS-5, No. 3 (1958) 127. 8) H . W . Koch and J.M. Wyckoff, J. Research NBS 56 (1956) 319. 9) A. Penfold and J.E. Leiss, Phys. Rev. 95 (1954) 637. 10) M.J. Berger and J. Doggett, J. Research NBS 5;6 (1956) 355. 11) J.S. O'Connell, P.A. Tipler and P. Axel, Phys. Rev. 126 (1962) 228. 12) J.W. Weil and B.D. McDaniel, Phys. Rev. 92 (1953) 391. la) j. Moffatt and M . W . Stringfellow, J. Sci. Instr. 35 (1958) 18. 14) D . K . Aitken, F . F . Heymann, R.E. Jennings and P.I.P. Kalmus, Proc. Phys. Soc. 77 (1961) 769. 15) G.R. Davies, R.E. Jennings, F. Porreca and R.E. Rand, N u o v o Cimento Suppl. 17 (1960) 202. 16) L.H. Lanzl and A. O. Hanson, Phys. Rev. 83 (1951) 959. 17) P.I.P. Kalmus, Vacuum 9 (1959) 147. 18) G . W . Hutchinson and G . G . Scarrott, Phil. Mag. 42 (195l) 792. 19) R. Miller, Rev. Sci. Instr. 30 (1959) 395. 2o) T . W . W . Stewart, P h . D . Thesis, London, 1962. 21) L.I. Schiff, Phys. Rev. 83 (1951) 252. 22) H . W . Koch and J.W. Motz, Rev. Mod. Phys. 31 (1959) 920.