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How much light from rectangular scintillation counters? (for whole body counting)
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1975 Phys. Med. Biol. 20 282 (http://iopscience.iop.org/0031-9155/20/2/010) View the table of contents for this issue, or go to the journal homepage for more
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PHYS. MED. BIOL., 1975, VOL.
20,
NO.
2, 282-295. @ 1975
How Much Light from Rectangular Scintillation Counters? T. SMITH,
M.SC., P H . D . ~
Medical Research Council Department of Clinical Research, University College Hospital Medical School, London, England
Received 12 June 1974 ABSTRACT. Theoretical calculations have been made of the efficiency and uniformity of light collection a t one end of a large rectangularliquidscintillationcounter (80 x 20 x 18 cm). Three different arrangements were considered; ( 1 ) asystem in which all surfaces of the counter other than thecollecting surface were transparent to permit total internal reflection, ( 2 ) a similar system with a mirror placed close to the end opposite the collecting surface and (3) a system in which external reflectors were placed close to surfaces other than the collecting surface. In all cases, the effects of attenuation of light in the counter medium, inefficient total internal reflection and the reflectivities of external reflectors were considered and illustrated for selected values of these parameters. I n an experimental counter of the same dimensions, the amplitude of the spectrum of a collimated 42K source was used to measure the relative efficiencies of light collection for various reflector arrangements. I n general the experimental values agreed with theoryandthenon-uniformity of response was within the predictedrange. I t is estimated that about 30% of scintillation light can be collected a t one end of such a counter, with a non-uniformity not greater than 1004, if magnesium oxide is used &S the external reflector a t all other surfaces.
1. Introduction
Rectangularorganicscintillationcountersare used inthe fields of high energy physics and nuclear medicine where they fulfil the need for relatively cheaplarge-area or large-volumeradiationdetectors. For some yearsthis laboratory has been concerned with the development of such counters suitable for measuring total body radioactivity in clinical tracer investigations, ultimately to provide adequate sensitivity to permit the estimation of the total body content of naturally occurring *OK. This requires the accurate measurement of about 10 nCi or less of a radionuclide which emits a 1-52 MeV y-ray pernucleardisintegration, and rectangular perspex tanks filled withcheap liquid scintillator (Barnaby and Jasani1968) have proved satisfactory for this purpose. The use of large detectors, however, poses the considerable optical problem of light collection andthis is of particularimportanceiny-ray counters where the energy deposition is usually small relative to that in high energyparticledetectors. Efficient light collection is essential not only t o produce a detectable pulse but also to provide the best possible energy resolution, even though this is necessarily limited in organic detectors due to the predominat'ingCompton effect. Resolutionshouldpreferablybesuch t'hat total body 40K (1.52 MeV) canbe efficiently measured to the exclusion of
t Present address : Radioisotopes Division, MRC Clinical Research Centre, Northwick Park Hospital, Watford Road, Harrow, Middlesex,HAL 3UJ.
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I3'Cs (0.66 MeV), a long lived product of nuclear fallout, of which some 2-5 nCi may be present in the body (Shukla, Dombrowski and Cohn 1973). Resolution, of course, depends not only on the efficiency, but also on the uniformity of light collection, a condition which has considerable bearing on the design of largedetectors.Although the use of many largediameterphotomultipliers ensuresadequatelight collection (Van Dilla and Anderson1962), the most economical system, in terms of the fraction of scintillation light collected per unit photocathode area, is that in which photomultipliers are placed on the small ends of rectangular counters designed to permit total internal reflection of scintillation light (Brini, Peli, Rimondi and Veronesi 1955). Our counters are of this basic type and therefore methods have been developed to allow theoretical prediction of the response of rectangular counters of various dimensions and configurations. This is clearly of value in the development of detectors designed to maximize light collection whilst retaining good uniformity. Review of earlier studies I n a n earlier paper (Smith and Jasani 1972a) theoretical predictions were made of the amount of light collected a t one surface of a rectangular scintillation counter (61 x 40.5 x 63 cm) from scintillations occurring a t various positions with it. The collecting surface was one of the two surfaces of smallest area and all the others were assumed to be transparent to allow total internal reflection to take place. The concept of light escape cones (Shurcliff and Clark Jones 1949) was used to simplify the calculation. In thearrangement considered, only the light in theescape cone directed towards the collecting surface and the remaininglightnotincludedwithinthe escape cones (trappedlight)can escape through the collecting surface to which photomultipliers are optically coupled, except for scintillations occurring in a region immediately in front of the collecting surface. The length of this region is such that, for a scintillation within it, parts of the four lateral escape cones can intersect the collecting surface directly. I n t h e calculations, the escape cones and trapped light were assumed to be divided into coaxial cones of differing apical angle so that, for each increment between cones, the path length from a point of scintillation to the collecting surface could be calculated for a known fraction of the emitted light. For this purpose, the angular distribution of trapped light was determined by the formula given by Smith (1971). Various simplifying assumptions were made and the only correction applied was for loss of light by absorption in the medium of the counter. Estimations were also made of the contributions of lateral escape cones arising from scintillations close to the collecting surface. Predicted response curves were thus obtained for points of scintillation along the central axis of the counter and along a line parallelto it butchosen close to the edge of the counter. Two striking results emerging from this analysis were the large and rapid increase in light collection as the point of scintillation approached the collecting surface and the very uniform response from a more distant region. The former effect is largely explained by the contribution of light in lateral escape cones and the latter is due partly to the highly efficient total internal reflection of light (assumed to be 1.0 in the above calculations) 2.
284
T.Smith
and partly to the fact that the contribution from trapped light reflected from the back surface tends to compensate for increased absorption of forward light as the scintillation point moves away from the collecting surface. The response from scintillations off the central axis was shown to differ from the central axis response for positions close to the collecting surface only, on account of the different geometry for lateral escape cones. Indeed, the response from scintillations at the extreme edge of the detector sets the minimum length of light guide necessary to ensure that, in the type of counter considered, there is no contribution to the collected light from lateral cones. This dimension is given by l tan C where 1 is the length of the longer side of the collecting surface and C is the critical angle for total internal reflection. Allowing for the use of a light guide the theoretical analysis of the above counter predicted a possible light collection efficiency of 23%, assuming the collecting surface to be completely an sensitive, and a non-uniformity of & 9%. The uniformity obtained with experimental system of the same dimensions was within the predicted range, but the fractional light collection was estimated to be about 10% less than the theoretical value, probably due in part to the assumption of lOOyo efficiency for total internal reflection. It is often advantageousto collect more light from scintillation countersthan the abovesystem allows, provided that the uniformity is not significantly worsened as aresult.Usuallytwooppositesurfacesareused;light being or after reflection a t one of them. Even collected either from both directly with such an arrangement, a large proportion of light is lost from the other four surfaces and this has led to the use of various types of external reflector adjacent to these surfaces in order to further increase the collected proportion (Millar, Hincks and Hanna 1958, Faissner, Perrero, Ghani and Reinharz 1963, Crabb, Dean, McEwen and Ott 1966, Grieder 1967). I n a recent report (Smith and Jasani 1972b) experiments were described in which various reflectors were applied to model counters and a large scintillation counter. A useful gain in light collection was achieved, with good uniformity, when an efficient reflector such as magnesium oxide (MgO) was pla'ced close to, but slightly separated from, the noncollecting surfaces. With thescintillation counter (80 x 20 x 18 cm), for example, in which light was collected from one end, the use of MgO in this way increased the amount of light collected by 70%, the non-uniformity being & 7.6%.
In the present paper a theoretical analysis is described which extends the previous treatment (Smith and Jasani 1972a) to demonstrate the dependence of both the efficiency and uniformity of light collection on various parameters in a scintillation counterused with and without externalreflectors. The analysis is made for a rectangular counter of the same dimensions as the experimental counter with which some data have been obtained, making possible a comparison between the predicted and observed light collection efficiencies. 3. Factors affecting the efficiency of light collection
Severalfactors, considered below, may influence the efficiencyof light collection to a greater or lesser degree, depending on the extent towhich they
How ,Vuch Light
from Rectangular ScintillationCounters?
285
vary with wavelength in relation to the spectral distribution of scintillation light. The fluorescence spectrum of the liquid scintillator used in our counters (Barnaby and Jasani 1966) extends from 370 nm to about 500 nm with a peak emission wavelength of 410 nm (fig. l(a)).
Wavelength (m)
Fig. 1. ( a )Fluorescence spectrum of the liquid scintillator used in the experimental counter. ( b ) Spectral response of photomultiplier (EM19623B).
3.1. Refractive index
The refractive indices of the scintillator and perspex container used in the present counter are both equal to 1.5 measured at 410 nm and are reasonably constant over the wavelength range of the fluorescence spectrum. It is estimated that small variations in refractive index with wavelength affect the calculated amount of light in escape cones by less than 3% and therefore the value 1.5 has been considered a reasonable average for the purposes of the present analysis. 3.2. Attenuation of light
There is some difficulty in measuring the intrinsic attenuation coefficient for the spectrum of scintillation light in the counter medium. As Barton, Crispin and Slade (1964) have pointed out, spectrophotometer measurements are not very satisfactory for this purpose as the effects of scattering and of re-emission of absorbed light are difficult to estimate, and therefore a method which more closely approaches the practical situation is required. These authors describe a method of measuring an effective attenuation length, which is the distance required to reduce the intensityof light in asingle escape cone by thefactor e-1. Since the mean distance travelled by light in an escape cone is about 15% larger than this value, an approximation to theintrinsic attenuation length can be found.Theabovemethod, however, involves multiple totalinternal reflections of light in the escape cone being measured and inefficient reflection leads to underestimation of the attenuation length. Thus theintrinsic attenuation length is not precisely known and for this reason two values, 200 and 400 cm, have been used in thetheoretical analysisto show the effect of differences in this parameter on the efficiency of light collection. The estimated value for the system described here is approximately 200 cm and attenuation lengths of 400 cm have been observed in plastic scintillators (Nicoll and Ewer 1971).
286
T . Xmith
3.3. Total internal rejection Ri The efficiency of total internal reflection, Ri, is theoretically 1-0 for scintillation light incident on the counter walls at angles greater than thecritical angle, and in a previous analysis (Smith and Jasani 1972a) only this maximum value was considered. However, such an efficiency requires a degree of optical perfection of the counter walls which is unlikely t o be attained in practice. Therefore in thepresent theory two values, 1.0 and 0.95, have been taken intoaccount. There is a small contribution t o internal reflection from light incident at angles less than the critical angle (JenkinsandWhite 1957),amountingtoabout 3-6% depending on wavelength, polarization and angle of incidence. Its effect is to increase slightly the estimated value of light collected by total internal reflection alone, but it is a relatively small effect which has been ignored in the present analysis. 3.4. Rejectivity of external re8ectors R,
Our investigations have involvedthe use of both specular and diffuse external reflectors, the most satisfactory of these being front surface aluminized mirrors, polished aluminium foil and MgO powder. Thereflectivity of evaporated aluminium on polished glass varies from 0.90 to 0.92 between 385 and 500 nm (Jelley 1958), that of aluminium foil is almost constant at 0.9 over the wavelength range 300-550 nm (Birks 1964),whilst dry MgO has superior reflectivity (almost 1.0) which is reasonably constant over the range 325-550 nm (Swank 1958). In the present theory, therefore, the effects of three different values of reflectivity (1.0, 0.95 and 0.90) have been compared. 3.5. Photomultiplier response The present theory calculates the proportion of scintillation light which is available a t a collecting surface of the counter assuming that the surface is completely photosensitive. It isthereforeindependent of theparameters associatedwith the photomultiplier which affect the counter response. I n order to compare the theory with the experimentalresults described later, corhowever, it is necessary to take these parameters into account. After a rection has been applied to allow for the proportion of the collecting surface occupied by the photocathode, the response of the systemdepends on the spectral response of the photomultiplier (fig. I@))in relation t o the spectrum of light collected. Although a secondary solute (POPOP)is used to match the fluorescence spectrum of our scintillator to an S-l1 photocathode, the modification of the spectrum of collected light in large detectors, dueto absorption and re-emission, tends to degrade this matching.To allow for this, a quantum efficiency of 12% has been assumed for the photocathode used in the experimental counter, although the peak value may reach1 7 % (EM1 data). 4. Extension of theoretical analysis
The amount of light incident on one small end (collecting surface) of a rectangular scintillation counter (80 x 20 x 18 cm) of uniform refractive index, 1.5,
How Much Light from Rectangular Scintillation Counters?
287
has been calculated for scintillations occurring at various positions within it. Three different systems (fig. 2) were investigated as described below. I n each case, light collections were calculated for various positions along the central
1 4 -
- - -- -- - - - ---
qX-??q 1 1 t r X
(C)
Fig. 2. Diagrams of the three counter systems analysed. (a)TIR system. ( b ) End reflector (mirror)opposite photomultiplier. (c) External reflectors at all surfaces other than the collecting surface. Points of scintillation were taken along the central axis and at one off-axis position (X).
axis and for one off-axis position a t a distance of 40 cm from the collecting surface and 1 cm in from each of two adjacent sides. External reflectors, when used, were assumed to be specular and situatedparallel to, but slightly separated from, the walls of the counter.
Basic total internal reflection (TIR) system In this arrangement (fig. 2 ( a ) ) ,all surfaces other than the collecting surface aretransparentandnoexternal reflectors are used.Thetheoreticallight collection was calculated by assuming the escape cone directed towards the collecting surface and all trapped light to be divided up into small elements (fig. 3(a))formed by coaxial cones, whose semi-apical angles differed by 5 O , and 4.1.
( U)
Fig. 3. ( a )Diagram illustrating the method of analysing individual elements of escape cones as described in text. ( 6 ) Cross-section of counter including the pointlof scintillation, showing the dimensions used in the calculations.
288
T.Smith
by radial planes 10' apart. It was then possible to estimate, for each element, the solid angle and hence the proportion of isotropically emitted scintillation light it contains, the meandistance from thepoint of scintillation to the collecting surface, and the number of tota'l internal reflections at the counter walls. Corrections were made for losses of light resulting from attenuation in the counter medium and from inefficient total internal reflection as specified in sections 3.2 and 3.3 above respectively. The total light collection was obtained by summing the individual contributions so calculated for each element. For each element of light in the quadrant defined by the distances p and s (fig. 3(b)), the fractional amount of light, f, reaching the photocathode was calculated using the formula
f = Qexp
xsec 0
where C? is the fractional solid angle of the element, x is the perpendicular distance from thepoint of scintillation to the collecting surface (for light reflected from the oppositesurface, x is the perpendiculardistance to that surface plus the length of the counter), Ri is the coefficient of total internal reflection, and Ni is the average number of internal reflections at the counter expressions x tan 0 sin + / ( p+ nd) walls. Ni is given by thesum of the terms in the and x tan 0 cos $/(S +nw)(where n = 0 , 1 , 2 , .. .) which are greater than or equal Do 1 ; W and d being the width and depth of the counter respectively. For light in the other three quadrants the combinations s with (d - p ) , (W - 8 ) with p and (W - S) with (d - p ) are used to calculatje Ni. For light reaching the collecting surface following a reflection at theopposite end, Ni was increased by 1. 4.2. Use of a rejlective surface opposite the light collecting surface
When a reflective surface is placed parallel to, and slightly separated from, the end of the box opposite the single light collecting surface (fig. 2(b)), it does not interfere with total internal reflection and hence does not affect the collection of trapped light and light in the escape cone directed towards the collection surface, as described above. It does, however, increase the amountof light collected by reflecting part of the light in the escape cone which is transmitted through the end of the box. This further contribution was calculated by a similar procedure to that used in section 4.1 above and an extracorrection was made t'o allow for the reflectivity of the end reflector as detailed in section 3.4 above. Thus, for light in this escape cone, eqn (1) was multiplied by R,, the coefficient of external reflection. 4.3. Use of rejectors around all noncollecting surfaces I n this situation (fig. 2 ( c ) ) there is a further contribution to the collected
amount from some of the light in the four lateral escape cones whose axes are parallel to the collecting surface, even beyond the region within which these cones areintercepted by the collecting surfacedirectly(Smith andJasani 1972a). A simple approximation was made by assuming each cone to be made
How Xuch Light from RectangularScintillationCounters?
289
up of small elements (fig. 3(a)) by first dividing the cone into a number of coaxial cones whose semi-apical angles were increased by 10". For a refractive index of 1.5,the critical angle for total internal reflection is 41' 48';thus each cone was divided arbitrarily into four coaxial cones, three with semi-apical angles of 10,20 and 30' and a fourth of 41" 48'.These cones were further subdivided by radial planesof 10" apart and hence each quadrant of each cone was divided into 36 elements whose solid angles and mean path lengths to the collecting surface were easily determined. The calculation of the contribution due to these elements is less straightforward than for those of the two cones whose axes are perpendicular to the collect'ing surface because of the different types of reflection involved. Thus light ina lateral cone which escapes from the surface t o which it is directed may, after specular external reflection, re-enter the detector and escape from the opposite surface t'o repeat the process:on the other hand, reflections at any other surface of the detector, either before or after external reflection, are totally internal. It was necessary, therefore, to calculate separately for each element the numbers of external and internal reflections, since the reflectivities considered differed inmostcombinations. Corrections for light losses were applied as in sections 4.1,and 4.2 above, and the total contribution from light in the four lateral escape cones was determined by summing the contributions due to all t'he elements comprising them. Light in four halves of these cones can only contribute after reflection at the opposite end of the box and this may imply large path lengths and many reflections. Fractional light' collection for elements in the four lateral cones was calculated from the formula
N , , the average number of internal reflections, is given by the number of terms ..) which are greater than in the expression x tana/(s+nw) (where n = 0,1,2,. or equal to l, and Ne, the average number of external reflections, is obtained similarly from the expression x cot 7 sec a / ( p+ nd). For light reaching the collecting surface after reflection a t the opposite end, Ni was increased by 1.
4.4. Experimental counter A perspex box measuring 80 x 20 x 18 cm was filled with liquid scintillator (Barnaby and Jasani 1966),except for a 20 cm light guide section at one end containing medicinal paraffin. Light was collected byan 18 cm diameter photomultiplier (EM1 9623B) on this end, firstly with no external reflectors, secondly with a front surfacealuminized mirror at theend opposite the photomultipler and thirdly with external reflectors (either aluminiumfoil or powdered MgO) on all surfaces other than the collecting surface. A collimated source of 42K photons (1.52MeV) was directed a t various points along the central axis of the counter and a Compton spectrum was obtained at each position,the peak being used as the criterion for light collection efficiency.
T.Smith
290
Results Curves showing the theoretical light collection a t one complete end of the scintillationcounter for the threesystemsinvestigated are shown in fig. 4. I n each case the solid lines represent values obtained using a total internal 5.
(0)
OJb
io
20
30
io
50
60
70
io
Distance fromcollectingsurface(cm)
Fig. 4. Curves showing theoretical light collection a t one end of the counter (80 x 20 x 18 cm) for various distances of scintillations from the collecting surface. The attenuation length of light in thecounter medium is 200 cm in ( a )and 400 cm in ( b ) . Solid curves were obtained using R,= 1.0 and broken curves using R,= 0.95. A = basic TIR system. B = use of end reflector. C = use of external reflectors on a11 non-collecting surfaces.
reflection efficiency of 1.0 and the dottedlines show the differences which result from reducing this value to 0.95. The effects of using two different attenuation lengths, 200 and 400 cm, are demonstrated in fig. 4(a) and 4 ( b ) respectively. The ratiosof the theoretical responses obtained with different reflector arrangements relativeto thebasic total internalreflection system, allconsidered for the central position of the detector, are given in table 1 for the different reflectivity and attenuation values. Using the above results, the predicted light collection eficiency for a similar counter, from which light is collected at two complete ends, has been determined (fig. 5 ) . The light collection efficiencies calculated for an off-axis scintillation, 40 cm from the collecting surface, were identical with those for a scintillation at the same distance along the central axis, for all the different systems analysed.
How MuchLight from Rectangular ScintillationCounters!
291
Table 1. Relative efficiency and non-uniformityof light collection from one end of the theoretical counter (80 x 20 x 18 cm) under various conditions, compared with the values obtained withan experimental counter of the same dimensions (all efficiency values refer to scintillations at the centre of the detector region and percentage non-uniformity is given in brackets) ~~
~~~
p = 400-1
p = 200"
1
Ri = 1.0
Ri = 0.95
Ri = 1.0
1
40
20
Ri = 0.95
Experimental counter
TIR
End mirror External reflectors
* Normalized. Values of R,: t 1.0,
0.95,
0.3b
5 0.90.
'
IO
20
30
30
O I
0
Distance from collecting surfaces (cm)
Rig. 5. Curves showing theoretical light collection at two opposite ends of the counter for various distancesof scintillation from thecollecting surface (details as for fig. 4).
Results obtained with the experimental counter (fig. 6 ) show how the amplitude of the 42K Compton peak obtained with various reflector arrangements, measured relative to that obtained for scintillations at the centre of the TIR system, varies withthe position of the source. The relative valueswhich apply
T.Smith
292
3'0r
q p
2.0
0
..
0
0
60
70
.
0
d
1.0 0
x
10 20 30 40 50 Distancefromcollectingsurface(cm)
80
Fig. 6. Relative amplitudes of 42K spectrum obtained using the experimental counter (80 x 20 x 18 cm) with various reflector systems. The amplitude obtained a t the centre of the detector using the basic TIR system has been normalized to 1.0.
for scintillations at thecentre of the detector are given in table 1 for the various systems investigated. 6. Discussion and conclusions The theoretical curves obtained for the single-ended detector (fig. 4) show that theuse of a reflector at theend oppositethe collecting surface improvesthe light collection efficiency of the basic TIR system by about 30%. It also gives better uniformity of response for this type of counter than any of the other configurations investigated(table 1). Uniformityis influenced by different conditions of TIR efficiency and light attenuation but the values obtained for an end reflector suggest that anon-uniformity of 10% or less is readily attainable. Values of non-uniformity given intable 1 arethe percentage variations from the mean response for that part of the counter (20-80 cm from the collecting surface) which, allowing for a 20 cm light guide, would constitute the scintillator region. The values are measured from central axis data shown in fig. 4 but, since it has been shown that the light collecting efficiency in this region is the same for off-axis scintillations, they represent the non-uniformity of the entire detector. If external reflectors are used on all free surfaces a highly reflective material must be chosen in order to achieve the best possible increase in light collection with good uniformity. For example, in the counter investigated, an external reflectivity approaching 1.0 improves the theoretical light collection by about looyocompared to thebasic TIR system, with a non-uniformityof 15% or less. On the other hand, a reduced reflectivity of 0.9 may result in only a 60% improvement with a non-uniformity of up to 20%. Apart from the reflectivity of external materials, inefficient internal reflection and increased attenuation of lightin the counterare seen t o have considerable effect on both the light collection efficiency and uniformity. Decreasing the efficiency of total internal reflection from 1.0 to 0.95, for example, reduces the amount of light collected by about 20% and decreasing theattenuationlength from 400 to 200 cm reduces it by 20-30y0. The theoretical results show that for a single-ended
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counter of these dimensions the use of efficient external reflectors provides a useful gain inlight collection with an acceptable non-uniformity, providedthat the reflectors are not in optical contact with the detector walls and hence do not interferewith the total internal reflection process. The degree of nonuniformity that is acceptable is determinedby its contribution to the broadening, and hence resolution, of y-ray peaks in relation to other sources of spectral broadening. The values derived indicate that, for the useful range of y-ray energies, the contributiondue to non-uniformity isless than that due to statistical variations in photoelectron production. It is clear that the combined effects of inefficient reflection and of light attenuation impose limitations on the dimensions of a counter in which light is collected from one end only and for a longer counter, collection from both ends must be considered. The results of doing this with the present counter (fig. 5) are to improve the light collection relative to the single-ended counter by 20-60y0, depending on the configuration used and the chosen values of reflectivity and attenuation length, and to reduce non-uniformity to 3% or less. These values have been derived for a shorter (40 cm) central portion of the counter allowing for the use of a 20 cm light guide at each end, and could only be attained at the expense of twice the number of photomultipliers and of a design which in several ways is less convenient than thesingle-ended type. The light collection improvement factors obtained using different reflector arrangements with the experimental counter are in good agreement with predicted factors and they suggest that aluminium foil has a reflectivity of 0.900.95 andMgO actslike a specular reflector of 0.95-1.0 reflectivity (the theoretical calculations apply strictly t o specular reflection). The non-uniformity obtained with external reflectors was in all cases less than 10%. It is not possible, by comparing the experimental and theoretical ratios, to estimate theproportion of scintillationlight collected, since values for the efficiency of totalinternal reflection and the attenuationlength of light in the counter cannot be specified. It is possible, however, to make an approximation using the resolution of the 42K Compton peak.Thethreemaincomponents which contributetothe resolution are (1) statistical variations in the number of photoelectrons produced for a given quantity of light incident onthe photomultiplier, (2) the nonuniformity of light collection and (3) the variation of Compton electron energy. Using the experimental counter with an end reflector only, the resolution of the 42Kpeak was found to be 30%. If a reasonable value of 10% is assumed for both the non-uniformity and the relativewidth of the Compton electron peak, the contribution to resolution due to the statisticsof photoelectron production ( A y o )is given by 30 = (A2+ l o 2 + lo2)*. Thus A = 27% and is clearly the major component. The contributionA is given by 268 N-* (Barnaby and Barton 1960) and hence the average numberof photoelectrons produced,N , is about 100. The scintillation efficiency of our liquid scintillatoris about 0.005 photons eV-1, the fraction of the collection surface occupied by the photocathode is 0.65 and the photocathode quantum efficiency is about 12% (section 3.5). Hence the fraction of the scintillation light that is collected by the complete collection surface, following a 42K y-ray interaction which yields a Compton electron of
294
T.Smith
maximum energy (1.30 MeV), is 0.2. Therefore, the fractional light collection, using an external MgO reflector on all free walls is estimated to be, by proportion, 0.31. These values are in reasonable agreement with theory if an attenuationlength of 200 cm and a total internal reflection efficiency of 95% are assumed. On account of the approximations used, the experimental results do not prove these values, but they do suggest, for instance, that an attenuation length as high as 400 cm is improbable in our system. I n conclusion it is shown how the use of the escape cone concept can be applied to predict the efficiency and uniformityof light collection in rectangular counters of large dimensions, in which the collecting surface is a small fraction of the total surface area. The use of external reflectors with an experimental counter increased light collection by factors very similar to those predicted by theory and the uniformity of light collection was within the predicted range. These results indicatethat no further significant improvement in lightcollection performance could have been achieved in this counter by practical modifications other than that of increasing the photosensitive area of the collecting surface. In the detector described, it is estimated that approximately 30% of the light emitted in ascintillationcan be collected a t one endsurface,withanonuniformity of 10% or less, if external MgO reflectors are used.
I wish to thank Miss J. Mackenzie for her assistance in the techniques and analysis described in this paper. RE SUM^ Combien de lumibre des compteurs a scintillation rectangulaires? On a exbout6 des calculs theoriques de l’uniformite et efficacit6 de la collection de la lumibre B une extr6mit6 d’un grand compteur a scintillations liquide rectangulaire (80 x 20 x 18 cm). On a pris en consid6ration trois arrangements diff6rents: 1” un systbme, dans lequel toutes les surfaces du compteur sauf la surface collectante Qtaient transparentes, defacon B permettre une r6flexion intbrieure totale, 2” un systbme pareil, avec un miroir, placb pres de l’extrAmit8 opposbe B la surface collectante, et 3” un systhme, dans lequel les rbflecteurs ext6rieurs btaient places pres des surfaces autres que la surface collectante. Dans tous les cas on a pris en consid6ration les effets d’att6nuation de la lumibre dans le milieu du compteur, l’inefficacit6de la rbflexion totale intbrieure et les r6flectivit6s des rbflecteurs extbrieurs, et on les a illustrbs pour des valeurs choisies de ces parambtres. Dans un compteur experimental ayant des dimensions identiques on a employ6 le spectre d’une source collimat6e de 42K pour mesurer les efficacit6s relatives de la collection de lumibre pour des arrangements diff6rents des r6flecteurs. E n g6n6rsl les valeurs experimentales s’accordaient avec la thborie, et la non-uniformit6 de la r6ponse btait dans les limites de l’bcart pr6dit. On appr6oie qu’environ 30% de la, lumikre de scintillations peutQtrecollect6 a une extr6mitb d’un telcompteur, avec une non-uniformit6 ne dbpassant pas lo:/,, si l’on emploie l’oxyde de magnesium comme rbflecteur extbrieur pour toutes les autres surfaces.
ZUSAMMENFASSUNG Wieviel Licht aus den rechtwinkligen Szintillationsziihlern? Es sind theoretische Berechnungen der Wirksamkeit und Gleichformigkeit der Lichtsammlung bei einem Ende ekes grossen rechtwinkligen flussigen Szintillationsziihlers (80 X 20 X 18 cm) ausgefuhrt worden. Es wurden drei verschiedene Anordnungen in Betracht genommen: ( 1 ) ein System, in welchem alle Ziihleroberflachen, ausser der Sammeloberfliiche, durchsichtig waren, um die totale Innenreflexion zu ermoglichen, ( 2 ) ein iihnliches System mit einem in der Niihe des Endes gegenuber der Sammelflache angebrachten Spiegel, und ( 3 ) ein System, in welchem die Aussenreflektoren in der Niihe der Fliichen angebracht wurden, mit der Ausnahme der SammelI n allen Fiillen zog man in Betracht und illustrierte die Effekte der Lichtschwhchung in
How Much Light from Rectangular Scintillation Counters?
295
dem Zahlermedium, die unwirksame totale Innenreflexion sowie die Reflexionsvennogen der Aussenreflektoren fur ausgewahlte Werte dieser Parameter. I n einem Experimentalzahler derselben Ausmasse ist die Spektrumamplitude einer ausgeblendeten 42K-Quelle angewandt worden, un die relativem Wirkungsgrade der Lichtsammlung fur verschiedene Reflektoranordnungen zu messen. I m allgemeinen stimmten die experimentellen Werte mit Theorie uberein, und die Sichtgleichfonnigkeit der Reaktion war innerhalb des vorhergesagten Bereichs. Es wird geschatzt, dass ungefahr 30% des Szintillationslichtes bei einem Ende eines derartigen Zahlers gesammelt werden kann, mit einer 10% nicht uberschreitenden Sichtgleichforigkeit, falls Magnesiumoxyd als Aussenreflektor bei allen anderen Flachen angewandt wird.
REFERENCES BARNABY, C. F., and BARTON, J. C., 1960, Proc. Phys. Soc., 76, 745-753. BARNABY, C. F., and JASANI, B. M., 1966, J . Sci. Instrum., 43, 220-223. BARNABY, C. F . , and JASANI,B. M., 1968, J. Phys. E : Sci. Instrum., 1 , 91-98. BARTON, J. C., CRISPIN, A.,and SLADE,M., 1964, J.Sci. Instrum., 41, 736-739. BIRKS,J. B., 1964, The Theory and Practice of Scintillation Counting (Oxford: Pergamon). P., 1955, NuovoCimento, Suppl. 2, BRINI, D., PELI,L., RIMOKDI,O., and VERONESI, 1048-1074. CRABB,D. G., DEAN,A. J., MCEWEN,J. G., and OTT, R. J., 1966, Nucl. Instrum. Meth., 45, 301-308. FAISSNER, H., FERRERO F., GHAXI, A.,and REINHARZ, M., 1963, Nucl. Instrum. Meth., 20, 289-293. GRIEDER,P. K . F., 1967, Nucl. Instrum. Meth., 55, 295-300. JELLEY,V.,J.1958, Cerenkov Radiation and its Applications (Oxford: Pergamon). JENKINS, F. A., and WHITE,H. E., 1957, Fundamentals of Optics (New York, London: McGraw-Hill). MILLAR, C. H., HINCKS,E. P., and HANNA,G. C., 1958, Can. J. Phys., 36, 54-72. NICOLL,D. R., and EWER,M. J. C., 1971, in Organic Scintillators and Liquid Scintillation Counting, Ed. D. L. Horrocks and C. J. Peng (New York: Academic Press) p. 279. SHUKLA, K. K., DOMBROWSKI, C. S., and COHN,S. H., 1973, Health Phya., 24, 555-557. SHURCLIFF, W. A., and CLARKJONES, R., 1949, J . Opt. Soc. Am., 39, 912-916. SMITH,T., 1971, Nucl. Instrum. Meth., 97, 409-411. SMITH,T.,and JASANI, B. M., 1972a, J . Ph.ys. E : Sci. Instrum., 5, 103-107. SMITH,T.,and JASANI, B. M., 1972b, J . Phys. E : Sci. Instrum., 5, 1083-1088. SWANK, R. K., 1958, in Liquid Scintillation Counting, Ed. C. G. Bell and F. N. Hayes (Oxford: Pergamon) p. 23. VAN DILLA,M. A., andANDERSON,E. E., 1962, in Proc.Symp.Whole-bodyCounting (Vienna: IAEA) p. 41.
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How much light from rectangular scintillation counters? (for whole body counting)
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PHYS. MED. BIOL., 1975, VOL.
20,
NO.
2, 282-295. @ 1975
How Much Light from Rectangular Scintillation Counters? T. SMITH,
M.SC., P H . D . ~
Medical Research Council Department of Clinical Research, University College Hospital Medical School, London, England
Received 12 June 1974 ABSTRACT. Theoretical calculations have been made of the efficiency and uniformity of light collection a t one end of a large rectangularliquidscintillationcounter (80 x 20 x 18 cm). Three different arrangements were considered; ( 1 ) asystem in which all surfaces of the counter other than thecollecting surface were transparent to permit total internal reflection, ( 2 ) a similar system with a mirror placed close to the end opposite the collecting surface and (3) a system in which external reflectors were placed close to surfaces other than the collecting surface. In all cases, the effects of attenuation of light in the counter medium, inefficient total internal reflection and the reflectivities of external reflectors were considered and illustrated for selected values of these parameters. I n an experimental counter of the same dimensions, the amplitude of the spectrum of a collimated 42K source was used to measure the relative efficiencies of light collection for various reflector arrangements. I n general the experimental values agreed with theoryandthenon-uniformity of response was within the predictedrange. I t is estimated that about 30% of scintillation light can be collected a t one end of such a counter, with a non-uniformity not greater than 1004, if magnesium oxide is used &S the external reflector a t all other surfaces.
1. Introduction
Rectangularorganicscintillationcountersare used inthe fields of high energy physics and nuclear medicine where they fulfil the need for relatively cheaplarge-area or large-volumeradiationdetectors. For some yearsthis laboratory has been concerned with the development of such counters suitable for measuring total body radioactivity in clinical tracer investigations, ultimately to provide adequate sensitivity to permit the estimation of the total body content of naturally occurring *OK. This requires the accurate measurement of about 10 nCi or less of a radionuclide which emits a 1-52 MeV y-ray pernucleardisintegration, and rectangular perspex tanks filled withcheap liquid scintillator (Barnaby and Jasani1968) have proved satisfactory for this purpose. The use of large detectors, however, poses the considerable optical problem of light collection andthis is of particularimportanceiny-ray counters where the energy deposition is usually small relative to that in high energyparticledetectors. Efficient light collection is essential not only t o produce a detectable pulse but also to provide the best possible energy resolution, even though this is necessarily limited in organic detectors due to the predominat'ingCompton effect. Resolutionshouldpreferablybesuch t'hat total body 40K (1.52 MeV) canbe efficiently measured to the exclusion of
t Present address : Radioisotopes Division, MRC Clinical Research Centre, Northwick Park Hospital, Watford Road, Harrow, Middlesex,HAL 3UJ.
How MuchLight from RectangularScintillationCounters?
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I3'Cs (0.66 MeV), a long lived product of nuclear fallout, of which some 2-5 nCi may be present in the body (Shukla, Dombrowski and Cohn 1973). Resolution, of course, depends not only on the efficiency, but also on the uniformity of light collection, a condition which has considerable bearing on the design of largedetectors.Although the use of many largediameterphotomultipliers ensuresadequatelight collection (Van Dilla and Anderson1962), the most economical system, in terms of the fraction of scintillation light collected per unit photocathode area, is that in which photomultipliers are placed on the small ends of rectangular counters designed to permit total internal reflection of scintillation light (Brini, Peli, Rimondi and Veronesi 1955). Our counters are of this basic type and therefore methods have been developed to allow theoretical prediction of the response of rectangular counters of various dimensions and configurations. This is clearly of value in the development of detectors designed to maximize light collection whilst retaining good uniformity. Review of earlier studies I n a n earlier paper (Smith and Jasani 1972a) theoretical predictions were made of the amount of light collected a t one surface of a rectangular scintillation counter (61 x 40.5 x 63 cm) from scintillations occurring a t various positions with it. The collecting surface was one of the two surfaces of smallest area and all the others were assumed to be transparent to allow total internal reflection to take place. The concept of light escape cones (Shurcliff and Clark Jones 1949) was used to simplify the calculation. In thearrangement considered, only the light in theescape cone directed towards the collecting surface and the remaininglightnotincludedwithinthe escape cones (trappedlight)can escape through the collecting surface to which photomultipliers are optically coupled, except for scintillations occurring in a region immediately in front of the collecting surface. The length of this region is such that, for a scintillation within it, parts of the four lateral escape cones can intersect the collecting surface directly. I n t h e calculations, the escape cones and trapped light were assumed to be divided into coaxial cones of differing apical angle so that, for each increment between cones, the path length from a point of scintillation to the collecting surface could be calculated for a known fraction of the emitted light. For this purpose, the angular distribution of trapped light was determined by the formula given by Smith (1971). Various simplifying assumptions were made and the only correction applied was for loss of light by absorption in the medium of the counter. Estimations were also made of the contributions of lateral escape cones arising from scintillations close to the collecting surface. Predicted response curves were thus obtained for points of scintillation along the central axis of the counter and along a line parallelto it butchosen close to the edge of the counter. Two striking results emerging from this analysis were the large and rapid increase in light collection as the point of scintillation approached the collecting surface and the very uniform response from a more distant region. The former effect is largely explained by the contribution of light in lateral escape cones and the latter is due partly to the highly efficient total internal reflection of light (assumed to be 1.0 in the above calculations) 2.
284
T.Smith
and partly to the fact that the contribution from trapped light reflected from the back surface tends to compensate for increased absorption of forward light as the scintillation point moves away from the collecting surface. The response from scintillations off the central axis was shown to differ from the central axis response for positions close to the collecting surface only, on account of the different geometry for lateral escape cones. Indeed, the response from scintillations at the extreme edge of the detector sets the minimum length of light guide necessary to ensure that, in the type of counter considered, there is no contribution to the collected light from lateral cones. This dimension is given by l tan C where 1 is the length of the longer side of the collecting surface and C is the critical angle for total internal reflection. Allowing for the use of a light guide the theoretical analysis of the above counter predicted a possible light collection efficiency of 23%, assuming the collecting surface to be completely an sensitive, and a non-uniformity of & 9%. The uniformity obtained with experimental system of the same dimensions was within the predicted range, but the fractional light collection was estimated to be about 10% less than the theoretical value, probably due in part to the assumption of lOOyo efficiency for total internal reflection. It is often advantageousto collect more light from scintillation countersthan the abovesystem allows, provided that the uniformity is not significantly worsened as aresult.Usuallytwooppositesurfacesareused;light being or after reflection a t one of them. Even collected either from both directly with such an arrangement, a large proportion of light is lost from the other four surfaces and this has led to the use of various types of external reflector adjacent to these surfaces in order to further increase the collected proportion (Millar, Hincks and Hanna 1958, Faissner, Perrero, Ghani and Reinharz 1963, Crabb, Dean, McEwen and Ott 1966, Grieder 1967). I n a recent report (Smith and Jasani 1972b) experiments were described in which various reflectors were applied to model counters and a large scintillation counter. A useful gain in light collection was achieved, with good uniformity, when an efficient reflector such as magnesium oxide (MgO) was pla'ced close to, but slightly separated from, the noncollecting surfaces. With thescintillation counter (80 x 20 x 18 cm), for example, in which light was collected from one end, the use of MgO in this way increased the amount of light collected by 70%, the non-uniformity being & 7.6%.
In the present paper a theoretical analysis is described which extends the previous treatment (Smith and Jasani 1972a) to demonstrate the dependence of both the efficiency and uniformity of light collection on various parameters in a scintillation counterused with and without externalreflectors. The analysis is made for a rectangular counter of the same dimensions as the experimental counter with which some data have been obtained, making possible a comparison between the predicted and observed light collection efficiencies. 3. Factors affecting the efficiency of light collection
Severalfactors, considered below, may influence the efficiencyof light collection to a greater or lesser degree, depending on the extent towhich they
How ,Vuch Light
from Rectangular ScintillationCounters?
285
vary with wavelength in relation to the spectral distribution of scintillation light. The fluorescence spectrum of the liquid scintillator used in our counters (Barnaby and Jasani 1966) extends from 370 nm to about 500 nm with a peak emission wavelength of 410 nm (fig. l(a)).
Wavelength (m)
Fig. 1. ( a )Fluorescence spectrum of the liquid scintillator used in the experimental counter. ( b ) Spectral response of photomultiplier (EM19623B).
3.1. Refractive index
The refractive indices of the scintillator and perspex container used in the present counter are both equal to 1.5 measured at 410 nm and are reasonably constant over the wavelength range of the fluorescence spectrum. It is estimated that small variations in refractive index with wavelength affect the calculated amount of light in escape cones by less than 3% and therefore the value 1.5 has been considered a reasonable average for the purposes of the present analysis. 3.2. Attenuation of light
There is some difficulty in measuring the intrinsic attenuation coefficient for the spectrum of scintillation light in the counter medium. As Barton, Crispin and Slade (1964) have pointed out, spectrophotometer measurements are not very satisfactory for this purpose as the effects of scattering and of re-emission of absorbed light are difficult to estimate, and therefore a method which more closely approaches the practical situation is required. These authors describe a method of measuring an effective attenuation length, which is the distance required to reduce the intensityof light in asingle escape cone by thefactor e-1. Since the mean distance travelled by light in an escape cone is about 15% larger than this value, an approximation to theintrinsic attenuation length can be found.Theabovemethod, however, involves multiple totalinternal reflections of light in the escape cone being measured and inefficient reflection leads to underestimation of the attenuation length. Thus theintrinsic attenuation length is not precisely known and for this reason two values, 200 and 400 cm, have been used in thetheoretical analysisto show the effect of differences in this parameter on the efficiency of light collection. The estimated value for the system described here is approximately 200 cm and attenuation lengths of 400 cm have been observed in plastic scintillators (Nicoll and Ewer 1971).
286
T . Xmith
3.3. Total internal rejection Ri The efficiency of total internal reflection, Ri, is theoretically 1-0 for scintillation light incident on the counter walls at angles greater than thecritical angle, and in a previous analysis (Smith and Jasani 1972a) only this maximum value was considered. However, such an efficiency requires a degree of optical perfection of the counter walls which is unlikely t o be attained in practice. Therefore in thepresent theory two values, 1.0 and 0.95, have been taken intoaccount. There is a small contribution t o internal reflection from light incident at angles less than the critical angle (JenkinsandWhite 1957),amountingtoabout 3-6% depending on wavelength, polarization and angle of incidence. Its effect is to increase slightly the estimated value of light collected by total internal reflection alone, but it is a relatively small effect which has been ignored in the present analysis. 3.4. Rejectivity of external re8ectors R,
Our investigations have involvedthe use of both specular and diffuse external reflectors, the most satisfactory of these being front surface aluminized mirrors, polished aluminium foil and MgO powder. Thereflectivity of evaporated aluminium on polished glass varies from 0.90 to 0.92 between 385 and 500 nm (Jelley 1958), that of aluminium foil is almost constant at 0.9 over the wavelength range 300-550 nm (Birks 1964),whilst dry MgO has superior reflectivity (almost 1.0) which is reasonably constant over the range 325-550 nm (Swank 1958). In the present theory, therefore, the effects of three different values of reflectivity (1.0, 0.95 and 0.90) have been compared. 3.5. Photomultiplier response The present theory calculates the proportion of scintillation light which is available a t a collecting surface of the counter assuming that the surface is completely photosensitive. It isthereforeindependent of theparameters associatedwith the photomultiplier which affect the counter response. I n order to compare the theory with the experimentalresults described later, corhowever, it is necessary to take these parameters into account. After a rection has been applied to allow for the proportion of the collecting surface occupied by the photocathode, the response of the systemdepends on the spectral response of the photomultiplier (fig. I@))in relation t o the spectrum of light collected. Although a secondary solute (POPOP)is used to match the fluorescence spectrum of our scintillator to an S-l1 photocathode, the modification of the spectrum of collected light in large detectors, dueto absorption and re-emission, tends to degrade this matching.To allow for this, a quantum efficiency of 12% has been assumed for the photocathode used in the experimental counter, although the peak value may reach1 7 % (EM1 data). 4. Extension of theoretical analysis
The amount of light incident on one small end (collecting surface) of a rectangular scintillation counter (80 x 20 x 18 cm) of uniform refractive index, 1.5,
How Much Light from Rectangular Scintillation Counters?
287
has been calculated for scintillations occurring at various positions within it. Three different systems (fig. 2) were investigated as described below. I n each case, light collections were calculated for various positions along the central
1 4 -
- - -- -- - - - ---
qX-??q 1 1 t r X
(C)
Fig. 2. Diagrams of the three counter systems analysed. (a)TIR system. ( b ) End reflector (mirror)opposite photomultiplier. (c) External reflectors at all surfaces other than the collecting surface. Points of scintillation were taken along the central axis and at one off-axis position (X).
axis and for one off-axis position a t a distance of 40 cm from the collecting surface and 1 cm in from each of two adjacent sides. External reflectors, when used, were assumed to be specular and situatedparallel to, but slightly separated from, the walls of the counter.
Basic total internal reflection (TIR) system In this arrangement (fig. 2 ( a ) ) ,all surfaces other than the collecting surface aretransparentandnoexternal reflectors are used.Thetheoreticallight collection was calculated by assuming the escape cone directed towards the collecting surface and all trapped light to be divided up into small elements (fig. 3(a))formed by coaxial cones, whose semi-apical angles differed by 5 O , and 4.1.
( U)
Fig. 3. ( a )Diagram illustrating the method of analysing individual elements of escape cones as described in text. ( 6 ) Cross-section of counter including the pointlof scintillation, showing the dimensions used in the calculations.
288
T.Smith
by radial planes 10' apart. It was then possible to estimate, for each element, the solid angle and hence the proportion of isotropically emitted scintillation light it contains, the meandistance from thepoint of scintillation to the collecting surface, and the number of tota'l internal reflections at the counter walls. Corrections were made for losses of light resulting from attenuation in the counter medium and from inefficient total internal reflection as specified in sections 3.2 and 3.3 above respectively. The total light collection was obtained by summing the individual contributions so calculated for each element. For each element of light in the quadrant defined by the distances p and s (fig. 3(b)), the fractional amount of light, f, reaching the photocathode was calculated using the formula
f = Qexp
xsec 0
where C? is the fractional solid angle of the element, x is the perpendicular distance from thepoint of scintillation to the collecting surface (for light reflected from the oppositesurface, x is the perpendiculardistance to that surface plus the length of the counter), Ri is the coefficient of total internal reflection, and Ni is the average number of internal reflections at the counter expressions x tan 0 sin + / ( p+ nd) walls. Ni is given by thesum of the terms in the and x tan 0 cos $/(S +nw)(where n = 0 , 1 , 2 , .. .) which are greater than or equal Do 1 ; W and d being the width and depth of the counter respectively. For light in the other three quadrants the combinations s with (d - p ) , (W - 8 ) with p and (W - S) with (d - p ) are used to calculatje Ni. For light reaching the collecting surface following a reflection at theopposite end, Ni was increased by 1. 4.2. Use of a rejlective surface opposite the light collecting surface
When a reflective surface is placed parallel to, and slightly separated from, the end of the box opposite the single light collecting surface (fig. 2(b)), it does not interfere with total internal reflection and hence does not affect the collection of trapped light and light in the escape cone directed towards the collection surface, as described above. It does, however, increase the amountof light collected by reflecting part of the light in the escape cone which is transmitted through the end of the box. This further contribution was calculated by a similar procedure to that used in section 4.1 above and an extracorrection was made t'o allow for the reflectivity of the end reflector as detailed in section 3.4 above. Thus, for light in this escape cone, eqn (1) was multiplied by R,, the coefficient of external reflection. 4.3. Use of rejectors around all noncollecting surfaces I n this situation (fig. 2 ( c ) ) there is a further contribution to the collected
amount from some of the light in the four lateral escape cones whose axes are parallel to the collecting surface, even beyond the region within which these cones areintercepted by the collecting surfacedirectly(Smith andJasani 1972a). A simple approximation was made by assuming each cone to be made
How Xuch Light from RectangularScintillationCounters?
289
up of small elements (fig. 3(a)) by first dividing the cone into a number of coaxial cones whose semi-apical angles were increased by 10". For a refractive index of 1.5,the critical angle for total internal reflection is 41' 48';thus each cone was divided arbitrarily into four coaxial cones, three with semi-apical angles of 10,20 and 30' and a fourth of 41" 48'.These cones were further subdivided by radial planesof 10" apart and hence each quadrant of each cone was divided into 36 elements whose solid angles and mean path lengths to the collecting surface were easily determined. The calculation of the contribution due to these elements is less straightforward than for those of the two cones whose axes are perpendicular to the collect'ing surface because of the different types of reflection involved. Thus light ina lateral cone which escapes from the surface t o which it is directed may, after specular external reflection, re-enter the detector and escape from the opposite surface t'o repeat the process:on the other hand, reflections at any other surface of the detector, either before or after external reflection, are totally internal. It was necessary, therefore, to calculate separately for each element the numbers of external and internal reflections, since the reflectivities considered differed inmostcombinations. Corrections for light losses were applied as in sections 4.1,and 4.2 above, and the total contribution from light in the four lateral escape cones was determined by summing the contributions due to all t'he elements comprising them. Light in four halves of these cones can only contribute after reflection at the opposite end of the box and this may imply large path lengths and many reflections. Fractional light' collection for elements in the four lateral cones was calculated from the formula
N , , the average number of internal reflections, is given by the number of terms ..) which are greater than in the expression x tana/(s+nw) (where n = 0,1,2,. or equal to l, and Ne, the average number of external reflections, is obtained similarly from the expression x cot 7 sec a / ( p+ nd). For light reaching the collecting surface after reflection a t the opposite end, Ni was increased by 1.
4.4. Experimental counter A perspex box measuring 80 x 20 x 18 cm was filled with liquid scintillator (Barnaby and Jasani 1966),except for a 20 cm light guide section at one end containing medicinal paraffin. Light was collected byan 18 cm diameter photomultiplier (EM1 9623B) on this end, firstly with no external reflectors, secondly with a front surfacealuminized mirror at theend opposite the photomultipler and thirdly with external reflectors (either aluminiumfoil or powdered MgO) on all surfaces other than the collecting surface. A collimated source of 42K photons (1.52MeV) was directed a t various points along the central axis of the counter and a Compton spectrum was obtained at each position,the peak being used as the criterion for light collection efficiency.
T.Smith
290
Results Curves showing the theoretical light collection a t one complete end of the scintillationcounter for the threesystemsinvestigated are shown in fig. 4. I n each case the solid lines represent values obtained using a total internal 5.
(0)
OJb
io
20
30
io
50
60
70
io
Distance fromcollectingsurface(cm)
Fig. 4. Curves showing theoretical light collection a t one end of the counter (80 x 20 x 18 cm) for various distances of scintillations from the collecting surface. The attenuation length of light in thecounter medium is 200 cm in ( a )and 400 cm in ( b ) . Solid curves were obtained using R,= 1.0 and broken curves using R,= 0.95. A = basic TIR system. B = use of end reflector. C = use of external reflectors on a11 non-collecting surfaces.
reflection efficiency of 1.0 and the dottedlines show the differences which result from reducing this value to 0.95. The effects of using two different attenuation lengths, 200 and 400 cm, are demonstrated in fig. 4(a) and 4 ( b ) respectively. The ratiosof the theoretical responses obtained with different reflector arrangements relativeto thebasic total internalreflection system, allconsidered for the central position of the detector, are given in table 1 for the different reflectivity and attenuation values. Using the above results, the predicted light collection eficiency for a similar counter, from which light is collected at two complete ends, has been determined (fig. 5 ) . The light collection efficiencies calculated for an off-axis scintillation, 40 cm from the collecting surface, were identical with those for a scintillation at the same distance along the central axis, for all the different systems analysed.
How MuchLight from Rectangular ScintillationCounters!
291
Table 1. Relative efficiency and non-uniformityof light collection from one end of the theoretical counter (80 x 20 x 18 cm) under various conditions, compared with the values obtained withan experimental counter of the same dimensions (all efficiency values refer to scintillations at the centre of the detector region and percentage non-uniformity is given in brackets) ~~
~~~
p = 400-1
p = 200"
1
Ri = 1.0
Ri = 0.95
Ri = 1.0
1
40
20
Ri = 0.95
Experimental counter
TIR
End mirror External reflectors
* Normalized. Values of R,: t 1.0,
0.95,
0.3b
5 0.90.
'
IO
20
30
30
O I
0
Distance from collecting surfaces (cm)
Rig. 5. Curves showing theoretical light collection at two opposite ends of the counter for various distancesof scintillation from thecollecting surface (details as for fig. 4).
Results obtained with the experimental counter (fig. 6 ) show how the amplitude of the 42K Compton peak obtained with various reflector arrangements, measured relative to that obtained for scintillations at the centre of the TIR system, varies withthe position of the source. The relative valueswhich apply
T.Smith
292
3'0r
q p
2.0
0
..
0
0
60
70
.
0
d
1.0 0
x
10 20 30 40 50 Distancefromcollectingsurface(cm)
80
Fig. 6. Relative amplitudes of 42K spectrum obtained using the experimental counter (80 x 20 x 18 cm) with various reflector systems. The amplitude obtained a t the centre of the detector using the basic TIR system has been normalized to 1.0.
for scintillations at thecentre of the detector are given in table 1 for the various systems investigated. 6. Discussion and conclusions The theoretical curves obtained for the single-ended detector (fig. 4) show that theuse of a reflector at theend oppositethe collecting surface improvesthe light collection efficiency of the basic TIR system by about 30%. It also gives better uniformity of response for this type of counter than any of the other configurations investigated(table 1). Uniformityis influenced by different conditions of TIR efficiency and light attenuation but the values obtained for an end reflector suggest that anon-uniformity of 10% or less is readily attainable. Values of non-uniformity given intable 1 arethe percentage variations from the mean response for that part of the counter (20-80 cm from the collecting surface) which, allowing for a 20 cm light guide, would constitute the scintillator region. The values are measured from central axis data shown in fig. 4 but, since it has been shown that the light collecting efficiency in this region is the same for off-axis scintillations, they represent the non-uniformity of the entire detector. If external reflectors are used on all free surfaces a highly reflective material must be chosen in order to achieve the best possible increase in light collection with good uniformity. For example, in the counter investigated, an external reflectivity approaching 1.0 improves the theoretical light collection by about looyocompared to thebasic TIR system, with a non-uniformityof 15% or less. On the other hand, a reduced reflectivity of 0.9 may result in only a 60% improvement with a non-uniformity of up to 20%. Apart from the reflectivity of external materials, inefficient internal reflection and increased attenuation of lightin the counterare seen t o have considerable effect on both the light collection efficiency and uniformity. Decreasing the efficiency of total internal reflection from 1.0 to 0.95, for example, reduces the amount of light collected by about 20% and decreasing theattenuationlength from 400 to 200 cm reduces it by 20-30y0. The theoretical results show that for a single-ended
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counter of these dimensions the use of efficient external reflectors provides a useful gain inlight collection with an acceptable non-uniformity, providedthat the reflectors are not in optical contact with the detector walls and hence do not interferewith the total internal reflection process. The degree of nonuniformity that is acceptable is determinedby its contribution to the broadening, and hence resolution, of y-ray peaks in relation to other sources of spectral broadening. The values derived indicate that, for the useful range of y-ray energies, the contributiondue to non-uniformity isless than that due to statistical variations in photoelectron production. It is clear that the combined effects of inefficient reflection and of light attenuation impose limitations on the dimensions of a counter in which light is collected from one end only and for a longer counter, collection from both ends must be considered. The results of doing this with the present counter (fig. 5) are to improve the light collection relative to the single-ended counter by 20-60y0, depending on the configuration used and the chosen values of reflectivity and attenuation length, and to reduce non-uniformity to 3% or less. These values have been derived for a shorter (40 cm) central portion of the counter allowing for the use of a 20 cm light guide at each end, and could only be attained at the expense of twice the number of photomultipliers and of a design which in several ways is less convenient than thesingle-ended type. The light collection improvement factors obtained using different reflector arrangements with the experimental counter are in good agreement with predicted factors and they suggest that aluminium foil has a reflectivity of 0.900.95 andMgO actslike a specular reflector of 0.95-1.0 reflectivity (the theoretical calculations apply strictly t o specular reflection). The non-uniformity obtained with external reflectors was in all cases less than 10%. It is not possible, by comparing the experimental and theoretical ratios, to estimate theproportion of scintillationlight collected, since values for the efficiency of totalinternal reflection and the attenuationlength of light in the counter cannot be specified. It is possible, however, to make an approximation using the resolution of the 42K Compton peak.Thethreemaincomponents which contributetothe resolution are (1) statistical variations in the number of photoelectrons produced for a given quantity of light incident onthe photomultiplier, (2) the nonuniformity of light collection and (3) the variation of Compton electron energy. Using the experimental counter with an end reflector only, the resolution of the 42Kpeak was found to be 30%. If a reasonable value of 10% is assumed for both the non-uniformity and the relativewidth of the Compton electron peak, the contribution to resolution due to the statisticsof photoelectron production ( A y o )is given by 30 = (A2+ l o 2 + lo2)*. Thus A = 27% and is clearly the major component. The contributionA is given by 268 N-* (Barnaby and Barton 1960) and hence the average numberof photoelectrons produced,N , is about 100. The scintillation efficiency of our liquid scintillatoris about 0.005 photons eV-1, the fraction of the collection surface occupied by the photocathode is 0.65 and the photocathode quantum efficiency is about 12% (section 3.5). Hence the fraction of the scintillation light that is collected by the complete collection surface, following a 42K y-ray interaction which yields a Compton electron of
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maximum energy (1.30 MeV), is 0.2. Therefore, the fractional light collection, using an external MgO reflector on all free walls is estimated to be, by proportion, 0.31. These values are in reasonable agreement with theory if an attenuationlength of 200 cm and a total internal reflection efficiency of 95% are assumed. On account of the approximations used, the experimental results do not prove these values, but they do suggest, for instance, that an attenuation length as high as 400 cm is improbable in our system. I n conclusion it is shown how the use of the escape cone concept can be applied to predict the efficiency and uniformityof light collection in rectangular counters of large dimensions, in which the collecting surface is a small fraction of the total surface area. The use of external reflectors with an experimental counter increased light collection by factors very similar to those predicted by theory and the uniformity of light collection was within the predicted range. These results indicatethat no further significant improvement in lightcollection performance could have been achieved in this counter by practical modifications other than that of increasing the photosensitive area of the collecting surface. In the detector described, it is estimated that approximately 30% of the light emitted in ascintillationcan be collected a t one endsurface,withanonuniformity of 10% or less, if external MgO reflectors are used.
I wish to thank Miss J. Mackenzie for her assistance in the techniques and analysis described in this paper. RE SUM^ Combien de lumibre des compteurs a scintillation rectangulaires? On a exbout6 des calculs theoriques de l’uniformite et efficacit6 de la collection de la lumibre B une extr6mit6 d’un grand compteur a scintillations liquide rectangulaire (80 x 20 x 18 cm). On a pris en consid6ration trois arrangements diff6rents: 1” un systbme, dans lequel toutes les surfaces du compteur sauf la surface collectante Qtaient transparentes, defacon B permettre une r6flexion intbrieure totale, 2” un systbme pareil, avec un miroir, placb pres de l’extrAmit8 opposbe B la surface collectante, et 3” un systhme, dans lequel les rbflecteurs ext6rieurs btaient places pres des surfaces autres que la surface collectante. Dans tous les cas on a pris en consid6ration les effets d’att6nuation de la lumibre dans le milieu du compteur, l’inefficacit6de la rbflexion totale intbrieure et les r6flectivit6s des rbflecteurs extbrieurs, et on les a illustrbs pour des valeurs choisies de ces parambtres. Dans un compteur experimental ayant des dimensions identiques on a employ6 le spectre d’une source collimat6e de 42K pour mesurer les efficacit6s relatives de la collection de lumibre pour des arrangements diff6rents des r6flecteurs. E n g6n6rsl les valeurs experimentales s’accordaient avec la thborie, et la non-uniformit6 de la r6ponse btait dans les limites de l’bcart pr6dit. On appr6oie qu’environ 30% de la, lumikre de scintillations peutQtrecollect6 a une extr6mitb d’un telcompteur, avec une non-uniformit6 ne dbpassant pas lo:/,, si l’on emploie l’oxyde de magnesium comme rbflecteur extbrieur pour toutes les autres surfaces.
ZUSAMMENFASSUNG Wieviel Licht aus den rechtwinkligen Szintillationsziihlern? Es sind theoretische Berechnungen der Wirksamkeit und Gleichformigkeit der Lichtsammlung bei einem Ende ekes grossen rechtwinkligen flussigen Szintillationsziihlers (80 X 20 X 18 cm) ausgefuhrt worden. Es wurden drei verschiedene Anordnungen in Betracht genommen: ( 1 ) ein System, in welchem alle Ziihleroberflachen, ausser der Sammeloberfliiche, durchsichtig waren, um die totale Innenreflexion zu ermoglichen, ( 2 ) ein iihnliches System mit einem in der Niihe des Endes gegenuber der Sammelflache angebrachten Spiegel, und ( 3 ) ein System, in welchem die Aussenreflektoren in der Niihe der Fliichen angebracht wurden, mit der Ausnahme der SammelI n allen Fiillen zog man in Betracht und illustrierte die Effekte der Lichtschwhchung in
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dem Zahlermedium, die unwirksame totale Innenreflexion sowie die Reflexionsvennogen der Aussenreflektoren fur ausgewahlte Werte dieser Parameter. I n einem Experimentalzahler derselben Ausmasse ist die Spektrumamplitude einer ausgeblendeten 42K-Quelle angewandt worden, un die relativem Wirkungsgrade der Lichtsammlung fur verschiedene Reflektoranordnungen zu messen. I m allgemeinen stimmten die experimentellen Werte mit Theorie uberein, und die Sichtgleichfonnigkeit der Reaktion war innerhalb des vorhergesagten Bereichs. Es wird geschatzt, dass ungefahr 30% des Szintillationslichtes bei einem Ende eines derartigen Zahlers gesammelt werden kann, mit einer 10% nicht uberschreitenden Sichtgleichforigkeit, falls Magnesiumoxyd als Aussenreflektor bei allen anderen Flachen angewandt wird.
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