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A prescription for the removal of Compton-scattered gamma rays from gamma-ray spectra
Nuclear Instruments and Methods in Physics Research A258 (1987) 111-118 North-Holland, Amsterdam
A PRESCRIPTION FOR THE REMOVAL OF COMPTON-SCATTERED GAMMA RAYS FROM GAMMA-RAY SPECTRA D.C . RADFORD
Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ IJO, and Argonne National Laboratory, Argonne, IL 60439, USA
I . AHMAD, R. HOLZMANN, R.V .F. JANSSENS and T.L . KHOO
Argonne National Laboratory, Argonne, IL 60439, USA
Received 20 January 1987 A method for the interpolation between and extrapolation from measured detector response functions is described. This technique enables the removal of the Compton-scattered background (i.e . "stripping" or "unfolding") of -y-ray spectra quickly, easily and accurately . Examples are presented, for both source and in-beam spectra, taken with Ge detectors. Some potential pitfalls in the accurate unfolding of spectra are discussed, and approximate remedies suggested.
1. Introduction In the study of high-spin states of nuclei, one technique that is employed is the analysis of quasicontinuum y-ray spectra emitted from heavy-ion fusionevaporation reactions [1]. In the past, such spectra have usually been measured using Nal(TI) crystals, but with the advent of large arrays of Compton-suppressed Ge (CSG) detectors, such as those currently under construction at Argonne National Laboratory [2], Chalk River [3], and other places, these measurements may now be made more accurately by taking advantage of the high resolution of the Ge counter. The peak-to-total ratio of the CSG is much improved over that of the bare Ge detector. Nevertheless, a substantial fraction (typically = 50%) of the counts in the spectrum arise from Compton-scattered y-rays . This contribution must be removed in order to determine the true (primary) spectrum of quasicontinuum radiation. Such a "stripping", or "unfolding", of spectra requires a knowledge of the response function of the detector as a function of the primary photon energy . That is, if one knows the shape of the Compton background (which for the purposes of this paper is considered to include the backscatter peak) and the peak-tototal ratio for all y-ray energies, one can in principle correct for Compton scattering and obtain a spectrum containing only full-energy counts . The detector response function can be measured for selected energies using monoenergetic y-ray sources, and also from sources emitting two y-rays, using fast coincidence tech0168-9002/87/$03 .50 C Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
niques . However, some means of interpolating between and extrapolating from these measurements, to photon energies that have not been measured, must be found. In the past, several such methods have been utilized [4-9]. Perhaps the simplest of these, used quite commonly at present [4,5], is to assume that the Compton spectrum is simply a flat background, extending from the photopeak down to the threshold. The only parameter that then needs to be interpolated as a function of energy is the height of this background, or the peak-tototal ratio. This method has the advantage that direct matrix inversion may be used to do the unfolding [4]. However, this approximation neglects the backscatter peak, and Compton edges or peaks, and is therefore usually inadequate for an accurate analysis of spectra taken with most detectors, especially when Compton suppression is not used . A second technique is to parametrize the response function, and to interpolate between the parameters as a function of y-ray energy. This method was used by Mollenauer [6] in the first unfolding program, and also by a number of other groups (e .g. refs . [7-9]) . It has the disadvantage that the fitting of the response functions to obtain the parameters can be very time-consuming. In addition, the accuracy of the unfolding depends very sensitively on the quality of the parametrization, and different parametrizations are generally required for suppressed and unsuppressed detectors . A different method was used by Heart et al . [10] for interpolating between Nal(TI) response functions. After subtraction of the photopeak, and normalization to a
D.C. Radford et al. /The removal of Compton-scattered -y -raysfrom 'y -ray spectra
fixed number of emitted photons, they divided the Compton background into several different segments, and adjusted the gain of each segment separately, so as to match up the position of common features such as the backscatter peak and Compton edge . They then used a sine-series fit to each channel of these gain-adjusted response functions to interpolate according to photopeak energy. By then reversing the gain transformations, they were able to generate a response function for any given -y-ray energy. This paper presents a new technique, similar in some ways to that of Heath et al ., for accurately interpolating between response functions of either suppressed or unsuppressed detectors. The Compton background is subdivided into three parts, and a separate transformation is applied to each of them, so that all three pieces are made to change slowly with photopeak energy . Interpolation or extrapolation may then be done on a channelby-channel basis, after which the reverse transformations can be applied . Although designed primarily for Ge detectors, this method also appears to work well for Nal(TI) counters .
Table 1 Gamma-ray sources used in response function measurements y-ray energy [keV]
Source
Halflife
122
57CO
271 d
145 166 171 245 279 392
'°' Ce 1 "Ce
32 d 137 d
514 662 835 898 1173 1333 1369 1836 2754 511
) ' ) In 203 Hg 113S n 85
Sr )3, Cs 54 Mn 89Y
bo C o 24
Na
88Y
24 Na
zz Na
2.8 d 47 d 115 d 65 d 30 yr 312 d 107 d 5 yr
15 h 107 d 15 h 2.6 yr
Coincidence Remarks (C) or singles (S) S
S S C(245) C(171) S S S S S C(1836) C(1333) C(1173) C(2754) C(898) C(1369) C(1275)
136 keV subtracted a)
255 keV subtracted a)
b) b) b) b)
Not used for interpolation `)
2. The method
a)
For the purposes of the unfolding, the response function is considered to consist of a single-channel photopeak, plus the Compton background and singlechannel escape peaks. Therefore the program interpolates the Compton background, normalized to a fixed photopeak area. It also uses an empirical formula for the relative intensities of the escape peaks as a function of energy . As the real photopeak is scanned during the unfolding, the program can then build up escape peaks with the same width and shape as the photopeak. The response function of a detector for a given -y-ray energy can be determined by measuring a spectrum for a monoenergetic -y-ray source . It is also possible to use sources that emit two -y-rays in coincidence, by using two or more detectors in fast coincidence . By gating on the photopeak of one transition in one detector, one then obtains the response of the other detector(s) to the second -y-ray . Naturally, any background must be subtracted from the response function spectra; the main contaminants are due to room background in the singles measurements, and random coincidences and Compton background underlying the photopeak gate in the coincidence measurements . Using these techniques, we have measured response functions for the -y-ray energies listed in table 1. By leaving one or more of these out of the input to the program, it was determined that they cover the range of energies from 122 keV to 2.75 MeV more than adequately. Other sources that could be used are given in ref. [10] .
Where applicable, weak contaminants peak were removed from the measured spectra by generating their response functions from a previous pass of the interpolation program. After normalization to the photopeak area, both this response function and the photopeak were subtracted . In addition, for -y-rays with energies above 1300 keV, annihilation peaks, and their associated Comptons, were removed by subtracting a 511 keV spectrum measured with a 22 Na source (table 1) . (This 511 keV spectrum was not used as a response function in the interpolation since summing can occur between two Compton-scattered 511 keV y-rays, and since positrons can annihilate away from the source location .) Finally, where X-ray peaks were evident in the spectrum, they were also removed. Before the response function is stored for interpolation, the photopeak and escape peaks (where applicable) are removed, and their areas noted, by subtracting a fitted lineshape. This leaves only the Compton background . Examples of these spectra, for a CSG detector, are shown in fig. 1. The main features evident in the spectrum are the backscatter peak, from y-rays which have been scattered through angles close to 180' into the detector, and the Compton edge, or in the case of
Weak contaminant peaks were removed by generating their response functions with a previous pass of the program. b) 511 keV annihilation radiation and the escape peaks were removed as described m the text . `) Used for subtraction of annihilation radiation from other spectra only.
D. C. Radford et al. / The removal of Compton-scattered -y -rays from -y -ray spectra
arately. The first of these is the region of the spectrum from the threshold to just below the Compton edge or peak, i .e . to the energy (somewhat arbitrarily) defined as E, = EP - 1 .2EB .
ZZ â U N Z
ôU
Fig. 1. Examples of spectra measured in a CSG detector, for monoenergetic photons of 279, 835 and 1836 keV. S.E. and D.E . denote single- and double-escape peaks, respectively . Compton-suppressed detectors, a Compton peak, due to y-rays which have been scattered through angles close to 180 ° out of the detector . These are at energies EB = Ep/(1 + 2Ep/mc2 ) , and
EC =E p -
EB
respectively, where Ep is the photopeak energy, and m = 511 keV/c2 is the mass of the electron . The method is illustrated schematically in fig. 2. For the purposes of interpolation, the Compton background is considered to be composed of three pieces, labelled 1, 2 and 3 in fig. 2, each of which is interpolated sep-
COMPTON PEAK (b) AFTER
J ZZZ
BACKSCATTER
Won
a F O U
(a)
COMPTON PEAK BEFORE
CHANNEL NUMBER
~ : (PHOTOPEAK) d n II
Schematic illustration of the segmentation and gain transformations of the measured response functions. Panel (a) shows the response spectrum as measured (in this case for a 54 Mn source), while (b) gives the spectrum after the transformation described in the text . This is the form in which the response functions are stored and interpolated . After interpolation, the reverse transformation is applied. Fig . 2
The second piece is from energy E, to the photopeak, and the third is the backscatter peak . The shape and area of the backscatter peak is entered by defining a smooth background beneath it, over the region 0.8E, to 1 .8E, . For Ep close to 255.5 keV, where the backscatter peak and Compton peak overlap, it has been found that the third component can be taken from an average of, or interpolation between, previously stored backscatter peaks, with very satisfactory results. These three components are independently adjusted in gain and offset by the program to align the lower and upper energy limits at common channels . That is, each piece of the spectrum is compressed (or stretched) and shifted, so that it will occupy the same range of channels for all of the transformed spectra. Thus, for example, the second components of all response functions are stored with the energies EA and Ep in channels 101 and 350 respectively. In addition to the gain adjustment and normalization to fixed photopeak area, the three components are also individually divided by their energy range (in keV), so that, for example, the counts per channel in component 3 (the backscatter peak) are divided by a factor proportional to A p (1 .8EB - 0.8EB )=A p E B , where A p is the photopeak area . This empirical renormalization produces a final spectrum in which the total number of counts changes only slowly with photopeak energy. The results of this partition and multiple transformation for a variety of response functions are shown in fig. 3. It is clear that, for each of the components, the number of counts per channel now changes slowly and smoothly as a function of - y-ray energy EP . This allows a channel-by-channel interpolation between (or extrapolation from) two transformed response functions to a given energy of interest . By applying the reverse transformation and adding the three components, one can then generate the Compton-background spectrum for a given photopeak area. The interpolation is done linearly with the value of log(EP), although an interpolation with Ep gives virtually identical results. Further smoothing of the spectrum beyond that provided automatically by the gain transformations is unnecessary, since, during the unfolding, each measured response function contributes to the calculated response for many different photopeak channels, but with different gain transformations . Thus any statistical fluctuations in the measured response spectrum appear at many different points in the subtracted Compton background, making it very smooth .
D. C. Radford et al. / The removal of Compton-scattered -y -rays from -y -ray spectra
Î
1 .3
I
1 .4
1 .5
1 .6 L6
E y (MeV) 1 .8 2 .0 2 .2 2.5
3.0 3 .5
0.05
I ~"
Y Q ar O F0
I
0.02 0.01
W
a 0.005 a U
I
i
I`
I
I\
I
V) W
J 0.002 0 Z
0.001 -1 .4
I
I
I
Fig. 3. The set of transformed response functions used m ref. [11] . The numbers at the top of the figure refer to the three components (see fig. 2 and text). The ratio of escape-peak areas to photopeak area may also be stored for Ep greater than 1100 keV. For energies lower than this, pair production is considered to be negligible . We derive an empirical formula for the ratio of pair-production to full energy absorption in a Ge detector, 2 935 , escape peak/photopeak = C [ln(Ey/2mc2 )] which is used to parametrize the areas of the escape peaks as a function of -y-ray energy Ey . This parametrization is shown compared to measurements of singleescape-to-photopeak ratio for a Ge detector in fig. 4. The program takes the proportionality constants CI and C for the two escape peaks from the escape-tophotopeak ratios in the highest-energy response function stored, since this is usually the one for which these ratios are best determined . Although this parametrization was taken from measurements in a = 100 cm3 coaxial Ge detector, to first order it should work well for all detectors, since it should depend only on the energy dependence of the ratio of the pair-production to photopeak cross sections .
-1 .2
-I .0 , -0 .8
-0.6
-0 .4
-0.2
In [in (E y/ 2rnc2 )]
0
0.2
Fig. 4. Ratio of single-escape peak to photopeak areas, as a function of incident y-ray energy, taken from a coaxial Ge detector . The solid line shows the energy dependence of the parametnzation for the escape-to-photopeak ratio (see text). In what follows, reference is made to the correction of spectra for the effects of detector efficiency . Where applicable, this is done by dividing the number of counts, channel-by-channel, by the relative efficiency. If the spectrum is to be unfolded, this must be course be done before the efficiency division . We use an empirical parametrization In(c) =[(A+Bx)-G+(D+Ey+Fy 2
-G
-11G
where e is the relative efficiency of the detector as a function of energy, and x and y are related to the y-ray energy by x = ln (E y/100 keV) , and y = ln( E .,/1 MeV) . The first term (A + Bx) describes the efficiency at low energies (EY < 120 keV for coaxial detectors), while the second term (D + Ey + Fy 2) describes the efficiency at high energies (E, :< 200 keV) . G is an interaction parameter that effectively connects the two regions; the larger G is, the sharper the join between the two curves . If the efficiency turns over very gently in the intermediate region, G will be small. A typical least-squares fit to obtain the parameters A, B, C, D, E, F and G is shown in fig. 5 . The fitted data are relative efficiencies measured using 133 Ba, 152E u and 56Co sources, for a coaxial high-purity Ge detector. The sharp turnover at Ey = 150 keV is caused by the slow-risetime rejection circuitry in the constant-fraction
D. C. Radford et al. / The removal of Compton-scattered y -raysfrom -y -ray spectra
1000
2000 ENERGY (keV)
3000
4000
Fig. 5 . A measured relative efficiency curve for a high-purity coaxial Ge detector. The solid line is a least-squares fit using the parametrization described in the text
discriminator. It can be seen that the above parametri-
zation allows a very accurate description of the efficiency .
500
ENERGY
1000 (keV)
1500
Fig. 6. (a) 152 Eu spectrum measured m a CSG detector, before and after unfolding. (b) As for (a), but with the vertical scale expanded by a factor of 45 . The peaks are common to both spectra; the figure is intended to show the effect of the unfolding on the background . The unfolded spectrum is the one lying along the baseline, thus illustrating the accuracy of this prescription .
3. The unfolding Once a set of response functions, such as that shown
in fig. 3, has been stored, channel-by-channel interpola-
tion and/or extrapolation enables one to obtain a response function for any given y-ray energy . By applying
the inverse of the transformation given above the ex-
pected Compton background and escape peak areas for a given photopeak area are then easily obtained . The
the 152 Eu spectrum the resultant peak-to-total ratio can be calculated to be in excess of 0.99, compared to = 0 .70 before the unfolding. Most of the residual back-
ground is believed to result from coincidence summing in the detector. A further test of the procedure, and especially of its ability to handle unsuppressed response functions, is
unfolding or "stripping" of spectra proceeds by beginning at the high-energy end of the spectra and proceeding channel-by-channel to lower energies . For each
channel, the background calculated according to the assumption that only photopeak counts are left in that
channel, is subtracted from the spectrum . This process
continues to as low an energy as the response functions can reliably be calculated (typically = 100 keV for coaxial Ge detectors) .
This procedure has been applied to many spectra,
with excellent results. In the study of quasicontinuum
or in-beam spectra, one needs to
be
sure that the
response is accurately subtracted . Thus the unfolding of test spectra taken with -y-ray sources, for which the unfolded spectrum should show no net background, is
an important test . We have unfolded several multi-y source spectra, including 133Ba, 152Eu and 56 Co . Beforeand-after examples, taken with a CSG detector, are shown in figs . 6 and 7. Some of these spectra have also been corrected for the detector efficiency . In the expanded segment of the 56CO spectrum (fig . 7) it can
easily be seen how accurately Compton peaks and escape
peaks are removed. By adding all of the peak areas in
Fig. 7. 56 Co spectrum measured m a CSG detector, before and after unfolding. The upper half shows an expanded portion of the spectrum to show more clearly the accurate removal of Compton and escape peaks . The spectra have also been corrected for relative photopeak efficiency .
D. C. Radford et al. / The removal of Compton-scattered -y-rays from -y -ray spectra
original (pre-unfolding) spectrum . Although about 4 times the number of counts have been subtracted, the systematic differences from the suppressed case are negligible, showing that the procedure does indeed work extremely well . An in-beam spectrum, taken with the '6 Ge(8° Se, 4n)IS2 Dy reaction using a CSG detector, is shown before and after unfolding in fig. 9. This was taken from a study of the IszDy quasicontinuum spectrum ; experimental details may be found in ref . [12] .
w â
U F Z ô
V
4. Further corrections and other considerations for inbeam spectra 600
700 800 ENERGY (keV)
900
1000
Fig. 8. Portions of unfolded IS2 Eu spectra from the same detector, taken for the same time ; (a) with Compton suppression, and unfolded using suppressed response functions, and (b) without Compton suppression, and unfolded using unsup pressed response functions . The spectra have also been corrected for photopeak efficiency . the comparison of unfolded spectra taken with Compton-suppressed and unsuppressed detectors. Fig. 8 shows segments of 152Eu spectra taken in the same detector, for the same length of time . The upper portion shows an unfolded spectrum taken with Compton suppression, while the lower portion was taken without suppression and unfolded using the unsuppressed response functions. Both have been corrected for detector efficiency . The increased statistical fluctuations in the lower half are due to the much larger number of counts in the 1000
(o) BEFORE UNFOLDING
Boo 600 400 w Z z 20C Ha.N a ci 0
Z
AFTER UNFOLDING
B00 60O 400
f
z 500
1000 E y (keV)
1500
Fig. 9. An in-beam spectrum (ref. [121) before (a) and after (b) unfolding. The '6Ge( s° Se,4n) reaction at 330 MeV was used . The spectra have also been corrected for the relative photopeak efficiency .
There are a number of potential pitfalls in the accurate unfolding of in-beam -y-ray spectra. Perhaps the most serious of these in most applications are the spectral distortions ansing from pileup, neutrons and coincidence summing. The contributions of these effects to the spectrum must be removed before the unfolding since not only do they change the shape of the spectrum, but they also have very different response functions from those calculated in the unfolding procedure. Pulse pileup and coincidence summing both increase the average pulse height in the spectrum. Pileup can be prevented during the accumulation of the spectrum by using low counting rates and/or pileup rejection circuitry. Coincidence summing occurs when two or more -y-rays emitted simultaneously from the same nucleus interact with the same detector . For an unsuppressed Ge detector = 10 cm from the target, and a high -y-ray multiplicity reaction, the fraction of counts in the spectrum affected by summing can be as large as 10-15%, although the effect is reduced by using Compton suppression and increasing the target-to-detector distance. It is relatively simple to correct the observed spectrum for coincidence summing under the assumption that is is "uncorrelated", i.e . the probability of detecting two y-rays in coincidence is simply the product of the probabilities of detecting each of them separately. Thus such effects as angular correations between pairs of -y-rays and the lack of autocoincidences between a y-ray and itself are neglected. The coincidence summing effectively removes some fraction of the counts from each channel of the "true" (corrected) spectrum and redistributes them at higher energies according to some coincidence probability . The above approximation assumes that this probability has the same shape as that of the spectrum itself. Thus one can write S, j
RPJ'P,_J /
k
Pk
where S,, is the contribution to channel t of the observed spectrum P, due to summing out of channel j of
D. C Radford et al / The removal of Compton-scattered y -raysfrom -y -ray spectra
the true (corrected) spectrum PJ', and R is the fraction of all counts in the spectrum affected by summing. R can be evaluated from the coincidence-to-singles ratio of a second detector in fast coincidence with the detector of interest . P,_,, rather than P,'_,, is used in eq. (1) since, for events with high -y-ray multiplicity M >> 1, this automatically corrects not only for 2-fold summing but also for 3-fold and higher order summing. From eq . (1), we derive P,=(1-R)P,'+ Y_S,J J <, =(1-R)P,'+RY_PJ' P,_J/Y_Pk . k J<1 Thus in order to evaluate (1-R)P, =P,-RI Y_ P"P_ '/ Y_ P" J<1 k one only requires P,' for j less than t. One may therefore begin at channel i = 1 and proceed incrementally to higher channels . Although the actual spectrum is clearly not uncorrelated (since certain -y-ray peaks are in coincidence while others are not) the above formula provides a correction almost indistinguishable from a correction which includes the correlations, and is easily within usual statistical errors . The more accurate correction may be taken from the measured coincidence matrix between two detectors, if it is available. Many reactions, especially heavy-ion fusion-evaporation reactions, produce neutrons which interact with Ge detectors mainly through the (n, n'y) reaction . The kinematics tend to focus these neutrons in the forward direction, so that they contribute more to the spectra of detectors close to 0 ° with respect to the beam axis . The resulting neutron spectrum is dependent on the energy of the neutrons and cannot easily be calculated, especially since the response of the detector to a 'y-ray emitted from within the detector itself will clearly be different to that of a y-ray coming from the target . However, the spectrum can be approximately measured using y-ray absorbers, and then subtracted from the total spectrum before unfolding. The effect can also be reduced by placing detectors at backward angles, and by using time-of-flight discrimination where applicable . Another possible source of error is the high-energy cutoff of the y-ray spectrum . If the unfolding of the spectrum does not begin at sufficiently high energies, some Compton and escape events will be treated as photopeak counts, leading to an undersubtraction at high energies and possibly an oversubtraction at low energies . To prevent this, either a spectrum extending up to at least 5 MeV should generally be taken, or a spectrum extrapolation method such as that of Sic et al . [12] should be used . Clearly, it is also important to measure the response
functions in the actual experimental setup wherever possible . The material surrounding the target or source can have a significant effect on the response, especially in the region of the backscatter peak . The effects of the low-energy discriminator and/or fast coincidence gate must also be carefully included, since they can change the relative detector efficiency and attenuate the Compton background at low energies . Use of slow-risetime rejection circuitry and generous fast-coincidence timing windows, together with the gating of all singles spectra by the constant-fraction discriminator are important to ensure that the correct response functions and relative detector efficiencies are measured . A further (very small) effect on the response functions, that has been neglected to our analysis, is the angular distributions of the inbeam -y-rays. Since the response function to the region of the backscatter peak depends on the angle of scattering into the detector, the response function of the nonisotropic in-beam -y-rays could be slightly different from that of the isotropic source spectra. 5. Conclusion We have described a method for the accurate removal of Compton background from y-ray spectra, by interpolating between measured detector response functions. This method, together with the prescriptions given here for treating other sources of distortion of the spectrum, has been used to analyse data from in-beam y-ray continuum studies (e .g. ref. [11]) with excellent results. Copies and/or listings of the computer code used may be obtained by writing to one of the authors (D .C .R .) . The program is currently being extended to be able to treat two-dimensional coincidence spectra, or y-y correlation matrices . Acknowledgements One of us (D.C .R.) acknowledges very helpful discussions with Dr . Y. Schutz. This work was partially supported by the US Department of Energy, Nuclear Physics Division, under Contract no . W-31-109-Eng-38. References [1] E.g . D. Ward, H .R. Andrews, B. Haas, P. Taras and N. Rud, Nucl . Phys . A397 (1983) 161; S.H . Sie, J.O. Newton, J.R. Leigh and R.M . Diamond, Phys . Rev. Lett. 46 (1981) 405. [2] R.V .F . Janssens, T.L . Khoo, E. Funk, U. Garg, J Kolata and J. Mihelich, unpublished (1983). [3] E. Hagberg, D. Horn, M.A . Lone, H Schmemg, D. Ward, P. Taras, J. Gascon, J.C . Waddington, G. Palameta, V. Koslowsky and O. Hâusser, Atomic Energy of Canada Limited report AECL-8329.
118
D. C. Radfordet al. / The removal of Compton-scattered -y -raysfrom -y -ray spectra
[4] D. Ward, H.R. Andrews, B. Haas, P. Taras and N. Rud, Nucl . Phys . A397 (1983) 161. [5] D.J .G. Love, A.H . Nelson, P.J . Nolan and P .J . Twin, Phys. Rev. Lett . 54 (1985) 1361 . [6] J.F . Mollenauer, Phys . Rev. 127 (1962) 867. (71 S.H . Sie, Nucl . Instr. and Meth . 155 (1978) 475. [81 W. Trautmann, IF Sharpey-Schafer, H.R . Andrews, B. Haas, O. Hausser, P. Taras and D Ward, Nucl. Phys . A378 (1982) 141. [9] See also Y. Jin, R.P . Gardner and K. Verghese, Nucl . Instr. and Meth . A242 (1986) 416.
[10] R.L . Heath, R.G . Helmer, L.A . Schnuttroth and G.A. Cazier, Nucl. Instr. and Meth . 47 (1967) 281. [11] D.C . Radford, I. Ahmad, R. Holzmann, R.V .F . Janssens, T L. Khoo, M.W. Dnggert, U. Garg and H. Helppi, Phys Rev. Lett . 55 (1985) 1727 . [12] R. Holzmann et al ., to be published. [13] S.H . Sie, R.M . Diamond, J.O . Newton and J.R. Leigh, Nucl . Phys . A352 (1981) 279.
A PRESCRIPTION FOR THE REMOVAL OF COMPTON-SCATTERED GAMMA RAYS FROM GAMMA-RAY SPECTRA D.C . RADFORD
Atomic Energy of Canada Limited, Chalk River Nuclear Laboratories, Chalk River, Ontario, Canada KOJ IJO, and Argonne National Laboratory, Argonne, IL 60439, USA
I . AHMAD, R. HOLZMANN, R.V .F. JANSSENS and T.L . KHOO
Argonne National Laboratory, Argonne, IL 60439, USA
Received 20 January 1987 A method for the interpolation between and extrapolation from measured detector response functions is described. This technique enables the removal of the Compton-scattered background (i.e . "stripping" or "unfolding") of -y-ray spectra quickly, easily and accurately . Examples are presented, for both source and in-beam spectra, taken with Ge detectors. Some potential pitfalls in the accurate unfolding of spectra are discussed, and approximate remedies suggested.
1. Introduction In the study of high-spin states of nuclei, one technique that is employed is the analysis of quasicontinuum y-ray spectra emitted from heavy-ion fusionevaporation reactions [1]. In the past, such spectra have usually been measured using Nal(TI) crystals, but with the advent of large arrays of Compton-suppressed Ge (CSG) detectors, such as those currently under construction at Argonne National Laboratory [2], Chalk River [3], and other places, these measurements may now be made more accurately by taking advantage of the high resolution of the Ge counter. The peak-to-total ratio of the CSG is much improved over that of the bare Ge detector. Nevertheless, a substantial fraction (typically = 50%) of the counts in the spectrum arise from Compton-scattered y-rays . This contribution must be removed in order to determine the true (primary) spectrum of quasicontinuum radiation. Such a "stripping", or "unfolding", of spectra requires a knowledge of the response function of the detector as a function of the primary photon energy . That is, if one knows the shape of the Compton background (which for the purposes of this paper is considered to include the backscatter peak) and the peak-tototal ratio for all y-ray energies, one can in principle correct for Compton scattering and obtain a spectrum containing only full-energy counts . The detector response function can be measured for selected energies using monoenergetic y-ray sources, and also from sources emitting two y-rays, using fast coincidence tech0168-9002/87/$03 .50 C Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
niques . However, some means of interpolating between and extrapolating from these measurements, to photon energies that have not been measured, must be found. In the past, several such methods have been utilized [4-9]. Perhaps the simplest of these, used quite commonly at present [4,5], is to assume that the Compton spectrum is simply a flat background, extending from the photopeak down to the threshold. The only parameter that then needs to be interpolated as a function of energy is the height of this background, or the peak-tototal ratio. This method has the advantage that direct matrix inversion may be used to do the unfolding [4]. However, this approximation neglects the backscatter peak, and Compton edges or peaks, and is therefore usually inadequate for an accurate analysis of spectra taken with most detectors, especially when Compton suppression is not used . A second technique is to parametrize the response function, and to interpolate between the parameters as a function of y-ray energy. This method was used by Mollenauer [6] in the first unfolding program, and also by a number of other groups (e .g. refs . [7-9]) . It has the disadvantage that the fitting of the response functions to obtain the parameters can be very time-consuming. In addition, the accuracy of the unfolding depends very sensitively on the quality of the parametrization, and different parametrizations are generally required for suppressed and unsuppressed detectors . A different method was used by Heart et al . [10] for interpolating between Nal(TI) response functions. After subtraction of the photopeak, and normalization to a
D.C. Radford et al. /The removal of Compton-scattered -y -raysfrom 'y -ray spectra
fixed number of emitted photons, they divided the Compton background into several different segments, and adjusted the gain of each segment separately, so as to match up the position of common features such as the backscatter peak and Compton edge . They then used a sine-series fit to each channel of these gain-adjusted response functions to interpolate according to photopeak energy. By then reversing the gain transformations, they were able to generate a response function for any given -y-ray energy. This paper presents a new technique, similar in some ways to that of Heath et al ., for accurately interpolating between response functions of either suppressed or unsuppressed detectors. The Compton background is subdivided into three parts, and a separate transformation is applied to each of them, so that all three pieces are made to change slowly with photopeak energy . Interpolation or extrapolation may then be done on a channelby-channel basis, after which the reverse transformations can be applied . Although designed primarily for Ge detectors, this method also appears to work well for Nal(TI) counters .
Table 1 Gamma-ray sources used in response function measurements y-ray energy [keV]
Source
Halflife
122
57CO
271 d
145 166 171 245 279 392
'°' Ce 1 "Ce
32 d 137 d
514 662 835 898 1173 1333 1369 1836 2754 511
) ' ) In 203 Hg 113S n 85
Sr )3, Cs 54 Mn 89Y
bo C o 24
Na
88Y
24 Na
zz Na
2.8 d 47 d 115 d 65 d 30 yr 312 d 107 d 5 yr
15 h 107 d 15 h 2.6 yr
Coincidence Remarks (C) or singles (S) S
S S C(245) C(171) S S S S S C(1836) C(1333) C(1173) C(2754) C(898) C(1369) C(1275)
136 keV subtracted a)
255 keV subtracted a)
b) b) b) b)
Not used for interpolation `)
2. The method
a)
For the purposes of the unfolding, the response function is considered to consist of a single-channel photopeak, plus the Compton background and singlechannel escape peaks. Therefore the program interpolates the Compton background, normalized to a fixed photopeak area. It also uses an empirical formula for the relative intensities of the escape peaks as a function of energy . As the real photopeak is scanned during the unfolding, the program can then build up escape peaks with the same width and shape as the photopeak. The response function of a detector for a given -y-ray energy can be determined by measuring a spectrum for a monoenergetic -y-ray source . It is also possible to use sources that emit two -y-rays in coincidence, by using two or more detectors in fast coincidence . By gating on the photopeak of one transition in one detector, one then obtains the response of the other detector(s) to the second -y-ray . Naturally, any background must be subtracted from the response function spectra; the main contaminants are due to room background in the singles measurements, and random coincidences and Compton background underlying the photopeak gate in the coincidence measurements . Using these techniques, we have measured response functions for the -y-ray energies listed in table 1. By leaving one or more of these out of the input to the program, it was determined that they cover the range of energies from 122 keV to 2.75 MeV more than adequately. Other sources that could be used are given in ref. [10] .
Where applicable, weak contaminants peak were removed from the measured spectra by generating their response functions from a previous pass of the interpolation program. After normalization to the photopeak area, both this response function and the photopeak were subtracted . In addition, for -y-rays with energies above 1300 keV, annihilation peaks, and their associated Comptons, were removed by subtracting a 511 keV spectrum measured with a 22 Na source (table 1) . (This 511 keV spectrum was not used as a response function in the interpolation since summing can occur between two Compton-scattered 511 keV y-rays, and since positrons can annihilate away from the source location .) Finally, where X-ray peaks were evident in the spectrum, they were also removed. Before the response function is stored for interpolation, the photopeak and escape peaks (where applicable) are removed, and their areas noted, by subtracting a fitted lineshape. This leaves only the Compton background . Examples of these spectra, for a CSG detector, are shown in fig. 1. The main features evident in the spectrum are the backscatter peak, from y-rays which have been scattered through angles close to 180' into the detector, and the Compton edge, or in the case of
Weak contaminant peaks were removed by generating their response functions with a previous pass of the program. b) 511 keV annihilation radiation and the escape peaks were removed as described m the text . `) Used for subtraction of annihilation radiation from other spectra only.
D. C. Radford et al. / The removal of Compton-scattered -y -rays from -y -ray spectra
arately. The first of these is the region of the spectrum from the threshold to just below the Compton edge or peak, i .e . to the energy (somewhat arbitrarily) defined as E, = EP - 1 .2EB .
ZZ â U N Z
ôU
Fig. 1. Examples of spectra measured in a CSG detector, for monoenergetic photons of 279, 835 and 1836 keV. S.E. and D.E . denote single- and double-escape peaks, respectively . Compton-suppressed detectors, a Compton peak, due to y-rays which have been scattered through angles close to 180 ° out of the detector . These are at energies EB = Ep/(1 + 2Ep/mc2 ) , and
EC =E p -
EB
respectively, where Ep is the photopeak energy, and m = 511 keV/c2 is the mass of the electron . The method is illustrated schematically in fig. 2. For the purposes of interpolation, the Compton background is considered to be composed of three pieces, labelled 1, 2 and 3 in fig. 2, each of which is interpolated sep-
COMPTON PEAK (b) AFTER
J ZZZ
BACKSCATTER
Won
a F O U
(a)
COMPTON PEAK BEFORE
CHANNEL NUMBER
~ : (PHOTOPEAK) d n II
Schematic illustration of the segmentation and gain transformations of the measured response functions. Panel (a) shows the response spectrum as measured (in this case for a 54 Mn source), while (b) gives the spectrum after the transformation described in the text . This is the form in which the response functions are stored and interpolated . After interpolation, the reverse transformation is applied. Fig . 2
The second piece is from energy E, to the photopeak, and the third is the backscatter peak . The shape and area of the backscatter peak is entered by defining a smooth background beneath it, over the region 0.8E, to 1 .8E, . For Ep close to 255.5 keV, where the backscatter peak and Compton peak overlap, it has been found that the third component can be taken from an average of, or interpolation between, previously stored backscatter peaks, with very satisfactory results. These three components are independently adjusted in gain and offset by the program to align the lower and upper energy limits at common channels . That is, each piece of the spectrum is compressed (or stretched) and shifted, so that it will occupy the same range of channels for all of the transformed spectra. Thus, for example, the second components of all response functions are stored with the energies EA and Ep in channels 101 and 350 respectively. In addition to the gain adjustment and normalization to fixed photopeak area, the three components are also individually divided by their energy range (in keV), so that, for example, the counts per channel in component 3 (the backscatter peak) are divided by a factor proportional to A p (1 .8EB - 0.8EB )=A p E B , where A p is the photopeak area . This empirical renormalization produces a final spectrum in which the total number of counts changes only slowly with photopeak energy. The results of this partition and multiple transformation for a variety of response functions are shown in fig. 3. It is clear that, for each of the components, the number of counts per channel now changes slowly and smoothly as a function of - y-ray energy EP . This allows a channel-by-channel interpolation between (or extrapolation from) two transformed response functions to a given energy of interest . By applying the reverse transformation and adding the three components, one can then generate the Compton-background spectrum for a given photopeak area. The interpolation is done linearly with the value of log(EP), although an interpolation with Ep gives virtually identical results. Further smoothing of the spectrum beyond that provided automatically by the gain transformations is unnecessary, since, during the unfolding, each measured response function contributes to the calculated response for many different photopeak channels, but with different gain transformations . Thus any statistical fluctuations in the measured response spectrum appear at many different points in the subtracted Compton background, making it very smooth .
D. C. Radford et al. / The removal of Compton-scattered -y -rays from -y -ray spectra
Î
1 .3
I
1 .4
1 .5
1 .6 L6
E y (MeV) 1 .8 2 .0 2 .2 2.5
3.0 3 .5
0.05
I ~"
Y Q ar O F0
I
0.02 0.01
W
a 0.005 a U
I
i
I`
I
I\
I
V) W
J 0.002 0 Z
0.001 -1 .4
I
I
I
Fig. 3. The set of transformed response functions used m ref. [11] . The numbers at the top of the figure refer to the three components (see fig. 2 and text). The ratio of escape-peak areas to photopeak area may also be stored for Ep greater than 1100 keV. For energies lower than this, pair production is considered to be negligible . We derive an empirical formula for the ratio of pair-production to full energy absorption in a Ge detector, 2 935 , escape peak/photopeak = C [ln(Ey/2mc2 )] which is used to parametrize the areas of the escape peaks as a function of -y-ray energy Ey . This parametrization is shown compared to measurements of singleescape-to-photopeak ratio for a Ge detector in fig. 4. The program takes the proportionality constants CI and C for the two escape peaks from the escape-tophotopeak ratios in the highest-energy response function stored, since this is usually the one for which these ratios are best determined . Although this parametrization was taken from measurements in a = 100 cm3 coaxial Ge detector, to first order it should work well for all detectors, since it should depend only on the energy dependence of the ratio of the pair-production to photopeak cross sections .
-1 .2
-I .0 , -0 .8
-0.6
-0 .4
-0.2
In [in (E y/ 2rnc2 )]
0
0.2
Fig. 4. Ratio of single-escape peak to photopeak areas, as a function of incident y-ray energy, taken from a coaxial Ge detector . The solid line shows the energy dependence of the parametnzation for the escape-to-photopeak ratio (see text). In what follows, reference is made to the correction of spectra for the effects of detector efficiency . Where applicable, this is done by dividing the number of counts, channel-by-channel, by the relative efficiency. If the spectrum is to be unfolded, this must be course be done before the efficiency division . We use an empirical parametrization In(c) =[(A+Bx)-G+(D+Ey+Fy 2
-G
-11G
where e is the relative efficiency of the detector as a function of energy, and x and y are related to the y-ray energy by x = ln (E y/100 keV) , and y = ln( E .,/1 MeV) . The first term (A + Bx) describes the efficiency at low energies (EY < 120 keV for coaxial detectors), while the second term (D + Ey + Fy 2) describes the efficiency at high energies (E, :< 200 keV) . G is an interaction parameter that effectively connects the two regions; the larger G is, the sharper the join between the two curves . If the efficiency turns over very gently in the intermediate region, G will be small. A typical least-squares fit to obtain the parameters A, B, C, D, E, F and G is shown in fig. 5 . The fitted data are relative efficiencies measured using 133 Ba, 152E u and 56Co sources, for a coaxial high-purity Ge detector. The sharp turnover at Ey = 150 keV is caused by the slow-risetime rejection circuitry in the constant-fraction
D. C. Radford et al. / The removal of Compton-scattered y -raysfrom -y -ray spectra
1000
2000 ENERGY (keV)
3000
4000
Fig. 5 . A measured relative efficiency curve for a high-purity coaxial Ge detector. The solid line is a least-squares fit using the parametrization described in the text
discriminator. It can be seen that the above parametri-
zation allows a very accurate description of the efficiency .
500
ENERGY
1000 (keV)
1500
Fig. 6. (a) 152 Eu spectrum measured m a CSG detector, before and after unfolding. (b) As for (a), but with the vertical scale expanded by a factor of 45 . The peaks are common to both spectra; the figure is intended to show the effect of the unfolding on the background . The unfolded spectrum is the one lying along the baseline, thus illustrating the accuracy of this prescription .
3. The unfolding Once a set of response functions, such as that shown
in fig. 3, has been stored, channel-by-channel interpola-
tion and/or extrapolation enables one to obtain a response function for any given y-ray energy . By applying
the inverse of the transformation given above the ex-
pected Compton background and escape peak areas for a given photopeak area are then easily obtained . The
the 152 Eu spectrum the resultant peak-to-total ratio can be calculated to be in excess of 0.99, compared to = 0 .70 before the unfolding. Most of the residual back-
ground is believed to result from coincidence summing in the detector. A further test of the procedure, and especially of its ability to handle unsuppressed response functions, is
unfolding or "stripping" of spectra proceeds by beginning at the high-energy end of the spectra and proceeding channel-by-channel to lower energies . For each
channel, the background calculated according to the assumption that only photopeak counts are left in that
channel, is subtracted from the spectrum . This process
continues to as low an energy as the response functions can reliably be calculated (typically = 100 keV for coaxial Ge detectors) .
This procedure has been applied to many spectra,
with excellent results. In the study of quasicontinuum
or in-beam spectra, one needs to
be
sure that the
response is accurately subtracted . Thus the unfolding of test spectra taken with -y-ray sources, for which the unfolded spectrum should show no net background, is
an important test . We have unfolded several multi-y source spectra, including 133Ba, 152Eu and 56 Co . Beforeand-after examples, taken with a CSG detector, are shown in figs . 6 and 7. Some of these spectra have also been corrected for the detector efficiency . In the expanded segment of the 56CO spectrum (fig . 7) it can
easily be seen how accurately Compton peaks and escape
peaks are removed. By adding all of the peak areas in
Fig. 7. 56 Co spectrum measured m a CSG detector, before and after unfolding. The upper half shows an expanded portion of the spectrum to show more clearly the accurate removal of Compton and escape peaks . The spectra have also been corrected for relative photopeak efficiency .
D. C. Radford et al. / The removal of Compton-scattered -y-rays from -y -ray spectra
original (pre-unfolding) spectrum . Although about 4 times the number of counts have been subtracted, the systematic differences from the suppressed case are negligible, showing that the procedure does indeed work extremely well . An in-beam spectrum, taken with the '6 Ge(8° Se, 4n)IS2 Dy reaction using a CSG detector, is shown before and after unfolding in fig. 9. This was taken from a study of the IszDy quasicontinuum spectrum ; experimental details may be found in ref . [12] .
w â
U F Z ô
V
4. Further corrections and other considerations for inbeam spectra 600
700 800 ENERGY (keV)
900
1000
Fig. 8. Portions of unfolded IS2 Eu spectra from the same detector, taken for the same time ; (a) with Compton suppression, and unfolded using suppressed response functions, and (b) without Compton suppression, and unfolded using unsup pressed response functions . The spectra have also been corrected for photopeak efficiency . the comparison of unfolded spectra taken with Compton-suppressed and unsuppressed detectors. Fig. 8 shows segments of 152Eu spectra taken in the same detector, for the same length of time . The upper portion shows an unfolded spectrum taken with Compton suppression, while the lower portion was taken without suppression and unfolded using the unsuppressed response functions. Both have been corrected for detector efficiency . The increased statistical fluctuations in the lower half are due to the much larger number of counts in the 1000
(o) BEFORE UNFOLDING
Boo 600 400 w Z z 20C Ha.N a ci 0
Z
AFTER UNFOLDING
B00 60O 400
f
z 500
1000 E y (keV)
1500
Fig. 9. An in-beam spectrum (ref. [121) before (a) and after (b) unfolding. The '6Ge( s° Se,4n) reaction at 330 MeV was used . The spectra have also been corrected for the relative photopeak efficiency .
There are a number of potential pitfalls in the accurate unfolding of in-beam -y-ray spectra. Perhaps the most serious of these in most applications are the spectral distortions ansing from pileup, neutrons and coincidence summing. The contributions of these effects to the spectrum must be removed before the unfolding since not only do they change the shape of the spectrum, but they also have very different response functions from those calculated in the unfolding procedure. Pulse pileup and coincidence summing both increase the average pulse height in the spectrum. Pileup can be prevented during the accumulation of the spectrum by using low counting rates and/or pileup rejection circuitry. Coincidence summing occurs when two or more -y-rays emitted simultaneously from the same nucleus interact with the same detector . For an unsuppressed Ge detector = 10 cm from the target, and a high -y-ray multiplicity reaction, the fraction of counts in the spectrum affected by summing can be as large as 10-15%, although the effect is reduced by using Compton suppression and increasing the target-to-detector distance. It is relatively simple to correct the observed spectrum for coincidence summing under the assumption that is is "uncorrelated", i.e . the probability of detecting two y-rays in coincidence is simply the product of the probabilities of detecting each of them separately. Thus such effects as angular correations between pairs of -y-rays and the lack of autocoincidences between a y-ray and itself are neglected. The coincidence summing effectively removes some fraction of the counts from each channel of the "true" (corrected) spectrum and redistributes them at higher energies according to some coincidence probability . The above approximation assumes that this probability has the same shape as that of the spectrum itself. Thus one can write S, j
RPJ'P,_J /
k
Pk
where S,, is the contribution to channel t of the observed spectrum P, due to summing out of channel j of
D. C Radford et al / The removal of Compton-scattered y -raysfrom -y -ray spectra
the true (corrected) spectrum PJ', and R is the fraction of all counts in the spectrum affected by summing. R can be evaluated from the coincidence-to-singles ratio of a second detector in fast coincidence with the detector of interest . P,_,, rather than P,'_,, is used in eq. (1) since, for events with high -y-ray multiplicity M >> 1, this automatically corrects not only for 2-fold summing but also for 3-fold and higher order summing. From eq . (1), we derive P,=(1-R)P,'+ Y_S,J J <, =(1-R)P,'+RY_PJ' P,_J/Y_Pk . k J<1 Thus in order to evaluate (1-R)P, =P,-RI Y_ P"P_ '/ Y_ P" J<1 k one only requires P,' for j less than t. One may therefore begin at channel i = 1 and proceed incrementally to higher channels . Although the actual spectrum is clearly not uncorrelated (since certain -y-ray peaks are in coincidence while others are not) the above formula provides a correction almost indistinguishable from a correction which includes the correlations, and is easily within usual statistical errors . The more accurate correction may be taken from the measured coincidence matrix between two detectors, if it is available. Many reactions, especially heavy-ion fusion-evaporation reactions, produce neutrons which interact with Ge detectors mainly through the (n, n'y) reaction . The kinematics tend to focus these neutrons in the forward direction, so that they contribute more to the spectra of detectors close to 0 ° with respect to the beam axis . The resulting neutron spectrum is dependent on the energy of the neutrons and cannot easily be calculated, especially since the response of the detector to a 'y-ray emitted from within the detector itself will clearly be different to that of a y-ray coming from the target . However, the spectrum can be approximately measured using y-ray absorbers, and then subtracted from the total spectrum before unfolding. The effect can also be reduced by placing detectors at backward angles, and by using time-of-flight discrimination where applicable . Another possible source of error is the high-energy cutoff of the y-ray spectrum . If the unfolding of the spectrum does not begin at sufficiently high energies, some Compton and escape events will be treated as photopeak counts, leading to an undersubtraction at high energies and possibly an oversubtraction at low energies . To prevent this, either a spectrum extending up to at least 5 MeV should generally be taken, or a spectrum extrapolation method such as that of Sic et al . [12] should be used . Clearly, it is also important to measure the response
functions in the actual experimental setup wherever possible . The material surrounding the target or source can have a significant effect on the response, especially in the region of the backscatter peak . The effects of the low-energy discriminator and/or fast coincidence gate must also be carefully included, since they can change the relative detector efficiency and attenuate the Compton background at low energies . Use of slow-risetime rejection circuitry and generous fast-coincidence timing windows, together with the gating of all singles spectra by the constant-fraction discriminator are important to ensure that the correct response functions and relative detector efficiencies are measured . A further (very small) effect on the response functions, that has been neglected to our analysis, is the angular distributions of the inbeam -y-rays. Since the response function to the region of the backscatter peak depends on the angle of scattering into the detector, the response function of the nonisotropic in-beam -y-rays could be slightly different from that of the isotropic source spectra. 5. Conclusion We have described a method for the accurate removal of Compton background from y-ray spectra, by interpolating between measured detector response functions. This method, together with the prescriptions given here for treating other sources of distortion of the spectrum, has been used to analyse data from in-beam y-ray continuum studies (e .g. ref. [11]) with excellent results. Copies and/or listings of the computer code used may be obtained by writing to one of the authors (D .C .R .) . The program is currently being extended to be able to treat two-dimensional coincidence spectra, or y-y correlation matrices . Acknowledgements One of us (D.C .R.) acknowledges very helpful discussions with Dr . Y. Schutz. This work was partially supported by the US Department of Energy, Nuclear Physics Division, under Contract no . W-31-109-Eng-38. References [1] E.g . D. Ward, H .R. Andrews, B. Haas, P. Taras and N. Rud, Nucl . Phys . A397 (1983) 161; S.H . Sie, J.O. Newton, J.R. Leigh and R.M . Diamond, Phys . Rev. Lett. 46 (1981) 405. [2] R.V .F . Janssens, T.L . Khoo, E. Funk, U. Garg, J Kolata and J. Mihelich, unpublished (1983). [3] E. Hagberg, D. Horn, M.A . Lone, H Schmemg, D. Ward, P. Taras, J. Gascon, J.C . Waddington, G. Palameta, V. Koslowsky and O. Hâusser, Atomic Energy of Canada Limited report AECL-8329.
118
D. C. Radfordet al. / The removal of Compton-scattered -y -raysfrom -y -ray spectra
[4] D. Ward, H.R. Andrews, B. Haas, P. Taras and N. Rud, Nucl . Phys . A397 (1983) 161. [5] D.J .G. Love, A.H . Nelson, P.J . Nolan and P .J . Twin, Phys. Rev. Lett . 54 (1985) 1361 . [6] J.F . Mollenauer, Phys . Rev. 127 (1962) 867. (71 S.H . Sie, Nucl . Instr. and Meth . 155 (1978) 475. [81 W. Trautmann, IF Sharpey-Schafer, H.R . Andrews, B. Haas, O. Hausser, P. Taras and D Ward, Nucl. Phys . A378 (1982) 141. [9] See also Y. Jin, R.P . Gardner and K. Verghese, Nucl . Instr. and Meth . A242 (1986) 416.
[10] R.L . Heath, R.G . Helmer, L.A . Schnuttroth and G.A. Cazier, Nucl. Instr. and Meth . 47 (1967) 281. [11] D.C . Radford, I. Ahmad, R. Holzmann, R.V .F . Janssens, T L. Khoo, M.W. Dnggert, U. Garg and H. Helppi, Phys Rev. Lett . 55 (1985) 1727 . [12] R. Holzmann et al ., to be published. [13] S.H . Sie, R.M . Diamond, J.O . Newton and J.R. Leigh, Nucl . Phys . A352 (1981) 279.