- Email: [email protected]
Evaluation of a method for the determination of accurate transition energies in the (n, γ) reaction
Nuclear Instruments and Methods 215 North-Holland Publishing Company
159
(1983) 159-165
EVALUATION OF A METHOD FOR THE DETERMINATION OF ACCURATE TRANSITION ENERGIES IN THE (n, -y) REACTION T.J. KENNETT, W.V . PRESTWICH, R.J. TERVO and J.S. TSAI Department of Physics, McMaster University, Hamilton, Ontario, Canada
Received 1 February 1983 The problems and limitations that are associated with energy determination of high-energy gamma-ray transitions are examined . The possibility of making use of the escape peaks arising as a result of pair production is explored, particularly with regard to sensing the form of the differential linearity of the spectrometer system . It is demonstrated that, provided appropriate techniques are employed to achieve spectral analysis, the spacing between escape peaks is moo` within an error of 15 eV . The fidelity_ of the pulse-height to energy transformation was assessed through the use of mixed (n, y) sources which had quite different reaction Q-values. Finally a constrained fitting procedure is presented which couples the channel number to energy transformation parameter% to the level energies for the mixed decay schemes studied. Energies are reported for levels in '° Be . ' 5 N. =QSi and "'Si as well as for the respective neutron separation energies.
1. Introduction
Gamma ray spectroscopy conducted through the use of a Ge detector and pulse height analysis is characterized by high energy resolution and because of the counter efficiency, usually high statistical precision. These factors combine to permit energy determinations to be attained that have uncertainties of only a few eV . Unfortunately, this high precision exists on a relative scale and to achieve a corresponding accuracy, calibration with standards that have been measured in a more fundamental way is required . Presently the gamma ray standards are concentrated in the energy region below 1.5 MeV so that attaining high accuracy in the energy range 4-11 MeV is made difficult because of the need to extrapolate . One approach to this problem has been to make use of a property of many high energy states in which deexcitation can proceed through one of many channels . This type of decay results in cascade cross-over transitions which, because of their lower energy, can be accurately calibrated. Summation then permits the value of the high energy state to be deduced. This approach which avoids the need for long extrapolation, but which does suffer from a compounding of errors, has been used in most of the recent precision energy determinations 11-31. As with all methods. the linearitv of the entire acquisition system must be determined since in most instances, deviation from the ideal far exceeds the magnitude of the uncertainty arising from statistical factors. The detector response function for high energy photons is dominated by a set of three peaks or probability 0167-5087/83/0000-0000/$03 .00 ,,tJ 1983 North-Holland
concentrations. These correspond to deposition of the full energy (FE), the full energy with the loss of one annihilation photon or simply the single escape process (SE) and the double escape (DE) process. The supposition that these peaks are approximately nt,,c l apart has been used to bridge otherwise extended energy spans within spectra (41 thereby "boot strapping" from the low to the high energy region . Recent studies sukest that the difference between these peaks ma-, not he exactly moo` . Thus for example Alkemade et al (`1 report a value of 510 .67 ± 0.09 rather than the 5 11 .001 ± 0.001 keV expected (6] when averaging the spacing between the FE and SE peaks. In contrast to this are the findings of Helmer et al . [71 who determined . after correction for the incremental field effect assoiated with planar detectors (8), the FE-DE difference to he 15 ± 25 eV less than the accepted value of 2rn~,c - . An examination of the current literature suggests that because of divergences such as this, there has developed a rather negative attitude concerning the use of all three peaks in precision energy measurements . It is apparent . when the overall task of precision energy determination is examined, that it is possible to device a procedure ; hat will fully utilize the information contained in such prominent spectral structure. With this in mind an experimental design to implement this was developed. At the outset we shall make the assumption that although the spacings he ,ween the FE . S E and 1)E: peaks may exhibit smal! 1 " ,.: t unknown deviations fron . the constancy expected, they should be . for a given detector. invariant under :nergy translation. If spectra are acquired that are rich in the number and disposition
t(A)T
.J. Kennett et at. / Accurate transition energies
of components, the cxmstancy of the spacing betwen the aforementioned peaks may be used to deduce the functional form of the system differential linearity. Until now this approach has been largely ignored, possibly loi4,' .iuse of the uncertainty associated with the absolute value of interpeak separations. The second factor which has changed during the past few years is that of accurately determined energies. While y-ray standards remain confined to the lower energy region, recently there has become available 181 a set of neutron separation energies ("N, II C and 15 N) which have estimated uncertainties in the region of 10 eV. When such information is used in conjunction with the proposed method for determination of scale linearity, a procedure results that has the potential to yield accurate energy estimates for high energy photon transitions. It should be recognized that successful attainment of the delineated goal requires a consistent and effective procedure for spectral analysis. Although not frequently recognized, the form of the peak shape and local background associated with the three peak types differ markedly . Failure to handle this problem in a proper way will produce what appears to be a deviation in the spacing between these components over and above any truly physical difference that exists. While the principal objective was to determine the degree of accuracy which could be consistently established, evaluation of the spacing between the FE, SE and DE peaks was also of interest. Both of these topics required that considerable effort be directed towards the problem of peak centroid evaluation . Finally. to provide a meaningful test of the con".istency with which energies have been determined, a rigorous method of evaluation is necessary. The collection of a set of transitions which are uniquely interrelated through a decay scheme certainly does provide ".ome constra iits upon the derived differential linearity. t_: nfortunately a single such decay scheme characterized by two steps cascades can only verify the accuracy of transitions in the region of one half the total energy . While the sum of other pairs may be correct, the value of the components are not well constrained and therefore can be in error. The looseness of the constraint associated with this approach can be tightened considerably by enlarging the scope of the measurement to include cases of other Q-values in addition to known standards. This extension provides a much enhanced possibility for assessing the degree of consistency attained . c.
Experimental arrangement
The irradiations were conducted using the in-core tangential facility at the McMaster University Nuclear Reactor 191 . Samples of silicon and melamine, which
were contained within capsules fabricated from either carbon or beryllium, could be changed while the reactor operated at the current power level of 1.5 MW . The entire irradiation system which is normally evacuated, can be backfilled with nitrogen gas for the purpose of calibration. The highly collimated radiation from the target passes through a 20 cm plastic neutron shield before impinging upon the detector. The detector used, a 155 efficient coaxial Ge device fabricated by Aptec, displayed a resolution of 1 .5 keV at 1 MeV and 5.5 , keV at 10 .8 MeV . Following a conventional pulse conditioning treatment, encodement was achieved through use of a Northern scientific 13 bit NS-635 ADC from which the digital output was communicated to a Nova2 computer (101 where the data logging function was performed. The acquisition system was stabilized against changes in both intercept and gain through use of a NS-409. All data were collected using the same spectral peaks for stabilization in order to assure .-t least minimally common gain . The composition of the spectra, determined by sample and container combination, permitted algebraic manipulation of data sets to be used to remove certain components and to thereby verify the degree of gain identity achieved . For the present study the reactions 9Be(n, Y)' ° Be, '° N(n, Y)' 5 N, 2KSi(n . y) 2 Si and 29 Si(n, y)-"'Si were selected . To ensure that adequate detail was retained throughout the energy range 0.5-11 MeV, each target was examined with system gains of 0.63 and 1 .23 keV/channel. In addition to the (n, y) components, all spectra contained background contributions from the decay of 54Mn . 60 Co . 109Ag and 152Eu which served as monitors of system stability and gain consistency . The objective of the present study concerned energy determinations and because of this, the counting time and sample size were selected to ensure that components of interest would have centroid precis:ions of less than 20 eV . Parameters were selected that would avoid high count rate problems and this led to acquisition times of approximately 50 h. 3. Spectral analysis While methods for the analysis of gamma ray spectra abound, most have been developed with the main thrust directed towards extraction of peak areas rather than peak centroids. The most common approach in use consists of modelling both the local background, or underlying continuum, and the peak in order that some form of regression analysis can be applied . Embellishments in the form of additional functions which can produce asymmetries and tailing of A peak, usually taken to he Gaussian, are commonly employed . Proceeding in this way requires that the energy dependence of all parameters used be determined if entire spectra
T.I. Kennett et a1. / Accurate tran .vrtlon cnergws
are to be analyzed . As alluded to earlier, analysis of spectra which encroach into the energy region where pair production is significant is made difficult because of both the differing peak shapes and background encountered. For example the full energy peak invariably displays a low energy tail attributable to incomplete energy deposition or collection . In contrast the double escape and to a less degree the single escape peak both display a high energy toe. This artifact arises from small angle Compton scattering of the annihilation photons within the detector itself . The marked dissimilarity of the peak types makes modelling and analyzing spectra, which span a wide energy range, a time consuming task that is not amenable to automation . The combination of all these factors pointed out the need to reexamine the whole question of spectral analysis, particularly when energy determination, hence peak centroid evaluation was of prime interest . A technique in which spectra are convolved with what corresponds to a smoothed second derivative [11], is known to yield centroids that show little dependence upon background variation. While this method is adequate in most instances, additional treatment is required when a highly accurate measurement of energy desired. In view of this it was decided to develop a method that would remove the background in a reliable and efficient way prior to application of the filter . The method that was devised (121 proceeds, without intervention, to sense local minima and from these to generate by interpolation an estimate of the background continuum . Following removal of the background, the resultant spectrum was filtered and centroids estimated by finding the mean of the positive lobe associated with each component . The entire process of background and centroid estimation was carried out in a completely automatic fashion finally producing a tabulation of channel centroids, peak areas and local background levels for all significant components . A series of tests were conducted in which different interpolation techniques were used in the construction of the background estimation . These revealed that centroid variations of 0.01 channels (12 eV) could occur for low peak to background ratio. Intense components showed a centroid consistency of 0.003 channels (3 cV) a value well within the statistical limit attainable in most instances. The evidence gathered in these and subsidiary studies clearly indicated that this procedure can yield centroid estimates that are characterized by a high degree of reproducibility . 4. Methodology used for energy determination The analysis procedure described above was used to obtain sets of centroid data which were oil a contnton, though not necessarily linear scale. Transformation of these data to a linear scale requires knowledge of the
system linearity . Some indication of this may he gicaned through use of standard pulser techniques but. becausr the detector portion of the spectrometer system is not included . it is ultimately necessary to make direct tie of gamma radiation. Retaining the notion 'hat the separa tion of the FE-- SF and SE DI: peaks is unknt un but constant permits, through examination of these differences as a function of channel, some insight to be gained about the form of the differential linearity. The set of experiments for which the gain was 1 .2 keV/chan revealed a total of some 40 transitions. Selection of peaks without spectral interference led to a total of more than 90 components which were suh%equentl% used to deduce the form of the differential linearity . -1n examination of the data from the measurements for which the gain was 0.62 keV/chan showed a similar form thereby indicating that the ADC was probably the .An principal contributor to the system non-linearity analytical expression was devised to describe the form of the differential linearity and the final transformation obtained through integration of this model function Inclusion of the fact that three peak type-, are present produces the function E,, = knt,c 2 + Y_ a, S,4 + h! + cl ` + d exp( -- tl ) . , Tu
1I)
where I is the channel number. EA the photon energy with the FE. SE and DE represented b% k = t). I respectively . (Note that any Jifferences bet%%een the FE-SE and DE--SE spacings and the equi%aleme of ni ts` will he revealed through a lack of tdentif% in the values of a, .) A constraint exists in that each component obtained from eq . (1) corresponds to a transition be tween bound states . Specif-ing the level enery as I . the states as land rn and nuclide as .4 permit, this to be written as Thus in seeking a method :o anal-we the data . :.m :urrent use should he made . for all transitions. of the conditions reflected in eqs. (1) and (21. .A reasonable approach to the problem of evaluating the parameter, of eq . (1) and the le%-el energies of eq . (2) can he am %ect at through minimization of the sum of the iseighted . with respect it, the desired residuals. (E,, - E,,)Z U1 quantities . Here the weighting should reflect the uncertainty associated with the estimation of the peak centroid . The use of such .t method will ensure that the level energies for the four nuclides are all on a common energy scale that is determined via the %clue .f m , Should it he later desirable to renormalire the kale in order to achieve greater accurac% . all results %, ill he adjusted in the same ,say thereby retaining the inte~:rw, of the set. The method of analysis that %%as actuall, used encompassed the principles outlined above hilt . to make
Results equations obtain of the parameters uncertainty two value entire decay 57 result number phase 91 10833 energy available then responseThe keV difference observations adistribution to of into Using an the within the be dispersion Because level for that average value data could values normal can was lead '-"N 0for atwo estimate used to energy implies about ue scheme Since differential consisted the eq experimental of solution the scale precision be which sets yields aand aall to found to and To account the eqs sets case 37 common are resolution, for of (1) tthat as keV derived sample standard probability error several between convergence possible the tractable, 0were energy 0do arrived range data cascades eq transition inputs a2°Si constraints of could particularly were the (1) of shown level Continuing spanned was of observed average sample to tothe this of can (1) and found separation 2"Si and linearity for and keV were be `N(n, of keV low these The using computed associated energies These mean transformation the points at t5N deviation the to combinations be for The be 0inbackground the were this (2) the since via The was standard ordinate energy eq cascade This energies strong used the to rms considered actual more the All parameterc after neutron fig An y)t`N these energies so observed The were in decay when the have size keV this will (2) formed §tandard represent full high subdivided energy deviation !5iv results from isthat 1because inability turn Iü0C2 via four with precisely anticipated and leverage standards transitions Solution in somewhat examined lead aiterative membership results This Clearly 1deviation The components precision reaction obtained schemes separation common the eq cumulative which are of from laor and model increase differences estimates to are function 1equations to Kennett single, deviations (1) nature curves the figure the five they to the MeV be of to an summed required on any of peak procedure the examined accurately transition into were and complete error the normally less 1131, to from transforestimate cycles involved energies of from through value increase reintrowere this aenergy et typical clearly in energy was assess multiof which rather 0these chancomat were form span iron area than emtwo was and disthe anset for an to of 2, Accurate transition /to cascades this large double DE energy 1 2vfactor energy I the that The 1case from certainly which line separation Gain the the and here all paints escape means energies gathered have of difference relation error kcV 511 scale from -=FE-SE, 12arise the asQ1are ref aeV f_,7 show keV be decay band of peaks astandard ref is8keV/channel, energy and indicates in Itr-V from on result error isbetween anticipated todifferences ref the 8the SE-DE expected encouraging icl this schemes be dispersions two cumulative 8associated deviation for present some orenormalized normally basis and decay theand tfi15N for range photo, range treated 0henceforth inaccuracy to The isschemes The study eV the of probability 100 distributed aAs FE-DE _1_associated only with 00-1 0span the keV kcV kcV same fact errors 6000% itindicated only 1000 4single using was The the keV ppm of size that kcV in all keV and The obtained errors quoted possible dotted normaliza10833 escape the used the with different energies the earlier, do values which value scale lines keV The and obnot are the for to
T.J. implementation phase.-,. bedded were first the set were of energies duced . to found
999 995 .
. .
.
98 tJ 9 m a ta 0 tr -5 a.
.
.
.
W
.
5. The application . the measurement . decay comparable components . range. peak obtain and . the 0.054 . this keV . combined error The . using bound ticipated reveals distributed. The evaluation should .018 . the .038 . model in the . nel linearity .32 small mation persion . The .a .t,.. uct2ittirled c was .343 .009 . mon the .
005 001 Fig . . 91 indicate straight in
.038 . . .
.
J' 1711 l -1oo I . I L ENERGY
move is must served from the 10833 .302 reported those include tion From find differences Table The the Case : Fe-SE SE-DE Fe('ace_ Fe-SE Se-DE Fe--DE
.. a
.
.
.054
.
. .
. .
.
. . .
. .23
511 .007 .012 511 .006 0.019 1022.016 .026 .,.a.^. ",I 511 .003 .017 511 .001 .022 1022.010 .021
T.J. Kennett et al. /Accurate transition energies
found for both gain settings used are presented in table 1. The results are consistent with each other and with the expected values of 511 .003 and 1022.006 keV. There has been no correction made for any field effect that might be attributable to the coaxial detector . Table 2 summarizes the results obtained for the '4 N(n, y )' 5 N reaction . Here the photon energies observed and deduced from the levels are given along with those reported in ref. 1 . The errors in the current results are smaller than those of ref. 1 throughout the full energy range. However it should be noted that with the exception of the weak doublet member at 1681 keV, the difference in means is well within the combined error. The same outcome holds true in the case of the level energy estimates which are also presented . Table 3 presents a summary of the results obtained for the 28 Si(n, y) 2l Si reaction . Here again the present findings have smaller uncertainties than the adopted level and transition energies (14,151. As in the case of
table 1, deviations of means are well within the error% reported . The separation energy of 8473 .944 t 0.(112 keV is in agreement with the adopted value of 8473 .9 + 0.5 keV. Table 4 gives a partial tabulation of the remult-, found for the 2"Si(n, Y)'°Si reaction . The errors a%soxtated with the energies given here reflect the fact that the contribution of this reaction to the spectrum is small compared to that of 2'Si . Regardless of this. the estimated quantities show marked improvement over previous measurements [16,171. The separation energy . found to be 10609.53 f 0.03 keV agrees well with the adopted value of 10609.5 t 0.6 keV. The transitions and level energies found for "'Be are given in table 5. Here as for the previous cases the precision in energy is markedly improved over earlier results (18,191. While no effort was made to determine the neutron separation energies of 2 H and "C . their presence in the
Table 2 A summary of the level and transition energies found for the reaction 14 N(n, -y Level energy Present
Ref. 1 5270 .116 5298 .800 6323 .904 7153 .064 7300 .886 8312 .689 9151 .607 9154 .936 10833.297
32 32 57 56 88 102 229 66 24
5270.159 5298 .805 6323 .889 7155 .123 7300.862 8312 .597 9152 .155 9155 .035 10833.302
14 20 20 21 17 14 35 50 12
Gamma ray energy Present (observed)
Ref . 1
1678 .260 1681 .599 1884.820 1999.729 2520.382 2830.745 3531 .964 3677.748 3855.604 3884.280 4508.665 5269.122 5297 .795 5533 .401 5562 .073 6322 .474 7298 .980 8310 .218 9149 .068 10829.101
N.
Deduced from le%el .
59 228 ,7
1678 .227 1681 .107 1884.853
38 40 22
101 87 86 56 73 73 52 32 32 32 32 57 M8 102 199 24
2520.478
-
3532 .001 3677 .721
27 22
4508 .720 5269 .146 5297 .780 5533 .397 5562 .020 6322 .473 7298 .975 8310 .126 9149 .181 10829. 112
27 22 27 27 22 31
w
38 31 31
16S1 046 1884 .837 1999,769 2520 .478 2830 .859 3531 .995 3677,696 38,'5.698 3884 .336 4508 .687 5269 .165 5297 .800 5533 .402 5562 .037 6322 .458 72,- .9 ~ 8310 .125 414a 15S 10829 IN
:" 21 54 2h 54 15 17 54 52 20 14 20 211 14 20 i 24 l5 12
T.J. Kennett et al. / Accurate transition energies
Ite1
Table 3 A summary of the level and transition energies found for the reaction 2R Si(n, y)29Si . Ref. 14
Level energy Present
1273.3 1 2425.6 2 4933.6 5 080 .7 4 8473.9 5
1273A 14 2426.030 4934 .563 6380 .836 8473.944
Ref. 15
1273 .4 2092 .9 425 .9 3538 .9 3660.9 3954.4 4933 .9 5106.6 6046.2 6379.9 199.5 8472.8
1273 .392 2093 .038 2425 .930 3539.111 3660.903 3945 .517 4934.075 5106.921 6047.245 6380.112 7199.586 8472.626
14 20 13 13 12
54 31 54 31 31 31 31 31 54 31 31 31
Deduced from levels 1273 .384 2093 .027 2425 .921 3539.148 3660.902 3954.517 4934.113 5106.939 6047.238 6380.083 7199.570 8472.615
14 12 20 12 13 16 13 13 19 13 12 12
Ref. 16
Level energy Present
.235 .37 13 3498 .7 2 3769.7 2 6641 .1 5 6744 .1 4 "SU7 .8 5 11898 .0 6 %19 .9 6 9792 .4 6 10609 .5 6
2235 .330 3498 .590 3769.573 6641 .442 6744 .319 7508 .041 8898 .637 9620 .176 9792 .635 10609 .529
Ref. 17
Gamma ray energy Present (observed)
1 2 4 3 4 3 3 3 4 4 5 17 5
2235.250 2359 .749 2446 .388 3101 .392 3498 .379 3769 .307 3864.975 3967 .817 4405 .744 5272.257 6743 .530 6839 .139 7110 .075
Ref. 17
3368 .0 2 5958 .3 3 6811 .8 4
Ref. 19
34 38 38 45 59 45 180 80 80 28
73 42 180 49 78 77 45 50 120 63 45 40 40
8 7
Present (observed) 8372.962 40 8897.221 150
6
9618.510
7 7
Deduced from levels 8372 .976 30
80
9790.919 40
10607.532 40
10607 .548 28
Table 5 A summary of the level and transition energies found for the reaction 9Be(n,y )'°He. Ref. 18
Table 4 A summary of the level and transition energies found for the reaction 29Si(n. y)"Si.
2234.3 2360.0 2446 .3 3101 .4 3498 .4 3769 .4 3864.8 3967 .9 4405 .3 5271 .8 6743.4 6839 .1 7110 .0
Gamma ray energy 8373 .0 8896 .3 9618.3 9770.5 10607 .3
Gamma ray energy Present (observed) (
2 3 4 3 4 5 4 6 7 7 8 12
Table 4 (continued)
853 .3 2589 .9 3367 .4 3443 .3 6809.4
3 3 2 3 4
Levelenergy Present
3368 .029 27 5958 .387 50 6812 .038 27
Gamma ray energy Present (observed) 853 .605 2589 .999 3367 .415 3443 .374 6908 .585
58 58 28 28 31
Deduced from levels 853 .612 2589 .998 3367 .420 3443 .386 6809 .546
40 40 27 27 27
spectra provided an independent way to monitor how well the energy measurements were achieved . For 2 H we obtained a value of 2224 .568 t 0.032 keV and for 1 C. 4946 .31 f 0.050 keV . The corresponding values given by Cohen and Wapstra [8J are 2224 .573 t 0.008 keV and 4946 .336 f 0.014 keV respectively . 6. Conclusions
Deduced from levels 2235 .241 34 3101 .348 3498 .371 3769.319 3864.963 3967.838 4405 .765 5272 .213 6743 .518 6839 .151 7110 .067
38 38 38 38 38 38 38 38 38 38
The objective of devising a technique to evaluate the consistency with which gamma -ay energies can be determined over a wide range has been met. Careful spectral analysis and full use of all prominent spectral features, specifically escape peaks, permits one to model the system linearity with the actual results that are to be treated. Clear evidence is presented to show that, for the detector used, the FE-SE-DE response triplet is characterised by equal spacings of value m,c2. A set of new secondary standards 29 Si, "Si and ß°ße have been established relative to 1SN . The uncertainty associated with level and transition energies for all nuclides studied is about an order of magnitude better r L. . -an rt :nutAUtc. ' TL ., t. ... t... : _. . _. au currently available. a ttc 'vatuc íà1 tnc tcäatntyuc~ presented here can best be demonstrated by independent study, particularly for the 2KSi(n, y)29Si reaction . Until that is done, the quality of the results can only be inferred by the overall consistency achieved .
T.J. Kernnett et al. / Accurate transition energies The authors would like to express their appreciation to the Natural Sciences and Engineering Research Council of Canada and to the Province of Ontario for financial assistance .
References (1] [2] [3] [4] [5] [6] [7]
R .C. Greenwood and R .E . Chrien, Nucl. Instr. and Meth . 175 (1980) 515 . H.H . Schmidt et al., Phys. Rev . 25C (1982) 2888 . B. Krusche et al., Nucl. Phys. A386 (1982) 245 . T.J. Kennett, N .P. Archer and L.B. Hughes, Nucl . Phys. A% (1967) 658. P.F. Alkemade, C . Alderliesten, P. de Wit and C . van der Luen, Nucl . Instr. and Meth. 197 (1982) 383. R.L. Kelley et al., Rev. Mod. Phys . 52 (1980) S33 . R.G . Helmer, R.C. Greenwood and R.J . Gehrke, Proc . Conf. on Atomic masses and fundamental constants, Teddington, England (Plenum, London, 1972) p. 112 .
165
[8] E.R. Cohen and A .H . Wapstra, Nucl . instr . and Meth . 211 (1983) 153 . [9] A .H . Colenbrander and T .J . Kennett . Nucl . Instr . an d Meth . 116 (1974) 251 . [10] G .C . Cormick, M .Sc . Thesis, McMaster Universitv . Hamilton, Ontario (1976) . [111 A . Robertson, W.V . Prestwich and T.J. Kennett. Nuct . Instr . and Meth . 100 (1972) 317. [12) R .J. Tervo, T .J. Kennett and W.V . Prestwich. Nucl. Instr . an d Meth . (in press). [13] T.J . Kennett, M .A . Islam and W .V. Prestwich, Can . J . Phys . 59 (1981) 93 . [141 P.M . Endt and V. van der Leun, Nucl. Phys . A310 (1978) 243 . [15] A.M.J. Spits, A.M . op den Kamp and H . Gruppelaar . Nucl. Phys. A145 (1970) 449. 1161 Ibid, 13, p . 271 . [171 A.M. Spits and J . de Boer, Nucl. Phys. A224 (1974) 517 . [ 18] F . Ajzenberg-Selove, Nucl . Phys . A320 (1979) 143. [19] R .C . Greenwood, Phys. Lett . 23 (1966) 482 .
159
(1983) 159-165
EVALUATION OF A METHOD FOR THE DETERMINATION OF ACCURATE TRANSITION ENERGIES IN THE (n, -y) REACTION T.J. KENNETT, W.V . PRESTWICH, R.J. TERVO and J.S. TSAI Department of Physics, McMaster University, Hamilton, Ontario, Canada
Received 1 February 1983 The problems and limitations that are associated with energy determination of high-energy gamma-ray transitions are examined . The possibility of making use of the escape peaks arising as a result of pair production is explored, particularly with regard to sensing the form of the differential linearity of the spectrometer system . It is demonstrated that, provided appropriate techniques are employed to achieve spectral analysis, the spacing between escape peaks is moo` within an error of 15 eV . The fidelity_ of the pulse-height to energy transformation was assessed through the use of mixed (n, y) sources which had quite different reaction Q-values. Finally a constrained fitting procedure is presented which couples the channel number to energy transformation parameter% to the level energies for the mixed decay schemes studied. Energies are reported for levels in '° Be . ' 5 N. =QSi and "'Si as well as for the respective neutron separation energies.
1. Introduction
Gamma ray spectroscopy conducted through the use of a Ge detector and pulse height analysis is characterized by high energy resolution and because of the counter efficiency, usually high statistical precision. These factors combine to permit energy determinations to be attained that have uncertainties of only a few eV . Unfortunately, this high precision exists on a relative scale and to achieve a corresponding accuracy, calibration with standards that have been measured in a more fundamental way is required . Presently the gamma ray standards are concentrated in the energy region below 1.5 MeV so that attaining high accuracy in the energy range 4-11 MeV is made difficult because of the need to extrapolate . One approach to this problem has been to make use of a property of many high energy states in which deexcitation can proceed through one of many channels . This type of decay results in cascade cross-over transitions which, because of their lower energy, can be accurately calibrated. Summation then permits the value of the high energy state to be deduced. This approach which avoids the need for long extrapolation, but which does suffer from a compounding of errors, has been used in most of the recent precision energy determinations 11-31. As with all methods. the linearitv of the entire acquisition system must be determined since in most instances, deviation from the ideal far exceeds the magnitude of the uncertainty arising from statistical factors. The detector response function for high energy photons is dominated by a set of three peaks or probability 0167-5087/83/0000-0000/$03 .00 ,,tJ 1983 North-Holland
concentrations. These correspond to deposition of the full energy (FE), the full energy with the loss of one annihilation photon or simply the single escape process (SE) and the double escape (DE) process. The supposition that these peaks are approximately nt,,c l apart has been used to bridge otherwise extended energy spans within spectra (41 thereby "boot strapping" from the low to the high energy region . Recent studies sukest that the difference between these peaks ma-, not he exactly moo` . Thus for example Alkemade et al (`1 report a value of 510 .67 ± 0.09 rather than the 5 11 .001 ± 0.001 keV expected (6] when averaging the spacing between the FE and SE peaks. In contrast to this are the findings of Helmer et al . [71 who determined . after correction for the incremental field effect assoiated with planar detectors (8), the FE-DE difference to he 15 ± 25 eV less than the accepted value of 2rn~,c - . An examination of the current literature suggests that because of divergences such as this, there has developed a rather negative attitude concerning the use of all three peaks in precision energy measurements . It is apparent . when the overall task of precision energy determination is examined, that it is possible to device a procedure ; hat will fully utilize the information contained in such prominent spectral structure. With this in mind an experimental design to implement this was developed. At the outset we shall make the assumption that although the spacings he ,ween the FE . S E and 1)E: peaks may exhibit smal! 1 " ,.: t unknown deviations fron . the constancy expected, they should be . for a given detector. invariant under :nergy translation. If spectra are acquired that are rich in the number and disposition
t(A)T
.J. Kennett et at. / Accurate transition energies
of components, the cxmstancy of the spacing betwen the aforementioned peaks may be used to deduce the functional form of the system differential linearity. Until now this approach has been largely ignored, possibly loi4,' .iuse of the uncertainty associated with the absolute value of interpeak separations. The second factor which has changed during the past few years is that of accurately determined energies. While y-ray standards remain confined to the lower energy region, recently there has become available 181 a set of neutron separation energies ("N, II C and 15 N) which have estimated uncertainties in the region of 10 eV. When such information is used in conjunction with the proposed method for determination of scale linearity, a procedure results that has the potential to yield accurate energy estimates for high energy photon transitions. It should be recognized that successful attainment of the delineated goal requires a consistent and effective procedure for spectral analysis. Although not frequently recognized, the form of the peak shape and local background associated with the three peak types differ markedly . Failure to handle this problem in a proper way will produce what appears to be a deviation in the spacing between these components over and above any truly physical difference that exists. While the principal objective was to determine the degree of accuracy which could be consistently established, evaluation of the spacing between the FE, SE and DE peaks was also of interest. Both of these topics required that considerable effort be directed towards the problem of peak centroid evaluation . Finally. to provide a meaningful test of the con".istency with which energies have been determined, a rigorous method of evaluation is necessary. The collection of a set of transitions which are uniquely interrelated through a decay scheme certainly does provide ".ome constra iits upon the derived differential linearity. t_: nfortunately a single such decay scheme characterized by two steps cascades can only verify the accuracy of transitions in the region of one half the total energy . While the sum of other pairs may be correct, the value of the components are not well constrained and therefore can be in error. The looseness of the constraint associated with this approach can be tightened considerably by enlarging the scope of the measurement to include cases of other Q-values in addition to known standards. This extension provides a much enhanced possibility for assessing the degree of consistency attained . c.
Experimental arrangement
The irradiations were conducted using the in-core tangential facility at the McMaster University Nuclear Reactor 191 . Samples of silicon and melamine, which
were contained within capsules fabricated from either carbon or beryllium, could be changed while the reactor operated at the current power level of 1.5 MW . The entire irradiation system which is normally evacuated, can be backfilled with nitrogen gas for the purpose of calibration. The highly collimated radiation from the target passes through a 20 cm plastic neutron shield before impinging upon the detector. The detector used, a 155 efficient coaxial Ge device fabricated by Aptec, displayed a resolution of 1 .5 keV at 1 MeV and 5.5 , keV at 10 .8 MeV . Following a conventional pulse conditioning treatment, encodement was achieved through use of a Northern scientific 13 bit NS-635 ADC from which the digital output was communicated to a Nova2 computer (101 where the data logging function was performed. The acquisition system was stabilized against changes in both intercept and gain through use of a NS-409. All data were collected using the same spectral peaks for stabilization in order to assure .-t least minimally common gain . The composition of the spectra, determined by sample and container combination, permitted algebraic manipulation of data sets to be used to remove certain components and to thereby verify the degree of gain identity achieved . For the present study the reactions 9Be(n, Y)' ° Be, '° N(n, Y)' 5 N, 2KSi(n . y) 2 Si and 29 Si(n, y)-"'Si were selected . To ensure that adequate detail was retained throughout the energy range 0.5-11 MeV, each target was examined with system gains of 0.63 and 1 .23 keV/channel. In addition to the (n, y) components, all spectra contained background contributions from the decay of 54Mn . 60 Co . 109Ag and 152Eu which served as monitors of system stability and gain consistency . The objective of the present study concerned energy determinations and because of this, the counting time and sample size were selected to ensure that components of interest would have centroid precis:ions of less than 20 eV . Parameters were selected that would avoid high count rate problems and this led to acquisition times of approximately 50 h. 3. Spectral analysis While methods for the analysis of gamma ray spectra abound, most have been developed with the main thrust directed towards extraction of peak areas rather than peak centroids. The most common approach in use consists of modelling both the local background, or underlying continuum, and the peak in order that some form of regression analysis can be applied . Embellishments in the form of additional functions which can produce asymmetries and tailing of A peak, usually taken to he Gaussian, are commonly employed . Proceeding in this way requires that the energy dependence of all parameters used be determined if entire spectra
T.I. Kennett et a1. / Accurate tran .vrtlon cnergws
are to be analyzed . As alluded to earlier, analysis of spectra which encroach into the energy region where pair production is significant is made difficult because of both the differing peak shapes and background encountered. For example the full energy peak invariably displays a low energy tail attributable to incomplete energy deposition or collection . In contrast the double escape and to a less degree the single escape peak both display a high energy toe. This artifact arises from small angle Compton scattering of the annihilation photons within the detector itself . The marked dissimilarity of the peak types makes modelling and analyzing spectra, which span a wide energy range, a time consuming task that is not amenable to automation . The combination of all these factors pointed out the need to reexamine the whole question of spectral analysis, particularly when energy determination, hence peak centroid evaluation was of prime interest . A technique in which spectra are convolved with what corresponds to a smoothed second derivative [11], is known to yield centroids that show little dependence upon background variation. While this method is adequate in most instances, additional treatment is required when a highly accurate measurement of energy desired. In view of this it was decided to develop a method that would remove the background in a reliable and efficient way prior to application of the filter . The method that was devised (121 proceeds, without intervention, to sense local minima and from these to generate by interpolation an estimate of the background continuum . Following removal of the background, the resultant spectrum was filtered and centroids estimated by finding the mean of the positive lobe associated with each component . The entire process of background and centroid estimation was carried out in a completely automatic fashion finally producing a tabulation of channel centroids, peak areas and local background levels for all significant components . A series of tests were conducted in which different interpolation techniques were used in the construction of the background estimation . These revealed that centroid variations of 0.01 channels (12 eV) could occur for low peak to background ratio. Intense components showed a centroid consistency of 0.003 channels (3 cV) a value well within the statistical limit attainable in most instances. The evidence gathered in these and subsidiary studies clearly indicated that this procedure can yield centroid estimates that are characterized by a high degree of reproducibility . 4. Methodology used for energy determination The analysis procedure described above was used to obtain sets of centroid data which were oil a contnton, though not necessarily linear scale. Transformation of these data to a linear scale requires knowledge of the
system linearity . Some indication of this may he gicaned through use of standard pulser techniques but. becausr the detector portion of the spectrometer system is not included . it is ultimately necessary to make direct tie of gamma radiation. Retaining the notion 'hat the separa tion of the FE-- SF and SE DI: peaks is unknt un but constant permits, through examination of these differences as a function of channel, some insight to be gained about the form of the differential linearity. The set of experiments for which the gain was 1 .2 keV/chan revealed a total of some 40 transitions. Selection of peaks without spectral interference led to a total of more than 90 components which were suh%equentl% used to deduce the form of the differential linearity . -1n examination of the data from the measurements for which the gain was 0.62 keV/chan showed a similar form thereby indicating that the ADC was probably the .An principal contributor to the system non-linearity analytical expression was devised to describe the form of the differential linearity and the final transformation obtained through integration of this model function Inclusion of the fact that three peak type-, are present produces the function E,, = knt,c 2 + Y_ a, S,4 + h! + cl ` + d exp( -- tl ) . , Tu
1I)
where I is the channel number. EA the photon energy with the FE. SE and DE represented b% k = t). I respectively . (Note that any Jifferences bet%%een the FE-SE and DE--SE spacings and the equi%aleme of ni ts` will he revealed through a lack of tdentif% in the values of a, .) A constraint exists in that each component obtained from eq . (1) corresponds to a transition be tween bound states . Specif-ing the level enery as I . the states as land rn and nuclide as .4 permit, this to be written as Thus in seeking a method :o anal-we the data . :.m :urrent use should he made . for all transitions. of the conditions reflected in eqs. (1) and (21. .A reasonable approach to the problem of evaluating the parameter, of eq . (1) and the le%-el energies of eq . (2) can he am %ect at through minimization of the sum of the iseighted . with respect it, the desired residuals. (E,, - E,,)Z U1 quantities . Here the weighting should reflect the uncertainty associated with the estimation of the peak centroid . The use of such .t method will ensure that the level energies for the four nuclides are all on a common energy scale that is determined via the %clue .f m , Should it he later desirable to renormalire the kale in order to achieve greater accurac% . all results %, ill he adjusted in the same ,say thereby retaining the inte~:rw, of the set. The method of analysis that %%as actuall, used encompassed the principles outlined above hilt . to make
Results equations obtain of the parameters uncertainty two value entire decay 57 result number phase 91 10833 energy available then responseThe keV difference observations adistribution to of into Using an the within the be dispersion Because level for that average value data could values normal can was lead '-"N 0for atwo estimate used to energy implies about ue scheme Since differential consisted the eq experimental of solution the scale precision be which sets yields aand aall to found to and To account the eqs sets case 37 common are resolution, for of (1) tthat as keV derived sample standard probability error several between convergence possible the tractable, 0were energy 0do arrived range data cascades eq transition inputs a2°Si constraints of could particularly were the (1) of shown level Continuing spanned was of observed average sample to tothe this of can (1) and found separation 2"Si and linearity for and keV were be `N(n, of keV low these The using computed associated energies These mean transformation the points at t5N deviation the to combinations be for The be 0inbackground the were this (2) the since via The was standard ordinate energy eq cascade This energies strong used the to rms considered actual more the All parameterc after neutron fig An y)t`N these energies so observed The were in decay when the have size keV this will (2) formed §tandard represent full high subdivided energy deviation !5iv results from isthat 1because inability turn Iü0C2 via four with precisely anticipated and leverage standards transitions Solution in somewhat examined lead aiterative membership results This Clearly 1deviation The components precision reaction obtained schemes separation common the eq cumulative which are of from laor and model increase differences estimates to are function 1equations to Kennett single, deviations (1) nature curves the figure the five they to the MeV be of to an summed required on any of peak procedure the examined accurately transition into were and complete error the normally less 1131, to from transforestimate cycles involved energies of from through value increase reintrowere this aenergy et typical clearly in energy was assess multiof which rather 0these chancomat were form span iron area than emtwo was and disthe anset for an to of 2, Accurate transition /to cascades this large double DE energy 1 2vfactor energy I the that The 1case from certainly which line separation Gain the the and here all paints escape means energies gathered have of difference relation error kcV 511 scale from -=FE-SE, 12arise the asQ1are ref aeV f_,7 show keV be decay band of peaks astandard ref is8keV/channel, energy and indicates in Itr-V from on result error isbetween anticipated todifferences ref the 8the SE-DE expected encouraging icl this schemes be dispersions two cumulative 8associated deviation for present some orenormalized normally basis and decay theand tfi15N for range photo, range treated 0henceforth inaccuracy to The isschemes The study eV the of probability 100 distributed aAs FE-DE _1_associated only with 00-1 0span the keV kcV kcV same fact errors 6000% itindicated only 1000 4single using was The the keV ppm of size that kcV in all keV and The obtained errors quoted possible dotted normaliza10833 escape the used the with different energies the earlier, do values which value scale lines keV The and obnot are the for to
T.J. implementation phase.-,. bedded were first the set were of energies duced . to found
999 995 .
. .
.
98 tJ 9 m a ta 0 tr -5 a.
.
.
.
W
.
5. The application . the measurement . decay comparable components . range. peak obtain and . the 0.054 . this keV . combined error The . using bound ticipated reveals distributed. The evaluation should .018 . the .038 . model in the . nel linearity .32 small mation persion . The .a .t,.. uct2ittirled c was .343 .009 . mon the .
005 001 Fig . . 91 indicate straight in
.038 . . .
.
J' 1711 l -1oo I . I L ENERGY
move is must served from the 10833 .302 reported those include tion From find differences Table The the Case : Fe-SE SE-DE Fe('ace_ Fe-SE Se-DE Fe--DE
.. a
.
.
.054
.
. .
. .
.
. . .
. .23
511 .007 .012 511 .006 0.019 1022.016 .026 .,.a.^. ",I 511 .003 .017 511 .001 .022 1022.010 .021
T.J. Kennett et al. /Accurate transition energies
found for both gain settings used are presented in table 1. The results are consistent with each other and with the expected values of 511 .003 and 1022.006 keV. There has been no correction made for any field effect that might be attributable to the coaxial detector . Table 2 summarizes the results obtained for the '4 N(n, y )' 5 N reaction . Here the photon energies observed and deduced from the levels are given along with those reported in ref. 1 . The errors in the current results are smaller than those of ref. 1 throughout the full energy range. However it should be noted that with the exception of the weak doublet member at 1681 keV, the difference in means is well within the combined error. The same outcome holds true in the case of the level energy estimates which are also presented . Table 3 presents a summary of the results obtained for the 28 Si(n, y) 2l Si reaction . Here again the present findings have smaller uncertainties than the adopted level and transition energies (14,151. As in the case of
table 1, deviations of means are well within the error% reported . The separation energy of 8473 .944 t 0.(112 keV is in agreement with the adopted value of 8473 .9 + 0.5 keV. Table 4 gives a partial tabulation of the remult-, found for the 2"Si(n, Y)'°Si reaction . The errors a%soxtated with the energies given here reflect the fact that the contribution of this reaction to the spectrum is small compared to that of 2'Si . Regardless of this. the estimated quantities show marked improvement over previous measurements [16,171. The separation energy . found to be 10609.53 f 0.03 keV agrees well with the adopted value of 10609.5 t 0.6 keV. The transitions and level energies found for "'Be are given in table 5. Here as for the previous cases the precision in energy is markedly improved over earlier results (18,191. While no effort was made to determine the neutron separation energies of 2 H and "C . their presence in the
Table 2 A summary of the level and transition energies found for the reaction 14 N(n, -y Level energy Present
Ref. 1 5270 .116 5298 .800 6323 .904 7153 .064 7300 .886 8312 .689 9151 .607 9154 .936 10833.297
32 32 57 56 88 102 229 66 24
5270.159 5298 .805 6323 .889 7155 .123 7300.862 8312 .597 9152 .155 9155 .035 10833.302
14 20 20 21 17 14 35 50 12
Gamma ray energy Present (observed)
Ref . 1
1678 .260 1681 .599 1884.820 1999.729 2520.382 2830.745 3531 .964 3677.748 3855.604 3884.280 4508.665 5269.122 5297 .795 5533 .401 5562 .073 6322 .474 7298 .980 8310 .218 9149 .068 10829.101
N.
Deduced from le%el .
59 228 ,7
1678 .227 1681 .107 1884.853
38 40 22
101 87 86 56 73 73 52 32 32 32 32 57 M8 102 199 24
2520.478
-
3532 .001 3677 .721
27 22
4508 .720 5269 .146 5297 .780 5533 .397 5562 .020 6322 .473 7298 .975 8310 .126 9149 .181 10829. 112
27 22 27 27 22 31
w
38 31 31
16S1 046 1884 .837 1999,769 2520 .478 2830 .859 3531 .995 3677,696 38,'5.698 3884 .336 4508 .687 5269 .165 5297 .800 5533 .402 5562 .037 6322 .458 72,- .9 ~ 8310 .125 414a 15S 10829 IN
:" 21 54 2h 54 15 17 54 52 20 14 20 211 14 20 i 24 l5 12
T.J. Kennett et al. / Accurate transition energies
Ite1
Table 3 A summary of the level and transition energies found for the reaction 2R Si(n, y)29Si . Ref. 14
Level energy Present
1273.3 1 2425.6 2 4933.6 5 080 .7 4 8473.9 5
1273A 14 2426.030 4934 .563 6380 .836 8473.944
Ref. 15
1273 .4 2092 .9 425 .9 3538 .9 3660.9 3954.4 4933 .9 5106.6 6046.2 6379.9 199.5 8472.8
1273 .392 2093 .038 2425 .930 3539.111 3660.903 3945 .517 4934.075 5106.921 6047.245 6380.112 7199.586 8472.626
14 20 13 13 12
54 31 54 31 31 31 31 31 54 31 31 31
Deduced from levels 1273 .384 2093 .027 2425 .921 3539.148 3660.902 3954.517 4934.113 5106.939 6047.238 6380.083 7199.570 8472.615
14 12 20 12 13 16 13 13 19 13 12 12
Ref. 16
Level energy Present
.235 .37 13 3498 .7 2 3769.7 2 6641 .1 5 6744 .1 4 "SU7 .8 5 11898 .0 6 %19 .9 6 9792 .4 6 10609 .5 6
2235 .330 3498 .590 3769.573 6641 .442 6744 .319 7508 .041 8898 .637 9620 .176 9792 .635 10609 .529
Ref. 17
Gamma ray energy Present (observed)
1 2 4 3 4 3 3 3 4 4 5 17 5
2235.250 2359 .749 2446 .388 3101 .392 3498 .379 3769 .307 3864.975 3967 .817 4405 .744 5272.257 6743 .530 6839 .139 7110 .075
Ref. 17
3368 .0 2 5958 .3 3 6811 .8 4
Ref. 19
34 38 38 45 59 45 180 80 80 28
73 42 180 49 78 77 45 50 120 63 45 40 40
8 7
Present (observed) 8372.962 40 8897.221 150
6
9618.510
7 7
Deduced from levels 8372 .976 30
80
9790.919 40
10607.532 40
10607 .548 28
Table 5 A summary of the level and transition energies found for the reaction 9Be(n,y )'°He. Ref. 18
Table 4 A summary of the level and transition energies found for the reaction 29Si(n. y)"Si.
2234.3 2360.0 2446 .3 3101 .4 3498 .4 3769 .4 3864.8 3967 .9 4405 .3 5271 .8 6743.4 6839 .1 7110 .0
Gamma ray energy 8373 .0 8896 .3 9618.3 9770.5 10607 .3
Gamma ray energy Present (observed) (
2 3 4 3 4 5 4 6 7 7 8 12
Table 4 (continued)
853 .3 2589 .9 3367 .4 3443 .3 6809.4
3 3 2 3 4
Levelenergy Present
3368 .029 27 5958 .387 50 6812 .038 27
Gamma ray energy Present (observed) 853 .605 2589 .999 3367 .415 3443 .374 6908 .585
58 58 28 28 31
Deduced from levels 853 .612 2589 .998 3367 .420 3443 .386 6809 .546
40 40 27 27 27
spectra provided an independent way to monitor how well the energy measurements were achieved . For 2 H we obtained a value of 2224 .568 t 0.032 keV and for 1 C. 4946 .31 f 0.050 keV . The corresponding values given by Cohen and Wapstra [8J are 2224 .573 t 0.008 keV and 4946 .336 f 0.014 keV respectively . 6. Conclusions
Deduced from levels 2235 .241 34 3101 .348 3498 .371 3769.319 3864.963 3967.838 4405 .765 5272 .213 6743 .518 6839 .151 7110 .067
38 38 38 38 38 38 38 38 38 38
The objective of devising a technique to evaluate the consistency with which gamma -ay energies can be determined over a wide range has been met. Careful spectral analysis and full use of all prominent spectral features, specifically escape peaks, permits one to model the system linearity with the actual results that are to be treated. Clear evidence is presented to show that, for the detector used, the FE-SE-DE response triplet is characterised by equal spacings of value m,c2. A set of new secondary standards 29 Si, "Si and ß°ße have been established relative to 1SN . The uncertainty associated with level and transition energies for all nuclides studied is about an order of magnitude better r L. . -an rt :nutAUtc. ' TL ., t. ... t... : _. . _. au currently available. a ttc 'vatuc íà1 tnc tcäatntyuc~ presented here can best be demonstrated by independent study, particularly for the 2KSi(n, y)29Si reaction . Until that is done, the quality of the results can only be inferred by the overall consistency achieved .
T.J. Kernnett et al. / Accurate transition energies The authors would like to express their appreciation to the Natural Sciences and Engineering Research Council of Canada and to the Province of Ontario for financial assistance .
References (1] [2] [3] [4] [5] [6] [7]
R .C. Greenwood and R .E . Chrien, Nucl. Instr. and Meth . 175 (1980) 515 . H.H . Schmidt et al., Phys. Rev . 25C (1982) 2888 . B. Krusche et al., Nucl. Phys. A386 (1982) 245 . T.J. Kennett, N .P. Archer and L.B. Hughes, Nucl . Phys. A% (1967) 658. P.F. Alkemade, C . Alderliesten, P. de Wit and C . van der Luen, Nucl . Instr. and Meth. 197 (1982) 383. R.L. Kelley et al., Rev. Mod. Phys . 52 (1980) S33 . R.G . Helmer, R.C. Greenwood and R.J . Gehrke, Proc . Conf. on Atomic masses and fundamental constants, Teddington, England (Plenum, London, 1972) p. 112 .
165
[8] E.R. Cohen and A .H . Wapstra, Nucl . instr . and Meth . 211 (1983) 153 . [9] A .H . Colenbrander and T .J . Kennett . Nucl . Instr . an d Meth . 116 (1974) 251 . [10] G .C . Cormick, M .Sc . Thesis, McMaster Universitv . Hamilton, Ontario (1976) . [111 A . Robertson, W.V . Prestwich and T.J. Kennett. Nuct . Instr . and Meth . 100 (1972) 317. [12) R .J. Tervo, T .J. Kennett and W.V . Prestwich. Nucl. Instr . an d Meth . (in press). [13] T.J . Kennett, M .A . Islam and W .V. Prestwich, Can . J . Phys . 59 (1981) 93 . [141 P.M . Endt and V. van der Leun, Nucl. Phys . A310 (1978) 243 . [15] A.M.J. Spits, A.M . op den Kamp and H . Gruppelaar . Nucl. Phys. A145 (1970) 449. 1161 Ibid, 13, p . 271 . [171 A.M. Spits and J . de Boer, Nucl. Phys. A224 (1974) 517 . [ 18] F . Ajzenberg-Selove, Nucl . Phys . A320 (1979) 143. [19] R .C . Greenwood, Phys. Lett . 23 (1966) 482 .