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Efficiency corrections for the γ-γ coincidence counting rates measured by the multi-detector correlation system
Nuclear Instruments and Methods in Physics Research North-Holland
A 336 (1993) 567-571
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Efficiency corrections for the y-y coincidence counting rates measured by the multi-detector correlation system D. Vénos and P. Tlusty t
Nuclear Physics Institute, Academy of Sciences of the Czech Republic, 250 68 Aez, Czech Republic
Received 7 May
1993
and in revised form 12 July
1993
A new method of determination of the efficiency corrections for the y-y coincidence rates measured by the multi-detector system is described. The method uses the random coincidence counting rates and is based on two assumptions: a) the counting rates ofboth true and random coincidences for a given pair of 7-quanta are proportional to the efficiencies of the registration of y-quanta in the detectors; b) there is no correlation between the gammas which coincide at random . Results of the test of the method applied to the multi-detector correlation system are presented . 1. Introduction Measurement of directional correlations of y-rays is performed by means of the multi-detector system . In refs. [1-14] one can find descriptions of various angular correlation apparatus with four or more detectors . The authors of ref. [ 15 ] present a procedure for measuring y-y correlations using a Compton suppressed array consisting of 20 Ge detectors (NORDBALL) . In conventional y-y directional correlations with two detectors, where the position of one detector is changed, the effects of different detector efficiencies are present only as a multiplicative factor which is the same for all angles. Therefore, evaluation of the measured angular distribution function can be made without knowledge of the detector efficiencies. Multi-detector arrangements have a number of known advantages in comparison with classical twodetector systems, especially for on-line or off-line short-lived activity measurements. But there are also disadvantages, particularly due to the different characteristics of the detectors and associated electronic circuits, which cause some problems in evaluation of the measured correlation . In this case only fixed detector positions are practically possible. Each pair of detectors represents one of the angles at which the coincident gammas emitted from the radioactive source can be measured. Therefore, the coincidence rate of this pair must be corrected by corresponding detector efficiency factors . Determination of the efficiency corrections strongly depends on the detector Present address : National Laboratory for High Energy Physics KEK, 1-1 Oho, Tsukuba, Ibaraki-ken 305, Japan . 0168-9002/93/$ 06 .00 © 1993 -
arrangement and the electronic design of the correlation apparatus in question, see refs. [1-6,9,11,14] . In this work we report on the method which allows us to involve the efficiency corrections deduced from the measurement of the random coincidence rates between particular detectors. The method uses the assumption that there is no correlation between gammas which coincide at random . So it is suitable for measurements of the radioactive source . On the other hand, it cannot be used in the case of in-beam experiments where the distribution of the measured random coincidences is not isotropic . The method was developed and tested on the multi-detector correlation apparatus (MCA) [12,13] . 2. The design of MCA The MCA was developed in the Laboratory of Nuclear Problems of JINR in Dubna, Russia for investigation of single y-spectra, prompt and delayed y-y spectra and directional correlations of y-rays emitted from radioactive nuclei in off-line and in on-line experiments as well. The apparatus consists of seven Ge (Li) detectors equally spaced on a circle; i.e. the angle between neighboring detectors is 51 .4° . The detectors used are true-coaxial type 40-49 cm3 in volume and with FWHM=2 .2-3.0 keV for 1332 keV 60Co y-rays . The source-detector cap distance is 75 mm which yields detector absolute efficiencies in the interval (2.9-6.3) x 10-4 for 1332 keV 60Co y-rays. The directional correlation function can be measured for angles of 51 .4°, 102 .8° and 154 .3° between the detectors, which is sufficient for the determination of the A22 and A44 parameters in the correlation formula.
Elsevier Science Publishers B.V. All rights reserved
568
D. Vénos, P. Tlusty / Efficiency measurements for y-y coincidence counting rates
The measurement proceeds on seven detector pairs for each correlation angle. The analog electronic circuits consist of the following units : preamplifiers, spectroscopic amplifiers, analog-digital converter with seven inputs - ADC (2048 channels), fast amplifiers, constant fraction discriminators, delay lines, special unit of coincidence and priority commutation of fast signals - CPC, and time to digital converter - TDC (2048 channels) . The unit CPC establishes the coincidence event within a preset resolution time (usually 150 ns) and commutes two corresponding fast signals to the START and STOP inputs of the TDC . The fast logical signal for the gating of ADC is also produced. The digital output of the CPC unit contains information about the two detectors which register the coincident y-rays (i.e. correlation angle) . The digital data for each coincident event from CPC, TDC and ADC (i.e. detector pair, time difference and two amplitudes) are transferred through a system of two flip-flop buffers and appropriate interfacing to a PC and then recorded sequentially on tape. Moreover, using a suitable computer program any set of coincidence events can be sorted off-line from the recorded tapes. Ignoring the time coordinate, the complete correlation data set from MCA can be imagined as the 21 E,-E, matrices, where E, and E, move in the region of energies registered in detector i and J, respectively . The coincidence measurement of 60 Co performed in the energy interval 50-1332 keV in both channels has shown that the FWHM of the prompt part of the peak in the time distribution summed over all detectors pairs is about 13 ns. As the apparatus is equipped with detectors of high resolution, directional correlations for a large number of cascades can be measured simultaneously in one experiment.
3. Efficiency corrections 3 .1 . Theoretical formulas and data processing
A schematic view of the correlation experiment performed by MCA is given in fig . 1 . The number of true coincidences in full energy peak for the Y, - Y2 cascade registered in detectors (i, j) (i 7~ J ; i, j = 1, 2, . .n) by means of MCA is given by the formula T(i,j,y,,Y2) _
r
aia2 WjEr(Yi)ej(Y2) a2 + A J 0
N (i) dT,
(1)
D
S
Fig. 1 . Fragment ofthe decay scheme with cascade Y,-Y2 and a corresponding correlation setup of the MCA . The a, and a 2 values represent the intensities of y, and Y2, respectively. where a,,
are the intensities of y,, Y2 per decay, the sum of intensities per decay for all transitions depopulating the intermediate nuclear state, F, (y, )F, (Y2) are the registration efficiencies of y, and Y2 quanta in detectors i and j, respectively, t is the time of the experiment in seconds, and N(i) the number of decays of the mother nuclei per second. In our case the number of detectors n equals 7 . In eq. (1) W, = A o [ 1 + A22Q22P2 (cos B,,) + A44Q44P4 (cos B ) J is the usual correlation function, where 0,, is the correlation angle, which corresponds to the detector pair (t, j) . Eq. (1) is derived under the assumption that the time window used at the determination of the true coincidence numbers can be taken sufficiently larger than the lifetime of the intermediate nuclear state. In the case of cylindrical symmetry of the detectors, the solid-angle correction factors (222, Q44 can be written as Q.U ( i, J, Yh Y2) = QA U, Y1 )QA G, Y2) . The quantities Q. (i, y) for Ge(Li) detectors are calculated according to Krane [ 16 J . In the calculation of the true coincidence counting rates T (t, j, y, , Y2 ), the subtraction of both the random coincidences and the contributions due to the coincidences with Compton background, which are present in coincidence spectra, has to be made accurately. The computer program sorts two twodimensional spectra (parts ofthe E,-E, matrix), containing the peak of interest and its near environment for the corresponding two time window setting. The first setting relates to the prompt peak and the second to the left and right wings of the time distribution. For both spectra the number of coincidences in the peak is obtained by subtracting the two-dimensional Compton background under the peak. A linear course of this background in spectra is assumed . Subtraction of the peak volume obtained for the second spectrum (normalized according to the time window widths) a2 + A is
a2
D . Vénos, P. Tlusty / Efficiency measurements for y-y coincidence counting rates
from the peak volume found in the first spectrum is needed to obtain the quantity T (i, j, yl, Y2) . Using the procedure described above, we assume that there are no tails on the time spectra that could give rise to a contribution of the true coincidences in the random part of the time distribution . The quantities T(i, j, YI , Y2) can be analyzed in terms of A zz and A 44 only if the fourteen numbers E, (YI ), El (Yz) - the efficiencies - are known. The number of random coincidences between y, registered in detector i and any y-ray detected in any other detector can be expressed by the relation A(i,Yi) = 2roa,E,(y,)
Jp
N (T)
EMm(T) dT, m=1
where ro is the time constant for the random coincidences in seconds, and Mm (r) the number offast signals per second from the mth detector chain reaching an input of the CPC unit. In full analogy with eq. (2) an equation for A (j, Y2) may be written. From such equations, written for all detectors, the efficiencies E, (y, ), El (Yz) will be taken into account in the following . Apparently there is no angular dependence in eq. (2) . This is not true for inbeam experiments where the distribution of the measured random coincidences is correlated in space. In fact, the quantity A (i, y, ) represents an area of the y, line observed in the summed spectrum of the random coincidences for detector t. In the calculation of the quantity A (i, y, ) the program sorts part of the spectrum of the coincidences between the y, -ray (and its near environment) registered in detector i and any y-ray detected in any other detector for the time window set outside the prompt peak. Of course, the analogous statement refers to the A (j, Y2) . The statistical accuracy of A (i, y, ) and A (j, Y2) is also determined by the widths of the time gates. The time gates for sorting must be reasonably large to avoid affecting the low y-ray energies due to the tails in the time spectrum . Let us define an experimental correlation function T (i, j, YI , Yz ) I(t,J,YI,Yz) = A(t,YI)A(j,Y2) and an expression
x,
=
n
J
N (r)
~M,n (i) dr.
(4)
For y, and Y2 registered in detectors i and j, respectively, and for each detector pair one can obtain an equation I(i,j,YI,Y2) = x,xJw(O,,,Ao,A22,A44),
where f N(T) dr W. . w(Brl,Aa,A22,A44) _ (2To)z(az+A)
(6)
From the set of 42 equations (5) the unknown supplementary efficiency corrections x, and quantities A., A22, A44 can be determined by minimization of the functional
[I(i,J,YI,Y2) - x,xJw(e ,Ao,A22,A44)] 2 , [AI (i,J,YI,Yz)12
(2)
m 7É i
569
where 4I ( i, j, YI, Y2) is the error which is calculated according to the definition of I (i,j, y,, Y2) in eq. (3) . The differences in the x, reflect some inequality of the detectors' chains . Due to the behavior of eq. (7) one x, can be set equal to 1 .0 before minimization. Moreover, the corrections x, are the same for all cascades measured in one experiment, see eq. (4) . Therefore, one usually uses the set ofx, obtained by means of a strong cascade for the weaker ones. In this case A., A22, A44 are determined by minimization of eq. (7) . There is a possibility ofa more straightforward approach - without introducing supplementary correction, viz. by using the random coincidence rate taken at full energy coincidence peak only. This rate depends directly on the product E, (YI )El (Y2) . However, the usual low rate in the peak for this case results in a large error of the efficiency correction making such a procedure unavailable . Use of the random coincidence rates between a given 7-quantum and the whole spectrum registered in one detector only, also does not tend to a satisfactory result . So, broadening by more detectors, as indicated in eq. (2), is important for the application of the proposed method. The relative errors of the experimental correlation functionâI(t,J,y,,Y2) =DI(t,J,Yl,Y2)/I(i,J,YI,Y2) propagate into the errors of the final result AA22, AA44 [17] . We define a value R as follows R
= 1: ST(i,J,YI,Y2)IE 8[A(i,Yl)A(J,Y2)1 .(8) 1,l=I 1711
1,l=I 101
Then R, which is actually the mean value taken over the detector pairs for the measured cascade, is a suitable factor for judging how the accuracies ofthe quantities A (i, y, ), A (j, Y2) contribute to the accuracy of the final result . For example, if R = 2, then the âi are larger on average by a factor of 1 .12 in comparison with the case of zero errors in A (i, y, ), A (J, Y2) .
D. VMos, P. Tlusty /Efficiency measurements for y-y coincidence counting rates
570
B)
A)
51 .4°
180 in
Fz r
160 - ++ 140
Fno
1
++
++
4 +
4 4
4
+
4 4
1,2 2,3 3,4 4,5 5,6 6,7 7,1 102.8°
O
+
U
r r 180
z w
z 160
4~
14
44
4
4
44
ij
140
1
1,2 2,3 3,4 4,5 5,6 6,7 7,1
m
102,8°
w
z 160 w U
w 0
a
160
a 180
w
z 140 ô U
z
r
g 180
51.4 °
180
+
++
z 160 ++
1,3 2,4 3,5 4,6 5,7 6,1
++
U
++
7,2
++
ô
++
140
++
1,3 2k
w 0
Z w
++
++
++
++
++
++
3,5 4,6 5,7 6,1
7,2
140
U r
154.3* ++
Ô U Z
1543°
180
z 160
++
++
++
++
++
++
++
140 1,4 2,5 3,6 4,7 5,1 6,2 7,3
1,4 2.5 3,6 4,7 5,1
DETECTOR PAIR
6,2 7,3
DETECTOR PAIR
Fig.
2. A) Dependence of the experimental correlation function I(i,l,y1,y2) for the 122-244 keV cascade at angles 51 .4°, 102.8° and 154.3° on the detector pair . B) the same as in A), but the quantities I(0, yl, y2) are divided by the product x,xj.
3.2. Experimental procedure
A liquid solution of 0.6 MBq 152 Eu was used to test the method . This source was chosen because its ylines cover a large energy range (122-1408 keV) and a useful comparison can be made as the angular correlations have been measured by many authors. The average singles counting rate for detectors reached up to 7000 s -1 . Altogether 5 x 10 8 coincidence events were detected by MCA in the 168 h experiment . During the evaluation the widths of the windows set on the peak and both wings of the time distribution were 66 and 210 ns, respectively . The rate of random coincidences for measured cascades was about 3% of that for the true ones . In table 1 a set of supplementary corrections x, for some cascades is listed. The differences between detector chains mentioned above can be seen . This effect is more obvious from fig. 2, which presents the dependence of both I (i, j, 122, 244) and I(i, j, 122, 244) / (x,x,) on the detector pairs corresponding to different correlation angles . It can also be seen that the correction within the group of equivalent detector pairs is correct (the values are constant
Table 1 Supplementary efficiency corrections x, obtained for some cascades observed in 152E u decay Cascade
xl
x2
x3
122- 244 122- 964 122-1112 122-1408 344- 778
1.000 1.000 1 .000 1.000 1 .000
1 .023 1 .013 1 .027 0.998 1 .039 1 .023 1 .010 0.998 1 .023 0.998
x4
x5
1 .018 0.976 1 .046 0 .986 1 .013 0.919 1 .007 0.964 1 .012 0.970
x6
x7
1 .013 1 .022 1.026 1.004 1.002
0.969 0.972 0.950 0.953 0.952
within uncertainties) . Table 2 presents the results obtained for coefficients A22 and A44 . The data of other authors concern only measurements made by means of high resolution detectors. Usually our results are consistent with existing data within experimental errors . Particularly, good agreement for the most intensive cascade 122-244 keV is observed . This also indicates that the quantities I (i, j, 122, 244) / (x,, x,) established for given correlation angles have the correct relative values. Values of R are also quoted in the table. For the
D. Vénos, P. Tlusty /Efficiency measurements for y-y coincidence counting rates Table 2 Results of angular correlations for cascades observed in 152Eu decay Cascade
Ra A 22
122- 245 2 .0 122- 964 2 .2 122-1112 2 .2 122-1408 2 .2 244- 867 1 .1 344- 411 1 .5 344- 778 1 .5 344-1090 6 .0 aFor
R,
A44
A22 [18] A44
A22 [11] A44
0 .099(3) 0 .004(9) 0 .000(9) 0 .287(14) -0 .290(8) -0 .109(16) 0 .235(5) 0 .000(13) 0 .148(10) -0 .194(17) 0 .113(11) 0 .006(19) -0 .073(4) 0 .004(8) -0 .206(19) 0 .020(39)
0 .1020 (15) 0 .003(3) 0 .017(5) 0 .320(22) -0 .260(10) -0 .082(14) 0 .227(8) 0 .003(8) 0 .101(5) 0 .015(9) -0 .063(10) -0 .010(25) -0 .243(19) 0 .015(33)
0 .090(13) -0 .012(15) -0 .020(20) 0 .340(54) -0 .239(30) -0.070(65) 0.172(31) 0.018(23) 0.144(8)b -0.185(13)b 0.104(9) 0 .015(13)° -0 .070(5)° 0 .002(7)° -0 .177(16)° -0 .062(24)°
see text . bValues from [19] . 'Values from [20] .
152Gd cascade 344-1090 keV, R = 6.0. Such a large value is caused by the presence of the y-line 1086 keV from' 52Sm, which was included in the number of random coincidences A (i, 1090) due to the close proximity to the 1090 keV y-line . In principle, for some cascades with small values of R a re-evaluation of the A22 and A 44 can be made . Instead of A (i, y) with a large error the relative efficiency is used in the usual manner. The necessary relative efficiencies are deduced from A (i, y) for the strong neighboring y-transitions . Corresponding y-ray intensities are taken from nuclear data tables . 4. Conclusions A new method for efficiency corrections applicable to measurement by means of multi-detector systems based on random coincidences is suggested and tested . The angular correlation data and the data necessary for efficiency correction are collected simultaneously, using the same detector chains and radioactive source for which the correlations are measured . Such a procedure has the following advantages : (1) it is not necessary to measure the efficiencies of the detectors in separate experiments ; (2) the appropriate part of the experimental data, obtained for the y-rays for which the directional correlation is measured, is used directly in the calculation of the registration efficiencies . In the example of 152 Eu decay it was shown that even at medium strength of the radioactive sources, where the number of random coincidences is rather small, the proposed method gives trustworthy results
571
for the angular correlation coefficients measured by the seven detector system . Acknowledgements The technical assistance of P . taloun, O .D. Kestarova and V .I . Stegailov is gratefully acknowledged . We wish to thank Dr. V.A. Morozov for helpful comments . Dr . Z. Dlouhy is highly appreciated for careful reading of the manuscript and for many excellent ideas improving the text . References [ 1 ] A.I . Belyaevskii and Yu.A . Gurjan, Izv. Akad. Nauk SSSR, Ser . Fiz . 25 (1961) 1291 . [2] B .E. Karlsson, E. Matthias and C .A . Ledefors, Ark . Fys . 22 (1962) 27 . [ 3 ] T . Hayashi, K . Okano, K. Yyasa, Y . Kawase and S .A . Uehara, Nucl. Instr. and Meth. 53 (1967) 123 . [4] H.K. Walter and A. Weitsch, Nucl . Instr. and Meth . 62 (1968) 189 . [5] J .T. Larsen, W.C. Schick, W .L. Talbert and D .I . Haddad, Nucl . Instr . and Meth . 69 (1969) 229. [6] T.R . Gerholm, Z .H. Cho, L . Erikson, L . Gidefeldt and B .G . Petterson, Nucl. Instr. and Meth . 100 (1972) 33 . [7] L . Eriksson and L . Gidefeldt, Nucl. Instr . and Meth. 114 (1974) 127 . [ 8 ] T. Hayashi, S. Uehara and T . Seo, Nucl . Instr . and Meth . 118 (1974) 541 . [9] G .J . Basinger, W .C . Schick and W.L . Talbert, Nucl. Instr. and Meth . 124 (1974) 381 . [10] E .W . Schneider, D.M. Glascock and W.B . Walters, Phys. Rev . C 19 (1979) 1025 . [ 11 ] A . Wolf, C . Chung, W.B . Walters, G . Peaslee, R .L. Gill, M. Shmid, V . Mazella, E. Meier, M.L . Stelts, H .I. Lisce, R.E . Chrien, Nucl . Instr . and Meth. 206 (1983) 397. [12] V.N . Abrosimov, I . Adam, D . Vasilev, D . Venos, Z . Gons, M . Gonusek, I. Gradec, K.Ya . Gromov, A.I. Kalinin, V .G. Kalinnikov, M .I . Krivopustov, G. Lizurei, S.V . Medved, S .I . Merzlyakov, A . Misiak, V.A. Morozov, F . Prazhak, V .I. Razov, D. Stuka, V .I. Stegailov, P. Chaloun and F. Foret, Preprint JINR, Dubna, P6-86-320 (1986) (in Russian) . [13] V.G. Kalinnikov, K.Ya . Gromov, M . Janicki, Yu.V . Yushkevich, A.W. Potempa, V .G . Egorov, V.A. Bystrov, N .Ya. Kotovsky and S .V . Evtisov, Nucl . Instr. and Meth . B 70 (1992) 62 . [14] I. Alfter, E . Bodenstedt, B. Hamen, J . van den Hoff, W . Knickee, H . Mi1nning, S. Piel, J . Schiith and R . Sajok, Nucl . Instr. and Meth . A 321 (1992) 506 . [ 15 ] L .P . Ekstr6m and A . Nordlund, Nucl. Instr . and Meth . A 313 (1992) 421 . [ 16 ] K.S . Krane, Nucl . Instr. and Meth. 98 (1972) 205 . [ 171 D. Vénos, PhD Thesis, Dubna, 1991 . [ 18 ] J . Barrete, M . Barrete, A. Boutard, G. Lamoureux and S . Monaro, Can. J . Phys . 48 (1970) 2011 . [ 19 ] J . Barrete, M . Barrete, R. Horoutunian, G . Lamoureux and S . Monaro, Phys . Rev . C 4 (1971) 991 . [20] H. Helpi, A . Pakhanen and J . Hattula, Nucl . Phys . A 247 (1975) 317 .
A 336 (1993) 567-571
NUCLEAR INSTRUMENTS & METHODS IN PHYSICS RESEARCH Section A
Efficiency corrections for the y-y coincidence counting rates measured by the multi-detector correlation system D. Vénos and P. Tlusty t
Nuclear Physics Institute, Academy of Sciences of the Czech Republic, 250 68 Aez, Czech Republic
Received 7 May
1993
and in revised form 12 July
1993
A new method of determination of the efficiency corrections for the y-y coincidence rates measured by the multi-detector system is described. The method uses the random coincidence counting rates and is based on two assumptions: a) the counting rates ofboth true and random coincidences for a given pair of 7-quanta are proportional to the efficiencies of the registration of y-quanta in the detectors; b) there is no correlation between the gammas which coincide at random . Results of the test of the method applied to the multi-detector correlation system are presented . 1. Introduction Measurement of directional correlations of y-rays is performed by means of the multi-detector system . In refs. [1-14] one can find descriptions of various angular correlation apparatus with four or more detectors . The authors of ref. [ 15 ] present a procedure for measuring y-y correlations using a Compton suppressed array consisting of 20 Ge detectors (NORDBALL) . In conventional y-y directional correlations with two detectors, where the position of one detector is changed, the effects of different detector efficiencies are present only as a multiplicative factor which is the same for all angles. Therefore, evaluation of the measured angular distribution function can be made without knowledge of the detector efficiencies. Multi-detector arrangements have a number of known advantages in comparison with classical twodetector systems, especially for on-line or off-line short-lived activity measurements. But there are also disadvantages, particularly due to the different characteristics of the detectors and associated electronic circuits, which cause some problems in evaluation of the measured correlation . In this case only fixed detector positions are practically possible. Each pair of detectors represents one of the angles at which the coincident gammas emitted from the radioactive source can be measured. Therefore, the coincidence rate of this pair must be corrected by corresponding detector efficiency factors . Determination of the efficiency corrections strongly depends on the detector Present address : National Laboratory for High Energy Physics KEK, 1-1 Oho, Tsukuba, Ibaraki-ken 305, Japan . 0168-9002/93/$ 06 .00 © 1993 -
arrangement and the electronic design of the correlation apparatus in question, see refs. [1-6,9,11,14] . In this work we report on the method which allows us to involve the efficiency corrections deduced from the measurement of the random coincidence rates between particular detectors. The method uses the assumption that there is no correlation between gammas which coincide at random . So it is suitable for measurements of the radioactive source . On the other hand, it cannot be used in the case of in-beam experiments where the distribution of the measured random coincidences is not isotropic . The method was developed and tested on the multi-detector correlation apparatus (MCA) [12,13] . 2. The design of MCA The MCA was developed in the Laboratory of Nuclear Problems of JINR in Dubna, Russia for investigation of single y-spectra, prompt and delayed y-y spectra and directional correlations of y-rays emitted from radioactive nuclei in off-line and in on-line experiments as well. The apparatus consists of seven Ge (Li) detectors equally spaced on a circle; i.e. the angle between neighboring detectors is 51 .4° . The detectors used are true-coaxial type 40-49 cm3 in volume and with FWHM=2 .2-3.0 keV for 1332 keV 60Co y-rays . The source-detector cap distance is 75 mm which yields detector absolute efficiencies in the interval (2.9-6.3) x 10-4 for 1332 keV 60Co y-rays. The directional correlation function can be measured for angles of 51 .4°, 102 .8° and 154 .3° between the detectors, which is sufficient for the determination of the A22 and A44 parameters in the correlation formula.
Elsevier Science Publishers B.V. All rights reserved
568
D. Vénos, P. Tlusty / Efficiency measurements for y-y coincidence counting rates
The measurement proceeds on seven detector pairs for each correlation angle. The analog electronic circuits consist of the following units : preamplifiers, spectroscopic amplifiers, analog-digital converter with seven inputs - ADC (2048 channels), fast amplifiers, constant fraction discriminators, delay lines, special unit of coincidence and priority commutation of fast signals - CPC, and time to digital converter - TDC (2048 channels) . The unit CPC establishes the coincidence event within a preset resolution time (usually 150 ns) and commutes two corresponding fast signals to the START and STOP inputs of the TDC . The fast logical signal for the gating of ADC is also produced. The digital output of the CPC unit contains information about the two detectors which register the coincident y-rays (i.e. correlation angle) . The digital data for each coincident event from CPC, TDC and ADC (i.e. detector pair, time difference and two amplitudes) are transferred through a system of two flip-flop buffers and appropriate interfacing to a PC and then recorded sequentially on tape. Moreover, using a suitable computer program any set of coincidence events can be sorted off-line from the recorded tapes. Ignoring the time coordinate, the complete correlation data set from MCA can be imagined as the 21 E,-E, matrices, where E, and E, move in the region of energies registered in detector i and J, respectively . The coincidence measurement of 60 Co performed in the energy interval 50-1332 keV in both channels has shown that the FWHM of the prompt part of the peak in the time distribution summed over all detectors pairs is about 13 ns. As the apparatus is equipped with detectors of high resolution, directional correlations for a large number of cascades can be measured simultaneously in one experiment.
3. Efficiency corrections 3 .1 . Theoretical formulas and data processing
A schematic view of the correlation experiment performed by MCA is given in fig . 1 . The number of true coincidences in full energy peak for the Y, - Y2 cascade registered in detectors (i, j) (i 7~ J ; i, j = 1, 2, . .n) by means of MCA is given by the formula T(i,j,y,,Y2) _
r
aia2 WjEr(Yi)ej(Y2) a2 + A J 0
N (i) dT,
(1)
D
S
Fig. 1 . Fragment ofthe decay scheme with cascade Y,-Y2 and a corresponding correlation setup of the MCA . The a, and a 2 values represent the intensities of y, and Y2, respectively. where a,,
are the intensities of y,, Y2 per decay, the sum of intensities per decay for all transitions depopulating the intermediate nuclear state, F, (y, )F, (Y2) are the registration efficiencies of y, and Y2 quanta in detectors i and j, respectively, t is the time of the experiment in seconds, and N(i) the number of decays of the mother nuclei per second. In our case the number of detectors n equals 7 . In eq. (1) W, = A o [ 1 + A22Q22P2 (cos B,,) + A44Q44P4 (cos B ) J is the usual correlation function, where 0,, is the correlation angle, which corresponds to the detector pair (t, j) . Eq. (1) is derived under the assumption that the time window used at the determination of the true coincidence numbers can be taken sufficiently larger than the lifetime of the intermediate nuclear state. In the case of cylindrical symmetry of the detectors, the solid-angle correction factors (222, Q44 can be written as Q.U ( i, J, Yh Y2) = QA U, Y1 )QA G, Y2) . The quantities Q. (i, y) for Ge(Li) detectors are calculated according to Krane [ 16 J . In the calculation of the true coincidence counting rates T (t, j, y, , Y2 ), the subtraction of both the random coincidences and the contributions due to the coincidences with Compton background, which are present in coincidence spectra, has to be made accurately. The computer program sorts two twodimensional spectra (parts ofthe E,-E, matrix), containing the peak of interest and its near environment for the corresponding two time window setting. The first setting relates to the prompt peak and the second to the left and right wings of the time distribution. For both spectra the number of coincidences in the peak is obtained by subtracting the two-dimensional Compton background under the peak. A linear course of this background in spectra is assumed . Subtraction of the peak volume obtained for the second spectrum (normalized according to the time window widths) a2 + A is
a2
D . Vénos, P. Tlusty / Efficiency measurements for y-y coincidence counting rates
from the peak volume found in the first spectrum is needed to obtain the quantity T (i, j, yl, Y2) . Using the procedure described above, we assume that there are no tails on the time spectra that could give rise to a contribution of the true coincidences in the random part of the time distribution . The quantities T(i, j, YI , Y2) can be analyzed in terms of A zz and A 44 only if the fourteen numbers E, (YI ), El (Yz) - the efficiencies - are known. The number of random coincidences between y, registered in detector i and any y-ray detected in any other detector can be expressed by the relation A(i,Yi) = 2roa,E,(y,)
Jp
N (T)
EMm(T) dT, m=1
where ro is the time constant for the random coincidences in seconds, and Mm (r) the number offast signals per second from the mth detector chain reaching an input of the CPC unit. In full analogy with eq. (2) an equation for A (j, Y2) may be written. From such equations, written for all detectors, the efficiencies E, (y, ), El (Yz) will be taken into account in the following . Apparently there is no angular dependence in eq. (2) . This is not true for inbeam experiments where the distribution of the measured random coincidences is correlated in space. In fact, the quantity A (i, y, ) represents an area of the y, line observed in the summed spectrum of the random coincidences for detector t. In the calculation of the quantity A (i, y, ) the program sorts part of the spectrum of the coincidences between the y, -ray (and its near environment) registered in detector i and any y-ray detected in any other detector for the time window set outside the prompt peak. Of course, the analogous statement refers to the A (j, Y2) . The statistical accuracy of A (i, y, ) and A (j, Y2) is also determined by the widths of the time gates. The time gates for sorting must be reasonably large to avoid affecting the low y-ray energies due to the tails in the time spectrum . Let us define an experimental correlation function T (i, j, YI , Yz ) I(t,J,YI,Yz) = A(t,YI)A(j,Y2) and an expression
x,
=
n
J
N (r)
~M,n (i) dr.
(4)
For y, and Y2 registered in detectors i and j, respectively, and for each detector pair one can obtain an equation I(i,j,YI,Y2) = x,xJw(O,,,Ao,A22,A44),
where f N(T) dr W. . w(Brl,Aa,A22,A44) _ (2To)z(az+A)
(6)
From the set of 42 equations (5) the unknown supplementary efficiency corrections x, and quantities A., A22, A44 can be determined by minimization of the functional
[I(i,J,YI,Y2) - x,xJw(e ,Ao,A22,A44)] 2 , [AI (i,J,YI,Yz)12
(2)
m 7É i
569
where 4I ( i, j, YI, Y2) is the error which is calculated according to the definition of I (i,j, y,, Y2) in eq. (3) . The differences in the x, reflect some inequality of the detectors' chains . Due to the behavior of eq. (7) one x, can be set equal to 1 .0 before minimization. Moreover, the corrections x, are the same for all cascades measured in one experiment, see eq. (4) . Therefore, one usually uses the set ofx, obtained by means of a strong cascade for the weaker ones. In this case A., A22, A44 are determined by minimization of eq. (7) . There is a possibility ofa more straightforward approach - without introducing supplementary correction, viz. by using the random coincidence rate taken at full energy coincidence peak only. This rate depends directly on the product E, (YI )El (Y2) . However, the usual low rate in the peak for this case results in a large error of the efficiency correction making such a procedure unavailable . Use of the random coincidence rates between a given 7-quantum and the whole spectrum registered in one detector only, also does not tend to a satisfactory result . So, broadening by more detectors, as indicated in eq. (2), is important for the application of the proposed method. The relative errors of the experimental correlation functionâI(t,J,y,,Y2) =DI(t,J,Yl,Y2)/I(i,J,YI,Y2) propagate into the errors of the final result AA22, AA44 [17] . We define a value R as follows R
= 1: ST(i,J,YI,Y2)IE 8[A(i,Yl)A(J,Y2)1 .(8) 1,l=I 1711
1,l=I 101
Then R, which is actually the mean value taken over the detector pairs for the measured cascade, is a suitable factor for judging how the accuracies ofthe quantities A (i, y, ), A (j, Y2) contribute to the accuracy of the final result . For example, if R = 2, then the âi are larger on average by a factor of 1 .12 in comparison with the case of zero errors in A (i, y, ), A (J, Y2) .
D. VMos, P. Tlusty /Efficiency measurements for y-y coincidence counting rates
570
B)
A)
51 .4°
180 in
Fz r
160 - ++ 140
Fno
1
++
++
4 +
4 4
4
+
4 4
1,2 2,3 3,4 4,5 5,6 6,7 7,1 102.8°
O
+
U
r r 180
z w
z 160
4~
14
44
4
4
44
ij
140
1
1,2 2,3 3,4 4,5 5,6 6,7 7,1
m
102,8°
w
z 160 w U
w 0
a
160
a 180
w
z 140 ô U
z
r
g 180
51.4 °
180
+
++
z 160 ++
1,3 2,4 3,5 4,6 5,7 6,1
++
U
++
7,2
++
ô
++
140
++
1,3 2k
w 0
Z w
++
++
++
++
++
++
3,5 4,6 5,7 6,1
7,2
140
U r
154.3* ++
Ô U Z
1543°
180
z 160
++
++
++
++
++
++
++
140 1,4 2,5 3,6 4,7 5,1 6,2 7,3
1,4 2.5 3,6 4,7 5,1
DETECTOR PAIR
6,2 7,3
DETECTOR PAIR
Fig.
2. A) Dependence of the experimental correlation function I(i,l,y1,y2) for the 122-244 keV cascade at angles 51 .4°, 102.8° and 154.3° on the detector pair . B) the same as in A), but the quantities I(0, yl, y2) are divided by the product x,xj.
3.2. Experimental procedure
A liquid solution of 0.6 MBq 152 Eu was used to test the method . This source was chosen because its ylines cover a large energy range (122-1408 keV) and a useful comparison can be made as the angular correlations have been measured by many authors. The average singles counting rate for detectors reached up to 7000 s -1 . Altogether 5 x 10 8 coincidence events were detected by MCA in the 168 h experiment . During the evaluation the widths of the windows set on the peak and both wings of the time distribution were 66 and 210 ns, respectively . The rate of random coincidences for measured cascades was about 3% of that for the true ones . In table 1 a set of supplementary corrections x, for some cascades is listed. The differences between detector chains mentioned above can be seen . This effect is more obvious from fig. 2, which presents the dependence of both I (i, j, 122, 244) and I(i, j, 122, 244) / (x,x,) on the detector pairs corresponding to different correlation angles . It can also be seen that the correction within the group of equivalent detector pairs is correct (the values are constant
Table 1 Supplementary efficiency corrections x, obtained for some cascades observed in 152E u decay Cascade
xl
x2
x3
122- 244 122- 964 122-1112 122-1408 344- 778
1.000 1.000 1 .000 1.000 1 .000
1 .023 1 .013 1 .027 0.998 1 .039 1 .023 1 .010 0.998 1 .023 0.998
x4
x5
1 .018 0.976 1 .046 0 .986 1 .013 0.919 1 .007 0.964 1 .012 0.970
x6
x7
1 .013 1 .022 1.026 1.004 1.002
0.969 0.972 0.950 0.953 0.952
within uncertainties) . Table 2 presents the results obtained for coefficients A22 and A44 . The data of other authors concern only measurements made by means of high resolution detectors. Usually our results are consistent with existing data within experimental errors . Particularly, good agreement for the most intensive cascade 122-244 keV is observed . This also indicates that the quantities I (i, j, 122, 244) / (x,, x,) established for given correlation angles have the correct relative values. Values of R are also quoted in the table. For the
D. Vénos, P. Tlusty /Efficiency measurements for y-y coincidence counting rates Table 2 Results of angular correlations for cascades observed in 152Eu decay Cascade
Ra A 22
122- 245 2 .0 122- 964 2 .2 122-1112 2 .2 122-1408 2 .2 244- 867 1 .1 344- 411 1 .5 344- 778 1 .5 344-1090 6 .0 aFor
R,
A44
A22 [18] A44
A22 [11] A44
0 .099(3) 0 .004(9) 0 .000(9) 0 .287(14) -0 .290(8) -0 .109(16) 0 .235(5) 0 .000(13) 0 .148(10) -0 .194(17) 0 .113(11) 0 .006(19) -0 .073(4) 0 .004(8) -0 .206(19) 0 .020(39)
0 .1020 (15) 0 .003(3) 0 .017(5) 0 .320(22) -0 .260(10) -0 .082(14) 0 .227(8) 0 .003(8) 0 .101(5) 0 .015(9) -0 .063(10) -0 .010(25) -0 .243(19) 0 .015(33)
0 .090(13) -0 .012(15) -0 .020(20) 0 .340(54) -0 .239(30) -0.070(65) 0.172(31) 0.018(23) 0.144(8)b -0.185(13)b 0.104(9) 0 .015(13)° -0 .070(5)° 0 .002(7)° -0 .177(16)° -0 .062(24)°
see text . bValues from [19] . 'Values from [20] .
152Gd cascade 344-1090 keV, R = 6.0. Such a large value is caused by the presence of the y-line 1086 keV from' 52Sm, which was included in the number of random coincidences A (i, 1090) due to the close proximity to the 1090 keV y-line . In principle, for some cascades with small values of R a re-evaluation of the A22 and A 44 can be made . Instead of A (i, y) with a large error the relative efficiency is used in the usual manner. The necessary relative efficiencies are deduced from A (i, y) for the strong neighboring y-transitions . Corresponding y-ray intensities are taken from nuclear data tables . 4. Conclusions A new method for efficiency corrections applicable to measurement by means of multi-detector systems based on random coincidences is suggested and tested . The angular correlation data and the data necessary for efficiency correction are collected simultaneously, using the same detector chains and radioactive source for which the correlations are measured . Such a procedure has the following advantages : (1) it is not necessary to measure the efficiencies of the detectors in separate experiments ; (2) the appropriate part of the experimental data, obtained for the y-rays for which the directional correlation is measured, is used directly in the calculation of the registration efficiencies . In the example of 152 Eu decay it was shown that even at medium strength of the radioactive sources, where the number of random coincidences is rather small, the proposed method gives trustworthy results
571
for the angular correlation coefficients measured by the seven detector system . Acknowledgements The technical assistance of P . taloun, O .D. Kestarova and V .I . Stegailov is gratefully acknowledged . We wish to thank Dr. V.A. Morozov for helpful comments . Dr . Z. Dlouhy is highly appreciated for careful reading of the manuscript and for many excellent ideas improving the text . References [ 1 ] A.I . Belyaevskii and Yu.A . Gurjan, Izv. Akad. Nauk SSSR, Ser . Fiz . 25 (1961) 1291 . [2] B .E. Karlsson, E. Matthias and C .A . Ledefors, Ark . Fys . 22 (1962) 27 . [ 3 ] T . Hayashi, K . Okano, K. Yyasa, Y . Kawase and S .A . Uehara, Nucl. Instr. and Meth. 53 (1967) 123 . [4] H.K. Walter and A. Weitsch, Nucl . Instr. and Meth . 62 (1968) 189 . [5] J .T. Larsen, W.C. Schick, W .L. Talbert and D .I . Haddad, Nucl . Instr . and Meth . 69 (1969) 229. [6] T.R . Gerholm, Z .H. Cho, L . Erikson, L . Gidefeldt and B .G . Petterson, Nucl. Instr. and Meth . 100 (1972) 33 . [7] L . Eriksson and L . Gidefeldt, Nucl. Instr . and Meth. 114 (1974) 127 . [ 8 ] T. Hayashi, S. Uehara and T . Seo, Nucl . Instr . and Meth . 118 (1974) 541 . [9] G .J . Basinger, W .C . Schick and W.L . Talbert, Nucl. Instr. and Meth . 124 (1974) 381 . [10] E .W . Schneider, D.M. Glascock and W.B . Walters, Phys. Rev . C 19 (1979) 1025 . [ 11 ] A . Wolf, C . Chung, W.B . Walters, G . Peaslee, R .L. Gill, M. Shmid, V . Mazella, E. Meier, M.L . Stelts, H .I. Lisce, R.E . Chrien, Nucl . Instr . and Meth. 206 (1983) 397. [12] V.N . Abrosimov, I . Adam, D . Vasilev, D . Venos, Z . Gons, M . Gonusek, I. Gradec, K.Ya . Gromov, A.I. Kalinin, V .G. Kalinnikov, M .I . Krivopustov, G. Lizurei, S.V . Medved, S .I . Merzlyakov, A . Misiak, V.A. Morozov, F . Prazhak, V .I. Razov, D. Stuka, V .I. Stegailov, P. Chaloun and F. Foret, Preprint JINR, Dubna, P6-86-320 (1986) (in Russian) . [13] V.G. Kalinnikov, K.Ya . Gromov, M . Janicki, Yu.V . Yushkevich, A.W. Potempa, V .G . Egorov, V.A. Bystrov, N .Ya. Kotovsky and S .V . Evtisov, Nucl . Instr. and Meth . B 70 (1992) 62 . [14] I. Alfter, E . Bodenstedt, B. Hamen, J . van den Hoff, W . Knickee, H . Mi1nning, S. Piel, J . Schiith and R . Sajok, Nucl . Instr. and Meth . A 321 (1992) 506 . [ 15 ] L .P . Ekstr6m and A . Nordlund, Nucl. Instr . and Meth . A 313 (1992) 421 . [ 16 ] K.S . Krane, Nucl . Instr. and Meth. 98 (1972) 205 . [ 171 D. Vénos, PhD Thesis, Dubna, 1991 . [ 18 ] J . Barrete, M . Barrete, A. Boutard, G. Lamoureux and S . Monaro, Can. J . Phys . 48 (1970) 2011 . [ 19 ] J . Barrete, M . Barrete, R. Horoutunian, G . Lamoureux and S . Monaro, Phys . Rev . C 4 (1971) 991 . [20] H. Helpi, A . Pakhanen and J . Hattula, Nucl . Phys . A 247 (1975) 317 .