An improved Description of Separation and Performance Capabilities of Liquid Scintillation Counters Used in Dual Isotope Studies PETER D. KLEIN and WILLIAM J. EISLER, JR. Division o f Biological and Medical Research, Argonne National Laboratory, Argonne, 111.
b A new figure of merit for liquid scintillation counters, expressing their ability to separate two beta spectra and thus to determine the concentrations of each isotope is derived. This separation efficiency is used to describe scintillation counters in an overall performance number which i s the product of the separation efficiency and each of the two isotope detection efficiencies at the point of optimum separation. The pure beta spectra of tritium and radiocarbon are used to predict the limiting performance numbers attainable in counter design and these numbers are compared with several examples obtained from commercial counters now on the market.
T
usual characterization of the separation of two isotope spectra by a liquid scintillation counter is given in the form of two efficiencies for each isotope: in the window or channel in which it is customarily measured as well as the channel in which the other isotope is measured. These parameters provide, at best, a qualitative indication of the separation ability because the efficiency with which one isotope can be measured is a complicated function of the contamination by the other isotope. Although there have been several extensive treatments of the error function and optimum counting conditions for dual isotope samples ( I , Z), no analysis appears to have been carried out of the theoretical performance that might be expected from the counter itself, nor has this performance been compared with that of present-day instruments. This report undertakes such an analysis and proposes two criteria; S , the separation efficiency and P , the performance number, a product of the efficiencies for each isotope and the separation efficiency. HE
THEOREllCAL
Separation. If a n instrument having two channels ( A $ ) is used to measure isotopes 1 and 2, the ultimate separation is attained when all of iso-
tope 1 appears in channel A (none appearing in channel B ) and all of isotope 2 appears in channel B with no contamination of the first channel. If the separation is incomplete, isotope 1 will appear in channel B; and isotope 2, in channel A . The mechanisms responsible for incomplete separation (physical overlap of spectra, quenching, instrument design, and operation) are of no consequence at this time: their effects are identical on the computation necessary to determine the content of each isotope. When there is incomplete separation, the concentration of each isotope must be computed from the total activity appearing in each channel and from the knowledge of the ratio of counts in the two channels for each isotope when they are present in pure form. This technique is usually called the “channels ratio” method (3, 4). In Table I are listed the definitions of parameters that are experimentally determined and their magnitude is compared for the situation where separation is complete with the case where separation is incomplete. The equations for the concentrations of each isotope and of their ratio are directly comparable to expressions previously
Table 1.
given by Okita et al. (4) and by Kabara and coworkers (S), but it is now desired to examine the manner in which the observed channel ratio varies with the isotope ratio of the sample. Where&s the observed channel ratio and the isotope ratio are identical when separation is complete, Equation 4 shows that for incomplete separation, the channel ratio changes in a complex manner with both the isotope ratio of the sample and with the channel ratios for the pure isotopes. Furthermore, complete separation leads to a proportionate response in the channel ratio to a change in the isotope ratio over all ranges of mixtures, while this is not the case for incompletely resolved spectra (Equation 5).
Figure 1 shows the consequence of incomplete separation upon the relationship of the observed channel ratio, R., to the true isotope ratio, Ri. The dotted line indicates the ideal-Le., complete-separation, while the solid curves represent various pairs of symmetrical separations. Under these circumstances, the separation of isotope 1 from isotope 2 is equal to the separation of isotope 2 from isotope 1, or: Rz = ~ / R I . Figure 1 indicates that the
Comparison of Computations When Separation Is Complete and Incomplete
d.p.m. Definitions:a Channel A Channel B Ri Isotope 1 a1 bi bdal Isotope 2 a2 bz Mixed sample a b Complete Quantity separation Isotope 1, d.p.m.
ai = a
Rz
R.
bzlap bla
Incomplete separation Rz - R, a1 =aRz - R I
Isotope 2, d.p.m. Isotope ratio, Rt Relationship of R, to Rt
4
The efficiency of detection is taken to be 100% for both isotopes.
VOL 38, NO. 1 1 , OCTOBER 1966
1453
10.0
1001
I
I
I
,
,
,
,
,
,
50
Rs 1.0
0
I I
10 Po 0.1 0.5
so
100 Re 0.01 RI
0.2
CHANNEL RATIO [Re* I/RI]
1
I
0.1 0.1
Figure 2. Change in per cent separation as a function of R 1 , 2 for symmetrical ratios
I.o
IO
Ri
Figure 1. Changes in the observed channel ratio of a mixed sample (R,) (taken as counts channel Blcounts channel A ) for a range of isotope ratios in the sample (isotope 2/isotope 1 ) Dotted line complete separation; solid lines: relationship between and R i for varlous Symmetrical pairs of R1, R1
change in R, is symmetrical about the point where both isotopes are present in equal concentrations-i,e,, the closer the isotope ratio is to unity, the more closely the observed ratio reflects changes in the isotope ratio. I t is also evident that over a range of isotope ratios from 1:lO to lO:l, the ideal response of the observed channel ratio changes 100 fold (from 0.1 to 10.0). If this is used as a basis of comparison one may express the response of the channel ratio from incompletely resolved spectra in the same manner and obtain a separation efficiency (S) which is expressed as per cent theoretical separation and is defined by the equation :
For example, with R1 = 0.1, Rz = 10.0, the observed channel ratios are 0.198 and 5.050 (from Equation 4) for isotope ratios of 1: 10 and 10: 1, respectively. Thus, for a 100-fold change in isotope ratio, the channel ratio changes by a factor of 25.50 and the value for S obtained from Equation 6 is 25.5'%. Figure 2 illustrates the changes in separation efficiency as a per cent theoretical separation for a range of symmetrical channel ratios from (1, 1) to (loo, 0.01). If the spectra of the two isotopes are not comparable in their energy distributions but differ considerably in range, the separations may not be symmetrical; that is, it may be possible to separate isotope 1 from isotope 2 but not isotope 1454
ANALYTICAL CHEMISTRY
R.
2 from isotope 1. Under these circumstances, one channel ratio may be obtained which approximates complete separation but the other isotope cannot be restricted to a single channel. Figure 3 shows the case where R1 is zero but Rz is a value less than infinite. Below isotope ratios of 1: 1, the behavior of the observed channel ratio approximates that of the isotope ratio of the sample, but there is a gradually increasing discrepancy as the proportion of isotope 2 increases. The separation efficiency
c
t
may be calculated in the same manner; however, in comparison to the example cited above, for R1 = 0.00, Rz = 10.00, S has the value 50.5. The complete exclusion of isotope 1 from channel B has thereby resulted in a two-fold improvement in the value of S. A table with 378 combinations of channel ratios, covering most of the commonly encountered values, has been computed and is shown in Table 11. Theoretical Values for Separation of H3and C14 Spectra. Because tritium and radiocarbon are the most commonly used pair of isotopes in liquid scintillation counting, an examination of the limiting separation efficiency to be expected is of interest. The distribution of 106 d.p.m. each of H3 and G I 4 were computed at 0.5-kv. intervals from 0 to 157 kv. This distribution makes the explicit
I
I
/an
I .o
Ri
Figure 3.
Relationship of R, to Ri for R, = 0, R2 variable
assumption that light output is directly proportional to the energy of the beta particle and that pulse height is directly proportionate to light output. The region above 18.5 kv. contains 86.08% of the C14 disintegrations and none of the Ha. Considering a spectrometer in which the low discriminator of channel B was set a t 18.5 kv. and the upper discriminator at 157 kv., the B channel would contain 860,800 d.p.m., all of which would arise from C". Channel A , having discriminators set at 0.0 and 18.5kv. would contain 1,000,000 d.p.m. H3 and 139,200 d.p.m. C14. From these values, R1 = 0.00, Rz = 6.18, and S is determined to be 38.83. The value of S increases as the upper channel of A is reduced, resulting in an increase in the relative proportion of H3 to C14 in channel A and (simultaneously) an increase in the ratio Ra. The value of S attained with the narrowest channel computed (0.0 to 0.5 kv.) was 99.32y0. Performance. The degree of separation between the H3 and C14 spectra can be controlled by the experimenter over a wide range by his selection of discriminator settings for each channel. Enhancement of the degree of separation has a price, however, which is paid in a loss of efficiency-ie., a decrease in the number of disintegrations occurring in the interval between lower and upper discriminators. If the performance of an instrument is to be described, the separation efficiency must be linked to the efficiency with which each isotope is detected at that degree of separation. We therefore define the performance number P as
1 500
[ ob'*
.:. . '
'
'
'
5
'
'
'
'
IO I '
'
'
'
15 I '
'
KILOVOLTS
'
'
0
Figure 4. Separation efficiency, tritium detection, and performance numbers for a hypothetical spectrometer as a function of the discriminator settings in the lower channel
For the hypothetical spectrometer with perfect detection ability, these efficiencies are the areas under the segments of the spectra considered. Figure 4 illustrates the performance of such a spectrometer in which the lower d i s criminator of channel A is set at 0 kv. and the upper discriminator is reduced from 18.5 kv. in 0.5kv. increments. Channel B remains constant at 18.5 to 157 kv. and, as indicated earlier, contains 86.0870 of the C14 disintegration. At the highest setting for the upper discriminator on channel A , the performance is given by
P -
(38.83)(100.0)(86.08) 1000
= 330.9
Progressive reduction in the setting of the upper discriminator results in an increase in P, largely as a consequence of the gain in S in a region where the
efficiency for H 3 is decreasing very slowly. Below 10 kv., however, the loss in H3 efficiency begins to grow a t a faster rate than the increase in S and the performance declines rapidly. The maximum performance number attained in this range is 436.1. Thi? number represents the optimum performance of a hypothetical instrument in which quenching, light losses, thermal noise, and background radiation have been eliminated and it is made up of the limiting values for S of 58.27, E l , 84.96, and E?, 86.08%. The corresponding channel ratios are R1 = 0.00; Ra = 13.73. Instrument Response Surface. I n any real spectrometer, the upper end of a beta spectrum is difficult to determine with accuracy and, indeed, the tail may extend beyond the theoretical energy limit. Under these circumstances, discriminator settings of the window for the upper channel
~~
Table II. Tables of Separation Values
Isotope 1 channel ratio B/A 0.1000
0.0900 0.0800
0.0700 0.0600
0.0500 0.0400
0.0300 0.0200 0.0100 0.0090 0.0080
0.0070 0.0060
0.0050 0.0040 0.0030 0.0020 0.0010 0.0005 0.0001
3.0 12.0 12.7 13.4 14.1 15.0 16.0 17.1 18.4 19.9 21.7 21.9 22.1 22.3 22.5 22.7 22.9 23.2 23.4 23.6 23.7 23.8
4.0 14.8 15.6 16.4 17.3 18.4 19.6 21.0 22.6 24.5 26.7 26.9 27.1 27.4 27.6 27.9 28.2 28.4 28.7 29.0 29.1 29.3
5.0 17.2 18.1 19.0 20.1 21.4 22.8 24.4 26.2 28.4 30.9 31.2 31.5 31.8 32.1 32.4 32.7 33.0 33.3 33.7 33.8 34.0
6.0
19.3 20.2 21.3 22.6 24.0 25.5 27.3 29.4 31.8 34.7 35.0 35.3 35.7 36.0 36.3 36.7 37.0 37.4 37.8 37.9 38.1
7.0 21.1 22.2 23.4 24.7 26.3 28.0 30.0 32.2 34.9 38.0 38.4 38.7 39.1 39.4 39.8 40.2 40.6 41.0 41.4 41.6 41.7
8.0
22.7 23.9 25.2 26.7 28.3 30.2 32.3 34.7 37.6 41.0 41.3 41.7 42.1 42.5 42.9 43.3 43.7 44.1 44.6 44.8 45.0
9.0 24.2 25.4 26.8 28.4 30.1 32.1 34.3 37.0 40.0 43.6 44.0
44.4 44.8 45.2 45.6 46.1 46.5 47.0 47.4 47.7 47.8
Isotope 2 channel ratio B / A 10.0 11.0 12.0 13.0 14.0 25.5 26.7 27.8 28.8 29.7 26.8 28.1 29.2 30.2 31.2 28.3 29.6 30.8 31.9 32.9 29.9 31.3 32.6 33.7 34.8 31.7 33.2 34.6 35.8 36.9 33.8 35.4 36.9 38.2 39.4 36.2 37.9 39.4 40.8 42.1 39.0 40.8 42.4 43.9 45.3 42.2 44.1 45.9 47.6 49.1 46.0 48.1 50.1 51.8 53.5 46.4 48.5 50.5 52.3 53.9 46.8 49.0 51.0 52.8 54.4 47.2 49.4 51.4 53.3 54.9 47.7 49.9 51.9 53.8 55.5 48.1 50.4 52.4 54.3 56.0 48.6 50.8 52.9 54.8 56.5 49.0 51.3 53.4 55.3 57.1 49.5 51.8 53.9 55.9 57.6 50.0 52.3 54.5 56.4 58.2 50.3 52.6 54.7 56.7 58.5 50.5 52.8 54.9 56.9 58.7
15.0 30.5 32.1 33.8 35.8 38.0 40.5 43.3 46.6 50.4 55.0 55.5 56.0 56.5 57.0 57.6 58.1 58.7 59.2 59.8 60.1 60.3
VOL 38,
16.0 31.3 32.9 34.7 36.7 38.9 41.5 44.4 47.8 51.7 56.4 56.9 57.4 57.9 58.5 59.0 59.6 60.1 60.7 61.3 61.6 61.9
17.0 32.0 33.6 35.5 37.5 39.8 42.4 45.4 48.9 52.9 57.6 58.2 58.7 59.2 59.8 60.3 60.9 61.5 62.1 62.7 63.0 63.3
18.0 32.6 34.3 36.2 38.3 40.6 43.3 46.4 49.9 54.0 58.8 59.4 59.9 60.5 61.0 61.6 62.2 62.8 63.4 64.0 64.3 64.6
19.0 33.3 35.0 36.9 39.0 41.4 44.1 47.2 50.8 55.0 59.9 60.5 61.0
61.6 62.2 62.8 63.4 64.0 64.6 65.2 65.5 65.8
NO. 1 1 , OCTOBER 1966
20.0 33.8 35.6 37.5 39.7 42.1 44.9 48.0 51.7 55.9 61.0 61.5 62.1 62.7 63.2 63.8 64.4 65.1 65.7 66.3 66.7 66.9 1455
Table 111. Tritium Efficiency, Carbon-1 4 Efficiency, and Separation Efficiency of a Liquid Scintillation Counter" Carbon-14 71.2 69.5 67.5 65.7 63.9 62.3 60.7 58.9 57.4 55.2 efficiency: 77.3 74.8 72.8 Tritium Separation efficiency efficiency 23.9 24.0 21.9 23.4 24.8 24.6 20.7 24.4 16.9 19.0 24.3 23.7 39.9 14.7 25.9 22.5 23.8 25.4 26.0 26.8 26.7 26.5 18.4 20.7 26.3 25.7 41.1 16.1 27.6 28.2 24.1 25.6 28.5 29.6 29.4 29.3 19.4 21.9 36.9 16.8 29.2 28.5 26.7 28.5 30.9 31.7 32.2 33.5 21.2 24.2 33.4 33.3 18.2 33.2 32.5 34.5 27.8 30.0 32.9 33.9 34.5 36.1 36.1 36.1 25.0 36.2 35.4 18.5 21.7 30.8 33.4 37.0 38.5 39.3 41.6 41.7 30.6 41.9 27.2 42.1 41.4 19.6 23.4 26.9 32.6 36.0 40.6 42.6 43.9 47.1 47.4 48.0 28.5 19.9 24.0 48.5 47.8 22.4 32.8 37.2 43.5 46.6 48.8 53.9 54.8 56.2 22.9 27.9 15.8 18.4 57.3 56.9 40.7 45.5 49.3 58.6 27.1 32.2 61.0 64.7 67.9 68.2 8.1 13.3 17.2 21.9 0 Instrument: Packard 314-EX2. Standards: Packard HH, HC series. Instrument settings: High voltage 7.00, lower channel: gain 1000, lower discriminator 100, upper discriminator variable from lo00 down. Upper channel, gain 100, lower discriminator variable from 100 up, upper discriminator, 1000. Monitor gain 1000, discriminator settings 100-1000. Individual settings shown in response surface table (Table IV). Tritium and carbon-14 efficiencies were measured in each channel for the settings shown in the row and column headings. From the channe! ratios and Equation 4 (Table I) the separation efficiencywas computed as the ratio of the observed channel ratios at isotope ratios of 10 1 and 1 : l O (C14/Ha) (see Equation 6).
may also be involved in the selection of the optimum point of performance. I t is then necessary to construct a matrix of settings in which the settings of the upper discriminator in channel A and the lower discriminator of channel B form the rows and columns, respectively. At each combination, there exists a separation value and a corresponding efficiency for each isotope contributory to the performance number. Computing this performance number for each combination gives rise to a twodimensional array which we designate the instrument response surface; this surface describes the experimenter's ability to alter the performance number by changing the settings on the instrument. RESULTS
Table I11 illustrates the relationship between efficiency for H3, C14, and separation efficiency obtained from a Packard Tri-Carb 314 EX2 instrument, using the H H and HC standards obtained from the Packard Instrument Co.
Table IV.
Performance Numbers of Commercial Instruments. I t was of interest to compare the theoretical numbers with the performance numbers of actual instruments now commercially available. Numbers so obtained would be illustrative of the state of
the a r t of current spectrometer design and production, and would indicate how much further improvement is yet possible. To date, ten instruments have been mapped in this fashion. Each instrument was tested with the same set of standards (Packard H H and HC) and was operated under balance point conditions. The response surface w&s obtained by progressive reduction of the two channel widths in the manner described and a minimum of 225 combinations were computed for each instrument. The values in Table V are those of the maximum performance number found on each surface and the contributory values of which it WM composed. Concurrently, the background count rate was recorded for each channel at that setting producing the maximum performance number. The numbers so obtained ranged from 72 to 143 and in those instances where more than one instrument of the same model were tested, there was good agreement between the performance numbers in each case. It has also been our experience that replicate measure-
Performance Numbers at Various Settings of a Liquid Scintillation Counter
Lower discriminator,
uca 100 110 120 Upper discriminator, LC 50.5 55.2 1000 45.5 61.8 51.1 56.6 900 58.9 47.8 53.5 800 54.7 60.7 700 48.5 50.1 56.1 600 44.0 47.0 53.4 40.8 500 46.4 34.3 40.2 400 27.1 32.2 22.5 300 12.9 8.3 10.3 200 ,. See information on instrument settings in 1456
Table IV lists the performance numbers resulting from the multiplication of the separation efficiency by the isotope efficiencies at that point; these are listed under the settings (row and column headings) a t which they were measured. The highest performance number on this response surface was 72.0, composed of the values SI 30.9, El, 34.5, and EP, 67.5. The response surface has a fairly flat contour in the region mapped; thus, it is possible to trade efficiency of either isotope for separation efficiency while maintaining approximately the same performance number. This discretionary ability of the experimenter cannot be reduced to a single number such as the maximum performance number but is best represented by the two matrices shown in Tables 111and IV.
ANALYTICAL CHEMISTRY
130 58.8 65.8 63.2 65.5 61.1 58.8 51.8 37.0 15.5 footnote
140
150
160
170
180
190
200
210
220
63.1 60.9 68.0 70.4 65.8 68.7 68.5 72.0 68.3 64.3 62.5 67.2 61.2 55.9 40.9 46.5 18.0 22.1 to Table 111.
62.6 69.8 68.5 71.9 68.6 68.0 62.8 48.5 24.0
61.4 68.4 67.3 70.9 67.9 67.7 62.8 49.4 25.4
61.7 68.7 67.9 71.9 69.3 69.7 65.6 53.1 29.4
59.7 66.5 65.9 69.9 67.5 68.2 64.4 52.7 29.8
57.5 64.0 63.6 67.6 65.5 66.4 63.2 52.3 30.6
55.8 62.2 61.9 65.9 64.0 65.1 62.3 52.1 31.4
52.3 58.3 58.1 61.9 60.2 61.5 59.0 49.7 30.3
ments of the same instrument under a variety of conditions will result in performance numbers within a range of 3 to 5 units of one another. DISCUSSION
Previous studies by Herberg (2) and by Bush (1) have been concerned with optimizing the precision and accuracy of double isot,ope counting over a wide variety of sample conditions of quenching, isotope ratio, absolute activity] and channel settings. These reports have focussed on the analysis of the sample as it is encountered in the experimental situation and the contribution of spectrometer performance, though implicitly involved in these analyses, has not been considered as an independent factor. I n this description of scintillation spectrometer performance, the problem was resolved into two components: first, that dealing with the separation efficiency and second, that of the efficiencies with which each isotope can be determined at a given separation efficiency. Two factorslimittheseparationability of a spectrometer used for double isotope determinations; these are the shapes and overlap of the pure spectra themselves and the distortion introduced by the liquid scintillation process and instrument design. The former represents the absolute physical limitation placed on the separation by the beta decay processes under study while the latter is composed of the interaction of the common scintillation process with the individual instrument designs. I n this study, the use of the same physical samples in each set of measurements permits comparisons to be made from instrument to instrument. The computation of separation ability requires a function with three terms: R1,Rz (the channel ratios for each isotope), and R,, the isotope ratio of the sample. Separation ability is also a symmetrical function so that the A-B separation must be taken into account as well as the B-A separation if the term is to be useful. For this reason, the separation efficiency was related to a
Table V.
Examples of Performance and Separation Numbers Obtained with Some Commercial Instruments
Instrument Ansitron Beckman Nuclear Chicago Packard 314-EX2
Picker Vanguard Theoretical
6725 6820 6820 2001 2321 3003 3003
P 116.2 120.1 78.1 112.9 118.3 72.0 78.7 101.7 140.9 143.1
426.1
range which extends one order of magnitude on either side of an isotope ratio of 1:1, that is, from 0.1 to 10.0. The separation efficiencyof a spectrometer is not, in itself, a unique number because it may be increased or decreased at will by paying the required price in detection efficiencies for either isotope. If the spectrometer performance is to be defined in an unambiguous fashion, the maximal product of separation and detection efficiencies attainable must be determined. This is illustrated in the twodimensional , or single discriminator variable, case in Figure 4 and in the three dimensional or two discriminator case by Tables I11 and IV. The response surface of Table IV illustrates the price which is paid for increased separation or for increased efficiency in the detection of either isotope for a given instrument setting. The maximum performance number attainable on that surface represents the Optimum performance Of which the instrument is capable under the most favorable circumstances. Either by choice or by circumstance the experimenter may not be able to operate the instrument at this performance level but this number is an inherent and unambiguous measure of the spectrometer performance and design. The utility of the performance number is
s
%
c.p.m.
Ha 42.0 40.7 41.7 39.3 34.4 36.4 38.3 41.1 37.8 42.6 30.9 34.5 43.7 31.9 45.5 32.3 41.6 47.3 44.9 44.7 Not available Not available 58.3 84.9
CI4 68.0 73.3 62.3 71.7 73.4 67.5 56.5 69.1 71.6 71.3
86.1
Bkg. 1 Bkg. 2 22.4 27.2 20.0 31.4 17.4 18.5 32.9 17.2 22.8 13.4 53.9 46.6 26.4 14.0 22.8 14.2 23.6 17.4 22.8 14.6
* * a
...
enhanced by the computation of the ultimate performance number for a hypothetical spectrometer in which all disintegrations occurring in the spectra of tritium and radiocarbon are detected without loss and scaled without distortion. This ultimate number of 426, resulting in a separation of 58,3y0while detecting 84.9 and 86.1% of the Ha and C14disintegrations, when compared with the highest number obtained in actual instruments, 143, indicates that t,he best instrument tested to date has a performance which is 33% of theory. ACKNOWLEDGMENT
The computer program for calculating the distribution of H3 and C14 was written by M. H. Dipert. LITERATURE CITED
(l!.FE?, E* T * ANAL. ~
CHEW
36, log2
(lYO4).
(2) Herberg, R. J., Zbid., 36, 1079 (1964). (3) Kabara, J. J., Spafford, N..R., MacKendry, M. -4.1 “Advances in Tracer Methodology,” Vol. I, S. Rothchild, Ed., p. 76, Plenum Press, New York, 1963, (4) Okita, G. T., Kabara, J. J., Richardson, F., Leroy, G. V., Nucleonics 15, 111 (1957). RECEIVEDfor review April 13, 1966. Accepted July 18,1966.
VOL 38, NO. 1 1 , OCTOBER 1966
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