4 (since efficiencies are considered errorless), 4% lower a t the tritium minimum and 1% lower at the carbon14 minimum. Since the complete and simplified equations give curves so much alike in shape, it is often satisfactory to use the simplified equations, at least as a preliminary means of determining settings. If sample rates are large compared to background, and the background error contribution is minimized by a long counting time, Equations 27 and 28 can be further simplified by commission of the terms involving background. It then is a very simple matter to calculate relative error curves for different instrument settings. Ratios of Isotopes t o Each Other. Equations 19 and 20 have been evaluated for many combinations of Cl4/H9 ratios, ratios of C14 to background and sets of counting efficiencies. For a given C14/background ratio, the C14 error decreases as C14/H3 increases. The tritium error increases as C*4/Hs increases. At some point the two errors are equal. This point occurs within the C14/H3 range 0.1 to 0.5 for a wide variety of conditions. As the C14/B ratio increases past 10, the errors decrease only very slightly within the C14/H3 range 0.1 t o 0.5. Miscellaneous. Equations 19 and 20 have been evaluated for a number of sets of counting efficiencies of carbon-14 and tritium and sample t o background ratios. For 1-minute counting times after internal standard addition and internal standard activities in the range 50,000-100,000 d.p.m., the errors in the four efficiencies ranged from 1.2% to 2.7%. Of this error, 1% is the contribution S A / A from internal
standard. I n all calculations throughout this paper a value of 0.01 has been used for S A I I . This is considered t o be an easily attainable accuracy. With care the variation can be kept in the range 0.5%4..7%. As with single isotope counting, standards used for drift correction should have the highest count rate consistent with other requirements. ACKNOWLEDGMENT
C. N. Rice of this laboratory and
R. E. Shultz, formerly of this laboratory, were the originators of an IBM program by which the errors were calculated. OF
LIST
SYMBOLS
C1 = Count rate a t condition 1. C2 = Count rate a t condition 2. R1 = Disintegration rate of isotope 1. R2 = Disintegration rate of isotope 2. el = Counting efficiency of isotope 1 at condition 1. e2 = Counting efficiency of isotope 1 at condition 2. e3 = Counting efficiency of isotope 2 a t condition 1. e4 = Counting efficiency of isotope 2 at condition 2. ml = Net count rate a t condition 1. = Net count rate at condition 2. bl = Background rate a t condition 1. b2 = Background rate a t condition 2. Cs = Count rate at condition 1 after addition of internal standard of isotope 1. C4 = Count rate a t condition 2 after addition of internal standard of isotope 1. C5 = Count rate a t condition 1 after addition of internal standard of isotope 2.
Ca = Count rate at condition 2 after addition of internal standard of isotope 2. A, = dpm added as internal standard of isotope 1. A, = dDm added as internal standard ~~of isotope 2. t = subscripted as counting time for that quantity. b = partial derivative symbol. s = standard deviation. 5 2 = variance. Cs- Cz = m7 cs - Cl = ?rl$ C4 - bz = mg C3 - bl = rn10 N 1 = numerator of (7) Nz = numerator of (8) N3 = numerator of (9) N4 = numerator of (10) D1= denominator of (7) D2 = denominator of (8) D3 = denominator of (9) D4 = denominator of (10) mJ = C3 - C1 = increase in count rate in channel 1 after addition of internal standard of isotope 1. = C4 - Cz = increase in count rate in channel 2 after addition of internal standard of isotope 1. me = C5 - C3 = increase in count rate in channel 1 after addition of internal standard of isotope 2. m6 = Ce - C4 = increase in count rate in channel 2 after addition of internal standard of isotope 2. LITERATURE CITED ’-)
Herberg, R. J., ANAL.CHEX.33, 1308
RECEIWDfor review November 14, 1963. Accepted February 17, 1964.
Liquid Scintillation Counting of Doubly-Labeled Samples Choice of Counting Conditions for Best Precision in Two-Channel Counting ELIZABETH T. BUSH Nuclear-Chicago Corp., Des Plaines, 111.
b The statistical precision with which two beta emitters may be determined in a single sample depends on the way the pulse height spectrum is positioned in the windows of the beta spectrometer. The best choice of the counter parameters, window width and gain, is discussed in relation to the isotope ratio, the absolute activity level, and the degree of quenching. Optimum counter settings are given for the isotope pairs H3 C14, Ha P32, and C14 P32. Settings for other
+
1082
+
ANALYTICAL CHEMISTRY
+
isotope pairs may be inferred by comparing the separation in the energies of those isotopes with the ones studied here.
S
IMULTANEOUS quantitative
determinations of two beta-emitting isotopes in a single sample by liquid scintillation counting have been practical for a number of years (9-4, 7, 8). The technique is most successful when the beta spectra of the two
isotopes have the greatest difference in average energy, and when the corresponding average pulse heights at the output of the detector are also well separated. The liquid scintillation counter, however, does not behave strictly as a proportional counter: Its output pulse height distribution differs significantly from the input spectrum. In particular, in the presence of quenching in the liquid sample, two pulse height spectra show more overlapping than the beta spectra which generated
them. I n Figure 1 are shown superimposed the hypothetioal spectra of two samples singly labeled with isotopes X and Y, respectively. The samples are assumed to have equal disintegration rates. Two (arbitrarily chosen) counting channels, each defined by two discriminator levels, me shown. The area under the curve of isotope X within channel 1 is proportiond t o the counting efficiency in that channel for isotope X, etc. Most commercial liquid scintillation counters permit the operator to count in two channels simultaneously, and to choose both the avemge pulse height (or gain) and the rang(#of pulse heights (or channel width) of each channel independently. These parameters fix the counting efficienvies of the two isotopes in the particular sample type. It is generally recognized that the choice of these paralrieters should depend in some way on the degree of overlapping of the spectra and the relative amounts of the two isotopes present. But the nature of this dependence does not seem t o have heen determined precisely, and most operators appear to make these decisions on an intuitive basis. The purposes of the present study are (1) to derive exact expressions to measure the statis1,ical precision of counting, as functions of the counting efficiencies of the isotopes in the two channels, the amounts of the isotopes, and the background count rates, and (2) to provide experimental information as to how the precision varies with these quantities, as a basis for a practical approach to the chcice of the two channels. This experimental study is important because the numerical computations of the expressions for statistical precision are so hngthy as to be impractical for the avtmge user. It is believed, however, thal; the data of this study provide the basis for generally useful conclusions. The information pi-esented here is also applicable to single-channel counters, in which doubly labeled samples must be counted at two consecutive rather than simultaneous channel settings. The treatment given here does not take into account the additional degree of frl3edom permitted by consecutive countings: the choice of a different counting time for each channel. It will be obvious that the total counting time for a single-channel counter need not alwzys be twice as long as that for a two-channel counter to give equal precision in the answer. The best division of total counting time between the coneective countinga can be found by applicibtion of the same principles which permit most efficient division of counting time between sample and background. The expressions whicsh are derived to m w u r e statistical precision in the
CHANNEL 2
*CHANNEL I
-
-I
W > W
I
0
I I
!
4
I
4
> w
-
\PULSE HEIGHT
Figure 1 . Idealized pulse height spectra of isotopes X and Y in liquid scintillation spectrometer Two counting channels defined by four discriminator levels
values of the quantities sought-the disintegration rates of the respective isotopes-are general for any two isotopes. Their form is suitable for calculations designed to show how the counting errors may be minimized by the best choice of the efficiencies in each channel. (Only one efficiency may be chosen independently in any channel; the efficiency of the other isotope is then determined by the degree of overlapping of the spectra, which is a function of quenching.) To set the counter for the desired efficiencies for the unknown sample it is convenient to employ “duplicate standard samples”Le., samples duplicating the unknown in quenching, but containing known amounts of radioactivity (preferably two singly labeled samples). Accurate duplication generally requires unlabeled material identical to the radioactive unknown; such material is not always available. Any difference in quenching between the unknown and standard samples will result in less than optimum counter settings. It is therefore of practical interest to know not only the optimum value of the efficiency but the rate of change of counting precision with efficiency in each channel. The preparation of duplicate standard samples is also, in principle, the simplest method of determining the efficiency of the unknown in double label counting. When used for this purpose the requirement of accurate duplication is very stringent. In case of doubt, the efficiency of the unknown may be checked after counting by internal standardization. One may add
standard solutions of both isotopes to the same sample or one isotope to each of two duplicate samples of the unknown. In the former method, the isotope of lower energy is added first, and the sample is recounted before adding the second isotope. Since the count rate in each channel should be several times its previous value for each recounting, to obtain good precision, the final activity tends to be very high. Better precision can be expected from the use of duplicate unknown samples. The original counting, before standards are added, provides a check on the reproducibility of preparation of the unknown. If triplicate unknowns are made up, two may be used for internal standardization, while one remains unaltered. This permits recounting if it should be desirable to improve channel settings, or if an error is suspected. If the activity of the added standard solution is very much greater than that of the sample material, the two internally standardbed samples may conveniently be used as the standard controls for setting up the counter. If the contribution of the sample material to their count rates is negligible, the approximate efficiencies may be calculated simply on the basis of the added known activities, If this condition is not met, the efficiencies can, of course, be calculated by counting all three samples and making appropriate corrections, but this is tedious for a set up procedure. The availability and level of activity of the sample material will determine whether replicate unknowns or control VOL 36, NO. 6, MAY 1964
1083
blank samples: provide the most convenient approach to finding optimum settings . It is necessary to know something about the relative activities of the isotopes in the sample, to choose optimum counter settings. It seems reasonable to believe that this ratio will usually be known to within an order of magnitude. If it cannot be guessed from the conditions of the experiment, the order of magnitude can quickly be established from a brief counting of the unknown, once the approximate efficiencieshave been determined. Since the isotope ratio in the counting sample strongly affects the precision of the determination, the investigator would presumably wish to know the optimum value of the ratio and t o choose his initial activity levels accordingly, in so far as he is not limited by other conditions of the experiment. In deriving the expressions for statistical precision, the ratio of the isotope activities is introduced as one of the variables because, first, this appears to be logical both experimentally and conceptually; second, the precision is particularly sensitive to this ratio; and third, it somewhat simplifies these expressions. DERIVATION
When an expression for e is found in terms of the efficiencies, experimental values may be substituted in order to determine a t what efficiencies E is minimized. Although numerical values of e so calculated will be smaller than observed values to the extent that counter instability and errors in determining efficiencies contribute to the total experimental error, these calculated values of e will show how precision varies with channel settings. Assuming that the error in measuring t is negligible, and taking uaz = G/t, and u B 2 = B/ts, we obtain from Equations 3, 4,and 5
Equations 8 and 9, and multiplying them by t,/Dyz we find CASEI. tX2ta =
(kFoDy)-'[Dy(kFi EY%
=
(FoDY)-'[Dr(kFi'
+ FP)+ + Fz') + Fa']
F3]
(10)
(11) We wish to find either the minimum value of the relative standard deviation, e, for a chosen counting time, or the minimum counting time which will give an allowed value of e. The form of Equations 10 and 11 shows that to minimize e (or e2) for fixed t. or to minimize 1, for fixed e2 we must, in either case, minimize the expression on the right-hand side, which is just equal t o &,. We shall therefore seek to minimize e2t, as a function of the F's (6). In the second case, more pertinent to the counting of single samples, a total counting time t -- t, ta is chosen. It is desired to divide this between t, and tb in such a way as to minimize u. We apply these conditions by taking dt = 0 and do = 0 (9) in Equations 6 and 7. This gives, for minimum uX,
+
Substituting values of G1 and G2 from Equations 1 and 2, and replacing D X by k D y gives
OF EXPRESSIONS FOR PRECISION
and for minimum
uy
We may write
GI
+ YlDy + B1 XZDX+ YPDY+ Bz
= XiDx
Gq =
(1)
(2) which may be solved to give the quantities sought in the measurement,
{9
[k(Z1222
+ 222x1) +
x12y2
+
It is apparent that the expressions of Equations 12 and 13 are not, in general, equal. Choice of counting times will thus depend on whether ux or uy is of greater concern. This is considered below. Substituting Equation 12 in 8, and Equation 13 in 9,
222Y11+
CASE11. The variances of Dx and DY may be found from the expression for the ... variance of any quantity au bv in which u, v , . . . are independent variables and a, b, . . . are constants:
+ +.
n2bu
+
bu
+
*.a)
=
U2U"Z
Fo
= XlYZ
Fi
=
+ b%,2 + . . . *
ANALYTICAL CHEMISTRY
Ex2t, = ( ~ F o D Y ) - ~
- 2zy1
+ Fz = yi2yz + Fa yi2Bz + ~2'B1 + Fz' xi2yz + ~ 1 ~ x 2yz2x1 ~
2
~
~
1
Fit
=
~
1
=
F3' =
~X2'Xi~
~
xi2B2
2
+ 2z2B1
+ F2')+
2
~
~
1
We shall consider two situations which determine the relationship between ta and ta. In the first, a single blank sample is used to determine the background rate for a group of similar unknowns, and it is counted once (or twice, pooling the counts) during the run. I t is conveniently counted for a time which is long compared with each individual sample counting. Then l/t, l/tb = l/&. Dropping l/ts from
+
-(Dy(kFl'
ey2t, =
=
(5)
This is an exact equation (6). We shall assume that efficiencies 21, y1, etc., are constants whose values may be determined to any desired precision by the counting of standards. The variance of D is thus calculated as a function only of the random nature of rates G and B. Uncertainty in the values of the efficiencies, erratic counter performance, etc., are not considered. We wish to find the channel settings which minimize the relative standard deviations, ex and cy, resulting from sample and background counting statistics alone, independently of the way in which the efficiencies may be measured. 1084
The following symbols are introduced for convenience:
EXPERIMENTAL
Three pairs of isotopes were studied 0 4 , Ha P32, experimentally: H3 and C14 P32. The labeled compounds used were toluene-Cl4, toluene-H3, and tri-n-butyl phosphate-P32, the latter in a final concentration in the sample of approximately 0.76 pg. per ml. (Tributyl phosphate is essentially nonquenching to liquid scintillation solutions even in much greater concentrations.) These labeled compounds were
+
+
+
added to scintillation sdution consisting of LiquiFlor (Pilot Chemical Co. concentrate of 2,5-diphenyloxazole and p-bis [2-(5-phenyloxazolyl) Ikenzene in toluene) diluted 20 times in reagent grade toluene. Acetone was added to some samples as quenching agent, as shown in Table I. The number 1 samp e of each pair is referred to as “unquenched.” All samples were in equilibrium with air. The samples were singly hbeled, and were prepared in identically quenched pairs. In addition to these 18 samples, some pairs of samples were prepared for the study of H3 and C14 ising ethanol or methanolic Hyamine hydroxide [Hyamine 10-X, Rohm & Haas trademark for p(diisobuty1cresoxyethqxyethyl) dimethylbenzyl ammonium chloride monohydrate] as quenching agents. The liquid scintillatim counters used were Kuclear-Chicago Corp. Models 703, 723, and 725 and Packard Instrument Co. 314EX-2. ‘The Models 703 and 725 are cooled and employ multiplier phototubes with “B-type” photocathodes. They differ in the geometry and reflective surface of the counting chamber. The 723 is an ambient temperature counter using “$-type” photocathodes. The 3 14EX-2 is cooled. Altogether, 11 differert counters were used. The lower discriminahor level in each channel was set a t 0.5 volt on the Kuclear-Chicago counters and the upper levels were varied to obtain data for different channel widths. The efficiencies in a channel were varied by varying either the upper discriminator a t fixed gain, or the gain at fixed discriminator settings. The gains in both channels were controlled by the high voltage applied to the detector phototube, and that in channel 2 was controlled also by an atte iuator when the high voltage was fixed. On the Packard counter the lower level discriminators were set a t 0070 and tke upper a t 1000. The total gain was varied by means of either the phototube high voltage or the channel gain control. Most of the data were obtained by varying the gains. E:tch sample of a pair was counted a t a series of gains, and graphs were made of x1 us. y1 and of x2 us. y2. The background count rate was plotted as a function of gain a t each channel width. Sets 0,. values of 2 1 , y1, and B1 and of x 2 , y2, and B2 were determined from these graphs and the recorded gain settings. The complete series of values for each pair of samples formed the matrix of input data for The calculations of ex?, and e&. calculations were prog;-ammed in Fortran and performed by an International Business Machines Model 1620 computer. RESULTS
A N D DISCUSSION
I t is assumed as a basis for this discussion that D X and D y are of equal interest to the investigator, and that he will, if possible, choose experimental conditions such that ex2 =
ey2. If the nature of his samples is such that the counting error is much larger for one isotope than for the other, he presumably will choose the channel settings so as to minimize the larger error; if the two errors are approximately equal but cannot be minimized a t the same settings, he will choose compromise settings. These assumptions also underlie the choices of tJtb in the derivations of Equations 14 and 15. These equations have been used to calculate all results presented here. At high activity levels the same results are obtained by using Equations 10 and 11: The values of 2 for Case I and Case 11 converge when G >> B , and the same counter settings are appropriate. At low activity levels it is moat important to optimize the counter settings for each sample in order to reduce statistical errors, and it is also important to determine the true background for ea,ch sample. Since background is affected by such factors as quenching and chemiluminescence, which are functions of sample composition, a blank of identical composition should be prepared for each different type of samplc. Conservation of counting time becomes an important consideration. Equations 14 and 15 are appropriate in this situation. In practice, the expressions of Equations 12 and 13 are not difficult to estimate from brief counts. It is apparent that e x z will decrease and c y 2 will increase as IC increases and that there will be some value of k , k , , for which ex2 = cy2. If X is the isotope of lower energy, k , will be greater than unity, since the interference from the “background” of the high energy isotope in the counting of the low energy isotope is always greater than the reverse-Le., y1 > 2 2 . When D x / D y = k,, minimum counting time is required to determine both isotopes to a given precision. The extent of the interference from each isotope in the counting of the other is measured by the quantities y1/xl and x2/y2. For any given values of y1 and $2 it is desirable to make these ratios as small as possible. Quenching is the principal factor in increasing the ratios, but smaller effects from other sources may be observed. Effects of Certain Instrumental Variables. VARYING CHANNEL WIDTH us. VARYINQGAIN. This comparison was made only with Nuclear-Chicago counters, but the effects of these two variables on yl/zi and x2/y2 are expected to be the same in other counters, since they were observed t o confirm the results predicted from a general knowledge of the pulse height spectra. It should be borne in mind that an increase in gain corresponds to a decrease in window width. The gain of channel 1 is set to .bring in the
Table 1.
Samples Prepared for Study
of Double Label Counting Sample % acetone Isotope pair no. by volume H3 CY4 1 None
+
C’4
H3
+ Pa2 + P33
2
3 1 2 3 1 2 3
1 33
None 10 30
STone 1 20
major portion of the spectrum of the lower energy isotope, and that of channel 2 is set for the higher energy isotope. Therefore the channel 1 gain is always higher. y1 decreases monotonically with an increase in gain in channel 1. 2 2 increases with gain in channel 2. The minimum values of yl/xl were obtained by using the widest possible channel 1. The effect of channel width on GJB1 must also be considered. Since the background spectrum varies with location as well as with the particular counter, no attempt was made to reach a general conclusion as to the best channel width for very low level counting. In the present study the maximum channel width yielded the minimum value of ex2 a t all activity levels. For channel 2 the maximum width was definitely indicated when counting Pas because of the better efficiency obtainable. For C14 with H a very little effect on C14 efficiency was observed in reducing the window width to 60% of the maximum, but the accompanying reduction in background gave slightly improved values of e y 2 a t low C14activity levels. An advantage in using maximum channel width stems from the fact that efficiency values change more slowly with gain (high voltage or attenuation) in a wide window. This results in decreased sensitivity of e2 to the gain control setting, near the minimum value of e2. Furthermore, if one wishes to count a series of samples which vary somewhat in quenching, use of a single gain for all samples will result in less deviation from optimum efficiencies if a wide window is used. I n all cases the channel width should be chosen on the basis of the above considerations, and the channel gain should then be adjusted to give the desired efficiencies. VARIATION IN COUNTERS.For any chosen values of y1 and x2 the values of yl/xl and x2/y2 were found to vary considerably between counters. Counters differ somewhat in their inherent efficiencies. When the same sample was counted in a higher and then in VOL. 36, NO. 6, MAY 1964
1085
0.05
0.10
0.15
0.20 CI
0.25
0.30
(4-
0.35
INCREASING GAIN)
Figure 2. Tritium error function for unquenched H3 sample vs. cb the channel 1 efficiency for C14
+ C14
Calculations for three values o f DH, tritium disintegration rate, with Dx = Dc
a lower efficiency counter, the difference in the graphs of x, us. y, was similar to the change produced by sample quenching as observed in a single counter. Counters of higher efficiency showed lower ratios yl/xl and xZ/yz. The effect on the optimum channel settings of a change in inherent counter efficiency is similar to that of a change in inherent sample efficiency. When the channel settings are expressed in terms of appropriate parameters, the optimum values are constant over a much wider range of quenching than that corresponding to the difference between counters. For this reason it is expected that the conclusions reached in this study will apply to liquid scintillation eounters generally. Isotope Pairs. The three pairs of isotopes studied here were chosen because they are probably the ones most frequently encountered in double label studies today, and because they range from the minimum spectral overlap which is likely to be encountered (H3 Ps2) to the maximum which is very practical to resolve (Ha C14). Some general results may be stated for all the isotope pairs studied. The optimum channel 1 setting was either at or somewhat above the balance point [the gain giving maximum efficiency in the window ( I ) ] for the lower energy isotope.
+
+
1086
ANALYTICAL CHEMISTRY
0.05
0.10
0.15
0.20
0.25.
0.30
Cl
Figure 3. Channel 1 efficiency data and calculated tritium error function for three variously quenched Ha f C1* samples, showing difference, Aht between maximum Haefficiency and optimum counting efficiency DE = l o * = 0.1 Do
This is easily understood from the fact that x1 goes through a fairly broad maximum as the gain is increased through the balance point value. The ratio yl/xl decreases rapidly in this region, where xl is nearly constant, and considerable improvement can be made in this ratio with very little sacrifice in x1 by staying on the high gain side of balance point. The amount by which x1 should be decreased from its maximum value a t balance point may be designated A x 1 = maximum zl-optimum zl. Axl increased withincreasing sample activity and with decreasing k.
The optimum channel 2 setting was, as expected, a t maximum y2 if the corresponding value of x2 was zero. When the spectra were not so well separated, the optimum gain for channel 2 was above the balance point gain. In the following discussion x and y are replaced by specific symbols for the isotope pair under discussion, taking x as the isotope of lower energythat is, z and y become, respectively, h and c for the pair Ha CI4, c and p for C14 P3*,and h and p for H3 P32. H3 WITH C14. Optimum Channel 1 Setting. A x 1 = Ah1 was found to
+
+
+
have values between 1.5 and 5.5 over the range of conditions lo2 5 D 5 106, 0.1 5 k 5 k., and quenching varying from sample 1 to 3. With Ahl set a t 2% for any sample, the error function ex%. was w,thin 10% of its minimum value. The tritium error shows a rather broad minimum about the optimum hl, c1 and the choice of efficiencies is not tcio critical. This may be seen in Figures 2 and 3, which show, respectively, how the tritium error varies with sample activity and with quenching. The relative standard deviation in the tritium determination, a/DH, may be found from the ordinate by assigning a convenient value to t,, such as 10 minutes, and taking the square root of the resulting value of ex2. The abscissa c1 is used because it is a single-valued function of the gain in this region, whereas hl goes through a maximum. The gain control setting itself is not used as the abscissa because i t is different for each counter. The optimum value of c1 for a particular sample differs slightly among counters, because of the difference in the curves of z1 us. y1 discussed above. Among all the counters used here, however, this difference was never as great as that between samples 1 and 2. Since the optimum c1 and Ahl show a remarkable constancy over a wide range of quenching, it may be assumed that the values given here are close to those for other counters of similar design. e1 is a more convenient parc;,meter than MI for setting up the counter, since only one gain setting need be found. One has only to place the duplicate standard C1* sample in the counter and adjust the gain in channel 1 to give the optimum c ~ . Each curve in Figures 2 and 3 is plotted for a constant value of hp: the optimum value. The two figures illustrate the range of values which &, may be espected to take at these values of k and DH. The value of k for Figure 3 is very unfavorable, and it is seen that impractically long counting times would be required to obtain reasonable precision for H 3 in highly quenched samples. When k: > k,, the tritium error is smaller than the C14 error, and the channel 1 setting should be chosen to minimize &,. The latter quantity, however, was insensitive to values of hl, c1 over a wide range. When k > k, it was only necessary tj3 have c1 2 0.16 in order that both €2and e H z be near their minima. Values of k, for samples 1, 2, and 3 were approximately 2.5, 4, and 10, respectively. Optimum Channel 8 Setting. For k 5 k,, was rath?r insensitive to values of h?, cz, except for highly quenched samples. The optimum value of hz was in the range of 0.003 5 hz 5 0.009 for all samples, and a t 0.005 _
10, the optimum values of c2 gradually decreased to 0.0001 a t k = 100, but the curve remained flat below c2 = 0.005. k , was slightly greater than unity. H8 WITH Pa%. This pair of isotopes differed from those preceding in that i t was possible to get the maximum efficiency for the higher energy isotope in channel 2 a t zero efficiency for the lower energy isotope. Counting errors for both isotopes were minimized when PSz was counted a t its maximum efficiency in channel 2-that is, a t balance point gain. The amount by which the channel 1 gain should be raised above balance point was negligible a t k = 1 but increased with decreasing k . At k = 0.1 the best operating point v a s that a t which H3 efficiency had fallen off by 0.5 to 1% (in absolute value) from its maximum. The best operating point was always at maximum hl for k 2 k,. Valuesof k , ranged from about 2 for an unquenched sample to about 10for the most quenched sample. VOL 36, NO. 6, MAY 1964
1087
c, Figure 6. P32 error function for quenched and unquenched CI4 P32 samples vs. cb channel 2 efficiency for C14
+
Calculations for
Pa* disintegration rate
Figure 5.
Carbon-14 error function for quenched sample vs. pl, channel 1 efficiency for P32, at various values of Dc and of k = Dc/Dp C14
+
P32
Insert.
CI VI.
PRACTICAL USE
OF
PI.
Balance point indicated
DATA
When appropriate parameters are used to describe liquid scintillation counter channels for double label counting, the optimum values of these parameters do not vary widely for large changes in sample quenching, isotope activities, and even isotope energies, By taking a compromise value within this limited range of optimum values for each parameter, one may expect to count any sample to a relative standard deviation, a / D , within about 10% of the minimum deviation which could be obtained in the same counting time under the best conditions for that sample. The counting channels described below represent the best compromises for a wide range of sample types of each of the isotope pairs studied. Optimization for any particular sample may be achieved through modifications suggested by the detailed results already presented.
1088
ANALYTICAL CHEMISTRY
Best counting conditions for other pairs of beta-emitters may be inferred from a comparison of their energy spectra with those of the three pairs studied here. Maximum channel widths should be used, with the following qualifications: The lower level discriminators on the Packard instrument should be determined in accordance with the procedure used for singly labeled samples; if the net sample count rate is close to the background rate, the optimum channel width may vary with the particular counter and location. The gain in each channel is adjusted to give a specified efficiency for one of the two isotopes for the sample to be counted. -4s sample quenching changes, the gain must be changed to maintain the same efficiency. This setting is best determined by counting a duplicate standard singly labeled sample. The gain settings should be as follows:
Dp
= 1 08,with Dc/DP = 10
H3 WITH C14. Channel 1 gain is above balance point for Ha, so that H3 efficiency has dropped 2% in absolute value from its maximum. An alternate, simpler, procedure is to set the gain for the equivalent optimum C1* efficiency in channel 1, which is about 15% over a wide range of quenching. Channel 2 gain is set for a H3 efficiency of 0.6%. C14 WITH P32. Channel 1 gain is above balance point for C14, so that 0 4 efficiency has dropped 2% from its maximum. Channel 2 gain is set for a C14efficiency of 0.4%. H3 WITH P32. Channel 1 gain is above balance point for H3, so that Ha efficiency has dropped 0.5 to 1% from its maximum. Channel 2 gain is a t balance point for P3*. NOMENCLATURE
B1
=
B2
background count rate in channel 2 = disintegration rate of isotope X = disintegration rate of isotope Y = gross count rate in channel 1 = gross count rate in channel 2 = Dx/Dy = sample counting time = background counting time = eficiency for isotope X in channel 1 = efficiency for isotope X in channel 2 = efficiency for isotope Y in channel 1 = efficiency for isotope Y in channel 2 = standard deviation of the &aintegration rate, D
Dx Dy Gl
G2 k t.
tb zl z2
yl y2
u E
background count rate in channel 1
=
=
a/D
ACKNOWLEDGMENT
The author is indebted to Paul Meier, University of Chicago, for helpful suggestions, to Walter :Kirk and David Zerfass, International Business Machines Corp., for programming and carrying out the cal(:ulations, and to David Hansen for many of the measurements taken. LITERATURE CITED
(1) Arnold, J. R., Science 119, 155 (1954).
(2) Blau, M., Nucleonics 15, No. 4, 90 (1957). (3) Dern, R. J., in “Liquid Scintillation Counting,” C. G. Bell and F. N. Hayes, e&., p. 205, Pergamon Press, New York, 1958. (4) Dern, R. J., Hart, W. L., J. Lab. Clin. M e d . 57, 322 (1961). (5) Feller, W., “Introduction to Probability Theory and Its Applications,” 2nd ed., pp. 214-16, Wiley, New York, 1960. (6) Greenfield, M. A., Koontz, R. L., Intern. J. A p p l . Radiation Isotopes 8,205 (1960).
(7) Herberg, R. J., ANAL. CHEM. 33,
1308 (1961). (8) Okita, G. T., Kabara, J. J., Richardson, F., LeRoy, G. V., Nucleonics 15, No. 6,111 (1957). (9) Price, W. J., “Nuclear Radiation Detection,” p. 60, McGraw-Hill, New York, 1958.
RECEIVED for review November 1, 1963. Accepted March 12, 1964. Division of Isotopes and Radiation, American Nuclear Society, 1963 Winter Meeting, New York.
A Computer Program to Optimize Times of Irradiation and Decay in Activation Analysis THOMAS L. ISENHOUR and GEORGE H. MORRISON Department o f Chernb’ry, Cornell University, Ithaca,
b To achieve increased selectivity in instrumental activation analysis, a computer program has been developed which rapidly determines the optimum times of irradiation and decay for activation analysis of any element in any mixture regardless of its complexity. The program is generalized for any combination of nuclear reactions, involving both elements for analysis and as interferemnces. Mathematical solution is accomplished through an iterative approximation process. The program is described in detail and examples of its application are given.
A
analysis is generally recogniEed as the most sensitive trace analytical techiique for many elements. Although chemical isolation of the activity of interest from interfering activities produced during irradiation has been emp!oyed successfully for many years, the more recent trend ha? been toward the use of direct instrumental methods which generally require less time and effort (3). However, the limited resolution of present counting equipment considerably restricts the applicability of this approach. One attempt to improve resolution has involved the application of spectral stripping to gamma iipectrometry including the use of computer techniques (4). Although several good computer programs have been developed to separate gamma spectra by “spectral unfolding,” it appearr; that the real problem lies in obtainng better input data. An aspect of activation analysis not fully exploited is the u:,e of appropriate times of irradiation and decay to gain CTIVATION
N. Y.
the optimum selectivity for the measurement of the sought activity. Although it is a relatively simple matter to calculate optimum conditions for samples producing one or two radionuclides, the task becomes exceedingly difficult as the number increases. For this reason, a computer program has been developed which rapidly determines the optimum irradiation and decay times for any element in any mixture regardless of its complexity. The program is written in a general fashion and can be applied to any method of irradiation-Le., thermal neutron, fast neutron, charged particle, or any combination thereof. Also, either gamma spectroscopy or measurements of total activity may be employed. The program is written in FORTRAN compiler language to permit its universal use. Thus, a method is provided which permits a rapid evaluation of the feasibility of applying activation analysis to any problem. MATHEMATICAL BASIS
- e-kTz)e-ATD
When the partial derivatives are determined and simplified, the following equations result: N
According to the equation for growth and decay of radionuclides : A = IXFu(1
The greatest selectivity (based on instantaneous counting rates) for isotope j will be obtained when R j is a maximum. The region of the maximum R i may be examined in detail where Ri decreases markedly during the counting interval. To arrive a t the maximum value of R i it is necessary to solve the following two equations:
C K { ( x -~ ~ j ) ( -l
e-’iTZ)e-’iTD
=
0
i=l
(4)
(1)
where A = usable activity I = fraction of events produced in the energy range to be measured N = number of target nuclei F = flux of bombarding particles u = reaction cross section A = decay constant T I = irradiation time To = decay time The ratio of activity of species j in a mixture is then:
e-AiTo =
0
where : Ki
= IiNiFi~i
These two equations must then be solved simultaneously for TI and TD, these values being the optimum irradiation and decay times. Since these VOL. 36, NO. 6, M A Y 1964
1089
