Evaluation of Radioactive Decay Data ROBERT G. MONK, ALAN MERCER,’ and TOM DOWNHAM Atomic Weapons Research Establishment, Aldermaston, Berks., England
b Methods of evaluating the individual component activities of a mixed source from radioactive decay data are compared b y applying them to counting data from sources prepared by mixing known amounls of M o g 9 and La1“. The most suitable equations are shown to be the “maximum likelihood” and “minimum chisquared,” and recommendations are made as to their practical application to minimize possible sources of ertor.
AS
THE REWLT of a radiochemical analysis, an active source containing one or more isotopes of a particular element is obtained. I n some cases the individual constituents may be determined by using the different characteristics of their radiations using techniques such as a- or y-ray spectrometry. Frequently, however, particularly in the cases of @-emitters, this is not possible, but if the components differ sufficiently in half life they may be resolved by means of decay data. The question then arises of how best to estimate the individual activities. I n this paper different methods of calculation are derived and compared by applying them t o data obtained from mixed sources containing known added amounts of two constituents each of known half life. The special case of a source containing a single radioactive species is also considered. The accuracy with which the individual components of an active source may be determined from decay data obviously decreases as the half lives approach one another. In practice, a half life ratio of 1.5 is about the minimum we have considered M orth using, an example being the palladium isotopes Pdlo9 (13,6h) and Pd”* (21h) produced in fission. For testing and comparing different methods of calculation, one needs data on a difficult case of this kind. If a radioactive source can be prepared by mixing known amounts of the different nuclear species, the true values are available against which any calculated figures may be compared. Radiochemically pure pal-
Present address, Armour and Co. Ltd., G.P.O. Box No. 250, International House, 193. St. John St., London, E.C. l., England. l
178
ANALYTICAL CHEMISTRY
ladium-109 and palladium-112 are not readily obtainable separately nor is there another similar pair of isotopes available which could be used for this study. However, for the present purpose any pair of radioactive nuclides with suitable half lives may be used, and i\Iog9(66.lh) and La1@(40.2h) were selected as they are readily obtainable in a pure state. STATISTICAL M E T H O D S
OF
ESTIMATION
.4 theoretical account of the procedures used in the paper is given in (1) ; for the statistical terminology readers are referred to ( 3 ) . The only errors considered in this section are the statistical errors due to the random nature of the decay process, and no account is taken of errors in the counting equipment. However, these cannot be ignored in practice, and some recommendations will be made later to minimize unknown errors from this cause. Notation. Suppose t h a t a source consists of c components each containing a n unknown number S,atoms, jvith knonm decay constants A,, j = 1, 2, . . . c, so that a t zero time the disintegration rates are given by
..
I , = X , S , , J = I, 2,
*
c
(1)
Let there be k counting intervals of duration T $ which start a t time t,, i = 1, 2, . . . k so that t, < t, Tt t ~ + i ,i =
+
1,2,
. . .k -
1
(2)
Then the probability of counting one particle of type j during the i t h interval isp,,, where p L 1 = exp ( - - X l t , ) ( 1 - exp ( - X , T % ) I (3)
If 7%is sufficiently small for A ) T ~to be small compared with unity--i.e., if the decay is negligible over t h e interval T % then , p , , may be written as p,, =
x , exp ~ (~- X , t , )
(4)
Suppose that during the it11 interval, the background rate is p c , so t h a t the expected number of background counts is p L 1 7 , . Denote the count in the i t h interval by n, and the observed counting rate by m t ,where 121 = ?nzTr, 2 =
1, 2,
. . .k
(5)
The dropping of a suffix implies t h a t it can only take one value as, for example, when there is only one component
present; this will be obvious from t h e context. Stopping Rules for Counting. \Then counting, a n experimenter might decide to count for a fised time so t h a t t h e number of counts is a random variable. Alternatively, he may decide to count until t h e next whole minute after he has recorded a given number of events, in which case t h e counting time is a random variable. The fundamental difference between the two stopping rules is that the first rule is independent, while the second rule depends on the number counted. S o w suppose that the experimenter intended to count for a given period of time but after he had started, he decided to count for a little longer. If he took the decision without examining the recorded counts at some stage during the elapsed period of counting, then the stopping rule falls in the first category. However, if he examined the counts and decided t o count longer, because perhaps he would only have had a small sample by the predetermined time, then the stopping rule falls in the second category, although both time and the final count may now be random variableq. The extra information supplied by knowing the stopping rules enables more accurate estimates of the activities t o be determined. However, when this information is used, it may lead to totally different estimators for different stopping rules. Thus it mag‘ be quite wrong to use an estimator derived for one stopping rule with a different stopping rule. Holvever, it can be shown that if the preassigned counts, for a stopping rule in n-hich the times are random variables, are sufficiently small for the counting intervals to be reasonably short for practical purposes, are small compared d h the initial number of radioactive atoms, and are not sinall when measured absolutely, then the maximum likelihood estimators and the covariance matrix are the same as for fixed time intervals. This conclusion will be unaltered for the intermediate case of a mived stopping rule. Hence in this paper, the counts are assumed to satisfy these three conditions, which is the practical situation, and the formulas have been derired assuming that the counting intervals are of fised duration.
Estimators When t h e Number of Counting Intervals Equals t h e Number of Components. The component activities cannot be estimated unless the number of counting intervals is a t least equal to the number of components. Then the only method is to equate the observed counts with their expectations. Thus the estimators are obtained by solving the c linear equations in A-,
have the Poisson distribution with mean N i l where
where the p L have been estimated It is obviously inindependently. advisable to reduce the number of counting intervals to the minimum for more than one source, when all the other estimators are the solutions of Equations 6. Maximum Likelihood Estimators. I n the absence of background, the distribution of counti for a single miirce is multinomial and the maximum likelihood estimator of i V is tlie integral part of
Equations 11 have t o be solved by iteration. Experience has shown that if the first approximation is obtained by selecting c of the k observations, which span the total counting time fairly uniformly, and solving Equations 6, then the iteration converges quickly. The information matrix can be shown to have the (u, u ) t h element
Then the maximum likelihood estimators of N i , j = 1, 2, . . . c, are the integral parts of the solutions of the c equations
These equations are less easy to solve than Equations 11; the covariance matrix cannot be obtained readily, and as they have no compensating advantage, they will not be considered further. If the variation in the term M,-l is ignored, then it is well known that this modified minimum chi-squared method of estimation leads to the maximum likelihood estimators. When the term Tif6-l is replaced by nj-l, then the estimators are the solutions of
(11)
so that the asymptotic value of the covariance of iYu and S,is AutIiAI
ant1 its variance is
This result is of practical importance because both the estimator and its and cpi. variance only depend on Thus, in theory, it is better to count a4 many particles as possible but no advantage is gained by taking intermediate counts or by counting after a long period of time has elapsed, n hen the particles will be omitted less frequently. In other words the best practical plan is to make one long count as soon as possible. This conclusion is also valid when the background is small compared with the espected source count so that the estimator is the integral part of
En,
K h e n the background is comparatively large, it may be better to use Equations 11 for one source; Equation 9 could be used as a first approximation. I n the general case, the probability distribution of the counts is extremely complicated. However, since the number of atoms of each component decaying in each counting interval is assumed to be small compared with the total number of the same component present, the distribution may be closely approximated by considering the counts in each interval to be independent and
(13)
where A,, is the cofactor of uuL and !AI is the determinant of the matrix whose elements are auL. For one source, these results will be seen to be in close agreement with Equations 7 and 8. X i t h c components, it is obviously necessary to make a t least c determinations, and it can be shown that there is a n advantage to be gained from counting in more than c intervals, even without background. This would be expected intuitively since the extra determinations provide information about the relative activities of the components. It is obviously desirable that as many counts as possible should be recorded. Minimum Chi-Squared Estimators. If the counts are random variables, the chi-squared statistic for the problem being considered here is
These equations have the advantage that they are linear and do not need to be solved by iteration. They are the equations which are frequently used by experimenters and called weighted least squares; then the standard expressions for the errors of weighted regression coefficients are used in the estimation of the variances. This procedure is incorrect because the weights used in the derivation of Equations 16 are theniselves random variables and not predetermined as in standard regression theory. Expressions for the correct errors have been derived, but these are too long and complicated t o present here; numerical values obtained from them are given in Table I. Minimum S u m of Squares. The minimum chi-squared estimators are derived by minimizing one weighted sum of squares. TFO other estimators which minimize sums of squares are considered below. First the unweighted sums of squares of rates is
If this is minimized, then the estimators are the solutions of the equations
i=l
This is obviously the weighted sum of the squares of differences between the observed and expected counts, so t h a t estimators may be obtained by minimizing chi-squared. When Equation 14 is minimized, the estimators are the solutions of the c equations
J =
1,2,.
..c
(15)
j
=
1, 2 , .
. .c
(18)
This method has been widely used by radiochemists in this establishment and elsewhere ( 5 ) . Compared with the minimum chi-squared estimators those given by equations 18 are weighted in favor of early counts. VOL. 35, NO. 2, FEBRUARY 1963
179
Second , consider t,he c weighted sums of squares
absence of background to that of the maximum likelihood estimator is k i=l
Then minimizing the hth weighted sum of squares with respect t o .VhJthe estimators are the solutions of the c equations
j
=
I, 2,
.. .c
(20)
This method has occasionally been used here in an effort to avoid the overweighting of early counts given by Equations 18. However, Equations 20 tend to overweight later counts. Thus, the first set of estimators will be sensitive to fluctuations early in the counting, and the second set will be sensitive to fluctuations late in the counting. Both sets of estimators are linear in the counts n,, so t h a t the covariance matrix may be derived easily; numerical results are given in Tables I and 11. Perkel (4) has used a least squares procedure with partial exponential weighting which is intermediate between the above two methods. If there is only one component, then Equation 20 leads to
which gives some insight into how the method of estimation leading t o Equation 20 was devised. The ratio of the variance of this estimator in the
i=l
It can easily be seen that this ratio is less than one unless all the p,’s are equal, and that it can become very small EO that Equation 21 could be a bad estimator. If, however, p , is maintained constant the ratio is equal to one and Equations 7 and 21 become equally good estimators. Expressions for Two-Component Systems. T h e expressions derived above are i n their most general forms. The equations for two coinponents are given below, in all cases k
being written for
C and
T~
being as-
%=l
sunied sufficiently small for Equation 4 to be true. The wffix i is dropped in all cases. Maximum Likelihood. From Equations 11 the values of I , and I 2 are given by
I n the other cases formulas for calculating Il are given. T h e expressions for I z are in all cases similar, hl and hz being interchanged. Minimum Chi-Squared. From Equations 16
Minimum Unweighted Least Squares. From Equations 18
Minimum Exponentially Weighted Least Squares. From Equations 20
180
ANALYTICAL CHEMISTRY
EXPERIMENTAL
High specific activity samples of &Io99 and Lala were obtained from Atomic Energy Research Establishment, Harwell. Ten milligrams of -11003 (AnalaR quality) were irradiated at about 1012 neutrons/sq.cm./sec. for a week in B.E.P.O. Because of the low (n,?) cross section of l I 0 9 8 , the irradiated material was given a purification treatment to remove any small amounts of activities arising from trace amounts of high cross-section impurities. The &Ioos was dissolved in ammonia, scavenged with ferric hydroxide, the solution made I J I with respect t o nitric acid, and molybdenum precipitated with a-benzoinoxime. The complex was oxidized with nitric and perchloric acids, and the solution fumed to dryness. The residual Moo3 mas dissolved in 4 V HCl and diluted to give a solution containing about 0.5 pc./’gram. Ten milligrams of La203 (Specpure) were irradiated a t about lo1?neutrons/ sq.cm./sec. for an hour in B.E.P.O. The material, used without purification, was dissolved in a few drops of IJI hydrochloric acid and diluted t o give a solution containing about 0.8 pc ./gram. The solutions were standardized by counting sources prepared from three n-eighed portions of each solution. h standard method of source preparation was used whereby the solution was evaporated on a polystyrene (Distrene) foil (thickness 1.2 mg./sq. em.) stretched over a circular hole of 23-mm. diameter in a n aluminum card 3 x 33i8 inches, insulin being used as a spreader. The hIog9 solution gave (3.64 -C 0.02) x 105 counts/min./gram and the Lala, (5.82 + 0.04) X 105 counts/ min./gram of solution. The errors are standard deviations obtained by the method of small number statistics ( 2 ) , and are typical for such a measurement under our conditions. Foiir sources, each containing both &Io99 and Lal40 mere prepared, tn.o
containing a total of 60,000 to 70,000 counts per minute and two a total of 1000 to 2000 counts per minute. I n each pair, the ratio of hIog9 to La140 was about 4 to 1 on one source and 1 to 3 on the other. The strong sources were prepared by weighing and mixing appropriate amounts of each stock solution. For the weak sources, dilute mixed solutions were prepared by weight from the stock solutions and portions of these weighed out. The weight of solid material on each source was less than 20 pg. and no corrections due to scattering or self-absorption of p particles were necessary. A11 sources were counted in the same standard position in a lead castle with a gas flow ,&proportional counter, the gas used being a 10 to 1 argon to methane mixture. All observations of count rate were carried out in triplicate t o obviate gross errors, and the time elapsed from zero was measured to midway between the beginning of the first period and the end of the third period; errors due to ignoring decay during counting were all negligible compared with statistical variations and Equation 4 is therefore valid throughout. Sources were counted a t approximately daily intervals (excluding weekends) over a period of 3 to 4 Lveeks in the case of the strong sources, and 2 t o 3 m-eeks in the case of the weak. Background counts were also determined before each set of observations. With the counting system used, no dead time corrections were necessary a t any count rate observed. The strong sources were used to determine the accuracy of the various procedures under the best possible conditions. Consequently, as the sources decayed, the periods of counting were increased to compensate partially for decay by observing as many events as possible. Initially about 150,000 counts per set of three observations were recorded, falling to about 3000 a t the end of the decay period. The weak sources were used to obtain data with very poor statistics and it was hoped thereby to emphasize differences in the various mathematical procedures used. Accordingly a fixed counting time of 3 minutes was used throughout the observations and the numbers of counts recorded per set of three observations ranged from about 10,000 down to about 130. RESULTS AND DISCUSSION
To conserve space, complete counting data are not given here but Table I shows the initial and final observations on each of the four sources. The corrected count rate is the observed rate minus the background rate. It is the corrected rate which has been used in comparing the formulas-Le. the corrected rate is taken to represent a set of counts in the absence of background. This device is not recommended for estimating unknown activities because, although the estimates are unlikely t o be altered, the errors
will be too low. However, no false conclusions will be reached when comparing formulas, since the corrected counts could, in fact, have been registered under background-free conditions. I n Table I1 the initial component activities and their errors are given. The errors quoted were obtained by combining the standard deviations of the specific activities of the stock solutions with those of source preparation, including Keighing errors. The maximum likelihood estimators, obtained by solving Equations 11, are given together with their standard deviations obtained from Equation 13. The minimum chi-squared estimators given by Equations 16 are also included. These are the estimators TT hicli are frequently used. The correct values of the standard errors of these estimators are given and the ~ e i g h t e dregression standard errors, which are incorrect because the weights are random variables, are also included. The minimum unweighted sum of squares estimators derived from Equations 18 and the estimators given by minimizing the sum of squares weighted with exp ( Z i t ) , obtained from Equations 20 are also given, together with their standard errors. That the maximum likelihood and minimum chi-squared estimators and their correct standard errors are equally good is shown in Table 11; the estimators are close to the true value, and the standard errors are less than those of the other two estimators. The incorrect weighted regression estimates of the standard deviations are too large; for sources A and C, the ratio of the two estimated standard deviations is almost two. The only difference between the two methods of estimation, provided that the correct standard deviations are used for the minimum chi-squared estimators, is
Table II.
Source Component Activity added, c.p.m. Error
Max. likelihood Std. error Minimum chisquared Std. error Regression std. error Unweighted last square Std. error Exponentially weighted least square Std. error
ease of computation. The maximum likelihood equations are less easy to solve, but it is much easier to find the standard errors of the estimates. When reasonably good first approximations to the estimates are obtained, in which case very few iterations are necessary, then the maximum likelihood method is preferable. The unweighted and exponentially weighted least squares estimators give some results which are appreciably different from the true values and have large standard errors. The unrreighted
Table I. Initial and Final Counting Data on Mixed Sources of M o g 9and La140
Mean time from zero, hr., ti
Counting
time, min., ri
Count rate in c.u.in. CorObrectrd
served,
771'-
Pz
7n1
1 38,135 38,105 1 38.513 38.483 1 39; 228 39; 198 35 71 45 20 69 43 3 928 898 3 973 943 3 973 943 3 44 17 3 44 17 3 36 9 1 51,085 51,055 1 51,021 50,991 1 50,848 50,818 20 70 46 20 67 43 20 73 49 3 1,382 1,352 3 1,446 1,416 3 1,368 1,338 3 44 17 3 52 25 3 53 26 Source A, 53 Observations. b Source B, 33 observations. source C, 60 observations. dSource D, 42 observations.
29.3" 29.3 29.3 560.3 560.3 29.8b 29.8 29.8 342.4 342.4 342.4 29.4~ 29.4 29.4 683.3 683.3 683.3 29. 6d 29.6 29.6 418.3 418.3 418.3 (2
Initial Component Activities and Their Errors
B
A
C
I)
La140
Mo9Q
357 3
14,550 133
56,920 398
370 4
1556 11
1121 26
353 15
14,895 241
57,715 117
375 37
1560 23
14,905 56
1139 26
340 I5
14,942 234
57,681 113
388 37
1550 25
259
90
34
19
433
210
43
26
45,157 450
15,548 281
1116 41
357 26
12,023 557
59,633 359
377 52
1561 33
45,653 353
15,090 103
1086 50
367 24
18,038 1,480
56,792 388
99 125
1662 51
La140
11099
LaI40
47,300 33 1
14,830 119
1187 10
46,076 162
14,922 55
46,115 165
~ 0 9 9
LaI40
VOL. 35, NO. 2, FEBRUARY 1963
11099
181
method, which favors the early counts, gives very poor answers for the two large sources, h and C. The exponentially weighted method, which favors the late counts, gives very bad answers for sources C and D, in which the component with the smaller decay constant predominates. The effect of overweighting the late counts when the exponentially weighted squares is minimized is clearly seen from the results for source D. The last three observations of count rate are 17, 25, and 26 counts per minute. These counts are obviously subject to considerable statistical variation, and the estimators have been obtained for the cases where these three counts might all have been either 17 or 25. The results are given in Table 111.
Table 111. Estimators for Source D Component La140 &fogs Activity added, c.p.m. 370 1556 Error 4 11 Actual result (exponentially weighted 99 1662 least squares) 125 51 Std. error All three observations 17 (exponentially weighted least 457 1524 squares) Std. error 121 50 All three observations 25 (exponentially weighted least -48 1719 squares) Std. error 137 52
The goodness of fit of the experimental data to their expectations has been studied by computing the chisquared statistic of Equation 14 for all four sources, where the values of the activities are the measured values and those obtained from the maximum likelihood equations. For sources B and D, with the smaller counting rate, there was good agreement, but for sources h and C, with the large counting rate, significant values of chi-squared were obtained. This suggests that for a large counting rate, there was a substantial counter error in addition t o the statistical error. How this affects the two estimates of the standard error of the minimum chi-squared estimators cannot be determined unless the form of the counter error is understood. However, even with counter error, the maximum likelihood and minimum chi-squared method give good estimators of the source strength although the standard errors are optimistic. Thus the main conclusions of the paper remain valid. Practical Application of Formulas. The maximum likelihood and minimum chi-squared formulas give results 182
ANALYTICAL CHEMISTRY
weighted in favor of observations which have the greater number of counts. Now if counter error, which will be a proportional and not a constant error, is present, a point will be reached when further increase in the number of counts observed will not significantly reduce the total error as the counter error will have become the dominant factor. If d is the total coefficient of variation of the observed count rate, and d, and d, are the respective coefficients of variation due t o statistics and to the instrument, then d = ddS2 d,2. Obviously when d, becomes small compared with d,, d = d,, and the error is no longer affected by the statistics of radioactive decay. We have now two mutually contradictory requirements. For maximum accuracy d should be a minimum; i t then becomes equal t o d, and the basis for the recommended formulas is no longer valid. For the formulas to be completely applicable, d should approximate ds-i.e., i t should be made large by restricting the number of counts observed so that di becomes negligible. The total error then increases with the applicability of the formulas. Obviously a compromise is necessary and it is suggested that the maximum number of counts observed should be that which makes d, about equal to d,. Then d will be 1.41 d,, and while such a n observation will be somewhat overweighted, it will not be excessively so. hleasurements over a period of 6 weeks using a long-lived source in 6 gas proportional p counters, involving about 300 observations on each counter, gave values of d, for each counter of 0.0, 0.5, 0.6, 0.4, 0.4, and 0.5%. The mean value of 0.4% corresponds to a total of 62,500 couiits, and i t therefore is reasonable to adopt 50,000 counts as a maximum per observation for this particular type of counter. I n practice, a n occasional measurement may be allowed t o carry on for a much longer time than was intended or perhaps it may be convenient to carry out the determination over a lunch hour or overnight. I n any event far more counts than the desired maximum will have been accumulated, and this observation will unduly weight the result finally obtained. A long count of this kind is far more likely to contain a large error due to drift in electronic components, fluctuation in mains voltage, etc. than a short one. I n such a case the figures must be adjusted before calculations are made by giving T* an arbitrary value, instead of its real value, so that T,TTZ; is about 50,000; for convenience 7, would be made an esact number of minutes. The above considerations also apply t o the case of the single component
+
source. I n this case the maximum likelihood Formula 7 gives
This is readily computed and is mathematically the soundest with the above precautions against 01-ercounting. The formula implies that as many counts as possible should be recorded but that there is no particular virtue in spreading the observations 01 er a period of timeLe., all the necessary counts could be recorded in one observation with a consequent simplification in computation. Practically, there are two reasons for making the measurements over a period: counter error and t h a t decay data give a check on the radiochemical purity of the source. Counter errors will tend to average out o\er a period. Concerning decay data, our normal practice has been to observe the constancy of (m, - p q ) exp(Xt,). T’isual inspection of the figures quickly s h o w whether a long or short lived impurity is present and also whether any of the observations contain gross errors. It has also been the practice to take the mean value of (m, - fit) exp ( A t , ) as the best value-Le., to use Equation 21, which overneights late counts and is therefore, mathematically, less sound. However, this procedure is quite valid under certain sets of conditions and both practical and useful over a wide range. The variance of the result of Equation 21 is equal to that of Equation 28 if p,(= e.tp(-iXt,)) is maintained constant. This means increasing T~ so that about the same number of counts are recorded per observation. There is a practical limit to doing this when count rates become very low. The other condition in which Equation 21 is valid is that in which elery observation includes so many counts that d, >> &,-i.e., d = d2, and every observation has the same coefficient of variation, that of the counting equipment. Practically i t has been found that even in circumstances where Equation 28 is clearly correct-Le., d, >> dL,there may still be negligible differences between the results of the two methods of calculation, Consider the following typical results given by two sources. Source 1 decayed from 121.8 to 11.4 c.p.m. Ten observations were made and 12% varied from 7311 to 916. Formula 7 gave I = 120.6 0.7 and Equation 28, I = 120.5 f 0.8. Source 2 decayed from 39.8 to 6.1 c.p.m. Nine observations were made and n, varied from 2385 to 369. Formula 7 gave I = 38.9 0.4 and Equation 28, I = 38.9 0.5. I n these cmes nL, and, hence. 7,
+
*
*
esp(-AXt,) laried over a maximum range of 8 to 1. From the above conqiderations it is reasonable to use Equation 28 for data in which 7 %esp(-AXt,) varies with a masimum range of 10 to 1. Care must be taken that the data being processed are those most suitable for the purpose in hand-Le., that they contain all the information required for determining the species of interest with the minimum of extra irrelevant data. For example, consider a I to 1 mixture of a pair of nuclides, X and Y , of widely differing half life, a system which is normally resolved graphically using a semilog plot. After a few half lives of the shorter-lived component A’, the remaining activity will be so largely due to Y that little further information about X will be available. On purely statistical grounds and assuming no change in counter behavior better values for both X and Y will be obtained by continuing the counting as long as possible and using all the data in the computation. However, counters are subjrct t o drift sometimes and if this occurs when all the X has decayed, late observations will adversely affect the accuracy of the X determination. I n such a case the best value of X will be obtained using early data only. On the other hand, all observations may be used for the determination of Y which is present in excess for most of the time. Whether i t is desirable t o split the data u p or t o use all of it will depend on having plenty of information on counter bFhavior. This may be obtained by repeated counting of a longlived source as a relative standard and observing whether the. mean count rate and d, change significantly over the period of decay of the unknown source. Such treatment is probably only necessary in the case of a small amount of short-lived activity mixed with com-
parable or greater amounts of much longer-lived species.
(c) If desk calculation is to be done, use the minimum chi-squared Equation
25. RECOMMENDATIONS
If maximum likelihood is being used for a single- or multicomponent system, and if some observations include so many counts t h a t the instrumental error exceeds the statistical, arbitrarily reduce the value of 7% t o the nearest whole number of minutes t o make the two about equal-Le., so that 100 = d, d T 2 ( r n ,
- P%)
Single Component. ( a ) Use t h e mean value of (m, - p , ) esp (At,) provided that the range in the values of T~ exp ( - A t , ) does not exceed 10 to 1, the number of observations is at least 6, and x~~(7n~ - p t ) is a t least 10,000. (b) I n other cases use the maximum likelihood equation
c
=
7
h
-
PL)
exp(-At,)
7%
Multicomponent System. If t h e amount of a short-lived component is comparable with, or less t h a n other much longer-lived constituents a n d i t is known t h a t counter drift has occurred, consider t h e d a t a carefully and divide into periods a s appropriate t o t h e half lives of t h e nuclides present, Otherwise treat all the data as in (b) or (c) below. For the higest accuracy activity determinations should be made as frequently, and over as long a decay period as possible. ( b ) Use the maximum likelihood Equations 23 and 24 if an electronic computer is available. A suitable program has been prepared for the IBhf 7090 and modified for the IBM STRETCH, nhich will resolve up to 10 components.
NOMENCLATURE
Aj
Ij
c ti
ii ni mi pi
pij
Mi
A
A,, d
d, di
Suniber of radioactive atoms of species j in source at zero time = Disintegration constant of speciesj = Decay rate of species j in source at zero time; I , = i , N , = Tqtal number of radioactive species in source = Time from zero at which zch observation of decay rate begins = Duration of i t h counting interval = Total number of counting intervals = Number of counts in it” interval = Count rate observed in ithinterval = Background count rate in ith interval = Probability of counting one particle type j during ith interval = Expected number of counts in ith iinterval = Information matrix, ccmposed of elements a,,,, = Cofactor of uuu = Coefficient of variation of observed count rate = Coefficient of variation of count rate due t o source = Coefficient of variation of count rate due to counter
=
Nj
ACKNOWLEDGMENT
The calculations were originally performed by Margaret Judd; the tensource program for the IBM 7090 computer has been written by I. C. Smith. LITERATURE CITED
(1) Cramer, H., “Mathematical Methods of Statistics,” Princeton University
Press, Princeton, 1951.
(2) Dean, R. B., Dixon, IT. J., -4x.k~.
CHEM.23,636 (1951).
(3) Kendall, 11. G., Buckland, IT. R., “A Dictionarv of Statistical Terms.”
Oliver and
RECEIVED for review June 22, 1962 Accepted November 16, 1962.
VOL. 35, NO. 2, FEBRUARY 1963
183
