very hard pyrosulfate fusion in a 250-ml Erlenmeyer flask, showing no detectable volatility under these conditions. Consequently, it is strongly rcommended that no operations in the quantitative determination of lead be carried out in a platinum dish. Reduction and Volatilization of Bismuth. During preparation of the bismuth-210 solution from lead-210 for use in standardization of the ,8 counter, both reduction and volatilization of bismuth must be avoided or the yield of product will be substantially reduced. If a solution of bismuth-210 containing sodium sulfate, hydrobromic acid, and excess sulfuric acid is evaporated to a pyrosulfate fusion, very little volatilization of bismuth occurs. Under the conditions recommended above for the volatilization of tin, loss of bismuth is only 2.5%. Clearly, sulfuric acid transposes any bismuth tribromide that might be present to the nonvolatile bismuth sulfate and hydrogen bromide which is volatilized completely before the temperature becomes high enough to cause volatilization of the bismuth tribromide. In contrast, if excess sulfuric acid is not added after the pyrosulfate fusion, as much as 35% of the bismuth-210 has been lost during a single evaporation with 5 ml of 48% hydrobromic acid. If excess sulfuric acid is not present, the solution will evaporate to dryness, permitting the temperature to rise above that required to volatilize bismuth tribromide while both bismuth
tribromide and hydrobromic acid are still present, and volatilization of substantial quantities of bismuth results. Sodium pyrosulfate alone is not sufficiently acidic to transpose bismuth tribromide before substantial volatilization of bismuth will have occurred. Quantities of bismuth-210 approaching 50% of the total present have been lost during the precipitation of metallic polonium from hydrochloric acid solution using either finely-divided metallic silver or stannous chloride in the presence of tellurium carrier. In the presence of hydrobromic acid as described, the oxidation potential of the bismuth ion-metallic bismuth half cell is so reduced because of the formation of the much more stable bromide complex that the reduction of bismuth by stannous chloride is reduced to about 3%. LITERATURE CITED (1) (2) (3) (4)
C. W. Sill and C . P. Willis, Anal. Chem., 37, 1661 (1965). C. W. Sill, Health Phys., 17, 89 (1969). C. W. Sill and R. L. Williams, Anal. Chem., 41, 1624 (1969). C. W. Sill, K . W. Puphal, and F. D. Hindman, Anal. Chern., 46, 1725 (1974).
(5) D. R. Percival and D. B. Martin, Anal. Chern., 46, 1742 (1974).
RECEIVEDfor review September 27, 1976. Accepted November 12,1976.
High Precision Pulse Counting: Limitations and Optimal Conditions J. M. Hayes*
and D. A. Schoeller‘
Departments of Chemistry and Geology, Indiana University, Bloomington, lnd. 4 740 7
It Is shown that, because uncertainties in the deadtime contribute to uncertainties in computed count totals, the maximum count rate for any measurement is given by uN/Nu, where uN is the required standard deviation of N, the number of counts, and up is the standard deviation of p, the deadtime. Further considerations define an optimal count rate, show that the minimum counting time varies inversely with the third power of the required precision, and discuss the cancellation of these uncertainties in ratios of count totals. It is concluded that, because this work shows that higher precisions require lower count rates, the maximum precision attainable from counting measurements of reasonable duration lies near 0.1%.
Using intricate and thorough considerations of the possible noise sources, Ingle and Crouch ( I ) have shown that pulse counting techniques, in general, will be characterized by lower signal-to-noise ratios than dc measurement systems under most conditions of analytical interest. The situation is reversed when signals become so weak that pulse overlap is insignificant, but this limitation, if accepted, has the effect of relegating pulse-counting techniques to applications where high precision (relative standard deviations 50.1%)is virtually unobtainable. We would generally agree that pulse counting is an inferior technique when high precision is sought, not only for the reasons put forward by Ingle and Crouch but for additional reasons which will be summarized in the conclusion Present address, Department of Medicine, University of Chicago, Chicago, Ill. 60637.
of this paper. Our experience, however, does certainly indicate that high precision can be obtained in ion-counting mass spectrometry (2). For this reason, and because experimenters are often confronted with the problem of extracting the highest possible precision from apparatus involving a pulsecounting detector system, it is useful to consider the aspects and possibilities of high precision pulse counting in some detail. Measurements which provide high precision in relatively short times will, in general, require the use of signal levels high enough that significant pulse overlap will occur in most counting systems. In these circumstances, some account of the effects of overlap must be taken, and we wish to focus on this in the present discussion, exploring the relationships between this process and the precision and accuracy attainable in practical counting measurements. The most common approach to accounting for pulse overlap is the application of a correction formula which will have the effect of restoring the counts lost due to pulse coincidence. A second approach involves the use of a recently developed instrumental technique (3, 4 ) , however, and it is necessary first to weigh these two alternatives, considering which is more appropriate to high precision measurements. The instrumental technique can be termed “dead time compensation” ( 3 ) ,and begins by deliberately excluding approximately half the true pulses from the counter. This exclusion is accomplished by setting the discriminator threshold about midway in the single-event pulse height distribution. Then, as the pulse rate increases and count losses begin to occur, a compensating pulse gain can be experienced with the effect that the linearity (observed pulse rate vs. true signal) of the system is improved over what would otherwise be ob-
306 * ANALYTICAL CHEMISTRY, VOL. 49, NO. 2, FEBRUARY 1977
served. The compensatory gain occurs because multiple-event pulses become important at high pulse rates. While less than half of the single-event pulses are counted (and pulse overlap further reduces this low efficiency), almost all of the multiple-event pulses are large enough to be counted, and this much higher efficiency tends to bend the count rate vs. signal line up at the same time coincidence losses are tending to bend it down. The effects can be quite well balanced, with the result that linearity is substantially improved ( 4 ) . While it is unquestionably ingenious, we have concluded that dead time compensation is less useful than count-loss correction techniques when high precision is sought. Ingle and Crouch ( 4 ) have reached a similar conclusion, noting that the exclusion of a large fraction of the pulses has the effect of decreasing the signal-to-noise ratio, and that compensation techniques are not applicable to all types of counting systems. T o these considerations we would add the following comments. The precision of the compensation technique must be subject to experimental verification. In practical terms, it must be possible to show that the apparent deadtime of the compensated system is zero in the signal range of interest, and it would seem that the adjustment and verification of discriminator levels would be a t least as tedious and involved as the determination of deadtimes and application of count-loss correction formulas. In our limited experience, the problem of system stability can be serious a t this point: the point of efficient compensation (apparent zero deadtime) requires placement of the discriminator threshold near the maximum of the pulse height distribution, and small drifts in the threshold level or in the pulse height distribution can thus cause large changes in the observed count rate, failure of efficient cdmpensation, and significant changes in the apparent deadtime. This sensitivity to drift goes against one of the strongest recommendations of conventional pulse counting techniques. Given that the use of count-loss correction methods provides the best approach to highly precise results, it is well to ask where the limitations on that technique must lie. Indeed, it was our observation that the anticipated precision could be quite elusive that prompted the recognition of the category of limitations which is described here for the first time. The essence of this argument is very simple: the use of count-loss correction expressions requires knowledge of the system deadtime; consequently, uncertainties in the deadtime will cause uncertainties, or imprecision, in the corrected count rate. We show here that this propagation of errors is subject to a straightforward treatment demonstrating that the precision with which the deadtime can be established is a critically important controlling factor in high speed, high precision counting measurements. Furthermore, it is shown that because uncertainties in collected count totals depend on the magnitudes of the associated count-loss correction factors, it follows that the uncertainty in a count total depends on the rate at which the events were collected. This leads to the possibility of defining optimal conditions for high precision counting measurements. A separate, but related question asks, what, exactly, is the deadtime of a counting system? How can it be defined and determined in a way which is both generally useful and consistent with the considerations introduced here? These topics are taken up in a separate paper ( 5 ) . THEORY OF COUNT-LOSSES The theory of coincidence-losses in counting systems has been elegantly summarized by Barucha-Reid (6), who begins by defining two sequences of events. The “primary sequence” is the sequence of true events, for example, the sequence of incoming ions in an ion-counting electron-multiplier system. The “secondary sequence” is the sequence of recorded events.
An understanding of count-losses due to pulse overlap has been achieved when the rule relating these two sequences can be precisely stated. In practice, two fundamentally different types of counting systems can be recognized. In one case, the system is characterized by a finite, fixed resolving time p, and a true event is recorded if an only if no recorded event has occurred within the preceding time interval p. Alternatively, the resolving time can be randomly variable, with true events being recorded if and only if no other true event has occurred within the preceding time interval p. If such a counter is subjected to a fast enough particle flux, all true events are separated by time intervals less than p , and no events are recorded. Such a system is said to be “paralyzable”. Less memorably, and much more confusingly, such systems have been designated as “Type 11”by Barucha-Reid (6)and as “Type I” by Ingle and Crouch ( 4 ) and Evans (7). Systems following the first rule given above would register events a t frequency p - l even if subjected to an infinitely fast particle flux, and are termed “nonparalyzable” (or “Type I” or “Type 11”). Paralyzable Counters. In general, the primary sequence is truly random, and the probability of observing x events in any given time interval t is given by the Poisson distribution (7):
mx P,(t) = - e - m X!
where m is the average number of events occurring in each interval t . For an event to be recorded by a paralyzable counter, we require that no true events occur within a time interval p. The probability of this occurring is given by
where F is the average frequency of events in the primary sequence and, thus, F p is the average number of events in any time interval p . As F p approaches zero, e-Fp approaches unity, the counter is consequently active nearly all the time, and the secondary sequence almost exactly duplicates the primary sequence. As F p becomes large, the probability of observing no events during a time interval p decreases, the counter is thus inactive for an increasingly large fraction of the time, and the secondary sequence excludes an increasing fraction of the true events. Quantitatively, the relationship is given by f =
e--FpF
(3)
where f is the observed frequency of events in the secondary sequence. Nonparalyzable Counter. A nonparalyzable counter is inactive for a time interval p after each event in the secondary sequence. The fraction of time during which the counter is inactive is thus given by the product f p ; the fraction of time during which the counter is active is (1- f p ) ; and the primary and secondary sequences are thus related by the expression
f
= (1 - f p ) F
(4)
Practical Measurements a n d Corrections. The analyst records the secondary sequence and needs to derive the primary sequence with some specified level of accuracy. Given only f and p , Equation 3 cannot be solved directly for F , although iterative techniques, such as Newton’s method (8),can furnish an exact solution. Given the somewhat cumbersome nature of the latter technique, it has been common in practical work to apply one of two first-order approximations. Series expansion of the exponential in Equation 3, followed by the dropping of quadratic and higher terms and insertion of the approximation Fp 4 f p , provides the approximate relationship given as Equation 5 in Table I. Note that this expression, ANALYTICAL CHEMISTRY, VOL. 49, NO. 2 , FEBRUARY 1977
307
Table I. Count-Loss Correction Expressionso
Approximate relationship
General error limit Relative Error Fp [between zero andb less than
)fori
Relative errorb,c
1
Maximum F p for accuracy of 0.01% 0.1% 1%
A ( l - B)-' - 1 -0.6(fp)' 0.08 0.014 0.044 0.13 ( 5 ) F = f/(l- f P ) A(l + B )- 1 -1.6(fP)' 0.08 0.008 0.026 0.086 (6) F = f(1 + f ~ ) (7) F = f ( 1 f p 1.5f2p2) -3.O(f~)~ 0.12 0.034 0.076 0.17 A ( l B 1.5B2) - 1 (8) F = f ( 1 f p 1.5f2p2 3.0f3p3) A ( l B 1.5B2 3.0B3) - 1 -5.0(f P ) ~ 0.20 0.091 0.14 0.26 O1 Statements regarding accuracy pertain to the application of these expressions to paralyzable counting systems. b Relative error = [ ( Fderived from correction equation)/(true F)] - 1.For example, relative error = -0.01 indicates that the corrected count rate is 1%less than the true count rate. A = e W F p , B = Fpe-Fp,
+ + + +
+ + + +
+
+
which is the more accurate of two first-order approximations for count-loss correction in a paralyzable system, is identical to Equation 4, which exactly describes the behavior of a nonparalyzable system. The second commonly applied correction formula, given as Equation 6 in Table I, is obtained by dropping quadratic and higher terms in the series expansion of (1 - f p ) - l . The approximations linking Equation 6 to both Equations 3 and 4 allow its application to a n y counting system, paralyzable or nonparalyzable, when count-losses are slight. This convergence is fortunate, because many practical counting syst,emsappear to have characteristics intermediate between the extreme paralyzable and nonparalyzable models. Albert and Nelson (9) have considered this fact and developed a theory applicable to the full range of intermediate types. If very accurate measurements are to be obtained, it is necessary to examine the limitations which the approximations in Equations 5 and 6 impose. General expressions for the relative errors resulting from use of these approximations have been derived and are presented in Table I together with some practical guidelines for the ranges of applicability of various equations. Equations 7 and 8 incorporate higher-order terms, not previously reported elsewhere, and can be employed when the circumstances merit.
UNCERTAINTIES IN COUNT-LOSS CORRECTIONS Single-Beam Measurements. In the simplest type of ionor photon-counting measurement, some total number of observed counts, n, is accumulated from a single ion or photon beam by integration over some time interval t . An expression for the number of true events which occur in the observation time can be written as follows (see Table I for limits on the accuracy of this expression). (In this and all following expressions, the effect of any background which might be present has been ignored. In general, the variance of the background and (signal background) count rates or totals must be separately assessed, with the variance in the signal alone being evaluated by routine statistical methods.)
+
N = Ft = ft/(l
-fp)
(9)
The variance in N is related to the variances in f and p as follows, assuming that p and f are not covariant.
UN2
= ( U N / N ) 2 = (ft)-l
Fmax
308
(12)
where U N is introduced to represent the relative standard deviation or coefficient of variation for N . The first term in this expression is not unexpected. Inasmuch as ft is simply the number of observed counts, we are not surprised to learn that the relative standard deviation depends on the inverse square root of f t . Indeed, it is most commonly considered that that is the whole story, and that, for example, to obtain a precision of one part in 103, it is necessary only to collect 106 counts. Equation 12 shows, however, that this kind of statement, which considers only counting statistics, ignores the second term in the equation and can represent a serious oversimplification. When observed count totals are corrected for coincidence losses, two quantities are required, both of which are subject to random errors. While it is never forgotten that ft will be variable, that is, that repeated measurements of the same particle flux can give different count totals in equal times, it is equally true that repeated observations of the deadtime ( p ) of a counting system will also describe a universe of results characterized by some standard deviation cr,. The calculation of N draws on both measurements, and the propagation of errors due to the uncertainty in p must be considered. (Note that p itself does not appear in Equation 12. This accurately represents the fact that the magnitude of p is less important than is up in determining the precision of counting measurements. In principle, the effects of a perfectly known deadtime can be perfectly taken inco account. I t must be borne in mind, however, that Equation &-on which Equation 12 is basedhas a restricted range of applicability.) M a x i m u m Count Rate. When up is nonzero, the second term in Equation 12 will contribute to the relative standard deviation to an extent dependent on the count rate. For any given required level of precision, the finite size of this second term will, in turn, require that the magnitude of the first term be reduced by the collection of a number of counts exceeding that predicted from simple counting statistics. At very high count rates, the second term can become so large that certain measurements would be impossible even if the first term tended to zero (collection of an infinite number of counts). This situation would occur if, for example, the required precision were one part in lo3,and the product Fa, exceeded It follows that, for any arbitrarily required precision, the maximum acceptable count rate will be given by
Recognizing that uf2 = f / t and evaluating the derivatives we obtain
Recasting Equation 11 in terms of the relative standard d3viation in N , and substituting f for F, we obtain the useful approximation
+ f2Crp2
=
(UN)reqd/up
(13)
T i m e Requirements. The time required for a given measurement can be determined by solving for t in Equation 12.
- -
-
As F a p 0, t ( f u > ~ * ) -the ~ , value predicted from simple F,,,, Fu,) will approach counting statistics. However, as F
ANALYTICAL CHEMISTRY, VOL. 49, NO. 2, FEBRUARY 1977
f t = U N - ~ but , Equation 14 shows that this quantity must be
1
'\ \ \ II
1x10~ 2
5
F , sec-'
Figure 1. Dependence of
F opt
4
F mox
measurement time ( t ) on count rate (FJ
It has been assumed that ( v ~= ) ~ and ~ ~ = ~2 ns. The variable %gives the factor by which the number of counts required exceeds the theoretical minimum. ( v N ) - ~ ~ ~ ~ ~
(U,AJ)reqd and t will be finite only for U N > (UN)reqd, in accord with the preceding discussion. Under many circumstances the experimenter has some control over count rates, and the question arises whether, in order to obtain some required precision, it is better to adopt a count rate near the maximum, and accept the need for substantial count-loss corrections, or to work a t some lower count rate. To answer this question, it is necessary only to differentiate Equation 1 4 with respect to count rate (adopting f F ) , setting the result equal to zero. In this way the optimum counting rate, the value of F a t which t is minimized, is found to be
increased, with n = (u.v2- F2u,2)-1. Under optimum condi~ @ ~ d , that the tions, for example, nopt = 1 . 5 ( ~ ~ ) - ~ indicating measurement which is most economical with respect to time requires a 50% "excess" in the accumulated count total. For convenience, a factor X = (count total acually required)/ (count total predicted from simple inverse square relationship) = (b'jjr)2reqd/[(U,AJ)2rqd - F2uP2]has been defined and is indicated at the top of Figure 1. Note that the deviation from simple theory becomes significant for Fu., > 2 X 10-4. Multiple-Beam Measurements. The measurement of a single photon- or ion-beam almost never stands alone. Instead, the observed quantity is nearly always compared to some other measurement. For example, a sample might be compared to a standard, one spectral feature compared to another, etc. Under these conditions, it is reasonable to ask whether the strict limitations described above still hold, or whether some of the errors in count-loss corrections might be expected to cancel, thus allowing some relaxation of the limitations. Ratios. In general, any multiple beam system can be reduced t o one or more pairwise comparisons, and it will be sufficient to consider this question in terms of simple ratios. We can define R = Fa/Fb and determine the effect of uncertainties in p on the precision of R . In terms of experimentally observed quantities, we can write
and, as above,
evaluating the derivatives, setting u2fa= f a & , n 2 f b= f b / t b , n, = f a t a ,n b = f b t b , and Af = f a - f b , we obtain with no approximations
-
The existence of this optimum is graphically demonstrated in Figure 1 , which plots t as a function of F for a typical case. The time requirements established by Equation 14 can vary in unexpected ways. No general, summarizing statement can be constructed, but one example is particularly informative: consider the minimum time required for any given measurement, which can be determined by substituting Equation 15 into Equation 1 4 Thus, a t the optimum, there is an inverse cubic dependence of counting time on required precision. If the precision of measurement is to be increased by a factor of ten, the counting time must increase by a factor of 1000. This comes about because, although the number of required counts is only increased by a factor of 100, the rate a t which these counts can be accumulated must be decreased by a further factor of ten (Equation 15) in order to reduce the uncertainty in the count-loss correction. This third power dependence is not revealed until the uncertainty in the deadtime is taken into account. For measurements not carried out under optimal conditions, the relationship between U N and t will not be strictly cubic, but the factor by which t must be increased will always exceed the square of the factor by which U N is to be improved. Count Accumulations. Simple count theory predicts n =
or: as an approximate form, accurate to within 10% for f, < 0.05:
Should it be, for some reason, useful to eliminate Af from this expression, use can be made of the approximation Af 0 Fb(R - 1). Maximum Count Rates. Equation 20, which applies to ratio measurements, is analogous to Equation 12, which applies to single-beam measurements. The first two terms resemble those determined from simple counting statistics in the absence of count losses and deadtime uncertainty (in that case, one finds U R = Nu t Nb -I). The third term represents the contribution which deadtime uncertainty makes to the overall ratio variance, and it can be seen that it is not the count rates themselves which determine the magnitude of this term, but, rather, the degree of mismatch between the two count rates. The maximum degree of mismatch is set by the requirement that the third term must not exceed the required precision, and we find
or
- [( R -
(Fb)max
(UR)2reqd ]1'2
l)'u,,'
ANALYTICAL CHEMISTRY, VOL. 49, NO. 2, FEBRUARY 1977
309
Thus, very high count rates can be employed as R approaches unity, while rates approaching the single-beam case are required when R is far from unity. T i m e Requirements. Ignoring any “overhead” necessary for spectrometer control, the total ratio-measurement time is given by
T
t,
(23)
tb
where t, = & / f a and t b = nb/fb, with the relative numbers of observed counts generally being controlled by the scanning pattern employed in data collection. If equal times are spent on each beam, na = Rnb. If observation times are divided optimally so as to minimize the overall ratio measurement time, n, = f i n b (10). To express T in terms of count rates, count totals, and the ratio of interest, we can write
where k n2,/nband is, thus, a factor allowing consideration of different patterns of observation. In order to eliminate nb from Equation 24, we can substitute n, = knb in Equation 20 and solve for n b , obtaining an expression which can be substituted in Equation 24
Substitution of this expression in Equation 24 provides an equation relating R and all the experimental variables to T , the total ratio measurement time:
This expression can be differentiated with respect to count rate (setting f b = Fb) in order to find the rate a t which R is minimized. In this case we obtain, independent of k , (27) This expression is the multiple-beam analog of Equation 15, and, exactly as we have done in the single-beam case, we need only insert it in the expression for total measurement time in order to determine the minimum possible total measurement time. In this case, it is necessary also to specify some value for k , and, consistent with the goal of determining the shortest possible observation time, we choose the most efficient value for k , namely k = R1l2.The result is
- 2 . 6 0 ~ ~d ( R - 1 ) 2 ( 1
--.(UR
l3reqd
+ 2R1’2 + R )
R
(28)
This expression is the analog of Equation 16. The inverse cubic dependence on required precision remains, but a new factor has been added in order to provide time for measurement of both beams. The value of the factor involving R in this ex(or lo?)to 15.6 at pression ranges from 1.06 X l o 3 at R = R = 0.1 (or 10) t o 2.91 a t R = 0.5 (or 2). If a precision of 0.1% is required and u[, = 2 ns, then any ratio differing from unity by more than a factor of five will require more than 1 min of counting time. Observation o f Complete Spectra. In mass spectrometry, for example, it is desirable to have some general approach to multiple-beam measurements which avoids consideration of individual ratios and allows the systematic determination of optimal observation conditions for an entire spectrum or large group of peaks. It is conventional to express the abundances of individual ion beams in terms of their magnitude in com310
parison to the “base peak”, or most abundant peak in the spectrum. Abundances, then, are expressed as ratios in which the base peak is the denominator. T o consider the count rate which can be accepted a t the base peak, we can refer to Equation 27. As the required dynamic range becomes large, R will tend to zero and the optimal counting rate will tend to 0.58 as in the single-beam case. The number of counts to be collected at the base peak will depend on the required dynamic range and on the distribution of observation time among the peaks in the spectrum. The possible variations are too numerous to consider here, and, in any case, follow in a straightforward way from the considerations introduced in the preceding section. In general, the number of counts required a t the base peak will increase with the square root of the required dynamic range. This fact, together with the low optimal counting rate and the much larger times required for data collection on the weaker beams, can lead to very long total observation times for high precision measurements of complete spectra. Differential Measurements. A final category of conceivable counting measurements takes in situations where the quantity sought is the relative difference between two ratios differing by 10% or less, and where the ratios have both been measured on the same counting system. In these cases, pairs of ratio measurements are made in order to compare the two ratios, with the absolute value of either ratio being of subordinate interest. This is exactly the situation which prevails in the measurement of natural abundance variations of stable isotopes, and, in an earlier paper on that subject ( 2 ) ,we have shown that a more complete cancellation of uncertainties in count-loss corrections will occur in such differential measurements. The controlling factor is the degree of mismatch between major beam intensities in the two ratios being compared. As this mismatch approaches zero, very high count rates can be used without loss of precision in the relative difference between the two ratios.
CONCLUSIONS Limitations on This Treatment. I t has been shown that the precision of a measurement made using a counting technique can depend not only on the number of counts accumulated but, in addition, on the rate a t which the events have been collected. As pulse overlaps become appreciable, there are optimal and maximal rates which depend on the precision sought and on the uncertainty in the deadtime. Should these conclusions be regarded as general, or are they in some way dependent on the models chosen? Equation 5, which has been used as a starting point, is exactly applicable to nonparalyzable systems, but is an approximation of limited accuracy when paralyzable systems are being considered. Therefore, while all aspects of the phenomena described above can be expected to characterize nonparalyzable systems over a wide range, it is reasonable to ask whether the same behavior can be expected in paralyzable systems at count rates high enough that the accuracy of Equation 5 is seriously degraded. What, for example, is the outcome of an error analysis based on Equation 7 rather than Equation 5? Although complicated, such an analysis is not difficult, and shows that maximal and optimal count rates will still be found a t values not differing greatly from those specified above. (Because use of Equation 7 or 8 brings in higher order f up terms, the variances predicted by the more detailed treatment are larger than those given above, and the optimal and maximal count rates are accordingly shifted to somewhat lower values.) Aspects of Pulse-Counting Techniques and Applications. The idea that minimizing p should be the primary design goal for high-speed counting systems should be modified. It appears equally useful and important to minimize up.
ANALYTICAL CHEMISTRY, VOL. 49, NO. 2, FEBRUARY 1977
(3) K. C. Ash and E. H. Piepmeier. Anal. Chem., 43, 26 (1971). (4) J. D. Ingle. Jr., and S . R. Crouch, Anal. Chem., 44, 777 (1972). (5) D. E. Matthews, D. A. Schoeller, and J. M. Hayes, to be submitted to Anal. Chem. (6) A. T. Barucha-Reid, "Elements of the Theory of Markov Processes and Their Applications", McGraw-Hill, New York, N.Y., 1960, p 299. (7) R . D. Evans, "The Atomic Nucleus", McGraw-Hill, New York, N.Y., 1955, Chap. 28; or in "Nuclear Physics", L. C. L. Yuan and C. S. Wu,Ed., Vol. 5, Academic Press, New York, N.Y.. 1963, pp 761-806. (8) S. D. Conte and C. de Boor. "Elementary Numerical Analysis". 26 ed., McGraw-Hill, New York, N.Y., 1972, p 33. (9) G. E. Albert and L. Nelson, Ann. Math. Stat., 24, 9 (1953). (10) J. M. Hayes, to be submitted to Biomed. Mass Spec.
Consideration should be given to means of stabilizing p and to techniques allowing its precise measurement ( 5 ) . The fact that higher precision requires lower counting rates places two parameters on a collision course. In the absurd extension, attainment of the highest precision would require the use of the lowest counting rate for an observation time approaching infinity. More practically, this opposition is expressed by Equations 16 and 25, which reveal the inverse cubic dependence of observation time on precision. Given present practical limitations ( p 2 10-8 s, up 2 x 10-9 s) it is apparent that the maximum precision obtainable from counting techniques is near 0.1% (relative standard deviation), and that improvements will require not only reduction of p but also creation of techniques for its precise measurement.
-
RECEIVEDfor review August 9, 1976. Accepted October 18, 1976. We appreciate the support of the National Aeronautics and Space Administration (NGR 15-003-118), which has principally funded this work; and of the National Institutes of Health (GM-18979),which has funded our related experimental work in isotope ratio mass spectrometry.
LITERATURE CITED (1) J. D. ingle, Jr., and S. R. Crouch, Anal. Chem., 44, 785 (1972). (2) D. A. Schoelier and J. M. Hayes, Anal. Chem., 47, 408 (1975).
Enrichment of Trace Metals in Water by Adsorption on Activated Carbon Bruno M. Vanderborght* and Rene E. Van Grieken" Department of Chemistry, University of Antwerp (U.I.A.), 8-2610 Wilrijk, Belgium
A combination of multielement chelation by 8-hydroxyquinoline with subsequent adsorption on activated carbon was developed for trace metal preconcentration. Adsorption characteristics of 8-quinolinol and metal quinolates on activated carbon were investigated in order to optimize the enrichment procedure. Interferences from alkali and alkaline earth ions were mhimized and working conditions for preconcentration from very differing samples were calculated. For about 20 elements simultaneously an enrichment factor of 10 000, precision of 5 to l o % ,and a recovery from 85 to 100% were demonstrated.
The merits of activated carbon (AC) for removal of organic compounds from water have been well documented, but the potential for enrichment of heavy metals has received little attention in the literature on water analysis. Sigworth and Smith ( I ) have reviewed some special applications of removing trace metals and inorganic compounds from aqueous solution by AC, but no quantitative study was performed. Kerfoot and Vaccaro ( 2 ) used AC for the extraction of copper from seawater and subsequent determination by atomic absorption spectrometry. A recovery of 70% was attainable. Better results are to be expected when the metals are complexed with organic chelating agents before adsorption on AC. Van Der Sloot et al. ( 3 ) determined uranium in sea and surface water by neutron activation analysis after complexation with 1-ascorbicacid and adsorption on AC. Jackwerth et al. ( 4 ) used different chelating agents in combination with AC to extract several elements from solutions. Contrary to most other chemical preconcentration techniques, naturally occurring organic metal complexes will also be adsorbed by AC. In view of the important complexation capacity, a characteristic of natural water (5),this feature should be a distinct advantage of enrichments through AC. In this study, the characteristics of multielement trace enrichment on AC were
evaluated and optimized for the trace metal analysis of water samples.
EXPERIMENTAL Reagents. All reagents were of analytical reagent grade. The water used was deionized and doubly distilled in quartz material. The chelating agent, 8-hydroxyquinoline (oxine),was available from Union Chimique Belge. Standard solutions of metals were prepared from Titrisol standards. Acids, bases, and salts were "pro analysi" reagents from Merck. The Activated Carbon, a "Baker Analyzed" reagent, was treated with concentrated H F and HC1, washed with water, and dried a t 110 OC to remove trace elements (6, 7). Metal contamination was reduced by a mean factor of five (8).Experiments were performed in glass material because polyethylene appeared to adsorb some oxine from the solution. Apparatus. The flasks were agitated on a temperature-controlled horizontal shaker. Spectrophotometric measurements for the oxine determinations were performed with a Beckman Cecil 202 double beam UV-spectrophotometer. Copper determinations by atomic absorption spectrometry (AAS) were carried out on a Perkin-Elmer 503 Flameless Atomic Absorption Spectrometer with Massmann oven and deuterium corrector. The micro samples were injected into the graphite rod of the oven using Eppendorf micropipets. The x-ray fluorescence (XRF) instrumentation, consisted of a Siemens Kristalloflex 2 high voltage power supply and an x-ray tube with tungsten target, a Mo secondary fluorescer and filter, a l6-position sample holder, and a Kevex Si(Li) semiconductor detector and related electronics. A 1024-channel analyzer coupled to a magnetic tape unit recorded the spectra which were analyzed by a PDP 11/45 computer. Instrumental neutron activation analysis (INAA) made use of the Thetis reactor a t Ghent University, usually at a neutron flux of ca. IO1?s-l cm-2. Gamma-ray spectroscopy was performed by means of a Ge(Li) detector and 4096-channel analyzer, connected to a PDP-11 computer in an on-line mode.
RESULTS AND DISCUSSION Adsorption of Chelated Metals on Activated Carbon. T o evaluate the possibilities of AC as an adsorber for free heavy metals, the adsorption of Zn2+and Co2+ions on AC was determined. A maximum adsorption capacity of less than 1 ANALYTICAL CHEMISTRY, VOL. 49, NO. 2, FEBRUARY 1977
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