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
311