Limits for Qualitative Detection and

Quantitative Determination
Application to Radiochemistry
Lloyd A. Currie
Analytical Chemistry Division, National Bureau of Standards, Washington, D, C. 20234

The occurrence in the literature of numerous, incon- 5000 T T T T T --

sistent and limited definitions of a detection limit has DEHNITIONS;
led to a re-examination of the questions of signal I -
BACKGROUND STANDARD DEVIATION
detection and signal extraction in analytical chemistry 2000 2- 10% OF THE BACKGROUND
(0*a)
and nuclear chemistry. Three limiting levels have 3- 2<TB
been defined: z.c-the net signal level (instrument re- 4- 3CTB
sponse) above which an observed signal may be reli- 1000 5- 37"g-F3(7"
ably recognized as “detected”; is-the “true” net (crg=SAMPLE STANDARD DEVIATION)
signal level which may be a priori expected to lead to 500
6-
7- iooo dpm
TWICE THE BACKGROUND

detection; and ¿c-the level at which the measurement 8- loodps

precision will be satisfactory for quantitative deter-
mination. Exact defining equations as well as series
of working formulae are presented both for the general 1 200
analytical case and for radioactivity. The latter, z
Q
assumed to be governed by the Poisson distribution, is ioo
treated in such a manner that accurate limits may be ti (Lq)
—

derived for both short- and long-lived radionuclides

either in the presence or absence of interference. 50
The principles are illustrated by simple examples of
spectrophotometry and radioactivity, and by a more
complicated example of activation analysis in which a 20 -!Ld)—-
choice must be made between alternative nuclear
reactions.
10 •(Lq)
o
in the of research dealing with photonuclear reactions
course '5
and activation analysis, it became necessary to determine limits 12 3 4 5 6 7 8

of detection of radiochemical procedures, to select among DEFINITION

alternative procedures, and to optimize given procedures with Figure 1. “Ordered” detection limits—literature definitions
respect to certain experimental parameters. Examination of The detection limit for a specific radioactivity measurement process
the analytical and radiochemical literature for an appropriate is plotted in increasing order, according to commonly-used alterna-
definition of the limit of detection revealed a plethora of tive definitions. Lc, Ld, and Lq are the critical level, detection
mathematical expressions and widely-ranging terminology. limit, and determination limit as derived in the text
One encounters, for example, terms such as lower limit of de-
tection (7), detection sensitivity (2), sensitivity (3), minimum that of the net signal. Some authors apply two-sided confi-
detectable activity (or mass) (4, 5), and limit of guarantee for dence intervals, while others use one-sided intervals. In addi-
purity (6)—all used with approximately equivalent meanings. tion, various “nonstatistical” definitions appear in which the
The nomenclature problem is compounded, however, because detection limit is equated to the background, 10% of the back-
other authors make use of the same, or very similar, terms to ground, 100 dps ( -radioactivity), or 1000 dpm (a-, 3-, -radio-
refer not to the minimum amount that may be detected, but activity). In order to compare some of the more commonly-
rather, to the minimum amount which may be determined with used definitions, “detection limits” have been calculated for a
a given relative standard deviation (such as 10%). Still other Hypothetical radioactivity experiment in which a long-lived
expressions, such as the “detection limit at the 95% confidence 7-emitter was counted for 10 min with an efficiency of 10%,
level” are used without explicit mathematical definition, which using a detector having a background of 20 cpm. The results,
leaves the meaning rather ambiguous. The various mathemati- plotted in increasing order in Figure 1, are obviously unsatis-
cal definitions of detection limit (or its equivalent) range from factory, for they encompass nearly three orders of magnitude!
one to twenty times the standard deviation of the net signal, In what follows, it will be seen that a complete discussion of
with the standard deviation of the blank sometimes replacing the (lower) limits for a measurement process requires the in-
troduction of three specific levels: (1) a “decision limit” at
which one may decide whether or not the result of an analysis
indicates detection, (2) a “detection limit" at which a given
(1) B. Altshuler and B. Pasternack, Health Physics, 9, 293 (1963).
(2) J. Wing and M. A. Wahlgren, Anal. Chem., 39, 85 (1967). analytical procedure may be relied upon to lead to detection,
(3) R. C. Koch, “Activation Analysis Handbook,” Academic and (3) a “determination limit” at which a given procedure will
Press, New York, N. Y., 1960. be sufficiently precise to yield a satisfactory quantitative esti-
(4) D. E. Watt and D. Ramsden, “High Sensitivity Counting mate. Following definition of the three levels, an attempt will
Techniques,” Macmillan Co., New York, N. Y., 1964.
be made to indicate their relations to the various definitions re-
(5) “A Manual of Radioactivity Procedures," Handbook 80,
National Bureau of Standards, Washington, D. C., 1961. ferred to previously, and to give asymptotic expressions which
(6) H. Kaiser, Z. Anal. Chem., 209,1 (1965). may serve as convenient working definitions.

586 · ANALYTICAL CHEMISTRY

 The general expressions for detection and determination will a real signal has been detected—i.e., whether ps > 0; (2)
be applied to radioactivity be means of the assumption of given a completely-specified measurement process, one must
“Poisson counting statistics.” In order to make the concepts estimate the minimum true signal, ps, which may be expected
generally useful, particularly in the fields of nuclear chemistry to yield a sufficiently large observed signal, 5, that it will be de-
and activation analysis, the formulae will be generalized to tected. The first aspect thus relates to the making of an a
take into account both long- and short-lived radioactivity, con- posteriori, binary (qualitative) decision based upon the ob-
tinuously-variable measurement parameters (such as dis- servation, S, and a definite criterion for detection. Following
criminator settings), and radioactivity measurements in the such a decision one should establish either an upper limit (if
presence of interfering radionuclides. “not detected”) or a confidence interval (if “detected”). The
second aspect relates to the making of an a priori estimate of
GENERAL PRINCIPLES the detection capabilities of a given measurement process.
Let us consider first the a posteriori problem. Following an
Part of the following discussion appears elsewhere in the experimental observation, one must decide whether or not that
literature in somewhat different form, for example, in Refer- which was being sought was, in fact, detected. Formally
ences (7 and 6). The purpose of this section is to bring together known as Hypothesis Testing, such a binary, qualitative deci-
the concepts of qualitative and quantitative analysis limits, to sion is subject to two kinds of error: deciding that the sub-
clearly show the relationships between an a posteriori decision stance is present when it is not (a; error of the first kind), and
with the related confidence interval and an a priori “detect- the converse, failing to decide that it is present when it is
ability,” and to lay the groundwork for general application to (ß; error of the second kind). The maximum acceptable
the detection of radioactivity, to be discussed in the following value for a, together with the standard deviation, , of the net
section. Readers who wish fuller explication of the underly- signal when ps = 0 establish the Critical Level, Lc, upon
ing statistical principles and methods—particularly those which decisions may be based. Operationally, an observed
relating to hypothesis testing, probability distributions, and signal, S, must exceed Lc to yield the decision, “detected.”
the estimation of statistical parameters—may find it helpful to Thus, the probability distribution of possible outcomes, when
consult a basic statistics text, such as Dixon and Massey (7). the true (net) signal is zero, intersects Lc such that the fraction,
Definitions and Notation. The process of measurement 1 —

a, corresponds to the (correct) decision, “not detected.”

(experimental procedure) must be completely defined— Once Lc has been defined, an a priori Detection Limit,
including the measuring apparatus, method of observation, Ld, may be established by specifying Lc, the acceptable level,
and sample nature—in order to draw valid conclusions with ß, for the error of the second kind, and the standard deviation,
respect to detection capabilities. In general, the physical aD, which characterizes the probability distribution of the
quantity of interest (mass, number of atoms, nuclear cross (net) signal when its true value (limiting mean), ps, is equal to
section, etc.) is not directly measurable, but is connected Ld. (Identification of the “true” net signal with the limiting
to that which is observed (digital counts, voltmeter deflection, mean assumes the absence of systematic errors. As with most
etc.) through a calibration constant. The statistics of detec- discussions of detection limits, we assume that the entire mea-
tion and determination apply directly to the observations surement process may be replicated, so that errors become
rather than to the underlying physical quantity, and therefore, random in nature. Systematic errors in a calibration factor,
the following discussions will deal specifically with the ob- connecting the net signal to the desired physical property, are
served (or observable) signal (meter reading) and its as- not involved at this stage.) LD is defined so that the prob-
sociated random fluctuations. Statistical conclusions drawn ability distribution of possible outcomes (when ps =
LD) in-
in terms of the net signal may be very simply extended to the tersects Lc such that the fraction, 1 /3, will correspond to the
—

related physical quantity by means of the calibration factor. (correct) decision, “detected.” The detection limit, defined in
Symbols will be defined, in general, as they appear in the this manner, is equivalent to the“limit of guarantee for purity”
text, but it may be helpful initially to list the following: of Kaiser (6) and the “minimum detectable true activity" of
Altshuler and Pasternack (7). The significance of this partic-
Blank: µ limiting mean (or “true” mean)
B observed value ular form for the definition is that it allows one to determine,
standard deviation for a given measurement process, the smallest (true) signal
which will be “detected” with a probability 1 —ß, where the a
Gross Signal: ps+B limiting mean posteriori decision mechanism has a built-in protection-level,
(S + B) observed value a, against falsely concluding that a blank observation repre-
+ b standard deviation sents a “real” signal. True signals, ps, lying between zero
and Ld will have larger values for ß, and therefore, although
Net Signal: ps =
ps+b -

Pb limiting mean
value derived from they may be “detected,” such detection cannot be considered
S = (S + B) —
B an
reliable.
observation pair
Mathematically, the critical level is given as
3 =
( 2+ + |)1/2 standard deviation
Lc —

kaao (1)
The blank is defined as the signal resulting from a sample
which is identical, in principle, to the sample of interest, except and the detection limit,
that the substance sought is absent (or small compared to ). Ld =
Lc -j- kpUD (2)
The blank thus includes the effects of interfering species.
where ka and kp are abscissas of the standardized normal
Qualitative Analysis. It is vital at the outset to distinguish
between two fundamental aspects of the problem of detection:
distribution corresponding to probability levels, 1 a and—

1
—

ß. The relationships between Lc, LD, and the probability

(1) given an observed (net) signal, S, one must decide whether
distributions for ps =0 and µ3 LD are depicted in Figure
-

2, where µ represents the limiting mean for the blank dis-

(7) W. J. Dixon and F. J. Massey, Jr., “Introduction to Statistical tribution, and ps+B represents the limiting mean for the
Analysis,” 2nd ed., McGraw-Hill, New York, N. Y., 1957. (observed) signal-plus-blank distribution.

VOL. 40, NO. 3, MARCH 1968 · 587

 REGION I REGION E REGION H
w(s) As =As+b ~Ab

J/ V
UNRELIABLE DETECTION DETECTION: QUALITATIVE ANALYSIS DETERMINATION: QUANTITATIVE ANALYSIS
H i As = 0

VLc
„
Lc =
kctO< k^cro

NET SIGNAL

H =
://$ Lg

L0 =
Lc+ kg^g
Lc ld
0 Lc Lq Lq
Figure 2. Hypothesis testing
Figure 3. The three principal analytical regions
Errors of the first and second kinds

“safer” in that it takes into account possible lack of knowledge

Confidence Interval and Upper Limit. The preceding dis- of . On the other hand, if in the region between zero and
cussion implied that one need do no more than compare an Ld is reasonably well-known (from “theory” or extensive ob-
experimental result with the decision level in order to draw a servation), and especially if it is approximately constant, more
qualitative conclusion. In most cases, however, one has realistic intervals and upper limits would be set through its
available an estimate of the net signal, S, and its standard de- use rather than s/\^n. (Of course, / /«, replaces , when
viation, , may be known. An interval may then be stated for S is used in place of S). If is not approximately known, and
µ5, based upon z^yw and corresponding to the confidence level, in control, in the region below LD, it is impossible, in principle,
1 —

y. (z 1—7/2 is the critical value for the standardized normal to estimate a limit of detection. In any case, replicates are
distribution such that Prob (z < zi-y2) = 1 y/2.) If the net
—

most desirable, if possible, and the estimate, sj '«, should be

signal and its standard deviation are estimated by means of consistent with / /«, if the measurement process remains “in
replication, 5 is replaced by S; , by s/ n; and - /2 , control.” [An exactly equivalent, and somewhat preferable,
by ti-yii-s/y/n. (Here, s represents the standard deviation scheme for treating replication, would include the number of
estimate computed from «-observations, and ii-y2 represents observations,«, as a part of the definition of the measurement
the critical value for the Student's t distribution correspond-
—

process. The standard deviation of the mean of «-observa-

ing to « —
1 degrees of freedom.) tions would then be simply (for the over-all process) rather
(1) If S(or S) > Lc, the decision, "detected,” should be re- than / 'n. The use of s/ « to test control rather than to
ported, and a symmetrical confidence interval should be set limits would be achieved by routine evaluation of 2/ 2,
given: 5 ± - /2 (or S ± -rw/Vn) where zi-y2 (or which should be within limits set by the distribution of 2/ .]
t\—y/2) refers to a two-sided confidence interval. Quantitative Analysis. Neither a binary decision, based
(2) If S(or S) < Lc, the decision, “not detected," should be upon Lc, nor an upper limit, nor a wide confidence interval
reported, and an upper limit should be given: 5 + z'i-yr may be considered satisfactory for quantitative analysis. One
(or S + t\-ysl /ñ) where z'i-7 (or t\-y) refers to the one-sided wishes instead a result which is satisfactorily close to the
confidence interval. (The prime is here used to emphasize true value (limiting mean). Therefore, for ps = Lq, the
one-sided.) Determination Limit, the standard deviation, aQ, must be but
Note that in the special case where - 0, and ' - is em- a small fraction of the true value. Such a definition is similar

ployed, Lc is numerically equal to ' -yr. This leads to the to that used by Adams, Passmore, and Campbell who defined
common, mistaken practice of equating \-7 with the “de- a “minimum working concentration” as that for which the
tection limit [(95%) confidence level]." Such a statement relative standard deviation was 10 % (S).
is seriously in error because the detection limit cannot be The Determination Limit so defined is,
characterized by a single “confidence level,” and because it
Lq =
IcqCq (3)
confuses the decision-making quantity ( ' -7 Lc) with LD, =

which is used to assess the a priori detectability. To the where Lq is the true value of the net signal, ps, having a stan-
extent that =
0, which may be satisfactory if is approxi- dard deviation, aQ, and 1 ¡kQ is the requisite relative standard
mately constant in the region between zero and LD, the use of deviation.
z'i-yC (one-sided) to test a given result is exactly equivalent to By way of summary, the levels Lc, LD, and Lq are deter-
the use of Lc for this purpose. mined entirely by the error-structure of the measurement
A second possible mistake is the confusion of an a posteriori process, the risks, a and ß, and the maximum acceptable
upper limit with the a priori detection limit. Here again, the relative standard deviation for quantitative analysis. Lc
two may coincide, if the net signal happens to be Lc, and if is used to test an experimental result, whereas LD and LQ
« cd and the one-sided confidence interval is employed. refer to the capabilities of measurement process itself. The
Such a coincidence is not accidental, for the detection limit relations among the three levels and their significance in
is, by definition, the maximum upper limit. It must be re- physical or chemical analysis appear in Figure 3.
membered, however, that, in general, the upper limit depends Special Cases (numerical results). In order to make the
upon the specific experimental result, S, whereas the detection significance of Equations 1-3 clearer, a number of specific
limit must be independent of S, depending, rather, upon the choices for a, ß, and the various ’$ may be helpful.
measurement process itself.
Finally, the difference between and / j/V« should be
(8) P. B. Adams, W. O. Passmore, and D. E. Campbell, paper No.
discussed. The use of the latter, which depends upon the 14, “Symposium on Trace Characterization—Chemical and
variance estimate resulting from «-observations, is clearly Physical,” National Bureau of Standards (Oct. 1966).

588 · ANALYTICAL CHEMISTRY

 Table I. “Working” Expressions for Lc, Ld, LQ.a
Lc Ld Lq
Paired observations 2.33 as 4.65 aB 14.1 as
“Well-known” blank 1.64 as 3.29 as 10 as
Figure 4. Extreme limits for Lc “
const.
Assumptions: a =
ß =
0.05; kq =
10; a =
» =

The left-hand diagram indicates coincidence of Lc with

zero; hence a = 0.50. The right-hand diagram indicates
coincidence of Lc with Ld! hence ß 0.50 =

cates detection (a =
0). LD = kgaD, and Lq kQaQ, where =

1. a = =
constant. If the risks of making both aD and aQ now depend upon the net signal only.
ß·,
5. a2s+B = ms + mb· Poisson statistics—to be discussed
kinds of mistake are set equal, then ka-= kg k, and
—

below under radioactivity. The standard deviation may not

Lc —
ka o (4a) be assumed constant; it increases with signal level.
6. a or ß = 0.50. If one is willing to tolerate a 50% error
Ld —
Lc + kad = k ( + aB) (4b) in wrongly identifying a false signal or in missing a true signal,
If, in addition, a is approximately constant, either Lc becomes zero or Lc = LD. That is, under these
circumstances, the pair of levels, Lc and LD, give the appear-
Lc = ka (5a) ance of just a single level. This is illustrated in Figure 4. If

Ld = 2k a = 2 Lc (5b) both risks are set at 50%, then both levels coincide with zero.
Therefore, any positive, net signal will be recognized as “real,”
In this case, the detection limit is just twice the critical level— and any nonzero “true” level in a sample will be “detectable.”
a situation which obtains in the large majority of (That is, the detection limit is in fact zero!) However, such
cases. Assuming that risks of 5% are acceptable, and that fantastic detection capabilities must be viewed with caution,
the random errors are normally-distributed, the constant, for regardless of whether ms 0, or if it is just above the =

k, takes on the value, 1.645. The standard deviation of the limit of detection, the conclusion will be wrong 50% of the
net signal is derived from time, and therefore the experiment could be performed equally
well by the flipping of a coin.
2 =
as+s + ß2 (6)
where <tJ+ b represents the variance of the “gross” (directly- RADIOACTIVITY
observed) signal and ß2 represents the variance of the blank.
If the standard deviation is approximately independent of the Signal Detection. Application of the foregoing considera-
tions to radioactivity involves the fact that the gross “signal”
signal level, then
and “blank” observations are in digital form which in most
2 =
a¿+B + ß2 = 2 ß2 (7) cases may be assumed to be governed by the Poisson dis-
tribution. (Extra, non-Poisson variability of the blank is
Making the additional requirement that kq =
10—i.e., that
discussed below, under “Interference and Background.”)
aQ =o 10%—we find that
If the numbers of counts are sufficiently large, the distribu-
Lq =
kqaQ = 10 a (8) tions are approximately Normal, and we may therefore
readily estimate the variance of the net signal and establish
The above results are summarized in Table I, which may be
approximate levels of confidence and significance. Under
used to provide convenient “working” formulae for the large
such circumstances, the variance of the net signal (number of
majority of problems. The first row is derived assuming
counts) is given by
equivalent observations of sample (plus blank) and blank,
while the second row, which differs by assumes that a l/V^ 2 =
as+b + ß —

(ms -p Mb) + —

(9)
long history of observations of the blank make the second «
term on the right in Equation 6 negligible.
2. Fixed Ld; varying ( ,ß). For a given measurement (B is assumed to have been derived from «-observations of
the blank.) Note that a is not independent of signal level
process specification of a fixes Lc. Similarly, specification
as was assumed in Table I. Its variation over the range
of Lc or a together with the specification of ß fixes LD.
For a given LD, however, an infinite set of combinations Ms
=0 to Ms Ld is trivial if mb is large, however. If
=

Mb =10 counts, aDlao 1.5 («-large). In the limit mb

~

0, =

( ,ß) exist depending upon the location of Lc. For example,

the value for LD resulting from the above choice, a ß = = ß/ oo , because a
= 0. This requirement, that a0
= 0 =

and Lc 0 if µ
=
0, is peculiar to the Poisson distribution,
=

0.05, would also obtain for all choices of Lc between zero

and represents one of the principal differences between the
and Ld. The extremes, Lc 0 and Lc Ld (still assuming = =

aD =
ao), correspond to ( ,ß) (0.5, 0.0005) and ( ,ß) = = “general” case and that of radioactive decay.
Remembering that 02 is the variance when m.s 0, and =

(0.0005, 0.5), respectively (see Figure 4).

3. µ 0. Because the above equations do not involve
=
that aD2 is the variance when ms LD, we obtain =

the magnitude of the background, per se, but only its stan- Lc =
ka 0 =
ka (mb + °~i)112 (10)
dard deviation, there is no change. Such a state of affairs is
reasonable, for a background of any magnitude can be set Ld —
Lq -f- kgao =
Lc + kg (Ld ao2) 1,2
(11)
equal to zero simply by a change of scale; such a change can- Solving Equations 10 and 11 for LD, leads to
not be expected to alter the detection limit.
0. In this case, the effect is profound. Lc is ke2 4¿c 4Lc2 1/2)
4. aB =
j
Ld —
Lc -j- 1 + , ,
(12)
necessarily zero, and any net positive signal definitely indi- kg2 ka2kg2

VOL. 40, NO. 3, MARCH 1968 · 589

 Table , “Working” Expressions for Radioactivity
Lc (counts)* Ld (counts) Lq (counts)
Paired observations

(| =

“Well-known" blank
µ) 2. 33 \/µ 2.71 + 4.65 V7b 50·
K
(7
Zero blank
=
0) 1.64 V7b 2.71 + 3.29 V7b 50
j +l -r}
(MB 0) = 0 2.71 100

Asymptotic ratio6»'
(S/tb) 1.64 3.29 10
*
Dimensions (counts) apply to the first three rows only.
6
“Well-known” blank case; for paired observations, multiply by /2.
'
Correct to within 10% if µ > 0, 67, 2500 counts, respectively, for each of the three columns. For paired observations, µ > 0, 34,
1250 counts, respectively.

Estimates of the mean value and the standard deviation of observing S =

Lc = 0 counts when µ8 Ln 2.71 is ap-
= =

the blank thus allow the calculation of Lc and LD for selected proximately 0.07 instead of the desired 0.05.
values of a and ß by means of Equations 10 and 12. A con- Interference and Background. Thus far, µ and have
siderable simplication takes place if ka k's k. Equation = =
been used to refer to the “blank.” In observations of radio-
12 then reduces to the form, activity one frequently approaches the situation where the
blank is due only to background radiation. When such is
Ld = k2 + 2LC (13) not the case, it may be desirable to decompose the blank into
Equation 13 differs by the term, k2, from that arising in the its separate components: background and interfering ac-
previous discussion in which the “reasonable” assumption tivities. Using b to denote background and I to denote in-
was made that 2 » const. Thus, even if 0 Lc,
—
= terference, the above quantities take the form,
we see that Ld may never be equal to zero. The determination
Ms =
µ» + µ/ (16)
limit, Lq, is given by
2 =
»2 + ,2 (17)
Lq =
kQ<TQ
—

kQ (Lq + 2)1/2 (14)

If the variances, 62 and 2, arise from the Poisson distribu-
which may be solved to yield
tion, they will be equal to the respective mean values, µ»
1/2 and µ ; if the interference has instead a fixed relative stan-
Lq 1 + (15) dard deviation, /z, its variance will be equal to (/ µ )2. The
ÍQ J
distinction between these two situations may sometimes be
Again, convenient “working” expressions may be derived quite critical in fixing the detection limit. Its significance in
from Equations 9, 13, and 15 for measurements in which choosing between alternative detection systems will be dis-
a=
ß 0.05, and kQ
—
10. These expressions are given in
= cussed in a separate publication. Note that the situation
Table II. may also come about where µ 0, but crz2 corresponds to an
=

Especially simple “working” expressions may be stated when extra (non-Poisson) component of variance. Such extra
the number of background counts, µ , is “large.” Such variance, which may arise from cosmic ray variations or in-
simplified expressions, presented in terms of the ratio of strument instability, must be included in the estimate of
the net signal to the standard deviation of the background 02 and, hence, in the estimates of Lc, LD, and LQ.
(“signal/noise”), appear in the bottom row of Table II. Limits for the Related Physical Quantities. In order to
Note the correspondence of the asymptotic ratios to the make the decision, “detected” or “not detected," one needs
“working” expressions in Table I. to know only the net number of counts resulting from a
The occurrence of nonintegral values for LD and Lq in specific experiment, and the critical number of counts, Lc.
Table II is not at all inconsistent with integral, Poisson dis- Limits for qualitative and quantitative analysis and upper
tributions, because LD and Lq represent the means of such limits or confidence intervals for actual results, however, are
distributions, and such means may take on any positive value, of value only when expressed in terms of the physical quan-
integral or nonintegral. Lc, on the other hand, represents a tity of interest, such as grams or atoms. The connection is
decision level against which an integral, experimental result simply made by means of the relevant calibration factor. For
must be compared. An exact, Poisson treatment would thus example, the detection limit, Ld, may be related to the mini-
lead to an integer for Lc, but only discrete values for a would mum detectable mass, mD(g), by means of Equation 18,
then be possible. The magnitude of the error in significance
Ld =
KmD (18)
level, due to the assumption of normality, is worth consider-
ing with respect to the data in Table II. For example, LD where K represents an overall calibration factor relating the
is there given as 2.71 counts for the zero blank case, and this is detector response to the mass present. Thus, K would be
supposed to correspond to ß 0.05. Examination of the =
equal to unity for direct (ideal) weighing; it would be equal
correct, Poisson distribution shows that the probability of to the absorbance per gram for spectrophotometry if the

590 · ANALYTICAL CHEMISTRY

 sample cross-section is fixed; it would be equal to the num- Relative detection capability has frequently been evaluated
ber of counts per gram for activation analysis. in terms of the “figure of merit.” Figure of merit, in the
Although the constant, K, is not involved directly in the limit of very small (net) sample counts, is usually defined as
statistics of the detection limit, its role is fundamental, and µ52/µ ; this expression is approximately the reciprocal of the
it must be included when choosing between experimental relative variance of the net signal. Use of the above expres-
procedures or in optimizing a given procedure. For the sion for comparing detector sensitivities generally involves
particular case of nuclear activation, the replacement of µ5 by the product of the sample disinte-
gration rate and the detection efficiency, and various detectors
K =
P(2)S(\,r)T(\,t,At)f(x) (19) are then compared by examining the respective ratios, €2/µ .

where P =
production rate (nuclei/g-sec) Such a procedure suffers from a number of limitations, when
S =
saturation factor 1 e~Xr = —
compared to the use of Equation 22 (or the analogous equa-
T =
(e_w/X) (1 _ ) (seconds) (a generalized
— tions for mc and mQ). The limitations include: (1) no al-
counting interval relating initial counting rate lowance is made for short-lived radioactivity; (2) inter-
to observed number of counts), ference—especially “decaying” interference—is not con-
e =
detection efficiency (counts/disintegration) (chem- sidered; (3) the formula may not be applied to the comparison
ical yield may be incorporated in e, when of critical levels or detection limits, because a- and ß- type
appropriate). errors have not been included; (4) the approximation, µ5
« µ , is built into the formula. This last factor, which
and X are characteristic of the nuclear reaction being would lead one to conclude that one particular detection
utilized; they represent the reaction cross section (cm2) and system is better than another, may lead to the wrong con-
product decay constant (sec-1), respectively, t, t, and At clusion for mD or mQ. That is, the exact equations, of the
are the times (sec) for activation, delay (decay), and counting, form of Equation 22, can lead to the conclusion that the one
respectively, x represents a variable detection parameter, detection system has the lower limit of detection, but that
such as absorber thickness, discriminator setting, etc. A the other has the lower limit of determination.
similar expression may be written for the mean number of
counts from an interfering radionuclide: ILLUSTRATIONS
µ =
m¡ [P(Xi)S(\i,T)T(X.¡J,At)f-i(x)] (20) In order to make clear the application of the preceding
formulae, three examples will be given, one selected arbi-
the mean number of background counts is given by
trarily from among “standard” methods of chemical analysis,
another dealing with the simple detection of radioactivity,
µ»
=
b(x)At (21)
and a third dealing with a more complex problem in activation
where b(x) is the background rate, which also may depend analysis.
upon the detection parameter, x. The preceding expressions (1) Spectrophotometry
may be incorporated into a single equation for the mass- The fundamental relation governing the absorption of light
detection limit: by matter may be written,
k2 + 2k [µ + + µ + (/;µ/)2]1/2 A =
µ + A (23)
mD = -
(22) o

( )5( , ) ( , , )<( >

where A and A0 are the absorbances for samples having con-
Equation 22, although somewhat complicated, allows one to centrations, c and 0, respectively; l is the cell path length;
calculate the minimum detectable mass for a given activation- µ is the absorptivity. In order to relate the problem to the
detection procedure, where there are no a priori restrictions preceding discussion, we make the following identifications.
placed upon the half-life of the product radionuclide (long-
vs. short-lived) and where interference may be considered in (a) Ao —

blank, µ , which is here set equal to zero, by

a completely general and unrestricted manner. In order to adjustment of the transmittance reading to 100% using a
include more than one interfering radionuclide, one simply “blank” sample.
replaces µ by ^µ/and (//µ/)2 by (//4/)2· If the variances (b) standard deviation of the blank absorbance
—
0.
of background and interference are governed by the Poisson (c) A net signal, µ5, which here equals the gross signal,
—

distribution and if they are determined by “equivalent” ob- because A0 has been set equal to zero.
servations, the radical in the numerator of Equation 22 takes (d) K =
µ/, the calibration factor.
the simpler form: (2µ + 2µ/)1/2. If the variances are negli- (e) as = standard deviation of the net signal, A.
gible, the radical becomes (µ + µ )112.
Experimental observations on the spectrophotometric de-
Systematic errors in calibration factors are not a part of the termination of thorium using thorin, yielded: 0.0020, =

present discussion. Such errors can in no way affect the crit- and K = 58.2 //g (9). The sample standard deviation, as,
ical level, Lc, because Lc refers only to the instrumental re-
was observed to be relatively constant and equal to , over
sponse at which the decision is made—“detected" or “not the concentration range studied. A particular sample an-
detected.” All physical quantities deduced from Ld, Lq,
alyzed was observed to give a response (absorbance) of 0.0060.
or an “observed” net signal, however, contain uncertainties
Using these data and expressions for paired observations from
due to calibration factor errors. Because the calibration
Table I, we find the following:
factor error is here considered systematic, while the observa-
Decision: Lc 2.33 =
0.0047, Thus, the observed re-
=

tion (response) error is random, they cannot be simply com-

sponse, 0.0060, leads to the decision, “detected."
pounded. As a result, the corresponding physical quantities
must be characterized by error limit intervals, the upper limits
of which might be used to provide “safe" estimates for mD (9) L. A. Currie, G. M. France III, and P. A. Mullen, National
and m6. Bureau of Standards, unpublished data (1964).

VOL. 40, NO. 3, MARCH 1968 · 591

 Table III. Detection of Potassium in Sodium
_Detection limit (g)_
Reaction Target nuclide0 Product6 Interference-free m( Na) = 1 g
(«,7) 41K (6.9%) 42K (12.4 hr) 1.3 X 10~10 0.26
23Na (100%) 24Na (15 hr)

( ,«) 8SK (93.1 %) 38K (7.7 min) 7.7 X 10-i° 6.0 X 10-7
23Na (100%) 22Na (2.6 yr)
0
Isotopic abundance listed in parentheses.
6
Half-life listed in parentheses

Confidence Interval: Because the signal is considered equal to 32 + 1.645 (25.4) =

73.8 counts; this is equivalent
detected, we may set a confidence interval, rather than an to a disintegration rate of 27.6 dpm.
upper limit. The absorbance interval for the signal at the The effect of half-life on the above limits is notable. For
95 % level of confidence is 0.0060 ± 1.96 (0.0020). The cor- example, if 38K had a very long half-life, T -* 15.4 min and
responding thorium concentration interval, assuming negli- aD would be reduced to 17.1 dpm. A long half-life, of course,
gible error in the calibration constant, is 36 to 170 >ug//, and would make useful a longer counting interval. If the interval
the estimate of the mean is 103 /xg//. were 103 min, for example, the minimum detectable activity
Detection Limit: LD = 4.65 0.0093. Thus, the = would be further reduced to 2.06 dpm. Thus, when half-
minimum detectable concentration (that which will give a life is the only variable, one can detect considerably smaller
signal exceeding Lc, 95 % of the time) is equal to 160 yug//. activities for long-lived species than for those which are short-
Determination Limit: Lq 14.1 = 0.0282. Therefore,
= lived.
in order to obtain a precision (relative standard deviation) (3) Activation analysis
as small as 10%, one must have a thorium concentration of The foregoing principles will next be applied to a somewhat
480 yUg//. more complex example in which one must consider both the
(2) Radioactivity effects of various product half-lives and the effects of inter-
Let us next consider the detection of the 7.7-min positron- ference arising from alternative nuclear projectiles. The
emitter, 38K. We shall assume that the radionuclide is de- problem to be examined is whether reactor neutrons or linac
tected by means of the 0.51-MeV positron annihilation quanta bremsstrahlung are better for the detection of potassium. The
using a sodium iodide crystal having a background of 20 cpm minimum detectable mass will be estimated for the inter-
and a detection efficiency (for 38K) of 32 %. Our aim will be ference-free situation, and also for that in which the potassium
to calculate the various limits in terms of activity (disintegra- is accompanied by 1 gram of sodium. Details of the calcula-
tion rate) instead of mass. As will be shown in a separate tion will not be presented here, but only the input data and
publication, the counting interval, At, which leads to the the results. The results were calculated directly from Equa-
minimum detection limit is approximately twice the half-life; tion 22. The input assumptions follow:
we shall therefore take At 15.4 min. Equation 18 here
=

(a) thermal neutron flux-1013 (n/cm2-sec).

takes the form, LD KaD, where aD represents the minimum
=

detectable activity, and K tT. Because the delay time, t,

=
(b) bremsstrahlung flux distribution-1014/E (quanta/MeV-
min-cm2) (electron beam energy well above the giant reso-
is not involved in the present example, T (1 _ )/ . = —

nance).
(For long-lived species, T, the “effective counting interval,"
(c) irradiation time and counting time-103 min or two
reduces to At, the physical counting interval.) Using the
half-lives (potassium product) whichever is less.
fact that µ 20 cpm X 15.4 min, one may calculate Lc,
=

(d)delay time-negligible.
Ld, and LQ directly from the formulae for paired observations nuclear cross sections-taken from the literature.
(e)
given in the first row of Table II. The above value for e
8.33 min, in order to de- (f) interference correction accuracy, f¡-\ %.
may then be combined with T =

(g) detection- paired observations, sodium-iodide count-

termine ac, aD, and aQ. The results are given below:
ing of the positron annihilation peak (bremsstrahlung-pro-
ac = 15.3 dpm duced activity) or the 1.51 MeV 42K -peak (neutron-pro-
duced activity). Interferences due to sodium activities were
aD =
31.6 dpm estimated from the respective gamma ray spectra. The back-
grounds and detection efficiencies are as follows:
act =
H4 dpm
Efficiency
Let us suppose than an observation of sample plus background ( -efficiency X branching)
gives a total of 340 counts. The net signal would then be Reaction Background K Na
32 counts (assuming that the background observation yielded 12 cpm 0.014 0.0097
(n,y)
308 counts), and its estimated standard deviation would be 0.32 0.29
(y,n) 20 cpm
25.4 counts. Lc, as calculated from Table II, is equal to
40.8 counts, and therefore such an observation would lead to Minimum detectable masses of potassium are given in
the conclusion, “not detected.” (The comparison might be Table III. Note that the minimum detectable activity cal-
made on the basis of the corresponding disintegration rates, culated for 38K in the preceding example applies also to the
12 dpm-observed, and 15.3 dpm ac, but the conversion
=
present example; the minimum detectable mass differs only
from counts to dpm is unnecessary for making the decision.) by the production factor, ( ), which here equals 4.11 X
The upper limit (95% confidence level) for the net signal is 1010 dpm/g for 38K.

592 · analytical chemistry

 The results given in Table III indicate that both a reactor ACKNOWLEDGMENT
and an electron linear accelerator provide excellent detection
Helpful discussions have taken place with a number of my
capabilities for potassium in the absence of interference, the colleagues, in particular: J. R. DeVoe, J. M. Cameron, . H.
reactor being somewhat better. The importance of Equation Ku, and K. F. J. Heinrich.
22 for detection limit estimation in the presence of inter-
ference, however, becomes quite clear upon examination of
the last column of Table III. Here, the detection limit by Received for review September 21, 1967. Accepted Decem-
bremsstrahlung irradiation has become about one thousand ber 15, 1967. The paper was presented in part at the Eighth
times poorer, and thermal neutron activation has become Eastern Analytical Symposium held in New York, November
practically useless. 1966.

Graphical Technique for Estimating Activity

Levels Produced in Thermal- and Fission-Neutron Irradiation
Jornia T. Routti
Lawrence Radiation Laboratory, University of California, Berkeley, Calif.

We describe a rapid and flexible graphical method for necessary to find the nuclear parameters in the literature
estimating the activity induced in neutron irradiations. Concise summaries of sensitivities for thermal-neutron activa-
This method, applicable to several activation problems,
is applied here to thermal- and fission-neutron irradia- tion have also been published (4, J). These generally apply
tions, which are widely used in activation analysis to selected counting and irradiation conditions.
with reactors. The calculated saturation activities of The techniques described here combine, in compact form,
the products of most activation reactions are repre- the essential nuclear data with a quick and flexible graphical
sented on two activation charts. We develop a graphi- calculation method. The various factors appearing in the
cal technique, using a transparent overlay, to obtain
corrections for the saturation, decay, and counting activity equation, including the saturation, decay, and count-
factors. The principal advantage of this method is ing-time corrections, can easily be determined. Also, quali-
that the corrections due to these factors can be applied tative information of optimal irradiation and decay times,
to any saturation activity of the chart, which already which are often desired in activation analysis, can be obtained
contains all the nuclear information required for
estimation of induced radioactivity. Most simple graphically rather than by using numerical computer tech-
activity calculations can be performed quickly and niques (6).
accurately; the technique is also useful in estimating
the best irradiation and decay times to enhance GRAPHICAL METHOD FOR COMPUTATION
selected activities in composite samples. We give the
numerical values for the construction of charts of The Activation Equation. In calculating induced radio-
saturation activities for thermal- and fission-neutron activity, we assume that the irradiated sample is so small
irradiation and for the overlay. Calculation tech- that it produces no significant attenuation of the flux of in-
niques are explained and clarified with typical ex- cident particles. The total disintegration rate of a radio-
amples.
isotope produced by irradiation of an element of natural
The calculation of induced radioactivity is required in many isotopic composition in a constant flux is then given by
activation problems. Generally, the same mathematical
formalism is applied, with different values of nuclear constants
and time parameters. The search for nuclear constants and
A =
"100^[(1
M
-

exp(—In 2 X /71/2)] (-1 2 X td/Tin) =

the solution of equations for composite samples is often time- AsatSD = N* In 2/7V2 (1)
consuming. We here develop a graphical method for rep-
where m is the mass of the irradiated element, / is the isotopic
resenting nuclear data and solving activation equations; this
method can be successfully applied to several activation abundance (%) of the isotope that undergoes the reaction,
No is Avogadro’s number, is neutron flux, a is the cross
problems, flux monitoring, and radioisotope production with
reactors and accelerators. (The two activation charts and section, M is the atomic weight of the irradiated element,
the transparent overlay are available as a courtesy from 7i/2 is the half-life of the induced activity, is the irradiation

General Radioisotope Processing Corp., 3120 Crow Canyon time, ti is the decay time (between the end of irradiation and
Road, San Ramon, Calif. 94583.) The specific applications the beginning of counting), Asat is the saturation
to thermal- and fission-neutron activation have been chosen 100 M
because of their wide and frequent use in activation analysis activity, S is [(1 —

exp (—In 2 X U/Tv·.)], the saturation factor,

with reactors. D is exp (—In 2 X ta/Tm), the decay factor, and N* is the
Several nomographs for calculating induced radioactivity number of decaying nuclei.
have been published (1-3). To use most of these, it is first
(4) W. Wayne Meinke, Sensitivity Charts for Neutron Activation
(1) Philip A. Benson and Chester E. Gleit, Nucleonics, 21, No. 8 Analysis, Anal. Chem., 792 (1959).
148 (1963). (5) Russell B. Mesler, Nucleonics, 18, No. 1, 73 (1960).
(2) E. Ricci, Nucleonics, 22, No. 8, 105 (1964). (6) Thomas L. Isenhour and George H. Morrison, Anal. Chem.,
(3) Edward C. Freiling, Nucleonics, 14, No. 8, 65 (1956). 36, 1089 (1964).

VOL 40, NO. 3, MARCH 1968 · 593