ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
Table 111. Correlation of Firefly CL Method with the Acetylacetone Colorimetric Method mg Triglyceride/lOO mL Acetylacetone method CL method 41.5 78.0 106.5 113.5 140.5 140.8 142.1 150.1 153.5 171.5 310.9
39.7 81.9 101.6 113.1 124.4
144.6 147.2 147.4 159.4 177.0 302.3
SloDe 0.98 * 0.03. Y interceDt = 0.0 -L 5.1 me1100 mL. .Correlation coefficient = 0.695. Confidence' interval ( Y )= 2 6.7 mg/100 mL. T h e acetylacetone (AcAc) colorimetric procedure was also used as a reference method. The correlation of CL data and AcAc data is shown in Table 111. Triolein standards were used in t h e study to establish calibration. Soloni's AcAc procedure (13) was followed with the exception of saponification. Alcoholic KOH was used in place of alcoholic sodium ethoxide which Soloni recommended. T h e AcAc procedure calls for acidification of the saponified sample (MgS04 is not added) before addition of NaI04. Assay buffer concentration was increased in t h e glycerol kinase reaction mixture to eliminate possible lowering of t h e desired glycerol kinase reaction pH. T h e AcAc method required 0.5 mL of the saponified serum solution while only 0.05 mL was required for t h e CL assay. T h e GK/firefly CL method has demonstrated its viability as a method for serum glycerol or triglyceride assay. Sensitivity is excellent as demonstrated by the glycerol standards and the assay of serum samples which were diluted by a factor of 10. This sensitivity permits use of smaller sample size or
25
dilution of compounds which might interfere with the firefly reaction assay. Conditions of time and reagent cost must be optimized t o meet the priorities of a given circumstance. Based on this research a glycerol sample range of 0.0014.009 pmol with an incubation period of 30 min was preferred. One individual can easily assay 15 samples/h (samples started every 2 min) with this incubation period. T h e quantity of A T P in the reaction mixture should be approximately 0.012 pmol. Glycerol samples which consume mort: than 75% of the initial A T P should be diluted or reduced in sample size. By maintaining a glycerol/ATP 17570, incubation times can remain fairly short without loss of linearity due to incomplete conversion of the glycerol.
ACKNOWLEDGMENT We thank W. R. Seitz and Peter W. Carr for helpful discussions.
LITERATURE CITED (1) R. A. Johnson, J. G. Hardman, A. E. Broadus, and E. W. Sutherland, Anal. Biochem.. 35. 91 (1970). (2) H. Holmsen, I , Holmsen, and A. Bernhardson, Anal. Biochem., 17, 456 (1966). (3) K . Van Dyke, R. Stitzel, T. McCellan, and C. Szustkiewicz, Clin. Chem. (Winston-Salem, N. C . ) , 15, 3 (1969). (4) P. E. Stanley and S. G. Williams, Anal. Biochem., 29, 381 (1969). (5) H. G. Hers, The Enzymes, 6 , 76 (19621. (6) R. T. Lee, J. L. Denburg, and W. D. McElroy, Arch. Biochem. Biophys., 141, 38 (1970). (7) W. D. McElroy and M. DeLuca in "Chemiluminescence and Bioluminescence", M. J. Cormier, D. M. Hercules, and J. Lee, Ed., Plenum Press, New York, N.Y., 1973, pp 285-308. (8) E. W . Chappelle and G.V. Levin, Biochem. Med.,2, 41 (1968). (9) S. Lin and H. P. Cohen, Anal. Biochem., 24, 531 (1968). (10) G. E. Lyman and J. P. DeVincenzo. Anal. Biochem., 21, 435 (1967). (11) 0. Holm-Hansen and C. R. Booth, Limnol. Oceanogr., 11, 510 (1966). (12) J. 8 . St. John, Anal. Biochem., 37, 409 (1970). (13) F. G. Soloni, Ciin. Chem. (Winston-Salem, N . C . ) , 17, 529 (1971).
RECEIVED for review July 28, 1977. Accepted October 7 , 1977. This work was supported by the National Science Foundation under Grant MPS 73-04680 and the National Institute of General Medical Sciences under Grant GM 13913-05
Effective Deadtime of Pulse-Counting Detector Systems J. M. H a y e s , " D. E. Matthews,' and D. A. Schoeller2 Departments of Chemistry and Geology, Indiana University, Bloomington, Indiana 4740 1
A practical paralyzabie counting system can be characterized as having an "effective deadtime" which can be very different from the detector output pulse width, even laking negative values. For cases in which 80% or more of the pulses are passed by discriminator threshold, the effective deadtime, p ' , is given by ( 2 A 1 A 2 ) p / A r Zwhere A I and A 2 are the discriminator efficiencies for single- and double-event pulses, respectively, and p is the minimum pulse-pair resolving lime of the counting system. I t is shown that p r can be effectively substituted for p in count-loss correction expressions involving relative count rates, and two independent methods for the measurement of p r are shown to give identical results. I t is further shown that p' depends upon A I in the expected way, with p r passing through zero and assuming negative values as A I is decreased.
-
Mathematical treatments of the behavior of counting systems almost invariably involve a parameter termed the "deadtime," generally leading to expressions giving the observed count rate as a function of the true count rate and the deadtime ( I ) . Very often, the deadtime is viewed in terms of the detector output pulse width, since i t is considered that the recording of one pulse must be complete before a second can be recognized. T h e precise definition and accurate measurement of the deadtime are matters of more t h a n theoretical interest to the analytical chemist involved in ion or photon counting. As a practical matter, the applicability of these detector systems can be significantly extended if count losses due to deadtime effects can be satisfactcrily modeled and taken into account. Any ion- or photon-counting measurement is the product of a comdicated svstem. involving not onlv the inevitable detector and amplifier but also timing gates, pulse-height discriminators, counters, and all the associated power supplies. All these elements, together with the normal aniilytical signal levels, comprise the measurement environment, which the I
'Present address, Department of Medicine, Washington University School of Medicine, St. Louis, Mo. 63110. 'Present address, Department of Medicine, Pritzker School of Medicine, University of Chicago, Chicago, Ill. 60637. 0003-2700/78/0350-0025$01 OO/O
1977 American Chemical Society
28
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
analyst wishes to perturb as little as possible when evaluating t h e performance of t h e system. T o t h e greatest extent possible, the conditions of the deadtime measurement should duplicate t h e conditions of the analytical measurement. In this practical context, it is straightforward and useful to define t h e deadtime as the constant which must be employed, together with an appropriate count-loss correction formula ( I ) , t o accurately convert t h e observed count rate into the true count rate. Conceptually, the deadtime could be determined by comparing an observed count rate and a known true count rate. Particle or photon beams with perfectly known absolute fluxes are rare, however, and t h e analyst more commonly requires a “bootstrap” procedure whereby the deadtime can be determined without reference to some perfectly standardized flux. Two such procedures [one new, the other previously reported ( 2 ) ] are described here. Although t h e procedures are independent in their concept and approach, they give t h e same results. T o introduce these techniques and explain their results, we begin by temporarily setting aside the conventional approach t o coincidence losses and further restricting our attention to paralyzable counting systems ( I , 3) in which the deadtime is controlled primarily at the discriminator threshold ( 4 ) (this category includes most high speed ion- and photon-counting systems). As the discussion proceeds, the reader will find that t h e proposed methods of deadtime measurement will, under certain conditions, provide negative results (e.g. deadtime = -2.1 ns). While this is inconsistent-to say the least-with t h e view t h a t t h e deadtime is given by t h e pulse width, detailed considerations show that such a result can be quite correct, accurately representing t h e behavior of a counting system.
THEORY The “Deadtime” in Systems Involving Discriminators. T h e principal insight in the approach to count-losses taken here is not due to the present authors but, instead, represents t h e theoretical articulation by Ingle and Crouch ( 4 ) of a concept introduced by Ash and Piepmeier ( 5 ) . The former authors consider the observed rate of resolved pulses in terms of a series, writing
where f represents the observed rate of resolved pulses, Le., t h e counter output, and the various terms represent the contributions of one-event, two-event, three-event...etc. pulses t o t h a t total observed rate. Each f,, represents the rate of resolved pulses containing i events, and each A , represents t h e average discriminator efficiency for such pulses. If, for example, the rate of resolved two-event pulses is f r 2 = 300 s-’, a n d if, on average, 90% of these two-event pulses are large enough to cross the discriminator threshold (that is, A2 = 0.9), then the contribution of two-event pulses to the observed pulse rate will be Adr2= 270 SS’. Rates of Individual Pulse Types. Ingle and Crouch ( 4 ) have shown t h a t t h e individual fIi values can be determined in a straightforward way by consideration of probabilities derived from the Poisson distribution, for which the distribution function is (3):
where P,(t) is t h e probability t h a t x events will be observed in time t , and m is t h e average number of events occurring in any time interval t . Following Ingle and Crouch, we can write:
probability of no pulse in time interval p prior t o pulse
probability of no pulse in time in-
(3)
where F is the rate of true events, and t h e multiplied probabilities have the effect of requiring that any given single pulse is not overlapped on either its leading or trailing edge. Note that the time interval p corresponds, in effect, to the minimum pulse-pair resolving time of t h e counting system. T h e probabilities can be assessed using Equation 2, noting that the average number of events occurring in any given time interval p will be Fp.
(4) Equation 3 then becomes
f r l = FPOpo = Fe-2Fp
(5)
Ingle and Crouch ( 4 ) have shown in addition t h a t
f r Z = F2pe-3FQ
(6)
and fr3 =
F3p2e-3FP/2 + F3p2e-4FP
(7)
In order to obtain the accuracy which will eventually be of interest in this discussion, it is necessary that we consider in addition the rate of resolved four-event pulses. Such pulses could occur in three distinct ways: (i) A single pulse could be triply overlapped. T h e rate which this will occur is given by FPJ‘3Po = F4p3e-3Fp/6. (ii) Some combination of double and single overlap could occur. Either the initial pulse could be singly overlapped, with the second pulse doubly overlapped (rate = FPflP,P2Po= F4p3e-4Fp/Z); or the initial pulse could be doubly overlapped, with the third pulse singly overlapped (rate = FP$,PlP0 = F ” ~ ~ e - ~ ~ p(iii) / 2 ) Finally, . three single overlaps might occur (rate = F P f l i P I P I P o = F4p3e-5Fp). Combining all these possibilities, we obtain: fr4 =
F4p3e-3F?/6 + FdpAe-4FP + F4p3e-SFP
(8)
Evaluation of the ratio f r 4 / Fshows t h a t more than 0.01 9’0 of t h e true events will be found in four-event pulses when Fp > 0.023, and more than 0.1% when Fp > 0.052. Overall Obserced Pulse Rate. An expression for f in terms of F , Q, and the discriminator efficiencies can be obtained by substituting Equations 5-8 into Equation 1. The resulting equation can be simplified by substituting power series equivalents for the exponentials, collecting terms, and dropping terms in which the product Fp appears to the fourth or higher power, thus obtaining:
F2p’(
4A1- 6A2 + 3A3 2A, 8A1- 2 7 A 2
+ 33A3 - 13A4
6 A1
)+”]
(9)
Complicated though it might be, this equation is one half of a very simple comparison t h a t demonstrates the concern of this paper. The other half of the comparison is the analogous equation derived from the conventional approach to deadtime effects. T h e equation which is conventionally regarded as exactly describing the behavior of paralyzable counting systems is (3):
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978 f =
Fe-FP
(10)
This equation can also be placed in a series form analogous t o t h a t of Equation 9: f = F ( l - Fp
+ F2p2/2
-
F 3 p 3 / 6+
’
.
(11)
Comparison shows that there are two very evident differences between Equations 9 and 11. First, the former is preceded by t h e coefficient A , , indicating t h a t the efficiency of the discriminator for single event pulses will exert a primary control on the observed pulse rate. Second, the higher order terms carry more complicated coefficients which depend on the discriminator efficiencies for higher order pulses. The practical significance of these coefficients can be shown to be substantial. T h e “Effective Deadtime.” Comparisons between Equations 9 and 11 will be facilitated if we define an “effective deadtime,” p ’ , as follows:
(12) Substituting, we can rewrite Equation 9 in the following approximate form:
where simplification of the higher order terms depends on establishing some functional relationship between AI and the multievent discriminator efficiencies. In general, a requirement for high precision and accuracy will lead t o t h e choice of high values (20.9) for AI. T h a t is, the experimenter will not attempt to obtain deadtime compensation by setting A , < 0.5, but will instead maximize the signal t o noise ratio by setting AI t o the highest practical value ( I , 4 ) . In t h e region Al > 0.8, Ingle and Crouch have shown that, for a typical system, A I and A2 are nearly linearly related, with A 2 = 0.885 + 0.115A1 (see Figure 1, Ref. 4 ) . Under these same conditions, it is safe to assume that all triple a n d higher pulses will pass the discriminator with perfect efficiency (A3 = A4 = 1.0). Making these substitutions in Equation 9, we obtain t h e following (an extended form of Equation 13):
f
=
A IF [l - A 1 Fp‘ + ( A , F P ’ ) ~ 3.31At
-
2.31A1
2 (1.885A1 - 0.885)’ ( A , F P ’ ) .-~4.895A: - 3.895A: 6
(1.885A1 - 0.885)3
t
(14)
Comparing this result with Equation 11, it can be observed that, here, A I F plays the role of F , and p’ plays the role of p . T h e latter parallel is especially significant because, while p must be finite and positive, p’ can be negative (when A2 > 2AJ or zero (when A 2 = 2A1). When p’ = 0, f is a simple linear function of F , and it can be seen that the definition of an effective deadtime is simply another way of looking a t the “deadtime compensation’’ or linearizing effects discussed by Ash and Piepmeier ( 5 ) and by Ingle and Crouch ( 4 ) . In addition, these considerations make it apparent that the quantity required for proper correction of count rates affected by pulse overlap is p’, not p , and t h a t methods for the measurement of p’, not the pulse width, are, thus, required. Count-Loss Corrections. If the approach introduced by Equations 9 and 14 is t o be preferred t o the conventional approach represented by Equations 10 and 11, exactly how can this advantage be realized in practice? In this regard, it is interesting to compare the result of Equation 14 with that
27
of Equation 15 given below: f =
A lFe-AlFp’
(15)
In making this comparison we are, in effect, asking whether Equation 15 is a good approximation of Equation 14. If t h e approximation is close, we will have shown that, while p’ and p might have different values, the effective deadtime controls the observed count rate in the same way t h a t p controls the observed count rate in the conventional model of count losses. T h e approximation can surely be made t o fail a t high values of Fp’ or a t low values of A,, but, as the highest value of potential interest, let us take Fp‘ = 0.076, the limit of 0.1% accuracy for the second-order count-loss correction formula (see Equation 7 , Table I, Ref. 1 ) . If we then calculate f / F as a function of A l using Equation 14, ‘we find t h a t i t is within 0.1% of Ale-”lFP (Le., Equation 15) for AI L 0.76 and within 0.01% for -4,I 0.96. We have, in effect, concluded that we ought to regard the behavior of practical counting systems as being controlled by Equation 15 rather than by Equation 10. Parenthetically, we can note t h a t for an ideal system, we have A , = A 2 = A3 = A, = 1.0 and that, under these conditions, Equation 15 reduces to Equation 10 and Equations 14 and 9 reduce to Equation 11. Inasmuch as the functional relationship between p’ and F is the same as t h a t between p and E‘, it is clear t h a t all the count-loss correction formulas developed in the conventional theory ( I ) can equally well be applied when the effective deadtime is used in place of p. When t h e appearance of the multiplicative factor A l in Equations 15, 14, and 9 is taken into account, it is seen that the correction formula conventionally written as F = f/ (1-fp) should instead take the form of Equation 16:
F
=
f / [ A l ( 1- f P ’ ) l
and that the conventional form F = f r 1 + f p be replaced by
f
F=-(l
+ fp‘ + 1 . 5 f ’ ~ ’ ~ )
(16)
+ 1 . 5 f p 2 ) should (17)
A1
Consideration of Equations 16 and 17 shows that absolute count rates cannot be known more accurately than the value of AI itself is known, a limitation which will condemn most absolute measurements to rather low accuracies. The accuracy of ratio measurements will depend upon the extent to which A , terms in the numerator and denominator of 1he ratio tend to cancel. When the ratio compares beams of identical particles, t h e A, values should cancel exactly. When ratios are formed between beams of different particles, however, the ratio will be biased by a factor of Aln/Al,,,, where n and m are introduced to represent the two particle types. Quantitative evaluation of this bias is hindered by the fact t h a t a second effect is inescapably linked with it. Not only will the discriminator efficiencies vary for different particle types, b u t also the detector efficiencies (defined as the fraction of incoming particles which actually produces pulses a t the multiplier anode) can be expected to depend on particle type (6). Although a detailed consideration of such systematic biases is well beyond the scope of this paper, it can be noted t h a t the combined effect of differential discriminator and detector efficiencies has been estimated ( 7 )to cause a relative error of 4 . 3 7 0 in ion-counting measurements of 13c/’2Cratios using C 0 2 + ion beams, and a relative error of greater than +1.0% in similar measurements of 128Xe/’36Xeratios using Xe+ ion beams. Measurement of the Effective Deadtime. Having shown that a measurement of the pulse width is likely to be unsat-
28
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
isfactory as a means of determining the effective deadtime, we are left with a requirement t o provide practical experimental techniques which can directly measure the effective deadLime. At least two methods appear satisfactory. Method 1. Although it was introduced as a means of isotope ratio measurement in ion-counting mass spectrometry by G. H. Riley in 1969 (8), the first method is new in this application. Consider two signals whose true count rates are in the constant ratio R = FJF,. Using the approximate form given by Equation 16, we can write:
If we define T , = f,-l, T , = fn-’, and R’as the observed ratio as affected by detector and discriminator biases, Equation 18 can be exactly rewritten as: T,
= p ’ ( 1 - R‘)
+ R’T,,,
(19)
This is qn expression of the form y = a + b x , and p‘ can be found by the regression of T, on T, (specific regression techniqdes are discussed in the Experimental section). In ion-counting mass spectrometry, data allowing the use of this method can readily be collected by observing an isotope ratio over a range of signal levels. Ratios differing from unity by a factor of 20 or more are required ( R = 0.01 is very satisfactory). Care must be taken t h a t the ion current ratio remains constant over t h e period of t h e measurement, and while, therefore, a molecular leak inlet system may not be satisfactory, we find t h a t a viscous inlet system (9) can easily provide Xe, Kr, N P ,or COPsample fluxes of sufficient stability. T h e signal level is conveniently varied by changing the pressure in the sample-inlet reservoir. One significant practical problem which arises in the use of this method has t o do with the background signals which might underlie the peaks of interest. More often than not, the observed background ratio Rb = f m b / f n b differs greatly from the ratio in the test gas, where fmband fnb are the background fluxes for m and n, respectively. In such cases, the use of an iterative procedure allows satisfactory isolation of the signals of interest. The background cannot simply be subtracted from t h e raw d a t a because it, too, must he affected by the count losses due t o pulse overlap. T h e steps in the iterative procedure are as follows: (1) Using the raw d a t a (to which we assign the subscript 0), calculate trial values for R’ (= Rb) and p’ (= P ’ ~ )using the least squares technique. (2) (Here we require t h a t R has been defined such t h a t f, is t h e smaller ion flux and we assume t h a t Rb > R ) . Assign new values to t h e less abundant fluxes in each pair of d a t a points according t o the expression
(3) Determine R’, and p’l using the new data set. If the values differ significantly from R b and p’,,, repeat the process, incrementing t h e subscript. The value for p’ determined by the iterative procedure must be still further corrected if systematic errors are to be avoided. T h e need for this correction has been traced t o an approximation in Equation 16, an expression which is derived by rearrangement of t h e exact series form f = A I F ( l - AIFp‘ + (AlFp’)*/2- (A1Fp’)3/6 + ...), after dropping quadratic and higher terms and substituting fp’ = A,Fp’ in the first order term. The effect of this approximation on p’ can be seen in t h e following way. T h e expression used to calculate p’ is:
where in and 7, are the average observed mean pulse intervals and the subscript i has been introduced t o denote the i t h iteration. T h e substitution o f f for A I F in Equation 16 has the effect of overestimating the T values, with the result that p‘, is, in turn, overestimated by a factor related to the average A,FIf ratio in the d a t a set. As a means of avoiding t h e need for correction, we have considered the alternative of adopting a nonlinear model based, for example, on Equation 17. The resulting profusion of higher-order terms, however, together with the requirement t o carry out nonlinear least-squares calculations, leads t o a conclusion t h a t correction of p’& offers the more practical approach. In making this correction, the factor AIF/ f can be approximated by 1 + fp’, and detailed considerations show t h a t the fully corrected value of p’ will be given by p
,
Pi
=zui
where
and f m L and f, are the mean count rates of the data used to determine p l r . I t has been previously shown ( 1 ) that the uncertainty which must be associated with the deadtime exerts a primary control on the applicability of any given counting technique. In the present case, the same regression analysis which provides slope and intercept values (denoted by b and a , respectively) for Equation 19 also allows assignment of standard deviations t o these quantities. A consideration of the propagation of errors in Equations 21 and 22 then gives the variance in p’ as follows
where sa and sb denote the standard deviations of the intercept and slope, respectively. Method 2. While the first method requires the availability of two variable signals related by a constant ratio, the second method (2) requires only a single variable particle or photon flux and is thus more broadly applicable, and concern over the composition or even the magnitude of the background is unnecessary. In this method, the assumption is made t h a t the anode current from the photomultiplier is proportional t o the true flux striking the first dynode of the multiplier:
I=kF
(25)
where Z is the anode current corrected for dark current (insignificant in electron multiplier work) and k is a constant proportional to the gain of t h e multiplier. When Equation 25 is substituted into Equation 16 (thus taking into account the “effective deadtime”) and rearranged, two useful forms arise:
where p’ is again subscripted in order to indicate that further corrections are necessary, in this case simply for the approximations inherent in Equation 16 and discussed above in connection with Equation 21. T h e fully corrected value of p’ will be given by:
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
where -
u= 1
+ p6f
where 7 is t h e mean count rate in t h e d a t a set used t o determine P ' ~ . Furthermore, Equation 25 can be substituted into Equation 15, a n d rearranged t o give:
(f/n
Regression of In on I allows determination of p' via the slope (A,p'/h) a n d t h e intercept [In ( A l / h ) ] . T h e approximations inherent in treatments based on Equation 16 d o not occur in this exact form, and no corrections t o p' are required. T h e uncertainties associated with p' values derived via Equations 26, 2 7 , or 30 are given by Equations 26a, 27a, and 30a, respectively: SP' = SP'
(26a)
sa
= §b
+ b2s,2)'
sPr = e-"(§;
(27a) (30a)
where a and b refer t o t h e linear intercepts a n d slopes, respectively.
EXPERIMENTAL Apparatus. The Varian CH-7 mass spectrometer and viscous inlet system used have been described previously (9). The ion counting sytem consisted of an I T T (Electro-optical Products Division, 3700 E. Pontiac St., Fort Wayne, Ind. 46803) Model F4074 16 stage Cu-Be electron multiplier followed by an SSK model 1120 pulse amplifier and discriminator. The pulse train from the SSK 1120 was transmitted to a Varian MAT 300 MHz 16-bit counter/gate interfaced to a Varian 620i computer. The standard Varian MAT CH-7 10' gain preamplifier (BDF) followed by a unity gain amplifier with a 10-Hz bandpass was used to measure the anode current from the electron multiplier. The amplifier output signal was monitored using a Tektronix DM-40 31/2 digit voltmeter. Because ion counting and analog current measurement cannot occur simultaneously, the BNC connection a t the anode of the electron multiplier was manually changed to select the measurement system desired. The CH-7 ion source was operated a t 70 eV with 30-pA emission. The ion source- and collector-slit widths were 0.040 and 0.500 mm, respectively, yielding peaks with flat tops more than 0.10 mass unit wide (at m / e 44) and a resolution of 270 (10% valley definition). The electron multiplier was normally operated a t a gain of 9 x 10'. Carbon dioxide diluted with helium was used as the sample gas in determinations of the system deadtime using both method 1 and method 2. The SSR model 1120 discriminator was normally set to accept more than 90% of the total pulses. However, for the data in Table VI, this threshold was varied between the minimum practical level and higher settings in order to control AI. The minimum practical level was taken as the threshold level below which noise pulses appeared even when no high voltage was applied to the multiplier. When the system was operated a t this level, observed ion beam count rates were independent of multiplier dynode voltage at gain levels greater than lo6, and it was assumed that the system was operating with AI 1.0. Under these same conditions, full-range variations of the threshold caused the observed count rate to vary by less than a factor of two, making it difficult to explore relationships between A I and p' a t this relatively high multiplier gain. When a multiplier gain of 1.8 X loJ was employed, however, threshold adjustment alone could cause the observed count rate to vary by nearly an order of magnitude, thus allowing the data in Table VI to be collected. Software. T o determine the flux a t mass 44 for method 2, the accelerating voltage was set by the computer to the centroid
-
29
of mass 44 and the flux was determined by observing the beam for one thousand 10.24-ms periods. When method 1 was employed, a general peak location and relative abundance (isotope ratio) data acquisition routine, GPLDA, was used to determine the fluxes a t masses 44 and 45. This program supervised beam switching via the digitally controlled high voltage power supply, employing a scanning function which specified that the time spent at any given mass should be inversely proportional to the square root of its flux. Typical observation times were 70 ms a t m / e 44 and 700 ms at m l e 45. Although the Varian 620i mini-computer was used for controlling the mass spectrometer and determining the fluxes, the Indiana University CDC 6600 computer was used to calculate all deadtime results. Two methods for a least squares fit to a straight line were used. The first was the standard regression of y on x . The second method, discussed by D. York ( I O ) , assumes that both the x and the 3' points are subject to error and that the deviations for both the x and y points should be minimized. T h e York method was used for calculating the rersults shown in the tables, although no significant difference was observed when the standard regression was used. Procedure. Method 1. A sample gas containing 99 mol % He and 1 mol 7~C 0 2 was placed in the viscous flow inlet system (9) and sample pressures were varied to obtain eleven different m / e 44 ion fluxes in the range between 2 X lo5 s-' and 1.5 X lo6 s-'. The pattern of observation was not fixed, with high and low flux observations being interspersed in order to confound any systematic effects which might otherwise affect the result. Approximately 4 x IO5 mlp 45 ions and 4 x IO6 m j e 44 ions were collected a t each flux level. Method 2. The sample gas used for method 1 was employed, and the observed fluxes were again varied randomly within the indicated range in order to confound systematic effects. Three measurements were made a t each flux :level: (i) the m / e 44 flux was determined by counting for 10.24 s; (ii) the ion-counting system was replaced by the analog amplifier system and the multiplier anode current determined (voltmeter reading averaged over -30 s): and (iii) the m / e 44 flux was redetermined by again counting for 10.24 s. The two counting observations were averaged.
-
-
RESULTS A N D DISCUSSION Experimental validation of t h e deadtime determination techniques is of obvious interest, h u t is a task fraught with subtle (though possibly highly consequential) complexities. Does the successful practical use of a given technique and its associated count-loss correction procedures prove t h a t t h e technique is accurate, or is model-dependent circular reasoning involved in such "tests?" Inasmuch as t h e theoretical development asserts t h a t t h e effective deadtime is a function of measurement conditions, what kind of truly independent test can be made? In t h e present case, fortunately, we have two techniques based on t h e same theoretical model b u t involving very different experimental measurements a n d significantly different assumptions. T h e best validation we can offer a t present is discussed in t h e following sections, which conclude by showing t h a t both methods give t h e same result when applied to t h e same experimental system. Method 1 . T h e system of Calculations outlined in Equations 19-24 was developed and initially test.ed using trial calculations in which dummy data generated from Equation 15 were processed. T h e problem of background correction was sidestepped in the first series of trial calculations, which used background-free data and tested the efficacy of Equation 22 as a means of correcting for imperfect approximations embodied in Equation 16. Varying (and often extreme) background ion fluxes were added t o t h e dummy d a t a in a second set of trial calculations designed t o test t h e background-correction procedure outlined in t h e discussion of Equation 20. Finally, t h e method was applied t o typical experimental d a t a to provide results comparable t o those obtained using method 2. Trial Calculations. T h e results summarized in Table I illustrate three effects. First, t h e values calculated using
30
ANALYTICAL CHEMISTRY, VOL. 50, NO. 1, JANUARY 1978
Table I. Background-Free Trial Calculations Employing Method 1 Nominal input values" Calculated results p ' ns RX 10' ~ , , s - lx 105 spi , ns P ' ~ nsb , p ' , nsc R ' , X 10' 50 1 2~ F , Q 1 0 0.9998 50.77 49.79 0.17 50 1 2 G F , G 2.8 0.9999 50.10 50.00 0.03 50 1 8 < F , < 8.8 0,9991 51.65 50.05 0.02 50 5 2
