PARAMETERS OF A RADIOCARBON INSTALLATION. where

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1 [RADIOCARBON, VOL. 38, No. 2, 1996, P ] PARAMETERS OF A RADIOCARBON INSTALLATION VLADIMIR Z. KIHAIT 85 Yermiyahy Street, Sderot 800 Israel ABSTRACT. I aim to define instrumental parameters a radiocarbon laboratory installation by one can estimate its precision a maximum age up to which its measuring results are reliable. The commonly accepted factor merit (FM) relates precision measurement to Poisson statistics. Unlike FM, proposed parameters show extent to which a 14C laboratory is affected by destabilizing factors that could cause additional measurement errors. Assuming that all destabilizing factors produce eir a change in counting efficiency or additional fluctuations background counting rate, I have derived two parameters for consideration. Several authors (Moljk, Drever Curran 1957; Vinogradov et al. 1961) have suggested estimating precision a radiocarbon age with help FM M = S/1fB ( S B st for counting rates stard background, respectively). Using FM, it is n possible to compare or 14C laboratories with respect to ir precision maximum age. Such a comparison is not appropriate in all cases. In some cases, investigators do not use FM to estimate precision measuring results (e.g., Ryabinin 1978; Sementsov 1970). Sementsov (1970) evaluated precision measurement results by estimating m based on experimental deviation counting rate stard Qxs: Si = a counting rate a stard in ith test n O 2 = [1/(n -1)] (Si _S)2 i=1 n s = Si/n = an average value counting rates in n tests. i=1 The FM was obtained under assumption that measurement is subject to Poisson statistics. Undoubtedly, process a radioactivity formation background signals obey Poisson statistics. Thus, according to properties Poisson distribution, one can determine a counting rate deviation called statistical deviation) (1) S = counting rate stard t = counting time. as2= s/r (2) The statistical deviation is not an independent value because it is determined from already known measured values S t. If, under experimental conditions for same sample, Equations (1) (2) are in good agreement n one may claim that 14C lab is optimal. Its instrumental errors are negligibly small may be neglected. The agreement errors calculated according to Equations (1) (2) is not always observed. Some investigators (Crevecoeur, Stricht Capron 1959) have preferred to calculate errors according to Equation (1) in spite fact that it is nec- 1We use word "deviation" hereafter implying "stard deviation" (1 Q). We do not use term "stard deviation" so as not to confuse it with term "stard", which has anor meaning. 375

2 376 V. Z. Khait essary to spend additional time making repeated tests before error calculation. Many investigators find that, according to ir measurements, summary (experimental) errors exceed statistical errors considerably (Crevecoeur, Stricht Capron 1959; Currie 1972; Zavelsky 1972; Khait 1982). Since it is necessary to decrease statistical deviation to an acceptable value, one must increase counting time. For long counting times, it is very likely that measurement process will be affected by external factors (e.g., temperature changes, supply voltage fluctuations), changes which will also result in additional fluctuations in counting rate as a result, instrumental errors occur. Summary fluctuations contain both instrumental statistical fluctuations. For future work, let us assume that all factors that influence measurement process cause eir a change in count efficiency E or an additional counting rate deviation background. Accepting this assumption, counting rate deviation for any 14C sample will be a function counting rate deviation stard that background (Khait 1982), given as 0r2 = [(R-B)2/(S-B)2lQ52 + [1-(R-B)2/(S-B)2lQh2 (3) R = counting rate any sample S = counting rate a stard B = counting rate a background Qr = counting rate deviation any sample QS = counting rate deviation a stard Qb = counting rate deviation a background. Comparison results calculated by Equation (1) those calculated by Equation (3) shows good agreement (Khait 1982). The counting rate deviations in Equation (3) are instrumental deviations. A summary (experimental) deviation consists instrumental deviation a statistical deviation QSt. To extent that se values are independent, n, in accordance with laws statistics with previous studies (Crevecoeur, Stricht Capron 1959; Currie 1972; Zavelsky 1972), following equations express summary deviations o, o,,. ose three counting rates S, R B incorporating instrumental statistical deviations Q 2 = O 2 + Sit (4.1) Q2 = 0r2 + R/t (4.2) 01b2 = Qb2 + Bit. (4.3) Equations (4.1)- (4.3) allow us to determine instrumental deviations ( first expression in right side equations). However, for o result given in Equation (3) is to be preferred to Equation (4.2). Equation (3) does not require repeated tests. As follows from Equations (4.1)-(4.3), statistical deviations ( second expression in right side equations) can be reduced by increasing counting time. Experience shows that with an increase counting time it is impossible to reduce summary (experimental) deviations o, Qsr 01b (Currie 1972; Zavelsky 1972). At present we cannot answer how instrumental deviation depends on counting time. To overcome this problem, we assume same counting time for stard, sample background is used that y appear in same order, i.e., stard-background-sample. By following se rules, a possible dependence instrumental deviation on counting time can be avoided. It is n possible to calculate O vb according to Equations (4.1) (4.3) if we determine preliminary summary deviations o 01b according to Equation (1). It may happen that a statistical deviation is much less than an instrumental one. In that event, we can neglect statistical deviation. In such a case, all furr reasoning will also be true for summary (experimental) deviation as well.

3 Parameters a Radiocarbon Installation 377 Equation (3) makes it possible to draw following important conclusions: 1. It is possible to save counting time because re is no need to use repeated tests to calculate counting rate deviation for every sample. 2. It is necessary to repeat counting stard background. These repeated measurements can be used in Equation (1) to calculate summary deviations. 3. It will suffice to measure counting rate any sample once n calculate its deviation according to Equation (3). There is no need to use additional time for measuring process. The following expression allows us to calculate deviation a 14C age. ot =.. os/(s-b)2 + o2/(r-b) + ob (S-R)2/(S-B)2 (R-B)2 (5) ti = average time life 14C. The investigators (Crevecoeur, Stricht Capron 1959) used this equation for calculating errors a 14C age. As is apparent from Equation (3), or is a value dependent on os 06. Substituting Or into Equation (5), according to Equation (3), we get following transformed expression ot = 4t o12 s/(s-b)2+ob(s-r)/(s-b)(r-b)2. (6) Also (R - B) = (S - B) exp (-TI ti) (7.1) R = (S - B) exp (-T / ti) + B. (7.2) After substituting (R-B) R (7) into (6), expression ot becomes ot = r os/(s-b) Ji + exp(t/t) [exp(t/i)-1] ob/os. (8) Equation (8) is a function instrumental deviation sample's age. The behavior Equation (8) is determined by its coefficients P = os/(s-b) Q=ob/oS. t Replacing os/(s-b) o,,/os with ir symbols, Equation (8) becomes QT = P 1 + exp(t/i) [ exp(t/t) -1] Q2. We may assume that nature instrumental deviation a 14C age for any 14C laboratory is given by Equation (). Two graphs in Figure 1 show deviation as a function age: graph 1 was carried over from a paper by Crevecoeur, Stricht Capron (1959); graph 2 was plotted according to Equation () for instrumental deviations liquid scintillation counter that was used in Moscow State University's Laboratory Recent Sediments Pleistocene Paleogeography in 1976.

4 378 V. Z. Khait 3 z io z Age in years Fig. 1. Stard deviation as a function age From similarity two graphs in Figure 1, it is very likely that our assumption mentioned above is correct. To verify this, coefficients first graph were determined by solving two equations to give 1) T = 0; ST =0 2) T = 30,000; ST = 400. Despite fact that Figure 1(Crevecoeur, Stricht Capron 1959) does not contain interval ages from T=0 to T=500, for purpose facilitating calculation its coefficients, we found it worthwhile to extend Figure 1 to intersect error scale. In accordance with se chosen points, we have following system two equations:

5 Parameters a Radiocarbon Installation 379 io 4 Fig. 2. Fragment Fig.1 Age in years 0 = f2-vp. J1+exp(O/t)[exp(O/t)-1]Q = f2 t P Ii + exp(30, 000/t) [exp(30, 000/t) -1] Q2. (11) The solution system gives results P = Q = Using se estimated coefficients, expression for measuring error on a 14C age for gas counter presented by authors (Crevecoeur, Stricht Capron 1959) over whole range ages is QT = exp(t/t) [exp(t/t) -1] 0.9. (12) From Equation (12), locations nine points were determined (Table 1) marked by crosses in Figure 1. As we see, locations se points do not deviate considerably from graph 1. The two functions with three deviations o, Qr 0b according to Equation (5) also paper (Crevecoeur, Stricht Capron 1959) with two deviations ors Qb according to Equation (12), as shown in Figure 1, coincide. This confirms above supposition also fact that

6 380 V. Z. Khait instrumental errors prevailed in results measurements presented in Crevecoeur, Stricht Capron (1959). It is possible that re are instrumental errors in most 14C measurements. Coefficients P Q are parameters Equation () are proposed parameters for measuring counting performance a i4c lab. Assuming that T«i, we obtain an approximate result in accordance with (), by neglecting second expression under root. ot =,[2 Pti. (13) Assuming that T>>-r, we get an approximate result for which first expression under root can be neglected QT = 1 P Q r exp (T/T). (14) Equation (13) does not contain variable T. Consequently, instrumental deviation depends very little on age for T«i. This is also seen in Figure 1(for first graph approximately up to 00 yr, for second graph up to 3500 yr). There is only one possibility to decrease instrumental errors over this range, namely lowering P. Equation (14) for T»t shows an exponential dependence on instrumental deviation a 14C age. As shown in Equation (14), product coefficients P Q characterizes precision age determination in this interval. According to Equation (9), this product is equal to P Q = Qb/(S -B). (15) We can consider this product P' Q as an individual parameter A (this is, A=PQ) on which precision measurements in range T»i depends. In any case, as follows from Eqs. (9.1) (15), to obtain a higher precision measurement over whole range 14C ages that can be measured in a lab, counting rate stard must be as high as possible, or, in or words, counting efficiency must be as high as possible. To obtain a higher precision measurements in interval young ages according to Equation (9.1), it is important to reduce counting rate deviation stard to a minimum. TABLE 1. Dependence Stard Deviation Age on Age Age (7) in years Stard deviation (T) in years No ,000 20,000 30,000 40, ,750 For a higher precision measurements in range T>> i, we must reduce counting rate deviation background to a minimum according to Equation (15). The precision measurement results depends on two instrumental parameters P Q over whole range a 14C age. These parameters show capability a 14C laboratory to resist effect destabilizing factors on a measurement process. Comparison se parameters various 14C laboratories enables one to ascertain which m can yield a higher precision measurement in a particular part age scale.

7 Parameters a Radiocarbon Installation 381 The availability three equations for calculation instrumental errors (Eqs. (), (13) (14)) allows us to divide whole range 14C measurements into three intervals: young, middle ancient ages. Equation () is generalized. The calculations instrumental errors by Equation () are always correct for whole range a 14C laboratory. We can consider Equation () as a function parameters P Q. The influence each se parameters on precision measurements varies in different places on age scale. Therefore, it is worth dividing whole range 14C ages into intervals. This permits us to estimate precision measurements with help approximating Equations (13) (14) showing eir a prevailing influence one se parameters (P ora) ir combined influence on precision measurement. So, precision measurement in range "young ages" depends solely on parameter P, but in range "ancient ages" it depends solely upon parameter A. In or cases, measurement error must be calculated according to Equation (). To determine when Equations (), (13) (14) apply, we solve following equations Ji + exp(t/t) [exp(t/t).4]. Q2-1 = 8 (16) 1 + exp(t/t) [exp(t/t) -1] Q2- Q. exp(t/t) = b Q exp T/t (17) 8 = relative permissible difference. The root Equation (16) is a bound between intervals young middle ages. We can find its value according to expression Tym = t ln(0.5 + JO /Q2). (18) The root Equation (17) is a bond between intervals middle ancient ages. We can find its value according to following expression Tma=tln b (19) b 2bQ2 Tym is a bound up to which we can use Equation (13) for calculation instrumental errors (deviations) bt. Tma is a bound over which we can use Equation (14) for calculation instrumental errors. Then difference in instrumental errors between each approximation equations generalized Equation () will not exceed a specified relative permissible difference S. Assuming relative permissible difference to be % (S = 0.1), Tym for each above-mentioned counters will be equal: Tym (gas counter) =1397 yr Tym (scintillation counter) = 4042 yr. Assuming relative permissible difference to be % (S = 0.1), bound Tma was determined only for scintillation counter that was used in 14C laboratory Moscow University in It is equal to 9520 yr. It was not possible to determine Tma for gas counter used by Crevecoeur, Stricht Capron (1959) because relative permissible difference between generalized

8 382 V. Z. Khait Equation () approximation Equation (14) could not be achieved at any points range this gas counter. As follows from Equation (13), precision measurements in range young ages depends almost entirely on parameter P = QS/(S B). The lower value for P, smaller error a measurement result in this interval. The bound Tym between interval young middle ages determines width interval young ages (O<T<Tym). Errors measurement in this interval are minimum almost constant. The higher value for Tym, better lab. The locations bounds Tym in graphs for two counters Figure 1 are also in agreement with fact that instrumental errors hardly ever change over range young ages. As follows from Equation (18), parameter Q = Qb/QS influences location Tym on age scale. Thus we can evaluate all capabilities a 14C laboratory using its two parameters P Q. The growth measurement errors becomes appreciable for T>Tym. This fact can also be observed on graphs Figure 1. The precision measurements in interval middle ages (Tym<T<Tma), as follows from generalized Equation (), depends on two parameters P Q. As age T grows, influence parameter Q on precision dating grows too. As Equation (19) shows, location bound between middle ancient ages also depends on parameter Q. As follows from Equation (14), instrumental error QT is an exponential function age T for T>Tma. The growth measurement errors in this interval is quicker than over central range ages. Therefore, higher bound between middle ancient ages, better lab. The product parameters P Q determines precision dating in interval ancient ages according to Equation (14). Still missing from our list parameters is maximum age TmAZ. It is possible to deduce this important additional parameter as a function parameters P Q. However, maximum age is subject separate investigation will be dealt with in future research. Table 2 presents parameters equations for ir calculation. The results calculations some se parameters are also presented in this table for two counters under discussion. We have used equations by Moljk, Drever Curran (1957) for calculation maximum age. Crevecoeur, Stricht Capron (1959) proposed that maximum age should be determined if re are summary (experimental) errors measurements. Moljk, Drever Curran (1957) suggest determining maximum age as a function FM. Table 2 lists results calculations Tmax performed in both se ways for scintillation counter used in 1976 at Moscow University's 14C laboratory. The counting efficiency E is also included in Table 2. It is possible to express parameters P A in terms counting efficiency d = specific radioactivity W = weight a sample. P = os /E d W (20) A = ab/edw (21) It is possible to determine weight a sample to obtain necessary values for P A, ensuring required precision measurements.

9 Parameters a Radiocarbon Installation 383 TABLE 2. Parameters a Radiocarbon Installation T'pe counter Gas Scintillation Parameter University Moscow No. symbol Parameter name Calculation formula Louvain University 1 E (%) Counting efficiency 2 M FM 3 P S B,0 dw S/ J QS/(S-B) x ' x '3 4 Q Qs/ QB A PQ = vb/(s-b) 8.36x '3 3.92x '3 Bound between 6 Tym (yr) intervals young ' 2.8 In middle ages Q Tma (yr) =+= Bound between intervals middle iln ancient ages 4b SQ2 8 Tmax (yr) Maximum age i In S = ' bb 43,360 s ln(6m,ft) 40,500 CONCLUSION We have obtained a system instrumental parameters for a 14C lab as a result an investigation instrumental errors measurements. Two se parameters, P A, are fundamental. They are fundamental because y will suffice to determine completely precision measurements over whole age range capability a 14C laboratory. We can replace parameter Q with parameter A according to formula A=PQ. Then two parameters P A will be fundamental. Additional parameters, such as bounds age ranges maximum age, enable us to evaluate quickly performance a 14C lab. The FM is also a fundamental parameter as precision measurements is directly dependent upon it. This parameter is suitable for an ideal case only, that is, for measurements without fluctuations in counting efficiency without additional fluctuations background, refore I propose that P A should also be considered fundamental. The rest parameters are additional, being related to fundamental ones in a certain way. The additional parameters enable us to quickly evaluate performance a 14C lab, for example, such parameters as bounds intervals maximum age. From above reasoning, FM is a fundamental parameter as precision measurements is directly dependent upon it. It is neces-

10 384 V. Z. Khait sary to allow for an essential precondition obtaining this FM. This parameter is suitable for an ideal case only, that is, for measurements without fluctuations count efficiency without additional fluctuations background. ACKNOWLEDGMENTS I am very obliged thankful to my friends for ir assistance, without which I could not have presented this paper. The figures in this paper were plotted with AutoCAD program by my friend Vladimir Chapsky. I am very grateful to Leonid Margulyn for correcting text. REFERENCES Crevecoeur, E. H., Stricht, A. V. Capron, P. S Precision dating method: Stardization calculation errors maximum age in 14C method. Science 45: Currie, L. A The limit precision in nuclear analytical chemistry. Nuclear Instruments Methods in Physics Research 0: Khait, V. Z Errors Radiometric Apparatus in a Definition Radioactivity. Moscow, Academy Sciences: (in Russian). Molj k, A., Drever, R. W. P. Curran, F. R. S The background counters radiocarbon dating. Proceedings Royal Society A 239: Ryabinin, A. L Simple radiocarbon installation with usage integrated circuits. In Kocharov, G. E. Dergachev. V. A., eds., Transactions 6th Scientific Society on Problem Astrophysical Phenomena Radiocarbon. Tbilisi, Tbilisi University: 267 (in Russian). Sementsov, A. A Stabilization parameters a scintillation installation for counting natural radiocarbon. In Kocharov, G. E. Dergachev. V. A., eds., Transactions Scientific Society on Problem Astrophysical Phenomena Radiocarbon Tbilisi, Tbilisi University: 75 (in Russian). Vinogradov, A. P., Devirts, A. L., Dobkina, E. I. Martishenko, N. G A Definition anabsolute 14C Age with Help a Proportional Counter. Moscow, Academy Sciences USSR (in Russian). Zavelsky, F. S About a precision limits radiocarbon method definition an absolute age. In Proceedings Academy Sciences USSR, Geographical Series: (in Russian).

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