Precision in X-Ray Emission Spectrography H. A. LIEBHAFSKY, H. G. PFEIFFER, and P. D. ZEMANY Research Laboratory, General Electric Co., Schenectady,
N. Y.
X-ray emission spectrography may be regarded as a random Process, but only when operating conditions are ideal. Under such conditions, the standard deviation, background assumed negligible, should be predictable and equal to the square root of the mean number of counts, as it is in radioactivity. This conclusion was verified experimentally. When the actual standard deviation significantly exceeds the predicted, operating conditions are not ideal. This conclusion was verified experimentally for the case in which evaporation of the sample spoiled the alignment.
sample by the action of t,he exciting beam is thus analogous to a therefore ought radioactive sample. The conclusion sC = to hold in the former case, provided that aln-ays the Same fraction of quanta emitted by the sample is counted by the detector. The potential significance of this conclusion for the analytical chemist is clear from Figure 1, in which idealized distribution curves are given for the usual analytical method and for x-ray emission spectrography. I n t,hese frequency diagrams, the expected frequency for a given result is plotted as ordinate against the absolute value of the result, and standard deviations have been laid off along the abscissa from the means as origins. For the usual (accurate) method, the mean .If is assumed identical with the true value, and observed errors are attributed to an indefinitely large number of small causes operating a t random. The standard deviation, sg, depends upon these small causes and may assume any value; mean and standard deviation are wholly independent,, so that an infinite number of distribution curves, three being shown in Figure 1, is conceivable. B u t x-ray emission spect,rography considered as a random process differs sharply from such a usual case. Under ideal conditions, the individual counts must lie upon the unique Gaussian curve for which the standard deviation is the square root of the mean. This unique Gaussian is a fluctuation curve. not an error curve in the strictest ~ ~a sense; there is no true value of -2'as there is of ~ f - t h is~only most probable value S.
45
I
?T X-R.%Y emission spectrography, the intensities of the charact'eristic lines are generally SO low that it is necessary to use a counter as detector, which is the usual procedure in studying radioactivity. I n radioactivity, a truly random process ( 1 , 5 ) , the counts reaching the detector during successive identical counting intervals fluctuate according to a unique Gaussian distribution--that distribution for which the standard deviation is the square root of the mean. T h e purpose of this Paper is to demonstrate that' x-ray emission under ideal operating conditions also may be regarded as a random process, and to indicate how this affects the precision of analytical results. The background is assumed t o be negligible throughout the study. Under certain conditions, the counts made either on radioactive samples or in x-ray emission spectrography should obey a binomial or a Poisson distribution ( I , 2). The analytical chemist normally is concerned only rrith conditions for which a Gaussian distribution should apply. The argument belox can be extended to cover the other distributions as well. The mathematical treatment of random fluctuations stems from probability theory ( 6 ) ,and has been verified experimentally for radioactivity ( 1 , 4 ) . The principal conclusion from this treatment appears below as translated to x-raj- emission spectrography.
I
M
PREC! S ! 0 N N
IIDEAL!ZEDI
1 5 0 L
Assume that the spectrograph is operating satisfactorily-Le. , no assignable causes of variations are present-with a sample in place t h a t has just given d\-l counts in 1 seconds. Repeat this counting experiment n - 1 times t o make available t h e series of , , , , , :Yn.Sow, if n and N are large enough, the indicounts LVl, vidual, counts 5,. , . . . . N, lie on a Gaussian distribution of mean n' and standard deviation sc =
5, s;. s . E X P E R I M E N T A L RESULTSU S U A L A N A L Y T I C A L HETHOO
45.
The experimental verification of this theoretical conclusion by Rutherford and Geiger ( 4 ) proved t h a t radioactivity is a truly random process. The theoretical conclusion that sc = can be valid for radioactivity only if the tot,al number, of radioactive atoms in the sample remains virtually constant from the beginning of the first to the end of the last counting interval for a sample. The concl,lsion 8C = is usually applied Iyithout comment or misgiving to x-ray enlission spectrography. which differs from process, T h e conclusion radioactivity in not being a spontaneous thus applied can be strictly valid only when operating conditions are ideal. This statement becomes clear if the Epectrograph is regarded as a means of maintaining in the ssmple a sensibly constant number Soof virtually identical excited atoms that emit the s-rav quanta being counted by the detector, The emission of such a quantum by an!: one of these S o atoms is a spontaneous process. This system of K O excited atoms maintained in the
4.T:
dz
X-RAY
E M I S S I O N SPECTRROGRAPHY
Figure 1. Contrast betw-een expected distributions of results for usual analytical method (error curves) and for x-ray emission spectrography (unique fluctuation curve) under idealized conditions
Inasmuch as sc results from fluctuations that cannot be eliminated so long as quanta are counted: this standard deviation is the irreducible minimum for x-ray emission spectrography under ideal conditions. S o t only is it a minimum but it is also a predictable minimum. K h e n the actual standard deviation, S A ! significantly exceeds the predictable standard deviation, sc, it is likely that errors resembling those the analytical chemist usually encounters are superimposed UP011 the random fluctuations. This reasoning \vas tested experimentally as follows: ivith a tungsten sample in t h e spectrograph, the goniometer was adjusted until a counting rate near 100 counts per second was obtained. This counting rate is high enough to give a convenient counting interval and low enough to eliminate significant coincidence errors. T h e time required to reach 1024 counts was then 1257
1258
ANALYTICAL CHEMISTRY Table I. Random Fluctuation in 393 Groups of AVCounts Each
.\Iid-point of interval" Frequency a
1100
1090
2
2
1080 3
1070 10
I000 18
1040 48
10.50 41
1030 50
1020 41
1010 49
990 33
1000 39
980 26
970 14
960 8
940
960 0
2
Interval defined by S (as listed) + 5 counts, except t h a t lowest and highest intervals include all extreme values. ~
measured 393 times in succession. For each individual counting interval, t h e number of counts recorded for 10 seconds wap ralculated by simple proportion. (This arithmetic was without influence on t,he statistics.) I n this way, a bod2 of data was obtained for which t = 1 ) seconds; n = 393; and S = 1018. These data are summarized in Table I according to frequency in the intervals identified by their mid-points. The dat,a of this table are plotted in Figure 2 about, the G:iussiaii curve for which the st,andard deviation is the square root of the mean. The data of Rut,herford and Geiger, which were obtained by counting alpha particles, are plotted about the Eanie curve. I n Figure 2, both sets of data fit the Gaussian about equally well.
to an error in SA; this seems more likely because sc exceeds S A . I n general, SC is more precisely known. When S A significantly exceeds SC, there is an assignable m u s e of variation-in other words, operating conditions are not good. It, is not possible to make the preceding statement quantitative, and this limits its nscfuliirss ivhen the differences hetween the standard deviations :$re not large. One simple example of poor operating conditions is that of a volatile sample evapoi.:tting rapidly enough to change the optical path during the counting period. On-ing to this change. the counts registered during t seconds decreases with time if the spectrograph is not readjusted. This drcrease is aiiprriniposed upon the random fluctuation9 disciissrd with the result thxt S A exccwis SC.
X 0
.1 series of experimeiits paralleling those of Table I \\-as carried out on an open cell containing toluene, with the goniometer set so as to give a counting rate near 100 counts per second a t the mid-point of the series. The results are given in T:tble 11.
RUTHERFORD 0 GEIGER X.RAY E M I S S I O N DATA
x
Table 11. Effect of Drift Owing to Evaporation Superimposed on Random Fluctuations" Mid-point of intervalb Frequency
1335 28
738 31
835
28
935 26
1036 33
1135 34
1233 11:
1338 4
200 groups of S coirnis each: t = 10 seconds. b Inten-a1 defined by .\- (as listed) A50 counts, e s c e p t i h n t lowest and highest intervals include all extreme values. STANDARD
DEVIATIONS
Figure 2. Both x-ray emission spectrography and radioactivity conform to unique Gaussian curve based on alone
Inspection of Table I1 shows that the distribution is far from Gaussian. Calciilatioii of the standard deviations (E(1uatioiis 1 and 2 ) gives
.v
SA =
111 both cases the fit is good enough so that thrre is little point in trying to guess the causes of the deviations. The conclusion is that the fluctuation treatment developed for radioactivity applies to an \-ray emission analysis being carried out under the best conditions. PREDICTABLE STANDARD DEVI4TION
A comparison of predictable and actual standard deviations can he valuable to the analytical chemist as an index to t'he operating conditions of his spectrograph. For Table I , Predictable ptandard deviation = sc =
and for Actual standard deviation SA
d 5 = 31.9 counts
(1)
198 counts >>sc
=
31.9 counts
(4)
\Thenever 8.4 thus exreeds sc, operating conditioiis sliould be improved immediately. Further, when S.4 >> SC, :tu i i i ( w ; w iii the number of counts is not necessarily reduced S A . The example of Table I1 illustrates one important tliflereiice between x-ray emisaioii spectrography and the coiintiiig of radioactive disintegrations. 111 the former. the risk is greater that a disturbance of some sort increasrs the uncertainty of the result, especially if t,he counting period is prolonged. In x-ray :ihsorpis also present, and it has been tion measurements, this I& reduced by using the comparative method ( 6 1. Finally, precision is pi,ed ble in x-ray emission spectrography when Equation :3 is oheyed. L-nder these conditions. confidence limits nxty iiicleed be used with confidt.ncr (3). LITERATURE CITED
=
= dZi(lj7{-
;1')'/n - 1 = 20.5 counts (2)
Therefore, SA
SC
(3)
which means that operating conditions for the spectrograph were satisfactory inasmuch as sc is the irreducible minimum standard deviation, It was encouraging t o discover that good operating conditions could be maintained over the entire period (nearly a day) required to amass the data of TableI. The small difference betn-een SA and sc is probably attributab!c
(1) Bothe, W.,"Handbuc,h der Physik," Vol, 22, chap. 3.L p. 179, Julius Springer, Berlin, 1926. (2) Friedlander, G.. and Kennedy, J. IV,, "Introduction to Rarliochemistry," JTiley. S e w York, 1949. (3) Liebhafsky. H. .I., Pfeiffer, H. G., and Balis, E. \V.,.isa~. CHmf., 23, 1531 (1951). (4) Rutherford, E., a n d Geiger, H., Phil. M a g . , 20, 698 (1910). ( 5 ) Schweidler, E. \-on, Premier Congr. intern. Iiatliologie, LiBge, 1905. (6) Zemany, P. D., Winslow, E. H., Poellmitz, G . S., and Liehhafsky, H. -I,, - i S . i L . CHCM.. 21, 493 (1949). RECEIVED for review December 20, 1955. Accepted .\lay 4, 19.55. Presented before the Division of .Inalytical Chemistry a t the 127th Meeting of the AMERICASCIIEIIICII,SOCIETY, Cincinnati, Ohio, hIarcll 195;.
