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Statistical factors in X-ray intensity measurements
Spectrochimica Bcta, 1958,vol. 12, pp. 109 to 178. Pergamon I’ressLtd., London
Statistical factors in X-ray intensity meMurements M. MACK and N. SPIELBERG Philips Laboratories, Irvin&n-on-IIudson, Hew York, U.S.A.
Abstract-In X.-ray intensity measurements, statistical factors result from the inherent randomness of the X-ray excitation process. The optimum division of counting times for given counting rates is discussed for various combinations of counting rates, with particular attention to the difference and the ratio of two counting rates. The influence of various confidence levels on the total time and the total counts that must be accumulated for a given precision is demonstrated. Graphical means of determining these quantities are exhibited. The relationship between the aeeuracy of an intensity measurement and the nominal accuracy of a spect~rochemical analysis is considered.
1. Introduction purposes of X-ray analysis, it is often desired to determine the net peak intensity of a line above background or some quantity that involves a combination of net peak intensities of two or more lines. The development of counter tubes has made possible the measurement of X-ray intensities with a precision that, in theory, is limited only by the amount of time available. In practice, however, there are various factors to be considered in measuring a net counting rate, n, whether by a ratemeter, fixed time, or fixed count method Cl]. Furthermore the time available for the measurements is limited by practical considerations. Thus if a large number of measurements are to be made, or if much time is involved in a measurement, it is desirable to know how best to divide the available time to obtain the greatest precision in the measurements. This problem and related statistical factors were discussed by PARRISH [l]. The purpose of this paper is to present an expanded treatment illustrated with examples and to present graphs that will facilitate calculations. It will be assumed that the X-ray emission process is random and fluctuates according to a Gaussian distribution for which the standard deviation is the square root of the mean 112,31. FOR
2.
Dete~~tion
of net peak ~te~i~
(A) General case The net peak intensity of a line is usually determined as the difference of two individually measured counting rates: n1 = peak plus background (P + B) and n2 = background (B), as shown in Fig. l(a). The net peak is but one example of the determination of any quantity A as a function of two measured counting rates, n, and n2 A = f(%
%)
(1)
The problem to be considered is: Given two counting rates, n1 and na, and a limited total time, T = f, + t,, available for their measurement, how should the 169
M. MACK and N. SPIELBERG
Fig. l(b). X-ray reflection with peak (2’) = %I and with background (B) = /In, due to overlapping “wing” of adjacent reflection.
Fig. l(a). Typical X-ray reflection with peak (I’), peak plus background (P + B), and background (B) indicated.
measurement time be divided so that a minimum relative statistical error, eA, in the quantity A may be achieved? aA is a function of the time spent on the measurements of n, and n2: aA = a_&,, t& ~(t,, tz) = T -
(2)
VI + ta) = 9
(3)
The problem is to find a minimum value of (2) subject to the restraining condition (3). This may be done by the method of LaGrange multipliers [4] which yields:
a&A
asA
at, = al,
(4)
(B) Simple differences Consider the special case for which A=d=n,-nn,
(5)
This problem has been considered in great detail by other authors [5-S, 111 but the solution and results are here presented in a somewhat different form. Moreover the results here exhibited will be used in section 3 of this paper. Let aI and Ed be the relative errors of n, and nz, respectively. The absolute errors are nlal and nZaZ. The relative error of the difference is [9]: E = [(W1)2 d
+ (W,)21”2 ml -
n2
(6)
The form of equation (6) shows that there is little point in expending the effort t’o make the absolute error in one measurement small in order to achieve a small value of E~ without also making the absolute error in the other measurement comparably small. 170
Statistical
If Q is a constant
factors in X-ray
determined
intensity measurements
by the confidence level desired [lo], then c1 = equation (6) may be rewritten as:
Q/Wl)112 andc2= &I(%)u2. Therefore Ed
=
~
Q
112
“+?
[4
121 -122
Substituting
equation
(7) into equation
(4) yields
n1 ‘I2 (-) = K112
t, <= where
1
(8)
n2
K is the ratio of the counting rates. Equations (3) and (8) give K*/2 t, =
T
1 + K1i2
It is often desirable to know how long a time. must be expended to achieve a stipulated precision. Because it is necessary to accumulate a specific total number of counts to attain a specific precision (regardless of the counting method used), the problem may be rephrased: given two counting rates, n1 and n2, how many total counts, N, = nlt, and Ni = n,t,, must be accumulated so that the relative error of the difference will have a stipulated value? By manipulation of equations (7) and (8) it can be shown that N
1
Q”~312 (K1’2+
=
N
=
-
1) _ Q2
Ed2 Q” W1’2+ 1) (K
2 ed2
(K -
1)2
1)2
~~2
Q2 -
2
z
2
z
’
(10) (11)
In Fig. 2 the values of 2, = K3/2(Kl/2 + l)/(K- 1)2 and of 2, = (K1'2+ l)/ (K - 1)2 are plotted as a function of K. In Fig. 3, N vs. Z is plotted for various confidence levels and percentage errors (E&in %). The relations 2, = K3'22,and The values of Q used for the N, = K312N2are useful in condensing calculations. 50, 90 and 99 per cent confidence levels are O-67, l-64, and 2.58, respectively. Thus for two given counting rates, the total number of counts, N, and N,, necessary for a stipulated accuracy of the difference may be determined either analytically using equations (10) and (11) or graphically using Figs. 2 and 3. Table 1 gives values of N,, N,, and T for some arbitrarily selected counting rates to illustrate certain typical situations for optimum division of counting times. It is clear from the table that to increase the confidence level from 50 per cent (probable error) to 90 per cent, the total time expended must be increased by a factor of (1*64/O-67)2 = 6. Note that in examples 1 and 4, K is unchanged and therefore the corresponding values of N, and N2 are identical. However, because the counting rates for the two cases differ by a factor of ten, the total time expended also differs by a factor of ten. Examples 7 and 8 demonstrate that for K greater than 50 only a very nominal determination of n2 is required. It is of interest to examine further the effect on the total time and on the accuracy when the time is not used in the optimum manner. Table 2 summarizes data for 171
M. MACK
and N. SPIELBERG
Fig. 3. Graphical solutions of equations (10) and (11) (see text) for various confidence levels and percentage errors. Confidence levels of 50, 90 and 99 per cent are indicated by the solid, dotted and dashed lines respectively. The percentage errors are given at the top of the chart. Note that for Z = 1, N = Q2/$.
Fig. 2. Graphical solutions of the relations .Z, = (K2 + K3ie)/(K - 1)2 and of 2, = (W/2 + l)/(K - I)Z.
Table
1. Effect
-
of selected
counting
rates on statistical
accuracy
T 1 yc accuracy
3 B
n1 -
w4
n2
P
(c/s)
nl -
K
3
50%
conf. level
90%
conf. level
nllnz
PTB
I:2
N,
N,
40,000 620 1,500 40,000
150 60 40 15
100 10 10 10 _--I_
50 50 30 5
5 :1 3 : 1 1:2
1.5 6.0 4.0 1.5
73,000 9,100 12,000 73,000
1000 500 500 1000
100 100 10 10
900 400 490 990
9:l 4:l 49 : 1 99 : 1
10.0 5.0 50.0 100.0
7,300 10,200 5,300 5,000
.-
!
T (=c)
N,
890 210 450 8900
440,000 55,000 72,000 440,000
N2 / 239,000 3,700 9,000 239,000
5300 1300 ~ 2700 i 53000
1400 5400 90 30
58 176 73 33
10 29 12 5.5
I
For (P + B) = n, = 150; B = nz = 100; From Fig. 2, 2, = 16.35; 2, = 8.90. From Fig. 3, N, = 440,000; N, = 239,000; Thus, T = t, + t, = 3
n1
+ 3
nz
172
i
-I-
230 910 15 5
44,000 61,000 32,000 30,000
Sample calculation:
Eec)
440,000 = ~ 150
K =
1.50
(1% za and 90% +
239,000 ~
100
C.L.).
= 5300 sec.
Statistical factors in X-ray intensity measurements Table 2. Effect on total time and on accuracy when time is not used in the optimum manner
such a situation for three different counting time ratios. For the data in column (A), the total time, T, is divided so that the same number of counts is accumulated in each measurement (N, = N2). This determines both n, and n2 to the same relative error. The resultant value of .zdis to be compared with the value of ed in column (B), which is obtained when the same total time is used in the optimum manner (eq. 8). Column (C) gives the total time in which the value of cd in column (A) could have been determined if the optimum time division had been used. Note that as K increases, the disparity between ed of columns (A) and (B) increases. This is to be expected, for as K increases in the first case (N, = N,) a greater proportion of the time is spent on the lower counting rate while in the second case (opt’ lmum time) a greater proportion of the time is spent on the higher counting rate. (C) Overlapping lines Occasionally the expression for net peak takes a slightly that of equation (5). For example, the line whose intensity may be situated on the “wing” of another line, and hence the have the constant value assumed in Fig. 1 (a) and equation situation may be represented by Fig. 1 (b), for which A = n, -
#?n,
different form than is to be determined background will not (5); but rather the (12)
where n2 is the counting rate at the peak of the other line and pn, is the value of the background at the position where n, is measured. The quantity 8, which is determined from the spectral line shape, is a function of the instrument and is known beforehand. &A
or
=
%
k2[(wJ2
+ (Bn2E2)211/2
(13)
(14) Substitution
of (14) in equation 4
<=j
where
(4) yields 1
n,
l/2
C-1 n2
=
G2
K’=F 122
173
K'1/2
(15)
B
(16)
Manipulation
of equations N, =
N
(14) and (15) yields
Q2 &2 K's/2 ,91/2(K'/p)"' + 1 = Elp22'1 (K' _ 1)2 &g2
2
_
Q2 (K'/p)1'2 + 1 _ &g2
(K' -
1)2
Q2 z, &42 2
(17)
(18)
It is possible then to plot graphs, analogous to Fig. 2, of Z’, and Z’, as functions of K’ for various values of the parameter /l. As an example, when n1 = 500 c/s, /?n, = 100 c/s and @ = 0.1, it may be calculated that for a, = 1 per cent at the 50 per cent confidence level, the values of N, and N, are 8200 and 2200 respectively, with a total time expenditure of 18 sec. Equation (12) assumes only small overlap, with no contribution to n2 from the overlap of the line represented by nl. If the degree of overlap is such that n2 must be corrected, equation (12) must be divided by 1 - a/?, where a is the overlap factor from the first line. However, the results given in equations (13) to (18) will be unaltered because (1 - a/?) is a constant multiplier of A and will not affect the relative error.
3. Combination of net peak intensities A number of net peak intensities may be combined together to determine some particular quantity of interest. For example, the analyst may be interested in the determination of the ratio of two line intensities. The problem to be considered is: given a quantity X = F (A, B, C, . . . )
(19)
whereA, B,C,. . . represent individually determined line intensities, each requiring the measurement of two rates (as in Section 2) and a limited total time available for the overall measurement, how should the total available time be distributed among the various ni, so that a minimum relative statistical error sx in the quantity X may be achieved? If there are M quantities A, B, C, . . . there will be 2M individual n, and 2M individual ti &_Tf= 9 (&A, &B, &C * * * ) 91(tt) = T -
(20)
(ti + t2 + t, + t, + t, + t, + * * * 1 = 0
[5] has shown how to derive explicit forms of ax. The method of LaGrange multipliers yields the following
(21)
JARRETT
a&A
a&A
a&, -=-
a&,
at,
at,
set of equations:
at, _- at, 3% -=at,
174
a+ at,
(22)
Statistical factors in X-ray intensity measurements
and
i-37aEB a&, at,
67 asA --_=-a&, at,
(23)
aga&, a9 a&, a&,at, =&K
There will be M equations of the type (22) and (M - 1) equations of the type (23). These equations plus equation (21) give sufficient equations to determine the 2M ti’s. Equation (21) may be rewritten: [T -
(t3 + t, + t, + t, . . . )] -
t, -
t, = 0
(24)
The quantity in brackets is independent of t, and t,. Therefore (24) has the same form as (3). The first of the equations (22) is identical with (4); therefore it is possible to determine the relationship between t, and t, as if they were independent of the other ti, and similarly for the other pairs of t,. This reasoning verifies the intuitive expectation that the time allotted to each of the quantities A, B, C, . . . should be used in the optimum manner that would be found considering each quantity independently. Consider the special case
(25) The relative error in X is Ex = .z7= [&A2 + Eg2)1’2
I
and T -
(tr + t, + t, + t4) = T
Assume that
Therefore,
A = n, -
n2
(27)
B = n3 -
n4
(23)
&A and cB may be found from equation
(7).
Equations
=
K,l’2
zz
Equations
K,
(22) result in
(29) 112
(30)
(23) yield lJ2 n3
(z2)
4
n1
g= Combining A and B:
G-33)
- PA + tB) = 0
-
n4
nl -
(29), (30), and (31) yields the division t, + t, t,-_=__= tB
t,
+
t,
n31t2 -
(31)
n2 of measurement
na1f2
n1112- n2112 175
time between
(32)
and??.
M.MACK
SPIELBERG
Equations (29) and (30) are the same as equation (8). Therefore, as already pointed out, once the division of measurement time between A and B has been determined, the determination of A and B is carried out as prescribed in Section 2. It is first necessary to determine eA and Ed from the stipulated E, and the known approximate values of n,, na, n3, n4. Combining equations (7), (26), (29), (30), (31), and (32) gives
&A2 -zz EB2
Substitution
in equation
n3 n1
l/2
l/2
n4
l/2 -
l/2 =
tA g
(33)
n2
(26) gives (34)
(35) Thus for given Ed, n,, n2, nn3and n4, eA and Ed are computed from equations (34) and (35) and N,, N,, and Na, N, are found from Figs. 2 and 3 or equations (S), (10) and (11). For example, if n,, n2, n3, n4 equal 550, 100, 150 and 100 c/s respectively, for E, = 1 per cent at the 50 per cent confidence level, the values of N,, N,, N, and N, will be 67,000, 6000, 86,000 and 47,000 respectively, with a total time expenditure of 2000 sec. If n4 and n2 may be neglected by comparison to n3 and nl, equations (33), (34), (35) and (10) reduce to
&A
=
&A2 -_=__=
tA
&B2
tI3
&,[I
+
123 (
112
n1 1
(nl/n2)1/2]-1/2
cB = E, [l + (n&1)1/2]-1’2
(36) (37) (33)
N, = Q2/~42
(39)
N3 = Q2/~B2
(40)
The derivation given above is not rigorous, because the quantity l/B does not have a Gaussian distribution. However, if Bt > 400 its distribution is close enough to Gaussian for these purposes. If Bt < 400, eB is already large enough that there is little point in discussing relative errors at a given confidence level, and one might just as well deal with the r.m.s. deviation from the mean. In such a case, even for Bt = 100 the r.m.s. deviation of l/B from the mean is not significantly different from the r.m.s. deviation of B from the mean (12 per cent compared to 10 per cent).
4. Use of statistical errors in estimating chemical composition In X-ray analysis, the composition of a given specimen is sought by means of the interpretation of characteristic line intensities. The relative error in the percentage composition is not equal to the relative error in the characteristic line 176
Statistical factors in X-ray intensity measurements
intensity measurement unless the composition is proportional to the line intensity. In Fig. 4, A and B represent two calibration curves of intensity (counts/see) vs. Curve A would result for “i” in a low absorption concentration of element “i”. matrix; curve B for “i” in a high absorption matrix. At point p of curve A, an error in the intensity, E, corresponds to an approximately equal error in concentration. However, at point r, a small relative error in intensity results in a larger
-Y.
CONCENTRATION
Fig. 4. Typical calibration curves of intensity (counting rate) vs. concentration.
At point q of curve B, an intensity error corresponds relative error in concentration. to an approximately equal error in concentration, while at point s a small relative error in intensity results in an even smaller relative error in concentration. It may be shown that the ratio of the relative error in intensity to the relative error in concentration at a point x is directly proportional to the ratio of the slope of the curve at the point x to the slope of the line from the origin to the point x. (Note that linear scales are assumed for the calibration curves.) This consideration assumes the greatest importance in the upper concentration region of the curve. It applies regardless of whether the intensity error is due to counting rate statistics alone or in combination with other factors. As a note of caution, it cannot be emphasized too strongly that discussions of counting statistics in a given application are of limited value if other errors (random or systematic) that result from the experimental arrangements are not considered. It is true that significant deviations of the counting statistics from a Gaussian distribution whose standard deviation is the square root of the mean reflect some of the experimental errors [Z]. However, it does not follow from the lack of such deviations that all errors have been eliminated, for there are other errors, such as systematic errors, which do not affect the distribution.
5. Conclusions (1) In X-ray
analyses that require two or more intensity measurements, it is often desirable to divide the available measuring time among the measurements in such a manner as to give a minimum error in the quantity of interest. (2) When the quantity of interest requires the determination of two or more net line intensities in the presence of varying backgrounds, the optimum division 177
M. MACK and N. SPIELBERQ:Statistical factors in X-ray intensity measurements
of the total time may be found by determining the optimum amount of time to be allotted to each individual net intensity measurement and then optimizing independently the use of the time for each individual intensity measurement. (3) For the difference of two counting rates the optimum division of the available counting time results when the ratio of the counting times is directly proportional to the ratio of the square roots of the counting rates. (4) For the ratio of two counting rates, the optimum division of the available counting time results when the ratio of the counting times is inversely proportional to the ratio of the square roots of the counting rates. Acknowledgemente-We are indebted to Dr. W. PARRISH, who suggested this problem and reviewed the manuscript, to Mr. T. KOHLER, who made several suggestions, and to several of our colleagues who reviewed the manuscript.
References [I] [2] [3] [4] [5]
[6] [7] [S] [9]
;lO] -111
PARRISH W. Philips Tech. Rev. 1956 17 206. LIEBHAFSKY H. A. PFEIFFER H. G. and ZEMANY P. D. Andyt. Chem. 1955 27 1257. RAINWATER L. J. and WV C. S. iVucZeonic8 1947 1 No. 2 60. SOKOLNIKOFFI. S. and SOKOLNIKOFFE. S. Higher Mathematics for Engineers and Physicirrts (2nd Ed) pp. 163-167. McGraw-Hill, New York 1941. JARRETT A. A. Statistical Measurenzents Used in the Measurement of Radioactivity. AECU262, MonP-126, U.S. Atomic Energy Commission Technical Information Division, Oak Ridge, Tennessee 1946. LOEVINUER R. and BERMAN M. Nucleon&s 1951 9 No. 1 26. BRO~~NO W. E. JR. NucIeonics 1951 9 No. 3 62. FREEDMAN A. J. and ANDERSON E. C. Nucleonics 1952 10 No. 8 57. BEERS Y. Introduction to the Theory of Error pp. 33-34. Addison-Wesley, New York 1953. to Mathematical Statistics pp. 32, 130. Wiley, New York 1947. HOEL P. G. Introduction HORTON W. S. AppZ. Spectrosc. 1955 9 173.
178
Statistical factors in X-ray intensity meMurements M. MACK and N. SPIELBERG Philips Laboratories, Irvin&n-on-IIudson, Hew York, U.S.A.
Abstract-In X.-ray intensity measurements, statistical factors result from the inherent randomness of the X-ray excitation process. The optimum division of counting times for given counting rates is discussed for various combinations of counting rates, with particular attention to the difference and the ratio of two counting rates. The influence of various confidence levels on the total time and the total counts that must be accumulated for a given precision is demonstrated. Graphical means of determining these quantities are exhibited. The relationship between the aeeuracy of an intensity measurement and the nominal accuracy of a spect~rochemical analysis is considered.
1. Introduction purposes of X-ray analysis, it is often desired to determine the net peak intensity of a line above background or some quantity that involves a combination of net peak intensities of two or more lines. The development of counter tubes has made possible the measurement of X-ray intensities with a precision that, in theory, is limited only by the amount of time available. In practice, however, there are various factors to be considered in measuring a net counting rate, n, whether by a ratemeter, fixed time, or fixed count method Cl]. Furthermore the time available for the measurements is limited by practical considerations. Thus if a large number of measurements are to be made, or if much time is involved in a measurement, it is desirable to know how best to divide the available time to obtain the greatest precision in the measurements. This problem and related statistical factors were discussed by PARRISH [l]. The purpose of this paper is to present an expanded treatment illustrated with examples and to present graphs that will facilitate calculations. It will be assumed that the X-ray emission process is random and fluctuates according to a Gaussian distribution for which the standard deviation is the square root of the mean 112,31. FOR
2.
Dete~~tion
of net peak ~te~i~
(A) General case The net peak intensity of a line is usually determined as the difference of two individually measured counting rates: n1 = peak plus background (P + B) and n2 = background (B), as shown in Fig. l(a). The net peak is but one example of the determination of any quantity A as a function of two measured counting rates, n, and n2 A = f(%
%)
(1)
The problem to be considered is: Given two counting rates, n1 and na, and a limited total time, T = f, + t,, available for their measurement, how should the 169
M. MACK and N. SPIELBERG
Fig. l(b). X-ray reflection with peak (2’) = %I and with background (B) = /In, due to overlapping “wing” of adjacent reflection.
Fig. l(a). Typical X-ray reflection with peak (I’), peak plus background (P + B), and background (B) indicated.
measurement time be divided so that a minimum relative statistical error, eA, in the quantity A may be achieved? aA is a function of the time spent on the measurements of n, and n2: aA = a_&,, t& ~(t,, tz) = T -
(2)
VI + ta) = 9
(3)
The problem is to find a minimum value of (2) subject to the restraining condition (3). This may be done by the method of LaGrange multipliers [4] which yields:
a&A
asA
at, = al,
(4)
(B) Simple differences Consider the special case for which A=d=n,-nn,
(5)
This problem has been considered in great detail by other authors [5-S, 111 but the solution and results are here presented in a somewhat different form. Moreover the results here exhibited will be used in section 3 of this paper. Let aI and Ed be the relative errors of n, and nz, respectively. The absolute errors are nlal and nZaZ. The relative error of the difference is [9]: E = [(W1)2 d
+ (W,)21”2 ml -
n2
(6)
The form of equation (6) shows that there is little point in expending the effort t’o make the absolute error in one measurement small in order to achieve a small value of E~ without also making the absolute error in the other measurement comparably small. 170
Statistical
If Q is a constant
factors in X-ray
determined
intensity measurements
by the confidence level desired [lo], then c1 = equation (6) may be rewritten as:
Q/Wl)112 andc2= &I(%)u2. Therefore Ed
=
~
Q
112
“+?
[4
121 -122
Substituting
equation
(7) into equation
(4) yields
n1 ‘I2 (-) = K112
t, <= where
1
(8)
n2
K is the ratio of the counting rates. Equations (3) and (8) give K*/2 t, =
T
1 + K1i2
It is often desirable to know how long a time. must be expended to achieve a stipulated precision. Because it is necessary to accumulate a specific total number of counts to attain a specific precision (regardless of the counting method used), the problem may be rephrased: given two counting rates, n1 and n2, how many total counts, N, = nlt, and Ni = n,t,, must be accumulated so that the relative error of the difference will have a stipulated value? By manipulation of equations (7) and (8) it can be shown that N
1
Q”~312 (K1’2+
=
N
=
-
1) _ Q2
Ed2 Q” W1’2+ 1) (K
2 ed2
(K -
1)2
1)2
~~2
Q2 -
2
z
2
z
’
(10) (11)
In Fig. 2 the values of 2, = K3/2(Kl/2 + l)/(K- 1)2 and of 2, = (K1'2+ l)/ (K - 1)2 are plotted as a function of K. In Fig. 3, N vs. Z is plotted for various confidence levels and percentage errors (E&in %). The relations 2, = K3'22,and The values of Q used for the N, = K312N2are useful in condensing calculations. 50, 90 and 99 per cent confidence levels are O-67, l-64, and 2.58, respectively. Thus for two given counting rates, the total number of counts, N, and N,, necessary for a stipulated accuracy of the difference may be determined either analytically using equations (10) and (11) or graphically using Figs. 2 and 3. Table 1 gives values of N,, N,, and T for some arbitrarily selected counting rates to illustrate certain typical situations for optimum division of counting times. It is clear from the table that to increase the confidence level from 50 per cent (probable error) to 90 per cent, the total time expended must be increased by a factor of (1*64/O-67)2 = 6. Note that in examples 1 and 4, K is unchanged and therefore the corresponding values of N, and N2 are identical. However, because the counting rates for the two cases differ by a factor of ten, the total time expended also differs by a factor of ten. Examples 7 and 8 demonstrate that for K greater than 50 only a very nominal determination of n2 is required. It is of interest to examine further the effect on the total time and on the accuracy when the time is not used in the optimum manner. Table 2 summarizes data for 171
M. MACK
and N. SPIELBERG
Fig. 3. Graphical solutions of equations (10) and (11) (see text) for various confidence levels and percentage errors. Confidence levels of 50, 90 and 99 per cent are indicated by the solid, dotted and dashed lines respectively. The percentage errors are given at the top of the chart. Note that for Z = 1, N = Q2/$.
Fig. 2. Graphical solutions of the relations .Z, = (K2 + K3ie)/(K - 1)2 and of 2, = (W/2 + l)/(K - I)Z.
Table
1. Effect
-
of selected
counting
rates on statistical
accuracy
T 1 yc accuracy
3 B
n1 -
w4
n2
P
(c/s)
nl -
K
3
50%
conf. level
90%
conf. level
nllnz
PTB
I:2
N,
N,
40,000 620 1,500 40,000
150 60 40 15
100 10 10 10 _--I_
50 50 30 5
5 :1 3 : 1 1:2
1.5 6.0 4.0 1.5
73,000 9,100 12,000 73,000
1000 500 500 1000
100 100 10 10
900 400 490 990
9:l 4:l 49 : 1 99 : 1
10.0 5.0 50.0 100.0
7,300 10,200 5,300 5,000
.-
!
T (=c)
N,
890 210 450 8900
440,000 55,000 72,000 440,000
N2 / 239,000 3,700 9,000 239,000
5300 1300 ~ 2700 i 53000
1400 5400 90 30
58 176 73 33
10 29 12 5.5
I
For (P + B) = n, = 150; B = nz = 100; From Fig. 2, 2, = 16.35; 2, = 8.90. From Fig. 3, N, = 440,000; N, = 239,000; Thus, T = t, + t, = 3
n1
+ 3
nz
172
i
-I-
230 910 15 5
44,000 61,000 32,000 30,000
Sample calculation:
Eec)
440,000 = ~ 150
K =
1.50
(1% za and 90% +
239,000 ~
100
C.L.).
= 5300 sec.
Statistical factors in X-ray intensity measurements Table 2. Effect on total time and on accuracy when time is not used in the optimum manner
such a situation for three different counting time ratios. For the data in column (A), the total time, T, is divided so that the same number of counts is accumulated in each measurement (N, = N2). This determines both n, and n2 to the same relative error. The resultant value of .zdis to be compared with the value of ed in column (B), which is obtained when the same total time is used in the optimum manner (eq. 8). Column (C) gives the total time in which the value of cd in column (A) could have been determined if the optimum time division had been used. Note that as K increases, the disparity between ed of columns (A) and (B) increases. This is to be expected, for as K increases in the first case (N, = N,) a greater proportion of the time is spent on the lower counting rate while in the second case (opt’ lmum time) a greater proportion of the time is spent on the higher counting rate. (C) Overlapping lines Occasionally the expression for net peak takes a slightly that of equation (5). For example, the line whose intensity may be situated on the “wing” of another line, and hence the have the constant value assumed in Fig. 1 (a) and equation situation may be represented by Fig. 1 (b), for which A = n, -
#?n,
different form than is to be determined background will not (5); but rather the (12)
where n2 is the counting rate at the peak of the other line and pn, is the value of the background at the position where n, is measured. The quantity 8, which is determined from the spectral line shape, is a function of the instrument and is known beforehand. &A
or
=
%
k2[(wJ2
+ (Bn2E2)211/2
(13)
(14) Substitution
of (14) in equation 4
<=j
where
(4) yields 1
n,
l/2
C-1 n2
=
G2
K’=F 122
173
K'1/2
(15)
B
(16)
Manipulation
of equations N, =
N
(14) and (15) yields
Q2 &2 K's/2 ,91/2(K'/p)"' + 1 = Elp22'1 (K' _ 1)2 &g2
2
_
Q2 (K'/p)1'2 + 1 _ &g2
(K' -
1)2
Q2 z, &42 2
(17)
(18)
It is possible then to plot graphs, analogous to Fig. 2, of Z’, and Z’, as functions of K’ for various values of the parameter /l. As an example, when n1 = 500 c/s, /?n, = 100 c/s and @ = 0.1, it may be calculated that for a, = 1 per cent at the 50 per cent confidence level, the values of N, and N, are 8200 and 2200 respectively, with a total time expenditure of 18 sec. Equation (12) assumes only small overlap, with no contribution to n2 from the overlap of the line represented by nl. If the degree of overlap is such that n2 must be corrected, equation (12) must be divided by 1 - a/?, where a is the overlap factor from the first line. However, the results given in equations (13) to (18) will be unaltered because (1 - a/?) is a constant multiplier of A and will not affect the relative error.
3. Combination of net peak intensities A number of net peak intensities may be combined together to determine some particular quantity of interest. For example, the analyst may be interested in the determination of the ratio of two line intensities. The problem to be considered is: given a quantity X = F (A, B, C, . . . )
(19)
whereA, B,C,. . . represent individually determined line intensities, each requiring the measurement of two rates (as in Section 2) and a limited total time available for the overall measurement, how should the total available time be distributed among the various ni, so that a minimum relative statistical error sx in the quantity X may be achieved? If there are M quantities A, B, C, . . . there will be 2M individual n, and 2M individual ti &_Tf= 9 (&A, &B, &C * * * ) 91(tt) = T -
(20)
(ti + t2 + t, + t, + t, + t, + * * * 1 = 0
[5] has shown how to derive explicit forms of ax. The method of LaGrange multipliers yields the following
(21)
JARRETT
a&A
a&A
a&, -=-
a&,
at,
at,
set of equations:
at, _- at, 3% -=at,
174
a+ at,
(22)
Statistical factors in X-ray intensity measurements
and
i-37aEB a&, at,
67 asA --_=-a&, at,
(23)
aga&, a9 a&, a&,at, =&K
There will be M equations of the type (22) and (M - 1) equations of the type (23). These equations plus equation (21) give sufficient equations to determine the 2M ti’s. Equation (21) may be rewritten: [T -
(t3 + t, + t, + t, . . . )] -
t, -
t, = 0
(24)
The quantity in brackets is independent of t, and t,. Therefore (24) has the same form as (3). The first of the equations (22) is identical with (4); therefore it is possible to determine the relationship between t, and t, as if they were independent of the other ti, and similarly for the other pairs of t,. This reasoning verifies the intuitive expectation that the time allotted to each of the quantities A, B, C, . . . should be used in the optimum manner that would be found considering each quantity independently. Consider the special case
(25) The relative error in X is Ex = .z7= [&A2 + Eg2)1’2
I
and T -
(tr + t, + t, + t4) = T
Assume that
Therefore,
A = n, -
n2
(27)
B = n3 -
n4
(23)
&A and cB may be found from equation
(7).
Equations
=
K,l’2
zz
Equations
K,
(22) result in
(29) 112
(30)
(23) yield lJ2 n3
(z2)
4
n1
g= Combining A and B:
G-33)
- PA + tB) = 0
-
n4
nl -
(29), (30), and (31) yields the division t, + t, t,-_=__= tB
t,
+
t,
n31t2 -
(31)
n2 of measurement
na1f2
n1112- n2112 175
time between
(32)
and??.
M.MACK
SPIELBERG
Equations (29) and (30) are the same as equation (8). Therefore, as already pointed out, once the division of measurement time between A and B has been determined, the determination of A and B is carried out as prescribed in Section 2. It is first necessary to determine eA and Ed from the stipulated E, and the known approximate values of n,, na, n3, n4. Combining equations (7), (26), (29), (30), (31), and (32) gives
&A2 -zz EB2
Substitution
in equation
n3 n1
l/2
l/2
n4
l/2 -
l/2 =
tA g
(33)
n2
(26) gives (34)
(35) Thus for given Ed, n,, n2, nn3and n4, eA and Ed are computed from equations (34) and (35) and N,, N,, and Na, N, are found from Figs. 2 and 3 or equations (S), (10) and (11). For example, if n,, n2, n3, n4 equal 550, 100, 150 and 100 c/s respectively, for E, = 1 per cent at the 50 per cent confidence level, the values of N,, N,, N, and N, will be 67,000, 6000, 86,000 and 47,000 respectively, with a total time expenditure of 2000 sec. If n4 and n2 may be neglected by comparison to n3 and nl, equations (33), (34), (35) and (10) reduce to
&A
=
&A2 -_=__=
tA
&B2
tI3
&,[I
+
123 (
112
n1 1
(nl/n2)1/2]-1/2
cB = E, [l + (n&1)1/2]-1’2
(36) (37) (33)
N, = Q2/~42
(39)
N3 = Q2/~B2
(40)
The derivation given above is not rigorous, because the quantity l/B does not have a Gaussian distribution. However, if Bt > 400 its distribution is close enough to Gaussian for these purposes. If Bt < 400, eB is already large enough that there is little point in discussing relative errors at a given confidence level, and one might just as well deal with the r.m.s. deviation from the mean. In such a case, even for Bt = 100 the r.m.s. deviation of l/B from the mean is not significantly different from the r.m.s. deviation of B from the mean (12 per cent compared to 10 per cent).
4. Use of statistical errors in estimating chemical composition In X-ray analysis, the composition of a given specimen is sought by means of the interpretation of characteristic line intensities. The relative error in the percentage composition is not equal to the relative error in the characteristic line 176
Statistical factors in X-ray intensity measurements
intensity measurement unless the composition is proportional to the line intensity. In Fig. 4, A and B represent two calibration curves of intensity (counts/see) vs. Curve A would result for “i” in a low absorption concentration of element “i”. matrix; curve B for “i” in a high absorption matrix. At point p of curve A, an error in the intensity, E, corresponds to an approximately equal error in concentration. However, at point r, a small relative error in intensity results in a larger
-Y.
CONCENTRATION
Fig. 4. Typical calibration curves of intensity (counting rate) vs. concentration.
At point q of curve B, an intensity error corresponds relative error in concentration. to an approximately equal error in concentration, while at point s a small relative error in intensity results in an even smaller relative error in concentration. It may be shown that the ratio of the relative error in intensity to the relative error in concentration at a point x is directly proportional to the ratio of the slope of the curve at the point x to the slope of the line from the origin to the point x. (Note that linear scales are assumed for the calibration curves.) This consideration assumes the greatest importance in the upper concentration region of the curve. It applies regardless of whether the intensity error is due to counting rate statistics alone or in combination with other factors. As a note of caution, it cannot be emphasized too strongly that discussions of counting statistics in a given application are of limited value if other errors (random or systematic) that result from the experimental arrangements are not considered. It is true that significant deviations of the counting statistics from a Gaussian distribution whose standard deviation is the square root of the mean reflect some of the experimental errors [Z]. However, it does not follow from the lack of such deviations that all errors have been eliminated, for there are other errors, such as systematic errors, which do not affect the distribution.
5. Conclusions (1) In X-ray
analyses that require two or more intensity measurements, it is often desirable to divide the available measuring time among the measurements in such a manner as to give a minimum error in the quantity of interest. (2) When the quantity of interest requires the determination of two or more net line intensities in the presence of varying backgrounds, the optimum division 177
M. MACK and N. SPIELBERQ:Statistical factors in X-ray intensity measurements
of the total time may be found by determining the optimum amount of time to be allotted to each individual net intensity measurement and then optimizing independently the use of the time for each individual intensity measurement. (3) For the difference of two counting rates the optimum division of the available counting time results when the ratio of the counting times is directly proportional to the ratio of the square roots of the counting rates. (4) For the ratio of two counting rates, the optimum division of the available counting time results when the ratio of the counting times is inversely proportional to the ratio of the square roots of the counting rates. Acknowledgemente-We are indebted to Dr. W. PARRISH, who suggested this problem and reviewed the manuscript, to Mr. T. KOHLER, who made several suggestions, and to several of our colleagues who reviewed the manuscript.
References [I] [2] [3] [4] [5]
[6] [7] [S] [9]
;lO] -111
PARRISH W. Philips Tech. Rev. 1956 17 206. LIEBHAFSKY H. A. PFEIFFER H. G. and ZEMANY P. D. Andyt. Chem. 1955 27 1257. RAINWATER L. J. and WV C. S. iVucZeonic8 1947 1 No. 2 60. SOKOLNIKOFFI. S. and SOKOLNIKOFFE. S. Higher Mathematics for Engineers and Physicirrts (2nd Ed) pp. 163-167. McGraw-Hill, New York 1941. JARRETT A. A. Statistical Measurenzents Used in the Measurement of Radioactivity. AECU262, MonP-126, U.S. Atomic Energy Commission Technical Information Division, Oak Ridge, Tennessee 1946. LOEVINUER R. and BERMAN M. Nucleon&s 1951 9 No. 1 26. BRO~~NO W. E. JR. NucIeonics 1951 9 No. 3 62. FREEDMAN A. J. and ANDERSON E. C. Nucleonics 1952 10 No. 8 57. BEERS Y. Introduction to the Theory of Error pp. 33-34. Addison-Wesley, New York 1953. to Mathematical Statistics pp. 32, 130. Wiley, New York 1947. HOEL P. G. Introduction HORTON W. S. AppZ. Spectrosc. 1955 9 173.
178