Simplified Procedure for Computing the Growth of Radioactive Decay Produds H. W. KIRBY and D. A. KREMER M o u n d Laboratory, Monsanto Chemical Co., Miamisburg, O h i o
-4 method is described for easing the burden of computing equations for the growth of radioactive decay products. A sample table and a sample calculation are included.
C
c,
active materials is a fairly simple matter when the decay chain is limited to two or three members. Equations for such calculations were originally proposed by Rutherford (Y),and can be found in textbooks of radiochemistry ( 2 , 4). When the decay chain is long, however, as in the radium, thorium, actinium, and neptunium series, the symmetrical expression derived by Bateman ( I ) is customarily used. The general form of a Bateman equation is
c, =
nJn=
-l
xj
-
xi)
j = l
(xi
Tl-T
1, ~
Tt - T ,
3 = -2
( Tn-2 T ;)
(3)
has the interesting property that, in
practice, it is usually either slightly greater than unity or slightly less than zero. This property makes it possible to evaluate many Bateman coefficients by inspection, for if any C, contains
at least one 2 I" which is approyimately equal to zero, then T , - TI that C, may be taken as equal to zero I n such cases, C,e-'St converges rapidly to eero and can be neglected for most practical purposes. I n the special case where the nth decay product is stable, a slightly different procedure is followed, since Equation 3 is not valid when one of the half lives is infinit?. I n this case, Tn is treated as a very large finite nuniher, and Equations 3 are rearranged as follows:
where s , ( t ) is the number of atoms of the nth member of the decay chain at time t, Nl(0) is the number of atoms of the parent a t t = 0, and Xi, the decay constant, is numerically equal to In 2 divided by the half life, Ti,of the ith member of the chain. Ci is a constant having the form
nn
(k) (+)
The function
ALCULATION of the growth of decay products from radio-
xi
=
(i
= 2, 3,
, ,
., 11
- 1) (4)
; ( j # i ; i = 2 , 3 , . . . ' ,n - 1 )
The present communication is ahstracted from a report (6) I"' for every possible T, Tj combination of two radionuclides in the four decay series. Table I is a portion of one of these tables. The nuclear data were taken from the table of isotopes compiled by Hollander et al. ( 5 ) . I n each case, the prrferrrd (first listed) half-life value was used. The follov ing sample calculation illustrates the use of the tables: Required: the equation of growth of radon-220 from thorium-282.
which includes tables of the function
When n is large, evaluation of C, may become tedious. Mechanical errors which are difficult to detect are not unconinion because of the number of operations involved. Flanagan and Senftle ( 3 )have made available tables from which the numerator and denominator of C, can be evaluated. Use of these tables affords a substantial saving in the time and number of operations required However, the authors have found that, by algebraic manipulation, Bateman coefficients can be expressed in a modified form, evaluation of which can, in some cases, be carried out by inspection. Equations 2 can be transformed as follon-s:
(jzi,i = 2, 3,
, , ,
TI
= 1.24217417
x
10-16
k l = 1.00000000 (by inspwtion of column under ThZ3*) ( - 1 ) (1.00010439) (1.39583333) (1.00118968) X KZ ( 1.00000026) = - 1 .?,!BO5897 (from column under RazzS)
, , n - 1)
~~
Table I.
1 . 3 9 X 1010 years 6 . 7 years 6 . 1 3 hours 1 . 9 0 years 3 . 6 4 days 5 4 . 5 seconds
0.00000000
TI1282
Ra228 A8228 Th228
Razz4 Rn22o
TA Thorium Series (4n) T A - TB'
-e
1.00000000 1.00000000 1.00000000 1.00000000 1.00000000
1 ,00010439 1.39583333 1 00148968 1 00000026
0.00000000 -0.00010439
0.00000000 -0.39583333 1.00036819
-0.00036819 -0.07546473 1.00247575
0.00000000 -0.00148968 1.07546473 - 0 00327292
1 .00i27292 1.00000091
I .'do017332
298
'
0 oooo0ooo - 0 00000026 -0.00247575 - 0 00000091 - 0 00017332
...
V O L U M E 27, NO, 2, F E B R U A R Y 1 9 5 5 kl = ( - 1 ) ( - 0 00010439) (-0 00036819) (-0 07516473) (1.00245575) = 0.00000000 (from column under *4czz8) = (4 X lo-*) (8 X or, by inspection, ( 1 X 3.2 x 10-9 o.oooooooo kr = ( -1) ( -0.39583333) (1.00036819) (1.00527202) (1.0oooo091) = 0.39806740 (from column under T h z B ) ks = ( - 1 ) (-0.00148968) (l.Oi546473) (-0.00527292) (1.00015332) = -0.00000845 (from column under Razz4) k8 = ( - 1 ) ( - 3 x 10-7) ( - 2 x 10-6) ( - 9 x 10-7) ( - 2 x 10-4) g ~ . ~ ~ O O(from O ~ column O under RnzZO) Therefore,
n’,(o)=
1.24247417 X
(e-X:t
-
1.39805897 ecAnt f
0.39806740 e - i l t - 0.00000845 C - A J ~
299 LITERATURE CITED
(1) Bateman, H., Proc. Cambridge Phil. Soc., 15, 423 (1910). (2) Cork. J. M.,“Radioactivity and Nuclear Physics ” 2nd ed., p. 27, Van Nostrand, Kew York, 1950. 26, 1595 (1954). (3) Flanagan, F. J., and Senftle, F. E., ANAL.CHEM., (4) Friedlander, G.. and Kennedy, J. W., “Introduction to Radiochemistrv.” DD. 109-10. Wilev. New York. 1949. (5) Hollander,”J. ii.,Perlman, I.,”and Seaborg, G. T., Rets. M o d . Phys., 25, 469-651 (1953). (6) Kirby, H. W., and Kremer, D. A . , “Simplified Procedure for Computing the Growth of Radioactive Decay Products,” Office of Technical Services, Department of Commerce, mashington 25, D. C., MLM-970, U. S. Atomic Energy Commission, May 10, 1954. (7) Rutherford, E., Chadwick, J., and Ellis, C. D., “Radiations from Radioactive Substances,” reprint ed., p. 13, Cambridge University Press, London, 1951. RECEIVED for relieu, J u n e 14, 1954. Accepted October 14, 1954. hlound Laboratory is operated by Monsanto Chemical Co for t h e United States Atomic Energy Commission under Contract Number AT-33-1-GES-23.
The Rank Correlation Method JOHN T. LITCHFIELD, JR., and FRANK WILCOXON Stamford Research Laboratories a n d Lederle Laboratories Division, American Cyanamid Co., Stamford, Conn., a n d Pearl River, N. Y.
The usefulness of the rank correlation method has been increased by providing a table of critical totals for two levels of probability and a nomograph which gives the rank correlation coefficient directlJ Use of the method is illustrated.
.
C
O R R F L I T I O S or association between two variables is
widely used RS a basis for product control and evaluation. Frequently it is difficult, time-consuming and, hence, C o d y to measure directly that property of a product which is of major interest. In such cases, search is made for some other property which can l)e measured quickly, easily, inexpensively, and which is closely related to the propert’y of major interest. Generally, a number of properties are esamined and that one showing the best correlation is selected. -1rapid test for correlat,ion is very useful, as it circumvents uiirieressary computations of correlation coefficients in cases where there is poor or insignificant correlation. The Spearman rank correlation method (4,6, 9) is generally satisfactor>- for thij purpose, and like other quick methods for detecting association ( 4 ) 7 ) is nonparametric-i.e., t,he normality of the distrihution sampled need not be assumed. However, the tables of critic:il totals given by Olds ( 6 ) , Bendall (4)) and Dixon and 3Iassey ( 1 ; : r e brief or incomplete. Furthermore, the values given by Dixon and 3Iaseey are for a probability of 0.05 and 0.01, only if the experimenter is interested in a one-sided test. I n many cases one is interested in the probability of exceeding the critical value of the coefficient in a or - direction, and wishes to work a t the 5 or 1% level of Fignificance. In order to facilitate use of the rauk correlation method. a table of critical totals of squared rank differences and a nomograph which permits direct reading of the rank correlation coefficient were prepared covering 6 t o 40 pairs of observations and two probability levels, 0.05 and 0.01. These probability levels are for the case described above in which the test is made for presence of correlation whether posit’ive or negative, and if in this test an observed total falls &hin the critical limits, the conclusion will be reached that the correlation may \vel1 be zero-Le., there is no correlation.
+
CO\IPUTATIOUS
The Spearman rank correlation coefficient, designated as has limits of + l to -1, and is obtained by solution of:
p,
where R.D. = difference between paired ranks, n = number of pairs, and Z = summation over n values of (R.D.jZ. Iiendall (4)calculated the exact, distribution of the sum of squared rank differences for values of n from 1 through 8 and found considerable departure from normalit,y, particularly in the tails of the curve. He showed that for values of n greater than 8 and less than 21, a good approximat,ion [vas given by making use of Student’s t distribution as: n - 2 ‘ = p t ; l l ; P ?
Substituting the espression for Z(R.D.)*gives:
p
(Equation 1 ) and solving for
Using the values of 1 for p = 0.05 or 0.01, and n-2 degrees of freedom, this equation v a s solved for the lower and upper critical totals of squarcd rank differences for all values of n from 8 through 40. Values of f were taken from the table in Snedecor ( 8 ) , except in the case of t for 31 through 34 and 36 through 38 degrees of freedom, where graphic interpolation was used. The accuracy of the interpolated figures was equal to that of the tabular values. I n Table I, the lower and upper critical totals of squared r m k differences for probabilities of 0.05 and 0.01 are given for n = 5 to n = 40. The values for n from 5 through 8 were taken from I
