Page 252 - Radiochemistry and nuclear chemistry

236 Radiochemistry and Nuclear Chemistry

followed by a decay within the time interval dt, probability rdt. The total probability is then
given by the product of the individual probabilities of the two processes, i.e. if the
combined probability is denoted P(t)dt then

P(t)dt = P(0) x r dt

By applying the Poisson distribution and noting that r is equal to SN 0 we obtain

P(t)dt = XIV 0 e-kNot dt = A 0 e -a~ dt (8.21)

In experimental work with radionuclides many other errors occur in addition to statistical
error in the count rate. Such errors may originate in the weighing or volumetric
measurements, pH determination, etc. Such errors must also be considered in presenting
the f'mal results. For such composite errors, the law of error propagation must be applied:

OF= [(% dF/dA) 2 + (a B dF/dB)2] V2 (8.22)

where o F is the (one standard deviation) error in F, which is a function of the uncorrected
variables A, B, ..., with the standard errors o A, o B, etc. For the product A.B and ratio A/B,
one obtains

F = (A.B) ( 1 -t- s) (8.23)

F = (A/B) (14-s) (8.24)

where

s = [(o A/A) 2 + (o B/B)2] ~A (8.25)

For the function/l x and log A the following relations are valid

F = A x + x.A x-1 o A (8.26)

F = log A +[o A/(2.303A)] 1'~ (8.27)
A useful technique for checking that the error in the measurements has a Gaussian
distribution is the so-called "x-square" test. The quantity X 2 is calculated from

M
X 2 - [ E (F - Fi )2 ]/[~'(k - 1)1 (8.28)
i=1

where M is the number of measurements (e.g. points on a curve) for which the function F
is (believed to be) valid. X 2 would have a value 0.5 - 1.0 when the Gaussian fit exceeds
50%. k is the number of fre~om, i.e. M plus the number of independent variables. This
relation is generally valid; for simple counting systems F is replaced by N, the number of
counts in a given time interval, and k = M.