JOHN D. TOMLINSON Minneapolis, Minnesota
chemists have found many uses for radioisotopes in the past few years. Radioactivity is definitely a valuable new tool for research and analysis. However, it must be used with a realization of the errors involved. If one does a radiation experiment without considering possible errors, one may get a meaningless answer, even if the work is otherwise done very carefully. In addition to any errors inherent in the experimental procedure, these factors may cause error: ( a ) background radiation, (b) randomness of decay, and ( c ) dead time of the counter. Each of these factors will be presented in a form useful to planners of most practical radiation experiments, without reference t o details needed only in special work. ANALYTICAL
calculating the error expected due t o randomness.%,The expressions of error probability usually used are (I) standard deviation, (2) probable error, and (3) reliable error.
BACKGROUND RADIATION
With a common Geiger tube, one usually obtains a reading of about 30 counts per minute (c. p. m.) with no source of radiation in the laboratory. This background connt is due mainly to cosmic rays.' Since the intensity of cosmic radiation increases with altitude, the background count may be several times as high in mountainous areas. Background also varies greatly for different Geiger tubes. Even with lead shielding, some high-efficiency gamma tubes have a background count of 400 c. p. m.2 Naturally, the presence of radioisotopes in the laboratory will increase the background count. To correct for background radiation, one should suhtract the background count from each reading. For accurate work, determine the background daily, since cosmic ray intensity and other factors vary from day to day. RANDOMNESS OF DECAY
Although one can accurately determine the average rate of decay of any radioisotope, the rate a t any instant is variable. Thus, the process is random, much like the dripping of a leaky faucet. The table shows that for a small number of counts, the variation in rate is large. Obviously, this randomness can cause large errors when one counts only a few rays. By taking longer counts, one can reduce the error to as low a value as one wishes. I n other words, one can design the experiment to give the desired accuracy, within certain limits. For this purpose, one has available simple formulas for
' JAFFEY,A.
H., T. P. KOBMAN, AND J. A. CRAWFORD, "A Manual on the Measurement of Radioactivity, MDDC-388, Oak Ridge, Tenn., Technical Information Division, Dec., 1943. Tracerlab, Inc., Catalog C, pp. 54-5.
= Standard
Deviation of reading from true value. d e ~ a t i o n ;P = Probable error; R = Reliable error.
Figu-
1.
F-9uency
/
Distribution Cum.
./"
( 1 ) Standard Deviation. Standard deviation, the commonest indication of probability, is given by the formula #
=
548
where N is the number of rays counted. Readings will lie within the range N* a 68 per cent of the time. As an illustration, suppose the true count is 900. Then n =
~ 4 =% 1 d w 0 = 3~30
I n about 68 out of 100 cases one would get readings between 870 and 930. (2) Probable Error. Probable error is aiven bv the formula P
=
*0.675
48
Since the probable error is exceeded in half of the readTypical Set of Counts from a Weak Sources Run numbw
1 2
-
Counts per 10 seconds
6
3 4
5 11 7
5 6
4 9
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JOURNAL OF CHEMICAL EDUCATION
probability of any error occurring can be found. For example, the per cent probability of an error exceeding +1.96u is 5 per cent, and the probability of exceeding +2.58u is 1 per cent.' I n some cases, one finds use for these or other values. Usually, the per cent error in a reading is more useful than the absolute error. Dividing by N and multiplying by 100, we obtain the following expressions of per cent error:
I 1000
2000
6000
10,000
20.000
5 0 , W 100.W
Number of oounts
One can use a graph (Figure 2) to quickly find values of these functions for any count. The graphs can be ings, it is sometimes called the 50 per cent error. For read with mtisfactory accuracy for error estimation and the case of 900 counts, P is *20. In 100 cases 50 of will save much calculation. The curves of Figure 2 can also be used in estimating the readings would be between 880 and 920. (3) Reliable Error. Reliable error is given by the the number of counts which must be taken t o get a desired accuracy. For example, suppose one wishes t o formula get a reliable error of *1 per cent. Reference t o Figure 2 shows that 30,000 counts will give this acErrors are less than the reliable error in 90 per cent of curacy. Obtaining a reliable error of +0.1 per cent, all cases. For 900 counts, R is *50. In 100 cases 90 however, requires 3 million counts. Thus, because of readings would be between 850 and 950. Although this the square root relationship, a ten-fold improvement in expression is not used as frequently as the first two, the accuracy requires a count 100 times as long. Because writer prefers it because it gives a less deceiving answer. of the time required for high counts, the accuracy of all For example, using probable error, one might think radiation experiments is limited. Reliable errors something was wrong with the experimental procedure below +0.1 per cent are rarely obtained. if one received readings of 850 and 950 in succession. DEAD TIME OF THE COUNTER Actually, even higher deviations sometimes occur, as The counting pulse of a Geiger tube temporarily shown by the reliable error. The frequency distribution curve of Figure 1 illustrates the meanings of the destroys the potential between the cathode and the anode. Until the potential is re-established, the tube three expressions of probability. does not detect entering rays. Thus, there is a dead While special names have been given t o these three expressions of error, one should realize that the per cent time or resolving time after each count, usually between 50 and 600 microseconds.' The commonest tubes have a dead time of 200 microseconds. Obviously, the dead time is not important a t low counting rates. However, the error becomes great a t high rates. For example, suppose a tube with a dead time of 200 microseconds records 100,000 counts in 1 minute. Since the tube is dead for 200 microseconds after recording each ray, the total dead time is 100,000 X 200 X or 20 see. Therefore, the number of rays which would be detected if the dead time were zero is
Observed count. per minute Fi-
3. Damd Tim. Co-&ion Facton for 1W. 3W. amd SW Microwcond Tube.,
or 150,000 rays. If one needs only relative values, a correction is not needed, provided the values to be compared are of about the same counting rate. However, one must correct for dead time if one plans to compare high-rate counts of different rates. See footnote 1.
' See footnote 2.
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APRIL, 1954
The calculation shown above suggests the method of correcting for dead time. Expressed in general terms,
vhere N , is the corrected c. p. m., N is the observed c. p. m., and d is the dead time in seconds. Figure 3 shows the results of applying this formula t o tubes with dead times of 100, 300, and 500 microseconds. To obtain the true count, multiply the observed count by the correction factor shown. In using the formula for dead-time correction, one must know the dead time of the Geiger tube. I n general, for rates below 5000 c. p. m., one obtains sufficient accuracy by using the approximate value given in the manufacturer's catalogue. However, a t high rates one often must determine the dead time experimentally by using paired sources. By counting each source separately and then counting them together, one can calculate the dead time. Since the dead time can never be determined exactly, one should not attempt to correct counts taken at extremely high rates, such as 100,000 c. p. m. I n other words, dead time limits the practical rate of counting when counts obtained at various rates are to be compared.
A WELaL-PLANNED EXPERIMENT
While it is hard to generalize on the subject of errors, the following example illustrates how a knowledge of errors enters into a well-planned experiment. Jim ordered enough radioactive material to give him about 10,000 c. p. m. in his experiment. Since he bought a tube with a low dead time, he merely made an approximate check by paired sources on the value given by the manufacturer. From the value of the dead time, he drew a curve similar to Figure 3 which he used for quickly correcting his readings. Since his purpose seemed to call for a reliable error of about + 1 per cent, he found from a curve like Figure 2 that he needed 30,000 counts. Hence, his runs were set a t 3 minutes. If he had chosen a much weaker source, his runs would have taken too long. Finally, by shielding the tube and keeping the laboratory free of contaminants, he got the background below 100 counts per 3-minute run. Since the reliable error was *300, the background count was negligible. This fact saved him the time and effort involved in taking and applying background counts. Thus, a consideration of the background count, randomness of radiation, and dead time of the counter gave Jim a smooth experiment.
