2G ean ak De fete da ol teded edchanaes arehtkalekeene 92 selection of y by “guessing” at e° would not in C. Influence of the Errors in g(x") Usually the selection of the columation width 1s re- lated to the limits of errors (statistical plus expermental) in g(z) due to the structure of the system, the quantity of radioactivity available in the distribution, the natural background, and sometimes the limitations on the time available for performing the measurements. Of significance in our mathematical methodis the re- lationship between y and the measure of the error e=Sd the results unduly. Various results for the distribution f(x) obtai using different y’s for the case FWHM = 8.5 shown in Figure 77; it is easily seen that for 5 y < 20-10°, the shape of f(x) changes verylit could be deduced from the corresponding cw Figure 70), whereas for y = 20-10°, the resu greatly influenced bvthe selection of y and an err estimation of the experimental value of e could uw meaningless f(x). t=1 shown by the curves y(e’) in Figure 70. Theselection of D. The Intervai h between Successive Readings« the type of collimation and other parameters affects the shape of y(e’) and consequently influences the determination of the computed distribution. In practice Whencarrying out an experiment, the proper of the interval h between two consecutive read g( z) is important for obtaining acceptable resul allowed in g(z’); henceit, is desirable to arrange the experimental parameters so as to have the acceptable values of the error e’ fall in the region of y(e’) where ¢” is a slowly varying function of y. A simple method to determine whether the experimental conditions are well chosen is to plot e” for a few different values of y, as in Figure 70 and check whether: (1) the form of the curve y(e’) is acceptable (as is the case in the three out of four curves in Figure 70, which have similar shape), and (2) the value of the error e = $7, e (experimental plus statistical) falls into role played by h in the computation of the tot it is difficult to define a general rule concerning the error the range of the curve which is insensitive to small variations in y (e.g., in Figure 70 for FWHM = 8.5 em the value of the e” should be less than 10°). The underlying reason is that normally the value of e’ is only roughly known and, with Jowsensitivity an error in the 0.70 oso [' partial activities under f(a2). Figures 78 and 75 the distribution f/(2) obtained using different in: h in the two extreme cases of FWHM = 8.5 ¢ FWHAMI = 15.0 cm, respectively; these data + that acceptable results can be achieved only for tain range of values of h. As a general rule, on propose that the upper limit for A be kept aroun: FWHM of the point response function K(2z,2° selection of its lower limit, on the other hand, se be strictly correlated with e(x), namely with the ure of statistical noise superimposed on every r in g(x’). In fact, if h is less than h, the valueof th e = » e; will eventually increase beyond accc levels. At such levels of e* the computation oft) tribution f(z) becomes sensitive to the choice of a wrong estimation of e° may makeit difficult to satisfactory value. If a very narrowcollimator, an sequently a very small h, is selected, better val f(x) can be obtained by smoothing the experi —-— y= 5:10° results* g;(x’), namely by reducing the statisti y = 20-105 cillations of the experimental data. This effect, f ——-—y: ia5 O.50- Section A above). We sawin Section A (Figure 7 ~aonoarm ¥ = 108 case of FWHM = 15.0 em, hf = 1, is shown in Fig: There, one sees that e” = >>; ej as a function of y absence of data smoothing is very sensitive 1 pd FE > - 040 change in y, which is an undesirable situation. Co <q Smoothing the data instead moves the corresp > 0.30 = cL “| curve in Figure 70 to the left (reducing e”), mal similar to the family of the other acceptable cur y as a function of e’. “] EK, Extension of Data Beyond the Limits of th: tribution ci J Lu Our oO 1 10 ! 20 ! 30 46 DISTANCE (cm) Fic. 77—Effect of y’s on f(z) l 50 60 Intuitively, one expects that a certain improv of the shape of f(z) near the extremities should *and K(z,2") if the kernel also has significant statist ror.