8342937122

SU PIAN BI N SAMAT AND C.J. EVANS

probabllity of obtaining the whole set of n data poilits )\,_f... y_n is proportional to the product ol n expressions of the form (2), and the best lit will be that which maximises this total probability. On replacing the product of exponentials by the exponential of the sum, we find (Samat and Evans 1989) that the condition for the best lit is that the expression

S =    - y,f = r₂    + c ~ ?,f

(3)

should be a minimum, where u/, is the statislical

weight of the observation y, and is defined to be

, |

a . The minimum value of 5musi be obtained by adjustingthe two parameters mand r. We therefore differentiatc the function .Swith respeet to w and c, and equate tfu* derivativcs to zero, to obtain

T^- = ^WI    ^r I *i - I«'i*Ji = 0

om

and    (4)

--- = m£aiiX| +    = 0.

dc

These are two linearequations which maybesolved for w and r. In a morę generał case, with morę than two coeflicients, the equations (4) would be best solved by an elimination technique. In the present case, the solulion can be written down explicitlv in terms of matrix inversion:

m	1	' Iw i	-IWiXi		Iw-^iYi . IWiYi
c	A		V ^2 Iw,x, J

where A is the determinant, £w. £w.x ‘ - (Xw.x.)² which can be shown to be a non zero quantity. Equation (5) gives

m =    - X    Wi>i)/A

C =    luj; - Ia<iXiIWjjcjiJ/A,

while the diagonal elements of the matrix on the RHSof (5) yield tłie errors in the parameters:

Am = ^(Z^j/A)

Ar = >/(Z ^U,\X^ / A)

Substituting the best values of the parameters into

equatian (3) gives

X²«at = 25 = Z^i(wjcj + c -

The calculations were carried out on a micrr, processor system (a Dragon 32 Computer connectet to a Radio-shack Trs-80 Computer cassette recorder NEC PC- 8023 BE-N printer and Philips TX B&V monitor), using a program written in BASIC. Tht order of calculation follows that of the equation derived above.

There are somc extra f acilities in this program As was mentioned earlier, if o is the error in y, thci xu. = o . To input these values of a. into thr program, there are options for a., namely (1)0^ constant (a given value), (ii) a = constant (urv known), (iii) o. =a constant percentage (given) 0; y. (iv)a = a constant percentage (unknown) of (v) a. = V a , and (vi) the vaiue of a. is supplied fo each point. In (i) and (iii) we only need to input one value of C. (or % a) while in (ii), (iv) and (v no input is needed in order to specify a.. Option (ii) and (iv) do not permit the calculation of X"

Two sets of dala from nuclear physics expei> ments were obtained and applied to test the pro gram.

The Rutherford ScatteringExpeńment The first set of data wasobtained from an experimem set up to verifv the angular dependence o( Rutherf ord scattering. Alpha-particlesemitted from

a radio-nuclide source (‘"Am, activity 50 fj( i energy 5.48 MeV, half-life 458 years) were passed through a gold foil (of thickness 2.5/im), and the resulting scattered a-particles were detected ai scattering angle 0 (varied between 10° and 52°) to a semiconductor device. The source, the gold foil and the detector were in a vacuum enclosure through the base-plate of which the output pulsci were brought out and fed into the counter/timet unit. If the observed number of a-particles ex-pressed as counts/sec is /, then / decreases ai

0    inereases, as given by the equation / = const *

sin (0/2). For the fuli derivation of thisequation see for example Evans (1982).    u

Attenuation oj fi-rays

1    he second set of experimental data was obtained by counting j3-rays (from a radioactive source of * Sr, activity 100 fiCi) which had been attenuated to differem numbers of sheets of alumunium foil using a GM-tube detector (connected to a scaler* timer type ST6 manufactured by Nuclear Enterprises). If the number of sheets is x and the observed number of /3-rays expressed as counts/ sec is /, then I decreases when x inereases, as given

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PERTANIKA VOI.. 14N0.1, 1991