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Description
The detection limit (DL) is the smallest value of the measurand which can be detected with a predefined probability. In terms of the indication, which is in gamma-ray spectrometry the number of counts in a peak occurring at an energy where the analyte radiates, the indication corresponding to the DL, np#, is the solution of the equation [1]
(np#- np)2=k1-β2[ur2(W)+u2(np= np#)] , ()
where np denotes the indication corresponding to the decision threshold (DT), u(np#) and ur(W) are the uncertainty of the indication corresponding to the DL and the relative uncertainty of the factor converting the indication to the observed value of the measurand, respectively and k1-β is the quantile of the normal distribution corresponding to the probability β for making the error of the second kind. By approximating u(np) with a quadratic function of np it is possible to calculate np#.
Since the DT is calculated from the spectrum in the absence of the indication, i. e. from the indication background, to arrive at the DL, the indication must be stripped off from the measured spectrum
n0i=ni-pi·np,
where np denotes the observed value of the indication, ni and n0i are the channel contents of the measured and the reduced spectrum respectively and pi describes the shape of the indication. The matrix elements of the variance-covariance matrix of the reduced spectrum, are
U0ij= ni·δij+pi·pj·u2(np),
where u(np) denotes the uncertainty of the indication observed.
By the LSQ method the variance of the indication is given by the corresponding diagonal matrix element of the variance-covariance matrix of the results, given as
Un=(P2T·U0-1·P2)-1,
where P2 denotes the two-column matrix holding in one column the shape of the indication normalized to unity, pi, and in the second column the indication background n0i. Then the matrix element Un11 holds the square of the null-indication uncertainty u(np=0).
To arrive at the uncertainty at an arbitrary value of the indication, np~, the matrix U0 is modified to
Uij= U0ij+pi·pj·np~.
It should be observed that adding-up an indication to the spectrum introduces an additional correlation among channel contents, although np~ itself is uncertainty free. By calculating the matrix Uij with e. g. np~=k1-α·u(np=0) and e. g. np~ =(k1-α+ k1-β)·u(np=0), the parameters of the quadratic parabola can be determined. By inserting the parabola into the Eq. (), the equation can be solved on np#, the indication corresponding to the DL.
The procedure described will be illustrated with the DLs as functions of the indication calculated for isolated indications and indications overlapping with close peaks. The results will be compared with results obtained with other methods.
Reference:
[1] M. Korun et al., Calculation of the decision threshold and detection limit in high-resolution gamma-ray spectrometry, Nucl. Instr. and Meth. A 1014 (2021) 165686.
