5.15. Approach of calculating Decision threshold and Detection limit for Gamspk1

The iterative calculation of the Decision threshold and the Detection limit is performed after the mean \(\overline{A}\) has been calculated from the single values \(A_{i}\). The iteration is done by variation of the mean where a varied value is \(A'\). Then, all single values of the source activity \(A_{i}\) are replaced by the new (fictive) value of \(A'\). From the equation defining \(A_{i}\) :

(5.15.1)\[A_{i} = R_{ni}\frac{f_{att,i\ } \cdot \ f_{coinsu,i}}{\epsilon_{i}{\ \cdot \ p}_{\gamma i}\ }\]

one obtains by inverting this equation

(5.15.2)\[R_{ni} = A_{i}\left( \frac{\epsilon_{i}{\ \cdot \ p}_{\gamma i}}{f_{att,i\ } \cdot \ f_{coinsu,i}}\ \ \right)\]

and from this with the replacement \(A_{i} = A'\ \) an equation for the (fictive) net counting rates associated with the varied value \(A'\) :

(5.15.3)\[R_{ni}^{'} = A'\left( \frac{\epsilon_{i}{\ \cdot \ p}_{\gamma i}}{f_{att,i\ } \cdot \ f_{coinsu,i}} \right)\]

The aim is now to determine the uncertainties of the \(R_{ni}^{'}\), then, via uncertainty propagation in accordance with (5.15.1), the uncertainties of the single activity values \(A_{i} = A'\ \)and, finally, with the chosen method for the mean to derive the uncertainty \(u(A')\) of the (iterated or fictive) mean value \(A'\).

The following ansatz (a separation) is chosen for the uncertainties \(u\left( R_{ni}^{'} \right)\ \)(see also):

(5.15.4)\[u^{2}\left( R_{ni} \right) = \left\lbrack \frac{R_{ni}}{tlive} \right\rbrack + \frac{R_{T}}{tlive}f_{B} + \frac{R_{bg}}{tlive} + u^{2}\left( R_{bg} \right)\]

Herein, only the first term is related directly to the contribution from the sample activity. The remaining terms represent uncertainty contributions of those parameters which characterize the background of the i-th gamma line including also a contribution from a “peak in the background”.

By using the last equation now with the equations (5.15.3), (5.15.4) and (5.15.1) the (iterated or fictive) activities of the single gamma lines and their uncertainties can be determined as indicated already above. After these calculations those values are available which are necessary to go to the next iteration step and to test also for convergence of the iteration.

Important note:

External“ influences may exist leading to calculated values \(A_{i}\) of the single gamma lines which may exhibit a spreading which may be larger than to be expected from the uncertainties of single values.

This effect can be found with the weighted mean if the “external” is significantly larger than the “internal” standard uncertainty or the value of the “reduced Chi-square” significantly larger than one is.

In the case of the arithmetic mean with additive correction this may be inferred if the correction \(C\) and particularly its uncertainty \(u(C)\) lead to a significant shift of the results compared to the arithmetic mean (uncorrected) or its uncertainty.

These influences usually are not considered in the evaluation model given by (5.15.1). Determining Decision threshold and Detection limit requires iteration of the activity values. The inversion of that equation, i.e. calculating the net count rates \(R_{ni}^{'}\) according to (5.15.3) to be expected for a given (iterated) value of the activity \(A'\), leads directly to the elimination of that “external” influence. Then, from the obtained net counting rates \(R_{ni}^{'}\) single activity values result from (5.15.1) having all the same identical value \(A'\), i.e. their spreading is equal to zero!

This means for the calculation of Decision threshold and Detection limit that the external effect which may have been found from the primary evaluation of the output quantity in this latter case does not come into effect. Insofar, the usability of the external standard deviation with the weighted mean or with the NIST-2004 method is low, at least regarding Decision threshold and Detection limit.