1.2.1. Structure of equations

The conditional equations defining the model for calculating the value of the output quantity/quantities y are written into a text field of the program. To improve the readability of equations auxiliary quantities may be inferred. The formula symbols are extracted automatically from the equations and are transferred into a table where they individually can be complemented by a unit and a meaning. After input of values and uncertainties of primarily measured quantities a “Function parser” is used for interpretation of the equations and calculation of the value and uncertainty of the output quantity y.

For the process of editing the equations it is important to infer a quantity representing a net counting rate, called \(\mathbf{R_n}\) in this help file, on which the output quantity depends linearly:

(1.2.1)\[y = R_{n} \cdot F_{L} + F_{C}\]

In equation (1.2.1) the proportionality factor FL represents the procedure dependent calibration factor. FC considers further interference contributions, as e.g. one originating from the addition of a tracer activity before beginning with the radiochemical analysis.

The net counting rate \(R_n\) shall be understood as that net counting rate (more precise: procedure dependent net counting rate) from which all those contributions to the gross counting rate \(R_g\) which are not derived from the source contribution itself have been subtracted. The latter are not only the detector-related background \(R_0\) but also blank contributions \(R_{bl}\) and, if applying a tracer solution, additional blank contributions due to impurities in the tracer solution. Additionally, a calculated contribution \(R_{int}\) may be included due to interference by another radionuclide. As an example, the procedure dependent net counting rate may then be:

(1.2.2)\[R_n = R_g - R_0 - R_{bl} - R_{int}\]

The constants \(F_L\) and \(F_C\) can be easily determined within the program for arbitrary types of equations, if these depend linearly on the net counting rate. This representation allows UncertRadio to solve Eq. (1.2.1) for a modified net counting rate value if the output quantity value were changed to y’:

(1.2.3)\[R_{n}' = (y' - F_C) / F_L\]

Similarly, the equation for a net counting rate \(R_n\) can be expressed more generally as a linear function of the gross count rate \(R_g\):

(1.2.4)\[R_n = F_B \cdot R_g - R_{0,total}\]

In most cases the factor \(F_B\) is equal to one; but FB may also differ from one. R0total is the sum of background contributions to be subtracted from the gross counting rate; see Eq. (1.2.2). At the beginning of computations, UncertRadio determines the values of \(F_B\) and from this the fixed value R0total:

(1.2.5)\[R_{0,total} = F_B \cdot R_b - R_n\]

A net counting rate value Rn’ obtained by iterations within the detection limit calculations, is associated with a modified value of the gross counting rate:

(1.2.6)\[R_{b}' = (R_{n}' + R_{0,total}) / F_B\]

Tip

All counting rates in equation (1.2.2) may also appear as to be multiplied with factors g, associated with uncertainties, as e.g.

\[R_n = g_b \cdot R_b - g_0 \cdot R_0 - g_{bl} \cdot R_{bl} - g_{int} \cdot R_{int}\]

1.2.1.1. Non-linear dependence

There may exist cases in which the dependence between output quantity and net counting rate, or, when using linear unfolding, between output quantity and the activity, is not linear. Consequently, the values \(F_C\) und \(F_L\) in Eq. (1.2.1) are only approximate ones and the inversion given by Eq. (1.2.3) is no longer correct.

Therefore, in addition to Eq. (1.2.1) and Eq. (1.2.3), two new internal functions are used in UncertRadio:

• As an alternative to Eq. (1.2.1) a function ActVal(\(R_{n}\)) for calculating the value of the output quantity is used based on the function RESULT (see below);

• For the reversion according to Eq. (1.2.3) a new function RnetVal(\(y'\)) is used as an alternative; it uses the numerically working secant method; it requires initial guess values for the lower and upper limit of the net counting rate values to be searched for, which are easily derived from the values of \(y'\), \(F_C\) und \(F_L\).