In practice, the measured gamma spectra often contain characteristic lines at energies that do not correspond to the energies of the calibration sources used, meaning that no values for the respective absolute efficiency are available for these.
The “missing” efficiency values must therefore be “calculated” from the existing efficiency values. Until now, there has been no theoretically founded model for this, which is why a variety of empirically found approaches have been adopted in practice, such as:
Polynomial in \(\log(\varepsilon)\) and \(\log(E_{\gamma})\)
\[\log(\varepsilon) = a_{0} +a_{1} \cdot \log(E_\gamma ) + a_{2} \cdot [\log(E_\gamma )]^{2} + \cdots + a_{5} \cdot [\log(E_\gamma )]^{5} \]
Polynomial in \(\log(\varepsilon)\) and \(E_{\gamma}\)
\[ \log(\varepsilon) = a_{1} \cdot E_{\gamma} + a_{2} + a_{3} \cdot E_{\gamma}^{-1} + a_{4} \cdot E_{\gamma}^{-2} + \cdots \]
Polynomial in \(\log(\varepsilon)\) and \(\log(\frac{1}{E_{\gamma}})\)
\[ \log(\varepsilon) = a_{0} + a_{1} \cdot \log{\frac{c}{E_{\gamma}}} + a_{2} [\log{\frac{c}{E_{\gamma}}} ]^{2} + \cdots + a_{5} [\log{\frac{c}{E_{\gamma}}} ]^{5} \]
Inverse Exponent
\[ \varepsilon = \frac{1}{a} {E_{\gamma}^{-x} + b \cdot E_{\gamma}^{-y}} \]
The various parameters a0, ... ai, c, x or y are determined through a fit to the measured efficiency values.
Example: Experimental Efficiency Calibration for a Measurement in Open Geometry
The goal of the efficiency calibration is to determine the efficiency curve, which is then used for evaluating the data of an open geometry measurement on a volume source filled with sand (the following detailed listing of all data of the efficiency calibration serves traceability).
For the efficiency calibration, a quasi-point 152Eu source with an activity A0 of 5.85·106 Bq at the measurement time was used. The dimensions of the radioactive material in the cylindrical enclosure of the source are a few millimeters and can be practically neglected compared to the distance S = 60 cm of the source from the detector or considered as an additional contribution of about 0.5% to the measurement uncertainty.
The following figure shows the measurement setup schematically in the upper section and as a photo below. Although the detector is collimated, this measurement arrangement can also be used to determine the efficiency for the open geometry measurement according to Filß,
- since a point source is used for calibration, which is positioned on the detector axis at a distance S from the detector and
- the inner diameter of the cylindrical collimator is greater than the diameter of the detector crystal
Thus, geometrically induced effects, such as absorption or scattering in the collimator, can be neglected in the determination of the efficiency values.
A larger contribution to the measurement uncertainty occurs due to the always present background radiation. For an experimental determination of the detector efficiency values, it is therefore advantageous to perform the measurement in the same arrangement as subsequently given for the measurements: an inactive object, which as far as possible corresponds in size and composition to the object to be measured later, should be positioned at the later measurement location. This way, the shielding effects of the later measurement object against the background radiation are approximately taken into account.
The calibration source (point source) was therefore mounted on an aluminum cylinder, which has the same dimensions as the later measurement object, filled with sand, on the side facing the detector, and the gamma spectrum (black line in the spectrum) was measured.
To estimate the influence of the background radiation on the values of the efficiency calibration, an identical gamma measurement was additionally conducted without calibration source (blue spectrum).
The resulting spectra, normalized to the respective measurement times (live time), are shown in the following figure. It is clearly visible that in this case, the contribution from background radiation is several orders of magnitude smaller, meaning that it can be largely neglected.
Note:
The gamma spectrum of the background radiation provides useful information for the qualitative evaluation of the gamma spectra of the later measurement objects for the assignment of characteristic peaks to radionuclides that are either present in the measurement sample or originate from the background radiation.
The peak areas determined from the calibration spectrum, the corresponding counting rates for various characteristic lines of 152Eu, and their emission probabilities are listed in the table. The diagram shows the efficiency curves determined from the table using various empirical approaches.
| Energy in keV |
Emission Probability in % |
Peak Area (counts) |
Counting Rate |
Efficiency εabs | Uncertainty Δεabs |
| 122.0 | 28.41 | 92890 | 61.086 | 3.67E-05 | 1.33E-07 |
| 244.8 | 7.55 | 20660 | 13.586 | 3.07E-05 | 2.64E-07 |
| 344.4 | 26.59 | 54490 | 35.833 | 2.30E-05 | 1.04E-07 |
| 411.2 | 2.24 | 3867 | 2.543 | 1.94E-05 | 4.53E-07 |
| 444.1 | 2.80 | 5202 | 3.421 | 2.08E-05 | 3.83E-07 |
| 779.1 | 12.97 | 14750 | 9.700 | 1.28E-05 | 1.16E-07 |
| 867.4 | 4.24 | 4054 | 2.666 | 1.07E-05 | 2.24E-07 |
| 964.2 | 14.50 | 14730 | 9.687 | 1.14E-05 | 1.00E-07 |
| 1086.2 | 10.13 | 10320 | 6.787 | 1.14E-05 | 1.24E-07 |
| 1112.2 | 13.41 | 12610 | 8.293 |
1.06E-05 |
1.00E-07 |
| 1408.0 | 20.85 | 16130 | 10.607 | 8.68E-06 | 6.94E-08 |
Self-Test Questions:
- Comparing the spectrum with the table, you can see that the spectrum shows numerous other lines as well. What is their origin?
- Why do the measured efficiency values fluctuate around the efficiency curves?
