Data analysis

The conventional approach described in the guidelines of the International Atomic Energy Agency (IAEA) for the spectral analysis of NaI(Tl) in-situ measurements is called Window Analysis Method (WAM) and is based on the monitoring of count rates N in broad spectral windows associated to the main gamma emissions of natural radioisotopes. These windows are centred in the 40K (1460 keV), 214Bi (1765 keV) and 208Tl (2614 keV) gamma emission energies and their width is chosen to embrace the corresponding photopeaks, which shape is governed by the intrinsic energy resolution of the equipment. The 238U and 232Th mass abundances are indirectly estimated by looking at γ-rays respectively produced by 214Bi and 208Tl. Therefore, the applied approach implicitly assumes that the secular equilibrium condition holds in both decay chains, which is the reason why U and Th mass abundances are generally expressed as equivalent Uranium (eU) and equivalent Thorium (eTh).

Radioisotope Gamma emission energy (keV) Energy range (keV)
K 40K 1460
 1370-1570
U 214Bi 1765 1660-1860
Th 208Tl 2614 2410-2810

The count rates NK, NU and NTh linearly depend on the mass abundances (concentration) of K, U and Th CK, CU and CTh according to the following equations:

N K = C K S K K + C U S U K + C T h S T h K + B K

N U = C K S K U + C U S U U + C T h S T h U + B U

N T h = C K S K T h + C U S U T h + C T h S T h T h + B T h

where the Sij term represents the sensitivity coefficient of the spectrometer for the detection of the i-th element in the j-th energy window in cps per unit of concentration and Bi is the background count rate in the i-th energy window. Background radiation components are atmospheric radon, cosmic background and instrument background and can be estimated by means of off-shore acquisitions or via measurements on blank calibration pads. The equations can be jointly written in matrix notation as follows:

N = C S + B

where:

  • Ni (cps) is the count rate in the i-th energy window (i = K, U and Th);
  • Sij (cps per unit of concentration) is the sensitivity coefficient for detecting the j-th element in the i-th energy window, i.e. the count rate in the i-th energy window for unitary concentration of the j-th element (j = K, U and Th);
  • Ci (10-2 g/g for K and mg/g for U and Th) is the concentration of the i-th element;
  • Bi (cps) is the background count rate in the i-th energy window.

The sensitivity matrix can be determined on the basis of calibration measurements on sources having known radionuclide mass abundances, which can be concrete pads enriched in K, U and Th, or alternatively natural calibration sites characterized for their radioactivity content by independent measurements on soil samples.

In the following table are reported the sensitivity matrix published by IAEA for a 3' NaI(Tl) detector.

S = ( 3.360 0.000 0.000 0.250 0.325 0.011 0.062 0.075 0.128 )

As expected from the physics point of view, the sensitivity matrix has an almost triangular shape: indeed, as 40K decay generates a single gamma with 1.46 MeV energy, 40K does not provide any count rate contribution in the U and Th energy windows, centered at higher emission energies.

The vector of unknown concentrations C can be computed from the background corrected count rates N = N* - B provided the inverse sensitivity matrix S-1 as follows:

C = N * S -1

Therefore, given the background corrected count rate for a specific measurement and the detector sensitivity matrix, the K, U and Th concentrations CK, CU and CTh can be separately estimated as well as properly combined to obtain the overall natural specific activity A in Bq/kg according to the following equation:

A = 313 C K + 12.35 C U + 4.06 C T h

The concentration-to-activity conversion coefficients are constants which can be determined on the basis of the decay physical parameters. The calculation of the K concentration-to-activity conversion coefficient is reported at the bottom of this page.

The presented WAM formalism is based on the monitoring of 3 energy windows associated to K, U and Th natural radionuclides but can in principle be extended to anthropogenic isotopes. This can be relevant for instance when performing environmental measurements in areas that experienced a traceable fallout after the Chernobyl accident as an additional photopeak centred at 662 keV can be recognized in the spectral shape due to the gamma emission of 137Cs.

Calculation of the K concentration-to-activity conversion factor

Let's assume to have a unitary concentration of K in the soil, corresponding to 10-2 g/g, i.e. mK = 10 g of potassium in 1 kg of soil.

Provided the:

  • Avogadro number NA=6.023 • 1023 atoms/mol
  • K molar mass MK = 39.098 g/mol
  • 40K isotopic composition i40K= 1.17 • 10-4 40K atoms/K atoms

We can estimate the number of 40K atoms (nuclei) as:

N 40 K = m K M K N A i 40 K = 10 g 39.098 g m o l 6.023 ⋅ 1 0 23 a t o m s m o l ⋅ 1.17 ⋅ 1 0 -4 40 K a t o m s K a t o m s = = 1.80 ⋅ 1 0 19 40 K a t o m s

The general formula A = N / τ can be applied to derive the activity of 10 g of K, knowing that the 40K mean life is:

τ = 1.83 ⋅ 1 0 9 y r = 1.83 ⋅ 1 0 9 y r ⋅ 365.24 d a y y e a r ⋅ 23.93 h o u r d a y ⋅ 3600 s e c o n d s h o u r = = 5.76 ⋅ 1 0 16 s e c o n d s

A = N 40 K τ = 1.80 ⋅ 1 0 19 n u c l e i 5.76 ⋅ 1 0 16 s e c o n d s = 313 d e c a y s s e c o n d