783620H / 1013 6. ANALYSIS METHODS
1) The option is enabled.
2) The spectrum is energy calibrated.
3) The peak was rejected for any test by the total summation method.
4) The peak is a singlet (multiplets allowed in ENV32).
If all these conditions are met, the spectrum region is fit for 3 times the calibration FWHM, centered on the peak energy. The function is the background plus a Gaussian peak shape. The five parameters are peak amplitude, peak centroid, and a quadratic background function. The initial values and uncertainties are taken from the spectrum.
The fit is iterated until the reduced chi square for the fit changes by less than 1% from the previous iteration, to a maximum of 10 iterations. Most cases will converge in 3 to 4 iterations.
If the fit fails, the peak values are set to the total summation values. If the fit converges, the background and peak area are calculated from the fit values.
6.3.2.3. ISO NORM Singlet Peak Method
The guidelines for singlet peak calculations are in Annex C of the ISO 11929:2010 standard, and are similar to the standard GammaVision total summation method discussed in Section 6.3.2.1.
When the peak background does not dominate, the peak width should be ~2.5 FWHM, per ISO
11929 Eq. C.9:
(32)
where h is the FWHM of the peak.
In case of a dominant background, the peak region width should be 1.2 FWHM, per Eq. C.10:
(33)
When t
g
is 2.5, almost the entire peak area is included in the region. However, when t
g
is 1.2, the peak region only covers about 84% of the whole peak area. Therefore, a correction factor for the peak area is needed, as discussed in the remainder of this section.
In GammaVision, the percent Peak Cutoff, entered on the Analysis tab, is used by default as the criterion for dominant background. Background is considered dominant except when the peak uncertainty is less than the Peak Cutoff, in which case the peak is flagged as “identified.”
Therefore, for all identified singlet peaks, the default region width should be about t
g
× FWHM for this method.
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If the library peaks are not found, the MDA values are calculated with a width of ~1.2 FWHM.
Since the exact criteria for dominant background are not specified in ISO 11929, GammaVision includes a “Dominant Background Peak Cutoff and Override Flag” in the b30winds.ini
file
(page 449) to enable you to define a dominant background cutoff. This entry is formatted as:
25.0 T
The first parameter is the user-defined dominant background peak uncertainty cutoff. The default value is 25%, and ranges from >0% to <1000%. The second parameter is a Boolean flag. If the flag is set to false (F), the user-defined peak cutoff for dominant background is used. If the flag is true (T), the Peak Cutoff value on the Analysis tab is used.
To calculate a correction factor when the peak region does not cover all the peak area, let the peak region width be
ν (in units of the peak FWHM):
(34)
Assume the peak shape is a perfect Gaussian. Let f be the ratio between the net counts in the region and the true total peak area, then f can be calculated as:
(35)
where
Φ is a Gaussian integral function defined as:
(36)
and c is a constant equal to:
(37)
The correction factor f is used as a divisor in the calculations of the peak area:
(38)
where N
0
is calculated with the region width chosen per ISO 11929. The f factor is not used when N
0
is zero or negative since no peak is present and a negative peak area is non-physical.
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The use of f is consistent with Eq. D.5 in ISO 11929:
6. ANALYSIS METHODS
(39)
The definition of the quantities involved in above equation can be found in section D.5.1 of
ISO 11929. The factor f in the denominator is defined in Eq. 35 above, and i is the branching ratio.
The f factor is not used to scale the peak background. The background does not scale as the peak area (that is, even though 84% of the peak area is covered, it is incorrect to state that only 84% of the background is included).
When x is positive, the Gaussian integral
Φ(x) can be calculated as:
(40)
where Erfc(x) is the Complimentary Error Function:
(41)
The Complimentary Error Function is calculated using the following analytic approximation:
30
(42)
where t and
φ are defined in the following equations:
(44)
(43)
30
Press, W. H., et al., Numerical Recipes in C: The Art of Scientific Computing, Cambridge University Press; 2nd ed. (October 30, 1992).
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(45)
where:
a
7 a
8 a
9 a
4 a
5 a
6 a
0 a
1 a
2 a
3
= 1.26551223
= 1.00002368
= 0.37409196
= 0.09678418
=
!0.18628806
= 0.27886807
=
!1.13520398
= 1.48851587
=
!0.82215223
= 0.17087277
The approximation work wells for the range of x values we are concerned with.
To calculate the actual peak region start and stop channels, the half width of the region is calculated first, e.g., 0.5 t
g
. This half width is converted by rounding to an integer, in channels, with a lower limit of 1 channel. For best precision, the analysis engine compares the rounded-down and rounded-up values and selects the best match to the desired overall regional width. For highenergy peaks, it may not matter whether the integral width is rounded down or up, however, there may be a big difference for some low-energy peaks. For example, if the energy calibration at low energies is ~0.5 keV per channel and the calibrated peak FWHM is ~1.0 keV, the difference between the two rounding methods could be as great as one FWHM. For a desired peak region width of 1.2 FWHM, the width obtained by rounding up could be about 2.2 FWHM, defeating the intent to use a narrow peak width when the background is dominant.
The ISO peak evaluation method is applied only for peaks that meet the following requirements:
1) The peak is in the library.
2) The peak range, including the designated number of background points, has no overlap with both lower or higher energy peaks.
3) The energy separation between the current peak and the nearest low- and high-energy peaks should be greater than the separation for deconvolution (the latter determined by the “Peak overlap range in units of peak FWHM” parameter in b30winds.ini
; see page 248, 449.
4) The peak shape is close to Gaussian. Peaks marked with * or @ on the report do not qualify.
5) The peak centroid cannot be too far from the library energy. Peaks marked with } on the report do not qualify.
6.3.3. Example Peak Area
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