3.5.5.6. Detector Resolution

The statistical process of pulse formation yields a Poisson distribution of voltages (see below). At sufficiently high count rates a Poisson distribution approaches a normal distribution and the relative standard deviation may be expressed as:

where K = constant (close to 16 for Ar-filled detectors), and E = energy of incident X-rays (keV). The pulse height distribution is wider for longer wavelengths (lower E). For example, the standard deviation is 13.1% for Al-Ka and 6.3% for Fe-Ka. A more refined way of calculating resolution is expressed by:

where E_FWHM = full width at half maximum of the energy distribution (eV), E = energy of incident X-rays (eV), ei = effective ionization potential (eV), and F = Fano factor that depends on gas type.

The Fano factor is required because it has been demonstrated that the resulting variance is only a fraction of what would be expected. This factor ranges from 0.5 to 0.22 for Ar/methane mixtures. The 2.355 factor in the formula relates 1 standard deviation (1 sigma) to the FWHM of the pulse-height distribution. For example, with F = 0.22, the Al-Ka energy distribution is 218 eV (14.7%) at FWHM.

It is more convenient to express the resolution as a percentage of the X-ray energy. The observed resolution is defined as the full width in energy units of the peak at half-maximum, E_FWHM, divided by the energy of the X-ray, E (Figure 3.5.5.6).

Figure 3.5.5.6. Peak shape and energy resolution of the detected X-ray peak (A) and of the proportional counter (B). Resolution of the X-ray peak includes the additional effects produced by the analyzing crystal, whereas, resolution of the proportional counter reflects the effects of only the detector electronics.

Once the resolution of a detector has been established for given energy the relative resolutions for other energies may be calculated:

where R₁ = observed resolution at E₁, E₂ = energy of interest, and R₂ = resolution at energy of interest.