I'd like to return to my
Mo Lβ3/Lα count rate dataset as an example and re-interpret a little...
While it is true that the origin of pulse pileup differs from that of dead time, corrections for the two are described by the same types of models. For instance, correction for pulse pileup requires a model equivalent to that for an extending dead time (see work by Pommé):
N’ =
Nexp(
-τN). As shown by
Müller (1991), the first order approximation (based on power series expansion) to the extending dead time correction is the non-extending correction,
N =
N’/(1 -
τN’) or
N’/
N = 1 -
τN’. The implication of this is that, even if JEOL pulse processing circuitry is in truth subject only to pulse pileup, then the non-extending (non-paralyzable, linear) model should still be applicable at relatively low count rates (certainly below 50 kcps). In contrast, for the more general case extending to high count rates, superposition of pulse pileup and dead time requires a more complicated treatment such as that proposed by
Pommé (2008). If a dead time is not enforced electronically, then correction for pulse processing count losses at high count rates could potentially be described by a simple model.
In the plot below (and in all of the data plots that I’ve shown in my application of the
Heinrich et al. ratio method), most of the correction and essentially all departures from linear behavior are due to one X-ray line (within the ratio). In the case illustrated below,
N’32 represents the Mo L
α count rate on channel 5/PETH, while
N’12 represents the Mo L
β3 count rate on channel 2/PETL. For measured count rates below 200 kcps, the ratio,
N’32/
N’12, is greater than 20:1. It appears that the plotted ratio (
N’12/
N’32), in which departure from linearity can be ascribed effectively solely to Mo L
α, is fit well by an exponential function. The same is true for my corresponding dataset for Ti, which extends to measured count rates up to 227 kcps. For my corresponding dataset for Si, an exponential fit works well for Si K
α count rates up to about 140 kcps (on channel 4), but the dataset is fit better as a whole by a quadratic. I believe that the reason for this may lie in the extreme degradation of resolution in the pulse amplitude distribution, which may have contributed to irrecoverable loss of peak X-ray counts above ~140 kcps.
I should note that I’ve restricted the range over which I’ve applied the linear model to the Mo data plotted below to about 63 kcps. This is lower than the maximum count rate at which I applied the linear model before, as later I became concerned that ratios calculated using values greater than this might display noticeable departure from linearity. Using the linear fit, I obtain
τ3 = 1.32 μs (channel 5). Although I’ve chosen a value of
τ3 = 1.30 μs for the exponential correction, the two values are essentially identical when considering propagation of counting error. Increasing the value of
τ3 produces a lower ratio value for either the exponential model or the Donovan et al. model. Note that the exponential correction requires an iterative solution. I'll let the plot do the rest of the talking.