Generalized dead times

[Probeman]

Increasing the dead time constant using the logarithmic expression for the Si Ka k-ratios would only cause an over correction at moderate count rates. And at this dead time and count rates the exponential expression fails...

Let me emphasize that the exponential expression can only work for cases in which the correction is due to an extending dead time or pulse pileup.  If an enforced, non-extending dead time is present (as in the Cameca pulse processing circuitry), then a more involved treatment such as that of Pommé (2008) must be applied.  Also, keep in mind that SEM Geologist has modeled the latter situation at high count rates using Monte Carlo simulation.

I should have explained this better.

I'm not saying the exponential expression is not accurate (we are still evaluating the expression using both JEOL and Cameca data). What I was saying was that at sufficiently high dead times and/or count rates the exponential expression fails mathematically.

That is, if we look at the exponential expression and solve for the predicted count rate we can see that the term -dtime * cps cannot be less than -1/e:



This is heavily dependent on both the count rate and the dead time constant. Here is a calculation at 1.5 usec dead time:



So at 1.5 usec we are limited to around 245k cps (which is actually pretty good!).  However, on a Cameca instrument (assuming 3 usec) we are limited to 123K cps:



Which is easily attained on any PET and especially LPET crystals.  Now it maybe that this expression is not applicable to a Cameca instrument if indeed it exhibits purely non-extending behavior. But based on some data it appears that the Cameca may exhibit extending behavior at sufficiently high PHA gain settings as shown here:

https://probesoftware.com/smf/index.php?topic=1489.msg11233#msg11233

But again, the dead time constant in the exponential expression is very sensitive so at 1.1 usec, we can handle count rates up to 334K cps, so pretty high count rates:



By the way, when you say the JEOL instrument exhibits extending behavior, do you mean that at sufficiently high count rates, the intrinsic dead time of the system is increasing to higher values?   Or do you mean a different dead time constant is dominant at these sufficiently high count rates?

I think that is partially what Almutairi, 2019 meant in this passage:



The Excel spreadsheet for these exponential examples are provided below as attachments if anyone is interested.

[Probeman]

We can see the limits of the exponential expression more clearly here at 1.5 usec:



And here at 3.0 usec:



Now this is not to say that this exponential expression is not useful in many situations (it's already been implemented in Probe for EPMA!) but it has some caveats as has been discussed. 

[Brian Joy]

By the way, when you say the JEOL instrument exhibits extending behavior, do you mean that at sufficiently high count rates, the intrinsic dead time of the system is increasing to higher values?   Or do you mean a different dead time constant is dominant at these sufficiently high count rates?

What I mean is that it appears that loss of X-ray counts in the JEOL pulse processing circuitry is dominated by pulse pileup and not dead time.  Pulse pileup is described mathematically in a manner equivalent to an extending dead time.  At low count rates, the pileup is described adequately by a non-extending dead time model, even though it arises due to a different mechanism.  The value of the time constant should not vary between the two models (and I’ll post more about this).  For Cameca proportional counters, the situation is more complicated.  Great care is required in examination and application of the extending dead time model (or any count rate correction model).

[Probeman]

What I mean is that it appears that loss of X-ray counts in the JEOL pulse processing circuitry is dominated by pulse pileup and not dead time.  Pulse pileup is described mathematically in a manner equivalent to an extending dead time.  At low count rates, the pileup is described adequately by a non-extending dead time model, even though it arises due to a different mechanism.  The value of the time constant should not vary between the two models (and I’ll post more about this).

OK.   Are you saying that what we call dead time exists solely in the electronics and not in the detector?

Or are you are saying (on JEOL systems) it appears that there is a non-extending component that is dead time and an extending component that is pulse pileup?  And that at low count rates it is dominated by dead time, but at high count rates it is dominated by pulse pileup? And by dead time do you mean photon coincidence?

[sem-geologist]

-What is the Airspeed Velocity of an Unladen Swallow?
-What do You mean? An African or European swallow?

Great care is required in examination and application of the extending dead time model (or any count rate correction model).

I have zero knowledge of the electronic mechanisms (and to be honest I really am not interested in all the gritty details!     ), I'm just trying to model the dead time effects mathematically (whatever they are) so we can obtain constant k-ratios for quantitative analysis on both JEOL and Cameca instruments!   

If You are going to construct mathematical model You need all these gritty details, even if You are not interested in them (which I don't blame anyone, it needs some nerdy passion to be interested in electronics - it is not for everyone). If You want to calculate the time the vehicle goes from point A to point B knowing the speed and straight line distance between A and B, You need to know what kind of vehicle it is. A plane will go in straight line, a Car will go on the road (thus You need then additional information about road network), Ship will go on the water, and Train will go on rails. And even if they will go at same velocity, they will need different kind of additional information to tell how much it will take to go from point A to point B and additional corrections (i.e. ship will need speed of river flow, plane the direction and speed of wind...).

Going back to the EPMA electronics. Is the dead time expandable or not expandable it is by design. EDS counting circuits implements extendable dead time where it is extended as much as needed so that pulse which is going to be counted and its amplitude measured would have no pulses before (not piled-us on a tail of preceding pulse, be it positive or negative tail). That is (expandable/extendable) by design so that energy of measured x-ray lines would not drift depending from count rate as it does on WDS PHA. Because it is extendable it is possible to observe on detector with increasing current and counting rate the the decrease of raw count rate at very high currents/count rates - which is the paralyzing behavior. It is possible to stall the counting by reaching 100% dead time.
We have non-extendable dead time on WDS on both Jeol and Cameca instruments. Why? 1) because we have PHA peak shifts - again extension of dead time is to prevent that, and as we see the shifts it is clear that there is no extension; 2) If I increase the current it the raw count rate increases, at high current it increase very little, but still it is the increase and no count rate decrease is observable even at >1µA beam at large crystal at most intense lines. It is clearly non paralyzable. 3) EDS for extendable dead time needs many different shapping amplifiers, where one fast shapping amplifier works constantly in parallel to the main high resolution (slower) amplifier. At least on Cameca WDS there is only and only one shapping amplifier integrated with Charge sensitive preamplifier in a single package, which is connected directly to the GPC - because of that there is no way to implement (and hide away in any possible means) EDS-like extendable dead time circuit.

Have pulse pile ups have anything with extendable vs not extendable? It depends what we mean with "pulse pile up". If it is pile up on the tail (imperfect pile up, recognizable with very fast shapping and sensing circuit on EDS) then EDS extendable dead time  introducing circuit is the response to that. Else if it is perfect pile up (within shaping time of fast circuit of EDS) - the extendable EDS counting circuit fails to recognise it and we can observe such pile-ups appearing on EDS at very high count rates. Can pulse pile up do anything to non-extendable circuit? No not at all, as it by design does not care. It is designed with profound superstition that it is fast enough (80es, there were still no large diffracting crystals) and will not come to such situation. Also integral counting method does not care about pulse pile ups.

Additionally it is important to understand multiplexing and consequences of that (in case of using such older kind of solutions/boards) where dead time will depend from count rate of other detector connected to same multiplexer (and ADC). The result of dead time correction (and dead time during real measurements) can be completely different when using single WDS, or loading all WDS detectors with high count rates. Fortunately new generation (last generation) of WDS boards on Cameca went away from that, but I am pretty sure there are still many SX100 with old WDS boards, and people should know this and importance of that.


Well, there is some still hard to answer questions, i.e. is Cameca integral mode real integral mode or same "pseudo-integral" as Jeol, and other questions which You are stimulating my head to come at. I came to a plan to check that out with injecting the deterministically generated pulse train with signal generator (unplugging the signal cable from detector and plugging it to such generator). Such equipment is expensive $$$$ and out of my budget, but I found out that with some resistor ladder improvised DAC I could do that with Raspbery pico board (4$) and few electronic components (fast opamp to drive the signal $$). I will open separate thread to show how to construct such a device, program it and use it for such a purpose. This board is able to output signals at its clock speed of 133Mhz, but saw someone overclocking it to 250Mhz, anyway that is more than enough to emulate nearly exact pulse shapes emitted and fed to WDS counting electronics from Shapping amplifier near detector. I think this experiment will prove or disprove some of my claims such as:
* GPC's has no dead time (in case it is true, we should see the exactly same rate of missing pulses with increased pulse rate from such generator).
* GPC's pulses are precise - WDS counting electronics introduce PHA spread (feeding the artificially precise pulses with exact same pulse height PHA scan should produce very narrow peak if that claim is wrong).
 

[Probeman]

We have non-extendable dead time on WDS on both Jeol and Cameca instruments. Why? 1) because we have PHA peak shifts -

I am beginning to wonder more and more if this is mostly a problem with PHA shifting...

https://probesoftware.com/smf/index.php?topic=1466.msg11247#msg11247

https://probesoftware.com/smf/index.php?topic=1489.msg11230#msg11230

But in any case you definitely win the "nerd" award!   

[Brian Joy]

I’ve attached a very readable paper by Lindstrom and Fleming (1995) in which the authors examine “intrinsic” dead time due largely to the ADC as well as pulse pileup effects in pulse processing circuits that do not contain an enforced dead time.  They note that the detector (HPGe in this case) itself contributes negligibly to the dead time.  Although the discussion focuses on behavior of a solid state detector, the principles should be applicable to proportional counters as well.

By the way, does anyone happen to have a schematic for the X-RAY CONT PB to which the JEOL pre-amplifiers send their signals?  It is missing from my book of JEOL schematics, but I fear that this might not be accidental.

[Probeman]

For those interested Aurelien found an early reference for the exponential dead time correction expression in a book from 1955:

R.D. Evans, The Atomic Nucleus, McGraw-Hill, New York, 1955, p 786, Eq. 1.1

Maybe there's an even earlier reference, but in any case this is referring this expression:



where x is the observed count rate, y is the predicted count rate and b is the dead time constant.  When solving for the predicted count rate, W is the log product which has to be solved iteratively.  We utilized the method of Lambert.

[Brian Joy]

For those interested Aurelien found an early reference for the exponential dead time correction expression in a book from 1955:

R.D. Evans, The Atomic Nucleus, McGraw-Hill, New York, 1955, p 786, Eq. 1.1

Maybe there's an even earlier reference, but in any case this is referring this expression:



where x is the observed count rate, y is the predicted count rate and b is the dead time constant.  When solving for the predicted count rate, W is the log product which has to be solved iteratively.  We utilized the method of Lambert.

One of the earliest references is Schiff (1936, Physical Review 50:88-96); I've attached it.

[Probeman]

One of the earliest references is Schiff (1936, Physical Review 50:88-96); I've attached it.

Nice find. 

Interesting that this exponential expression appeared a year before the linear expression (Ruark, 1937) that is traditionally utilized today.

[Probeman]

We have non-extendable dead time on WDS on both Jeol and Cameca instruments. Why? 1) because we have PHA peak shifts - again extension of dead time is to prevent that, and as we see the shifts it is clear that there is no extension; 2) If I increase the current it the raw count rate increases, at high current it increase very little, but still it is the increase and no count rate decrease is observable even at >1µA beam at large crystal at most intense lines. It is clearly non paralyzable. 3) EDS for extendable dead time needs many different shapping amplifiers, where one fast shapping amplifier works constantly in parallel to the main high resolution (slower) amplifier. At least on Cameca WDS there is only and only one shapping amplifier integrated with Charge sensitive preamplifier in a single package, which is connected directly to the GPC - because of that there is no way to implement (and hide away in any possible means) EDS-like extendable dead time circuit.

So you agree that WDS dead time is non-extending? I agree this would seem to be true by definition, since all WDS systems count only for exactly as long as the specified count time.  But then why does Brian make this claim:

What I mean is that it appears that loss of X-ray counts in the JEOL pulse processing circuitry is dominated by pulse pileup and not dead time.  Pulse pileup is described mathematically in a manner equivalent to an extending dead time.

How can pulse pileup in a (JEOL) WDS system equate to an extending dead time model when the count time is fixed? Is the (JEOL) pulse processing electronics "saving" pulses to be counted later?  But then you go on to say:

Have pulse pile ups have anything with extendable vs not extendable?... Can pulse pile up do anything to non-extendable circuit? No not at all, as it by design does not care. It is designed with profound superstition that it is fast enough (80es, there were still no large diffracting crystals) and will not come to such situation. Also integral counting method does not care about pulse pile ups.

I agree with this, but then why does Brian say the JEOL WDS system is extending?  How could it be different from the Cameca?  Because of the "enforced" dead time of the Cameca electronics?  I am somewhat confused by these seemingly conflicting statements.

I would very much like to see you and Brian to discuss this question!

Well, there is some still hard to answer questions, i.e. is Cameca integral mode real integral mode or same "pseudo-integral" as Jeol, and other questions which You are stimulating my head to come at. I came to a plan to check that out with injecting the deterministically generated pulse train with signal generator (unplugging the signal cable from detector and plugging it to such generator). Such equipment is expensive $$$$ and out of my budget, but I found out that with some resistor ladder improvised DAC I could do that with Raspbery pico board (4$) and few electronic components (fast opamp to drive the signal $$). I will open separate thread to show how to construct such a device, program it and use it for such a purpose. This board is able to output signals at its clock speed of 133Mhz, but saw someone overclocking it to 250Mhz, anyway that is more than enough to emulate nearly exact pulse shapes emitted and fed to WDS counting electronics from Shapping amplifier near detector. I think this experiment will prove or disprove some of my claims such as:
* GPC's has no dead time (in case it is true, we should see the exactly same rate of missing pulses with increased pulse rate from such generator).
* GPC's pulses are precise - WDS counting electronics introduce PHA spread (feeding the artificially precise pulses with exact same pulse height PHA scan should produce very narrow peak if that claim is wrong). 

These experiments should be performed on both the Cameca and JEOL electronics so we can gain a better understanding of these "black boxes" that we depend on so much!

[sem-geologist]

So you agree that WDS dead time is non-extending? I agree this would seem to be true by definition, since all WDS systems count only for exactly as long as the specified count time.  But then why does Brian make this claim:

It is not that I agree or don't agree (That is not a matter of an agreement).
1) on Cameca instruments I am 100% sure it is non-extendable as I am fully aware how the hardware is built, and on Jeol I argue that by seeing secondary observations (strong shifts of PHA) that it is designed with very similar hardware (and missing hardware part which is needed for extension of dead time - else there would be no PHA shifts). However, probably on Jeol it does have some unintended pralysable behavior misidentified as "extension" by some mechanisms/processes, which are not on Cameca instrument.
2) Before we dwell further we need to distinguish that most of dead time we observe on these instruments is intentionally designed to be there and it cover-over (with huge overlap) the unintentional dead-time (the missing counts from other processes which creeps into signal process depending from count rate)... I think confusion comes from that all (EDS and WDS) systems enforce some dead time in different ways, but I think it does that for a bit different reasons and thus it is more (EDS) or less (WDS) complicated/advanced.
3) lets look to EDS "enforced" dead time. The main reason for EDS enforced dead time is energy accuracy. The counting system looks for all pulses (with very fast but low resolution pulse shapping amplifier) in parallel to the main (high resolution) shapping amplifier and rejects the currently processed pulse if any (accepted or rejected) pulse was close enough before to overlap anyhow with currently processed pulse. As such counter is keeping track of all incoming pulses it will keep rejecting pulses perpetually, unless there is enough of space before the current pulse and its amplitude then can be guaranteed to be accurate. That ability to keep rejecting pulses, unless the height of incoming pulse can be guaranteed to be accurate, - that is what makes the dead time extendable by hardware design.
4) WDS could look similar on the first glimpse, as it "enforces" some dead time. But, it does it differently: a) it enforces dead time after the sensing pulse and blinds itself from sensing any incoming pulses during the dead time (see the difference: EDS does not blind itself at the fast track - so it could keep a note of all incoming pulse, where WDS blinds itself completely) b) it could look that the reason is similrar to EDS: a simplified attempt to prevent to count the pulse coming after the sensed pulse (As there is normally negative tail of pulse, thus preventing overlapped pulse with tail) - thus only the pulse with accurate height would be counted. I initially thought that would be the reason - but, it fails completely, as system have no idea what happened before the sensed pulse (and thus we see PHA shifts on both Cameca and Jeol). Basically if it sees a pulse it holds the pulse and blinds itself (it is accepted or rejected by PHA) for "enforced" amount of time. c) I think the main reason for WDS "enforced" dead time is not accuracy (which we know fails miserable) but to have predictable dead time and overcome the bottleneck of sharing the part of pipeline by few spectrometers. In example on Cameca SX old WDS boards that is up to 3 spectrometers, where analog pulse signal is multiplexed to single shared ADC - The multiplexer requires 1µs for switch! setting dead time anything below 3µs with all three spectrometers on high count rate would not decrease the dead time! On new WDS boards multiplexing is shifted to digital domain (switching can be done at 50 MHz) on single digital bus (all five spectrometers); There setting the "enforced" dead time below 3µs shows the huge difference in count rates, even when all spectrometers are near fully saturated. Still because of multiplexing it should not be set below 1µs (and thus it is blocked from doing that) as the dead time would start to "float" depending from count rate of other spectrometers.

So it is not that "WDS systems count only for exactly as long as the specified count time" - You actually can force most of EDS systems to count for realtime and not live time, which would make it the same from that perspective. No, it is so because of the different counting design and hardware. But, why Brian brings in extending dead time? probably there is misunderstanding what is extending vs non-extending and paralyzing vs non-paralyzing. I think Jeol is at disadvantage and I think You had uncovered the reason in your other thread showing that Jeol is much more affected with PHA shifts than Cameca instruments, which introduce paralysable behavior where more and more pulses are rejected by baseline of PHA. I actually could simulate paralysable behavior at my Monte-Carlo simulation for diff mode (which demonstrates it (diff mode) is very unsuitable for high count rates), --the rejection by baseline would be a similar.

How can pulse pileup in a (JEOL) WDS system equate to an extending dead time model when the count time is fixed? Is the (JEOL) pulse processing electronics "saving" pulses to be counted later?  But then you go on to say:

Again "fixing counting time" has nothing to do with extending - non-extending. As JEOL sees more rejected pulses by PHA baseline with increasing count rate, due to severe broadening and shifting of the PHA it starts to observe paralysable behaviour. It have nothing to do with extension of dead time as hardware is blind for any pulse-pileup and don't care (same as I had wrote before).

I agree with this, but then why does Brian say the JEOL WDS system is extending?  How could it be different from the Cameca?  Because of the "enforced" dead time of the Cameca electronics?  I am somewhat confused by these seemingly conflicting statements.

First, I believe the Jeol has "enforced" dead time - the difference from Cameca is that on Jeol it is "cut-in-the-stone", where on Cameca it is "user-settable" with low boundary of 1µs (to prevent from "floating" dead time by multiplexing) and high 255µs (max of 8-bits). Anyway, as by default it is set to 3µs and most user don't change it -- that will produce less of PHA shift than on Jeol. Other reason is Jeol gain circuit looks rubish (sorry), and setting the PHA peak position centrally by changing the bias is not the best idea (the countermeasures of PHA shift by increasing bias just increase the very cause of the shift). Lastly I am not sure about that, I am near ready to test it out, but I think Jeol has "pseudo"-integral mode, where Cameca has real integral mode for counting, and that would do the huge difference introducing the paralysable behaviour for Jeol and no paralysable behaviour observed on Cameca.

The biggest confusion comes from mixing the "extending" and "non-extending" with "paralysing" and "non-paralysing" terminology.
It is not synonimous, but misunderstanding originates due to extending deadtime producing paralyzable behaviour. But it is not the same other way arround!

If it goes about mathematics: extendable dead time will revert the input count rate vs observed count rate curve at some point, and it will drop and drop untill will reach the 0 output counts at extremely high input count rate - which would be 100% dead time on the EDS.
Clearly this is extendable and paralysing.
To compare with EDS, on WDS with non-extendable deadtime, we can also see paralyzable behaviour at some point - in particular if using diff PHA mode. However, that paralyzable behaviour won't lead to 0 cps at very extremely high input count rates - it will never drop there, as that is only additional mechanism blocking some but not all pulses. It will start dropping but after some time then will go into plato. If paralysing behaviour is noted on any detector, going above that point is absolutely bad as it is not possible to calculate the real count rate (as it can be from both sides of parabolic curve). EDS gets away with that as due to tracking dead time (as it measures all incoming pulses) it knows on which side of such parabolic curve it is. WDS by not tracking the total number of pulses is blind and resolution is impossible.

As for experiment, I have access only for Cameca instruments. As soon I will have something to share I will do, hopefully someone owning Jeol probe will feel adventurous and knowledgeable enough (connecting the earth/ground clip of oscilloscope to wrong place can instantly fry the boards! be warned!) to do such experiments for Jeol.

P.S. above described "double track" pipelines on EDS was on previous detectors. Newest generation of EDS detectors most probably has no more of double tracking but resolve the pileups with terrific beefy Digital Signaling Processors on FPGA's (I am aware that some EDS vendors had moved there - the outcome is terrific: You wont see any pulse-pile ups even with 90% dead time!!!). That is what I would like to go with for WDS too.

[Probeman]

First, I believe the Jeol has "enforced" dead time - the difference from Cameca is that on Jeol it is "cut-in-the-stone", where on Cameca it is "user-settable" with low boundary of 1µs (to prevent from "floating" dead time by multiplexing) and high 255µs (max of 8-bits). Anyway, as by default it is set to 3µs and most user don't change it -- that will produce less of PHA shift than on Jeol. Other reason is Jeol gain circuit looks rubish (sorry), and setting the PHA peak position centrally by changing the bias is not the best idea (the countermeasures of PHA shift by increasing bias just increase the very cause of the shift). Lastly I am not sure about that, I am near ready to test it out, but I think Jeol has "pseudo"-integral mode, where Cameca has real integral mode for counting, and that would do the huge difference introducing the paralysable behaviour for Jeol and no paralysable behaviour observed on Cameca.

The biggest confusion comes from mixing the "extending" and "non-extending" with "paralysing" and "non-paralysing" terminology. It is not synonimous, but misunderstanding originates due to extending deadtime producing paralyzable behaviour. But it is not the same other way arround!

If it goes about mathematics: extendable dead time will revert the input count rate vs observed count rate curve at some point, and it will drop and drop untill will reach the 0 output counts at extremely high input count rate - which would be 100% dead time on the EDS. Clearly this is extendable and paralysing.

Thank-you for your thoughts on this very complicated topic.  It will be interesting to see further results from your investigations. 

We have additional recent PHA data from Anette's JEOL instrument that I will be posting soon.  This data should be compared to the PHA data from my Cameca instrument which is here:

https://probesoftware.com/smf/index.php?topic=1466.msg11271#msg11271

But would be nice to see some PHA data from your instrument as well as some constant k-ratio measurements!

As for the fact that Cameca users tend to adjust their PHAs by setting the bias to a fixed number and adjusting the gain to center the PHA peak, while JEOL users tend to do the opposite (set the gain and adjust the bias to center the PHA peak- I don't know how to modify that behavior as the gain settings on the JEOL are in 2x increments which is very coarse.

Anette is also going to post more detailed information on JEOL PHA behavior as she tried centering her PHA peaks at various gain settings.  She has a lot of data to share.

[Probeman]

Maybe we can test some of these ideas regarding paralyzable/non-paralyzable with just looking at the raw data?  I plotted raw count (observed) rates from my LTAP and Anette's TAPL on Si metal and of course the Cameca is "topping out" at a lower count rate due to its higher nominal dead times (JEOL = ~1.5 usec vs. Cameca = ~3 usec) but perhaps if we keep going...



Clearly this graph doesn't take it far enough, but perhaps if we continue to increase the beam current we can see if the behavior at even higher count rates produces a different response?

But remember, one must carefully adjust their PHA settings to keep their PHA peak above the baseline to avoid affecting the measurement.  I suggest performing PHA scans at several beam currents over the range being utilized for these tests.  The JEOL seems to be more susceptible to pulse height depression so it is especially important for that instrument to monitor the PHA peak as the beam current is increased.

Another problem specific to JEOL instruments is that if one is adjusting the bias voltage to compensate for pulse height depression (as is usually the case) we have to ask ourselves, could different bias voltages produce different dead times, or are these effects minor compared to the dead times of the pulse processing electronics?

[sem-geologist]

Maybe we can test some of these ideas regarding paralyzable/non-paralyzable with just looking at the raw data?  I plotted raw count (observed) rates from my LTAP and Anette's TAPL on Si metal and of course the Cameca is "topping out" at a lower count rate due to its higher nominal dead times (JEOL = ~1.5 usec vs. Cameca = ~3 usec) but perhaps if we keep going...

Now about "topping out", that is a wrong term - it should be "flatting out".
As Far I remember for other posts Your SX100 have a new type of WDS card. Set DT (hardware) to 1µs - You will see it gives more benefits at integral mode for high count rate than drawbacks (it is not the same as old board - It is not the same experience). It will move the slope-flattening toward much higher currents (and higher count rates) - and thus will produce lesser uncertainty at 100-200nA range. With 1µs set as DT (hardware) at least on our SXFiveFE I had not seen it go down even going way up to 1000nA. The problem is that it starts to be useless at such conditions, as due to very flat slope the recalculation of measured count rate  to the input count rate even with the most perfect equations would have a very huge uncertainty (due to flat slope - as the calculated input count rate would get extremely sensitive to measured raw count rate).

Another problem specific to JEOL instruments is that if one is adjusting the bias voltage to compensate for pulse height depression (as is usually the case) we have to ask ourselves, could different bias voltages produce different dead times, or are these effects minor compared to the dead times of the pulse processing electronics?

I often forget that we talk about different dead times. From Your "dead time" (as a single constant including everything) position - Yes, absolutely. From my point of view looking into dead time of "this part" and dead time of "that part" the answer is - not at all. The missing counts (which makes us to think the detector is blind) are not due to bias, but due to PHA baseline, the reduction is due to rejected pulses, not due to not seen pulses. The PHA is not dead at all when it does it - it does it with a complete premeditation. The problem is that increasing bias will allow to center the pulse, which baseline (the bottom of the pulse) had shifted to negative voltages - by increasing bias the PHA spectrum is not only shifted, but also zoomed-in which enhance the pulse broadening (there are also other causes making pulse broadening, and increasing bias just make it more pronounced) and with increasing bias for centering at high count rates its left side (PHA distribution) is getting more and more being rejected by PHA baseline.

"rejected"... For a moment I thought - "wait a minute, maybe I am wrong about the extending vs paralysing as counter on EDS is also rejecting" - this looks like some surface for "extension" bit to to reaper, but no. After WDS PHA pulse rejection there is still a follow-up of fixed time blinding-of itself (counting electronics), where on EDS PHA despite pulse being rejected the counting system stays focused on all incoming pulses. So math for extending dead time (borrowed from EDS) is completely not fit to be tossed-in/mingled-into the equation for Jeol WDS counting dead time. It should be something else.