New method for calibration of dead times (and picoammeter)

[John Donovan]

With regards to Probeman's post above we've updated our Constant K-Ratio method document which is attached below (remember to log in to see attachments), to emphasize the importance of constructing the k-ratio with both the primary and secondary standards being measured at the same beam current for each beam current measurement.

[Probeman]

We are pleased to announce the publication of our new paper on improving dead time corrections in WDS EPMA:

John J Donovan and others, A New Method for Dead Time Calibration and a New Expression for Correction of WDS Intensities for Microanalysis, Microscopy and Microanalysis, 2023

https://academic.oup.com/mam/advance-article-abstract/doi/10.1093/micmic/ozad050/7165464

How to get the "best" dead time constant which produces the "best" zero slope "k-ratio v. current" line in your constant k-ratio method? I don't have the PROBESOFTWARE, so can I still perform dead time calibration by applying the constant k-ratio method?

Yes, you can perform the dead time calibration using the constant k-ratio method for general use as described here:

https://smf.probesoftware.com/index.php?topic=1466.msg11102#msg11102

But if you do not have the Probe for EPMA software, you will not be able to take advantage of the new non-linear expressions described in our paper for count rates above 30 to 50 kcps. In other words, you could perform the dead time calibration using the constant k-ratio method using the traditional linear expression, but you probably won't see a huge difference in the dead time value you obtain below 30 to 50 kcps.  Above those count rates, the tradition linear expression fails as shown in the linked post below.

That said, I think the constant k-ratio method is much easier to use and more precise for dead time calibrations whatever dead time expression is being used...

See here also:

https://smf.probesoftware.com/index.php?topic=1466.msg11173#msg11173

I like to know how the ideal dead time constant is produced. In the example of "TiO2/Ti"  shown in your post, when you found the DT constant (1.32 us) was too high, you dropped it  to 1.28 us and got a more "flat" line. So my question is why it is 1.28, but not 1.27 or 1.29. How is the exact number produced, using some algorithm or by repeated iterate and trial? If the later is the case, how do you determine the observed "k-ratio v. current" line， which depends on the exact DT constant, is the best?

It is easy. One simply adjusts the dead time constant (depending on which dead time expression is being utilized (because the dead time constant is actually a *parametric* constant as described in the paper) in order to obtain a trend with a slope close to zero.

Therefore, the goal is to obtain a dead time constant (with a suitable dead time expression) that yields a flat (zero slope) response from low count rates to the highest possible count rates.  One can do this visually by inspection because a zero slope is easy to evaluate.

Obviously at some point (above several hundred kcps) the dead time correction becomes very large and quite sensitive to small changes in the dead time constant.  But we found the logarithmic dead time expression can yield quantitative results from zero to several hundred kcps count rates once the dead time value was properly adjusted.

Don't forget, as described in the paper one can also utilize the same constant k-ratio dataset to check their picoammeter linearity and also the agreement of (simultaneous) k-ratios from one spectrometer to another (and even from one instrument to another given the same materials), which can be used to check one's effective take-off angle for each (WDS and EDS) spectrometer.

It should be pointed out that one can switch simply to the logarithmic WDS dead time correction method and keep using the dead time constant that was originally obtained using a linear extrapolation (limited to 30 to 40 kcps throughput):



Why do I say this?

Because even though you might be slightly over estimating your dead time at very high throughput (> 50 kcps) using the logarithmic expression, your accuracy will still be significantly better than using the old (traditional) linear dead time correction method:



And also, below 50 kcps the linear and logarithmic expressions give essentially identical results.

[dcoulthard319]

Hello,

Last week during our first PFE lab visit, we were getting in the weeds and gathering some data for calibration of dead time constants. I am still in the process of sourcing our standard blocks here at Colorado State, so I'm left to work with the Smithsonian suite of minerals and the JEOL maintenance block. Our channel loadout was initially Ch1 and Ch2 = Si on TAP, Ch3-5 = Zr (la) on PET (PETJ on Ch3 and PETLs on 4 and 5). We programmed a run to last a couple hours. Since we decided to use Si ka to calibrate channels 1 and 2, we focused our work on the NMNH Quartz and San Carlos olivine reference materials; setting the beam to 15 kV, with current ranging between 5- and 200 nA.

The attached spreadsheet contains the data from this initial run as well as the one that followed, which focused on the JEOL reference materials Zirconium and Zirconia. Contained in sheets 1 and 2, respectively. Data gathered using Channel 1 formed k-ratios that formed a linear array that was fairly easy to correct back to constant k-ratios using the methods described here and using the on-board pdf available in PFE. However, when we plotted the k-ratios for Channel 2, we observed a non-linear shape to the line with low calculated k-ratios at low current, rising to a maximum at ~40 nA and decreasing again towards our maximum current setting. Compare the signals from Channels 1 and 2 here:



We decided to take a closer look at the counts from these spectrometers, and they appeared to be linear throughout the session:



After scratching our heads for a little bit, we decided to repeat the entire procedure using PET crystals, which are available on all spectrometers. We again observed a hook-shaped curve in the low-current k-ratios on channel 2 that is not observed on any other channel. Note that the following figure was made after dead time constants were estimated for this spectrometer:



Again, the counts themselves (as cps) appear to be linear:



Now, I recognize this is not a substantial effect. Using the average k-ratios available, I calculated a 0.8% relative effect (max) from our observations. The critical part is that these effects are occurring where a vast majority of our analyses are going to be taking place (i.e., below 50 nA), and I'd like to understand this seemingly unique behavior better. Any input on this matter would be greatly appreciated!

~Danny C

[John Donovan]

It's amazing how sensitive the constant k-ratio method is for seeing instrumental artifacts. The above plots showing raw counts vs. beam current look linear but of course the k-ratio plots show they are not. You read our paper attached below?

https://smf.probesoftware.com/index.php?topic=1466.msg12465#msg12465

Your data is very interesting.  The artifacts you are seeing are relatively small but clearly significant. 

I might have said that a hump like that could be induced from a non-linearity in the picoammeter, but since it only affects a single spectrometer, it can't be that since that would affect all spectrometers.  Same for the composition of the standards. If they were heterogeneous, you would see that trend on all the spectrometers.

It could be from a PHA adjustment issue, for example  if the PHA depression was worse in one spectrometer than than another.  That is why I suggest making these constant k-ratio measurement using a completely wide open PHA in INTEGRAL mode where the PHA peak is *above* the baseline at the high beam current in the highest concentration material:

https://smf.probesoftware.com/index.php?topic=1466.msg11636#msg11636

As the count rate decreases at lower beam currents the PHA peak will shift up, but if you are in integral mode you will still see all the intensities even as the PHA peak shifts beyond the visible range to the right:

https://smf.probesoftware.com/index.php?topic=1466.msg11450#msg11450

The good news is that your variation between spectrometers is pretty small. Worth trying to figure out, but not anywhere as bad as some instruments:

https://smf.probesoftware.com/index.php?topic=1466.msg11531#msg11531

First, check your PHA adjustments and make sure they are wide open (increase gain until all PHA peaks are above the baseline at the highest current in the highest concentration material) and that you are in INTEGRAL mode and try again.

A difference in the effective takeoff angle between spectrometers would explain the vertical differences between the spectrometers, but again, why one spectrometer's k-ratios would be sloped positive and then negative I can't explain.  It's almost as though the counting electronics has two different pulse processing ranges (dead times).

Would it be possible for your tech to swap the spectrometer electronics with another spectrometer to see if the problem moves with the electronics?

[John Donovan]

As mentioned previously, the constant k-ratio method is surprisingly sensitive to instrumental artifacts. Danny kindly sent me his k-ratio measurement data files and I took a look, and while I can't quite explain everything I see, it is interesting.

First let's start with his Si Ka k-ratio measurements on San Carlos olivine (secondary std) and SiO2 (primary std) from 5 to 200 nA.  I noticed that he had not turned off the standard intensity drift correction, so I did that, though it didn't change things much because he had acquired his primary and secondary standards in "pairs" with the primary standard measured first for every k-ratio pair. Here's the unaveraged data for Si Ka on SP1 (TAP) and SP2 (TAP):



The k-ratios are pretty close, but as Danny mentioned, it is bothersome that the greatest divergence in the k-ratios occurs at beam currents 40 nA and below.

Next I disabled all the primary (SiO2) stds except the first one to look at the picoammeter linearity, because by measuring our primary and secondary stds in pairs, we null out any effects from picoammeter non-linearity. Now we see this:



So the picoammeter isn't quite as linear as we would like, but there's more going on here than just that.

I then wondered if the downward trend in the intensities for sp2 could be due to the dead time constants, because I noticed that all the spectrometer dead times were the JEOL default 1.1 usec.  Adjusting the sp2 dead time from 1.1 usec to 1.0 usec I obtained this plot:



The dead time constant primarily affects the high intensity count rates, so this makes sense.  My best guess at this time is that there is a very small alignment problem in sp2 that is compounded by some non-linearity in the picoammeter.

Interestingly, Danny also sent me some Zr La k-ratio measurements he did on all his PET crystals (all 5 spectrometers), going up to 1000 nA(!) and the results are actually quite (amazingly) good:



This is actually what we want to see for all spectrometers for all Bragg crystals!

Can your instrument perform as well?  Measure some constant k-ratios on your own spectrometers (both EDS and WDS) over a range of beam currents, and find out!

[Probeman]

I know I've been "going on" about proper adjustment of the PHA electronics for quantitative analysis for a bit, but the following may shock some of you.  But, it's absolutely correct. 

In our MgO, Al2O3 and MgAl2O4 FIGMAS analyses:

https://smf.probesoftware.com/index.php?topic=1823.0

we are measuring O Ka in addition to Mg and Al.  But tuning the oxygen PHA is problematic due to the extremely wide PHA peak as seen here:



You can see it better if I zoom in here:



And here's the shocking part. The PHA scan here (below) is correct once the gain is increased slightly:



But it sure doesn't look it!  How can this be correct?

It's because in Integral mode all the counts shown as "cut off" to the right of the scan are NOT cut off, they are still counted.  If we tune our PHA peak so that it is being cut off to the left by the beaseline level, changing count rates will affect our electronics linearity as the PHA peak shifts.

But if we tune the PHA peak by increasing (or decreasing) either the gain (Cameca) or the bias (JEOL) on a material with the highest concentration (proxy for count rate) at the highest beam current we will be utilizing and make sure that our PHA peak is completely above the baseline level, we will obtain a totally linear response from our counting electronics.

Yes, when we go to a lower concentration or a lower beam current, we will see the PHA peak shift, but ONLY TO THE RIGHT. And all those photons will be counted in Integral mode!

Now some might say don't we need to be in differential mode to use the window level to eliminate higher order interferences?  It is true that a portion of higher order interferences (with their 2x, 3x, etc energy levels) will be screened by the window level (though no first order interferences will). But only at the cost of a non-linear response of the counting electronics. It is far better in my opinion to run in integral PHA mode and correct any spectral interferences quantitatively in software:

Donovan, John J., Donald A. Snyder, and Mark L. Rivers. "An improved interference correction for trace element analysis." Proceedings of the Annual Meeting-Electron Microscopy Society of America. San Francisco Press, 1992.

That the photon count rate does not change when the PHA peak is not visible to the right is demonstrated by the following plot which plots counts versus PHA gain, where at the highest gain levels the PHA peak is completely shifted off to the right but the count rate stays the same!

Again, we do not care that the peak is being "cutoff" on the right side of the plot because in INTEGRAL mode the PHA system still counts those pulses as previously demonstrated using a gain test acquisition on Ti Ka on Ti meta:

The same practice applies to all elements when tuning PHA. That is, adjust the gain or bias so that your PHA peak is completely ABOVE the baseline level (when the beam is on a material with the highest concentration you will be measuring (usually the primary standard), at the highest beam current you will be utilizing). If part of the PHA on the right is not visible, that is OK when in Integral mode as those photons will still all get counted.

See here (and subsequent posts) for more details on PHA tuning:

https://smf.probesoftware.com/index.php?topic=1475.msg11330#msg11330

[aducharme]

Reading the deadtime correction paper, the logarithm appears to have fallen out by extrapolating the x + x^2/2 terms as the first two terms in the Taylor series of log(1-x)=\sum x^n/n. To me, those first two terms are hallmarks of the Taylor series for e^x = \sum x^n/n!. This gives another possible deadtime correction i/(2-e^(i*\tau)). It is surprisingly similar to the logarithm-based correction:


Maybe this would get rid of the squiggle? Or perhaps it's not a good model. John, would you be able to give it a shot?

[John Donovan]

Reading the deadtime correction paper, the logarithm appears to have fallen out by extrapolating the x + x^2/2 terms as the first two terms in the Taylor series of log(1-x)=\sum x^n/n. To me, those first two terms are hallmarks of the Taylor series for e^x = \sum x^n/n!. This gives another possible deadtime correction i/(2-e^(i*\tau)). It is surprisingly similar to the logarithm-based correction:
Maybe this would get rid of the squiggle? Or perhaps it's not a good model. John, would you be able to give it a shot?

Yes, we implemented this dead time correction, but the problem is that it "blows up" at relatively low count rates depending on the magnitude of the dead time constant. See the last dead time option here in Probe for EPMA:

[aducharme]

No, that has to be a different definition of "exponential." The plot in this post
(https://smf.probesoftware.com/index.php?msg=11246) shows the PfE-implemented exponential deadtime correction is larger than the logarithmic correction, but my exponential correction is smaller than the logarithmic correction. Also, the PfE correction is shown to have a discontinuity at 1/e, before the logarithmic correction's discontinuity at 1 - 1/e (approx 0.631). 1/(2-exp^x) has its discontinuity at ln(2) (approx 0.693), which is further right than the logarithmic correction.

[John Donovan]

No, that has to be a different definition of "exponential." The plot in this post
(https://smf.probesoftware.com/index.php?msg=11246) shows the PfE-implemented exponential deadtime correction is larger than the logarithmic correction, but my exponential correction is smaller than the logarithmic correction. Also, the PfE correction is shown to have a discontinuity at 1/e, before the logarithmic correction's discontinuity at 1 - 1/e (approx 0.631). 1/(2-exp^x) has its discontinuity at ln(2) (approx 0.693), which is further right than the logarithmic correction.

OK, I stand corrected. 

Let's work on implementing this when you get a chance and see how it does with some data.

[Probeman]

Reading the deadtime correction paper, the logarithm appears to have fallen out by extrapolating the x + x^2/2 terms as the first two terms in the Taylor series of log(1-x)=\sum x^n/n. To me, those first two terms are hallmarks of the Taylor series for e^x = \sum x^n/n!. This gives another possible deadtime correction i/(2-e^(i*\tau)).

Maybe this would get rid of the squiggle? Or perhaps it's not a good model. John, would you be able to give it a shot?

It IS surprisingly similar to the Log (Ln) dead time correction, but it doesn't appear to be much better.  Here's the "squiggle" plot you mentioned using the Log (Ln) dead time correction:



The variance of the k-ratio is 0.00176

Here is the Ducharme Exp dead time correction with the same dead time constant of 2.67:



Not too good, but if the dead time constant is adjusted to be a little higher to 2.79 usec, it looks like this:



A bit better (standard deviation = 0.00129) but I think we are at the limit of fixing things with math. What we really need is faster pulse processing electronics!  Fortunately the JEOL has dead times around 1.1 to 1.3 usec (compared to the above Cameca dead times of 2 to 3 usec) and that certainly helps. But because of large area crystals on modern instruments, we're still seeing very large dead time corrections on JEOL TAPL crystals. For example, count rates of around 700 kcps for Si Ka in Si metal at 120 nA:

St    5 Set   8 Si (JEOL)
TakeOff = 40.0  KiloVolt = 15.0  Beam Current = 120.  Beam Size =   15

On and Off Peak Positions:
ELEM:    Si ka   Si ka   Si ka
CRYST:     TAP    PETL    TAPL
ONPEAK 77.6720 228.237 77.3930
OFFSET -.16721 -.16292 .111794
HIPEAK 82.6720 233.237 82.3930
LOPEAK 70.6720 223.237 70.3930
HI-OFF 5.00000 5.00000 5.00000
LO-OFF -7.0000 -5.0000 -7.0000

PHA Parameters:
ELEM:    Si ka   Si ka   Si ka
DEAD:    1.230   1.270   1.200
BASE:      .70     .70     .70
WINDOW    9.30    9.30    9.30
MODE:     INTE    INTE    INTE
GAIN:      32.     16.     32.
BIAS:    1690.   1745.   1740.

On-Peak (off-peak corrected) or EDS (bgd corrected) or MAN On-Peak X-ray Counts (cps/1nA) (and Faraday/Absorbed Currents):
ELEM:    Si ka   Si ka   Si ka   BEAM1   BEAM2
BGD:       OFF     OFF     OFF
SPEC:        1       3       4
CRYST:     TAP    PETL    TAPL
ORDER:       1       1       1
   71G 2820.56 1334.98 5891.48 120.104 120.104
   72G 2823.54 1334.85 5895.98 120.104 120.091
   73G 2823.42 1338.11 5895.73 120.066 120.054
   74G 2819.48 1338.63 5891.42 120.091 120.079
   75G 2810.41 1330.26 5890.46 120.029 120.041

AVER:  2819.48 1335.36 5893.02 120.079 120.074
SDEV:     5.37    3.34    2.63    .032    .026
1SIG:      .52     .39     .63
SIGR:    10.43    8.58    4.19
SERR:     2.40    1.49    1.17
%RSD:      .19     .25     .04
DEAD:    1.230   1.270   1.200
DTC%:     49.7    22.3   110.0

A dead time correction of 110%! That be crazy!