An alternate means of calculating detector dead time

[Brian Joy]

As I noted elsewhere, Heinrich et al. (1966, attached) proposed an alternate method for calculating the non-extendible dead time of proportional X-ray counters; the method does not rely on measurement of beam current.  Their approach, which was adapted from that of Short (1960, Review of Scientific Instruments 31:618-620) for XRD systems, relies on calculation of ratios of apparent count rates of two X-ray lines measured simultaneously on two spectrometers over a range of beam currents.  Two datasets must be collected, with the chosen X-ray lines measured alternately on each spectrometer.  For instance, if Si Kβ and Si Kα are used in turn, then Si Kβ (observed count rate = N'₁₁) is collected on the first spectrometer while Si Kα (N'₂₁) is collected simultaneously on the second.  In the next dataset, Si Kα (N'₁₂) is collected on the first spectrometer while Si Kβ (N'₂₂) is collected on the second.  The dead time for each spectrometer can then be calculated as follows:

spectrometer 1:  τ₁ = (m₁ – m₂)/(b₂ – b₁)
spectrometer 2:  τ₂ = (b₂m₁ – b₁m₂)/(b₂ – b₁)

In each equation, m₁ and m₂ are the respective slopes on plots of N'₁₁/N'₂₁ versus N'₁₁ and N'₁₂/N'₂₂ versus N'₁₂, while b₁ and b₂ are the respective intercepts.  Note that the expressions as presented may only be applied in the range in which each plot shows linear behavior.

I’ve attached a spreadsheet that can be used as a template for the calculations (“method 2”).  The dead times calculated for Si Kα/Kβ (elemental Si, uncoated, 15 kV) on my channels 1 and 4 (TAPJ each) are similar to those obtained from the more typical beam current-dependent calculation (“method 1,” modified from Paul Carpenter's spreadsheet).  I haven’t yet compared results of the two methods for other cases.

[Brian Joy]

Rather than worrying about accurate dead time corrections at extraordinarily high count rates, I’d prefer to focus first on accurate characterization of dead time within the range of count rates within which I typically work while running quantitative spot analyses.  I rarely exceed 10 kcps on either standard or unknown and always work within the range in which the measured X-ray count rate divided by the true rate is an essentially linear function of the measured rate.  On the instrument that I operate, I see such linear behavior up to about 70 kcps using the P-10 gas-flow counters and up to at least 80 kcps using the sealed Xe counters.

Picoammeter inaccuracy can be a major obstacle when characterizing dead time using the relation, N’/I = k(1-N’τ) (as well as expansions of it).  On a plot of N’/I versus N’, the intercept of the regression line gives the value of k, and the negative of the slope divided by the intercept gives the dead time, τ.  Obviously, since measurement of current is required for the calculation, inaccuracy in that measured value will produce inaccuracy in the calculated dead time.  In contrast, not only does the “ratio method” not even require measurement of current, it should also minimize the effects of any problem that affects each spectrometer equally or at least roughly so.  This could include, for instance, accumulation of contamination around the beam.  Consequent absorption of electron energy would cause transition metal Kα and Kβ count rates to fall over time, yet the ratios of those count rates measured on different spectrometers would likely not be affected noticeably.

I’ve used the ratio method to determine dead times using both Cu and Fe metal; I’ve attached a spreadsheet containing my data and plots for Fe.  For this latest set of measurements, I used high purity Fe metal mounted in Buehler Konductomet conductive resin; I did not apply a conductive coat.  I lightly polished (0.25 μm diamond) and thoroughly cleaned the mount surface prior to the two data collection runs.  For each of the two measurement sets for Fe, I counted at either the Kα or Kβ peak position (details below) on three spectrometers simultaneously at 55 values of beam current as measured on the PCD.  The high voltage supply had been on for perhaps 30 hours when I started, and so the beam current was extremely stable.  Each set of measurements at a given beam current was collected at a different spot.  I made all measurements using the JEOL X-ray counter built into PC-EPMA.  I aimed for at least 30000 apparent counts for any given X-ray line and current and adjusted the count time accordingly down to a minimum of 10 seconds.  Collecting the data took me perhaps four hours (for a given metal).  In reflected light, I saw no obvious accumulation of contamination around the beam during the runs (though I’m sure I would have seen some in a secondary electron image – I neglected to check).  I collected the data for Cu on the JEOL 13-element mount in a similar manner.

Prior to making the measurements, I collected PHA scans at the Kα and Kβ peak positions at 200 nA (though the maximum current I used was 320 nA during the second run).  I was particularly concerned with ensuring that the Xe escape peak did not collide with the baseline at high count rate.  Additionally, I had to make certain that any “ghost peaks” produced by my aging counters would remain within the window.  Generally, I opted to increase electronic gain rather than anode bias and centered the 200 nA distribution around 5 volts (with baseline at 0.7 V).  (Also, working at relatively low anode bias should help to minimize shifts in the pulse amplitude distribution and should extend Xe counter lifetime.)  When I decreased the beam current to ~5 nA at these new detector settings, the center of each distribution shifted to between 6 and 6.5 V (roughly), and counts fell to near-zero by 8-8.5 V.

As I noted before, I counted X-rays simultaneously on three spectrometers.  During the first measurement set, I counted at the Kα peak position on channel 2/LIFL and at the Kβ peak position on channel 3/LiF and channel 5/LiFH.  During the second measurement set, I counted at the Kβ peak position on LiFL and at the Kα peak position on LiF and LiFH.  Due to this arrangement, I obtained two values for channel 2 dead time.  For both Fe and Cu, the ratio method produces larger dead time values than the current-based method:

Fe Kα/Kβ ratio method [μs]:
channel 2/LiFL:  1.44, 1.48
channel 3/LIF:  1.41
channel 5/LIFH:  1.41
Current-based method, Fe Kα [μs]:
channel 2/LiFL:  1.32
channel 3/LiF:  1.13
channel 5/LiFH:  1.31
Current-based method, Fe Kβ [μs]:
channel 2/LiFL:  0.97
channel 3/LiF:  0.22
channel 5/LiFH:  0.85

Cu Kα/Kβ ratio method [μs]:
channel 2/LiFL:  1.50, 1.46
channel 3/LIF:  1.45
channel 5/LIFH:  1.38
Current-based method, Cu Kα [μs]:
channel 2/LiFL:  1.37
channel 3/LiF:  1.06
channel 5/LiFH:  1.25
Current-based method, Cu Kβ [μs]:
channel 2/LiFL:  0.96
channel 3/LiF:  -0.12
channel 5/LiFH:  0.76

The following plots illustrate the data collected for Fe.  For the plots constructed using the ratio method, I've omitted data collected at greater than about 85 kcps.  For the plots related to the current-based method, I've included all data collected and, on the Fe Kα plot, have highlighted data used in the regression with black borders.





A pattern is apparent in the numbers, noting especially that a negative dead time (positive slope) is physically impossible (Cu Kβ/LiF).  On my instrument, when current is high but count rate is relatively low, such as when measuring at the Kβ peak on any spectrometer and when measuring at the Kα peak on channel 3/LiF, the dead time is clearly underestimated using the current-based approach.  The dead time values appear to progress monotonically downward as count rate at given current decreases.  One potential explanation for this is that the picoammeter is reading systematically higher (relative to the true current) with increasing current.  This would either 1) accentuate the departure from linearity on plots of N’/I versus N’ at high current (Kα/channel 5/LIFH/N’₃₂) or 2) give the appearance of a departure from linearity at anomalously low count rate (Kα/channel 3/LiF/N’₂₂).  It could also account for the near-zero slope on the plot for Fe Kβ on channel 3/LiF/N’₂₁ (also true for Cu Kβ).  In the past, I believe that I’ve systematically underestimated the dead time on my channel 3 (LiF/PETJ) due to its low count rate at given current.  Of course, if my idea is correct, then I should obtain a larger apparent dead time value via the current-based method when counting X-rays using PET rather than LiF, and I haven’t tested this yet.  Like I’ve noted before, I always minimize problems due to inaccurate dead time correction by either 1) working at relatively low count rate (<10 kcps) or 2) when count rates are higher, roughly matching the count rate on the standard with that on the unknown at given current. 

As a final note, I haven’t yet propagated counting error through my dead time calculations.  I’ll get to this eventually, as the use of ratios increases the contribution of random error in estimated uncertainty.  As I’ve noted above, though, systematic error appears to be more influential than random error in the current-based calculation.

[Brian Joy]

And here are the corresponding plots for Cu (collected at V = 15 kV):

[Probeman]

After thinking about this a bit, it seems to me that this Heinrich Ka/Kb ratio method is essentially similar to the k-ratio method that John Fournelle and I came up with recently!  Think about it...  they're sort of doing the same thing.

The point being that the ratio of the Ka and Kb lines on a single material should be constant on two spectrometers, just as the (k-)ratio of a single emission line of any two materials with different concentrations should also be constant as a function of beam current on a single spectrometer!  Both methods assume that the ratios of the emission line(s) (at significantly different count rates) remain constant!

The Heinrich method is cool because it doesn't depend on the picoammeter, but the k-ratio method does not either as I will explain.  Both methods require measurements at fairly high beam currents in order to obtain information on the dead time effects which will be negligible at low beam currents. The higher the count rates (beam currents), the better the handle we obtain on our determination of our dead time constants.

However, the constant k-ratio method can also reveal any hidden problems with ones picoammeter (when plotted differently as mentioned in the next paragraph), which is also important since we would like to utilize different beam currents for quant analysis.

The k-ratio method does in fact have an independence from beam current and picoammeter (mis)calibrations because the k-ratios that one produces, are for a primary standard and a secondary standard both measured at the same beam current!  I neglected to emphasize this enough. The point being that the k-ratio should be the same given that the count rates are significantly different for the two materials (at each of the different beam currents).  Any miscalibration of the picoammeter only reveals itself when plotting the k-ratios for the secondary standards (at multiple beam currents) using a single primary standard measured at a low beam current!  I should have emphasized that point also.  See this post here:

https://probesoftware.com/smf/index.php?topic=1466.msg10972#msg10972

But still, it's interesting that you found that the Ka/Kb ratio method gives such different answers compared to the "current" method. I suspect that this is because you're not using the expanded dead time expression.  Did you try the six term Taylor expansion series expression?  I find the traditional single term expression works well up to 50K cps or so, the two term expression up to about 100K cps, and the six term expression seems good to 200K cps and more.  Why limit our counting rates when we have a better dead time expression now?

The point is that performing these ratio tests using high beam currents is not necessarily because we will actually be running at such high count rates, but rather to ensure that our dead time correction is robust over a large range of beam currents. That is to say, if the dead time correction expression works at very high count rates, it will work even better at low count rates. 

That said, in our lab we often measure minor and trace elements using relatively high beam currents so this expanded dead time expression seems just the ticket to maintain k-ratio accuracy in all possible combinations of high concentrations and/or large area crystals and/or high beam energies and/or high beam currents.

See this post for all three dead time correction expressions:

https://probesoftware.com/smf/index.php?topic=1466.msg10909#msg10909

A question about picoammeters: Cameca has different ranges of adjustment and we are working towards obtaining a high accuracy current source to calibrate our picoammeter, but I head from someone that JEOL uses a different system for their current measurements. Do you know anything about the JEOL picoammeter electronics and or its adjustments? 

You should also try all three dead time expressions and let's see what you get. I bet the range over which you obtain a linear response from your spectrometers is greatly extended by using the six term expression.

[Brian Joy]

Yes, if all you’re doing is measuring the two materials on the same spectrometer at the same current, then this would help address the problem, but it should also allow you to use an expression for dead time that is independent of current.  When you use the current-based expression to determine the dead time for this case, at the very least you are making an unnecessary calculation.  Heinrich et al. (1966) present the proper approach.

I also emphasized that I like to keep things simple.  I have no interest in using a non-linear expression at high count rate; instead, I’d rather avoid those high count rates.  I do in fact operate at relatively high current (say, 50 or 100 nA) when analyzing for elements in low concentration, but the dead time correction should be simple because the count rate is relatively low.  Also, keep in mind that Jercinovic and Williams (2005, Am Min 90:526-546) pointed out that operating at high current can cause ablation of the carbon coat and subsequent accumulation of static charge.  How closely do you monitor absorbed current when operating at high current?

When you suggest that I try all three of the current-based calculations, you’ve totally missed my point.  In my plots using the simplest (linear) current-based calculation, I show that dead time appears to be calculated incorrectly at low count rate (< 50 kcps) when current is high.  This suggests that systematic error is present in the picoammeter reading (as it cannot be expected to be perfectly accurate).  Adding terms to the current-based dead time expression will not fix this problem and is not necessarily a “better” approach.

[Probeman]

Yes, if all you’re doing is measuring the two materials on the same spectrometer at the same current, then this would help address the problem, but it should also allow you to use an expression for dead time that is independent of current.  When you use the current-based expression to determine the dead time for this case, at the very least you are making an unnecessary calculation.  Heinrich et al. (1966) present the proper approach.

The constant k-ratio method is essentially independent of beam current because the two materials are always measured at the same beam current.  If the beam current reading is off by a few percent, that would merely slide the points along the x-axis slightly. The dead time correction (traditional or expanded!) would handle this automatically because these expressions are *not* based on beam current, they're based on count rate.  You seem to be missing this essential point.

You do realize that the traditional dead time correction is an incomplete mathematical treatment of dead time because it only utilizes the first term of the Taylor expansion series which is an infinite probability series?  The expanded dead time expression merely takes that into account by including a few extra terms to deal with extremely high count rates.

The Heinrich method still requires a dead time correction in order to obtain a linear response for the slope calculation.  By using the expanded form of the dead time correction, you would be able to utilize a wider range of beam currents and therefore higher count rates and still maintain a linear response for your slope calculations.

I also emphasized that I like to keep things simple.  I have no interest in using a non-linear expression at high count rate; instead, I’d rather avoid those high count rates.  I do in fact operate at relatively high current (say, 50 or 100 nA) when analyzing for elements in low concentration, but the dead time correction should be simple because the count rate is relatively low.  Also, keep in mind that Jercinovic and Williams (2005, Am Min 90:526-546) pointed out that operating at high current can cause ablation of the carbon coat and subsequent accumulation of static charge.  How closely do you monitor absorbed current when operating at high current?

Well then you should love the constant k-ratio method because there's nothing simpler in our field than a k-ratio measurement.  Which, by the way, is the essential measurement we make for quantitative analysis. These instruments are k-ratio measurement tools. That's all they are.

I also like the fact that constant k-ratio method is a completely intuitive approach in that one merely observes the variation in the k-ratios as a function of count rate (beam current) and one immediately obtains a quantitative appreciation of the magnitude of these effects.  As for the various dead time correction expressions, see this post here for constant k-ratio measurements on an old JEOL 8200, where their count rates are so low, it doesn't matter which expression they utilize for the dead time correction:

https://probesoftware.com/smf/index.php?topic=1466.msg10943#msg10943

The expanded form gives almost exactly the same results as the traditional expression at low count rates, but it gives improved results at high count rates simply due to being a more complete form of the Taylor expansion probability series. It's all comes down to the statistics of how often will a photon pulse be missed due to another photon pulse coming into the detector within the dead time.

When you suggest that I try all three of the current-based calculations, you’ve totally missed my point.  In my plots using the simplest (linear) current-based calculation, I show that dead time appears to be calculated incorrectly at low count rate (< 50 kcps) when current is high.  This suggests that systematic error is present in the picoammeter reading (as it cannot be expected to be perfectly accurate).  Adding terms to the current-based dead time expression will not fix this problem and is not necessarily a “better” approach.

You need to think about this a bit more.  The constant k-ratio method is not current based. Yes, Paul's spreadsheet method is current based because he's fitting counts vs. beam current, but the constant k-ratio method is not current based. Why?  Because the primary standards and the secondary standards are measured at the same beam current for each k-ratio!  It's all about the count rate differences between the primary and secondary standards, which are more affected by dead time at higher beam currents.

Yes, the traditional form of the dead time expression starts to fall apart around 50K cps, but the wheels could come off at lower count rates on some detectors, particularly sealed Xe detectors I expect which have become contaminated or pumped out over time.

Please take a look at this post and maybe it will start to make more sense to you:

https://probesoftware.com/smf/index.php?topic=1466.msg10972#msg10972

[Brian Joy]

You need to think about this a bit more.  The constant k-ratio method is not current based. Yes, Paul's spreadsheet method is current based because he's fitting counts vs. beam current, but the constant k-ratio method is not current based. Why?  Because the primary standards and the secondary standards are measured at the same beam current for each k-ratio!  It's all about the count rate differences between the primary and secondary standards, which are more affected by dead time at higher beam currents.

Yes, the traditional form of the dead time expression starts to fall apart around 50K cps, but the wheels could come off at lower count rates on some detectors, particularly sealed Xe detectors I expect which have become contaminated or pumped out over time.

Please take a look at this post and maybe it will start to make more sense to you:

https://probesoftware.com/smf/index.php?topic=1466.msg10972#msg10972

No, you’re the one who needs to think about this a bit more.  All that I’ve emphasized is a need for accurate estimation of dead time for the simplest possible case, and Heinrich et al. (1966) present a better way to do it using measurements made simultaneously.  I don’t really care what happens at excessively high count rates, as interactions between the X-ray counter and counting electronics become more complicated.

Your argument that the “wheels could come off” at lower count rates using Xe counters is weak, as it is unsupported by any actual evidence.  In truth, I see the same pattern with my P-10 gas-flow counters, and you can see it in the data that I posted for Si (for the ratio method).

Also, keep in mind that a precision current source will incorporate the same kinds of inaccuracies (non-linearity, for instance) as a picoammeter, which is a precision circuit due to necessity.  In fact, a current source can be used to construct an ammeter – I’ve done it myself with nice results using the OPA192 op amp from TI.

I’ll post more data and plots soon.  Until then, I’m not commenting further on the subject.

[Probeman]

I will give it a bit more thought!   

And the first thought that comes to my mind is that I'm pleased that you find the Heinrich ratio method to work for you.  I myself prefer a method that tests the instrument over the full range of the beam currents I regularly utilize.

By the way, you mentioned that you do use high beam currents for minor and trace elements. I'm sure it has occurred to you that running a primary standard for a trace element, say Ti Ka using Ti metal or TiO2 (for maximum analytical precision) with the same beam current (100-200 nA?) as ones unknowns, will result in a very large dead time correction on the standard intensities, which although not the largest source of error for a trace element in an unknown (Donovan, 2011), is still worth correcting for accurately.  That is where the expanded dead time correction expression really comes into its own.

On the other hand if one chooses to run their primary standard at a lower beam current (say 30 nA) than ones unknowns, in order to avoid a large dead time correction, then the accuracy issue will be with the picoammeter calibration.  The accuracy issue cannot be avoided either way.

The nice thing about the constant k-ratio dead time constant calibration method is that when one acquires k-ratios using the same beam current for the primary and secondary standard (or unknown), one avoids any dependency on the picoammeter accuracy.  That's the beauty of a k-ratio measured at the same beam current!  And of course the whole point of the constant k-ratio method is that one should obtain the same k-ratio at *any* beam current.  One then simply adjusts the dead time constant until the k-ratios are as consistently constant as possible over the range of beam currents (which could be plotted in any order of beam current, even randomly!), of course using the high precision expanded dead time correction expression for best accuracy.

Not only that, but if one wants to then evaluate their picoammeter accuracy, then one can take their previously acquired constant k-ratio data and simply plot the secondary standard k-ratios against any *single* primary standard, as I showed in the post here:

https://probesoftware.com/smf/index.php?topic=1466.msg10972#msg10972

Try it, you'll like it!   

[Brian Joy]

I will give it a bit more thought!   

And the first thought that comes to my mind is that I'm pleased that you find the Heinrich ratio method to work for you.  I myself prefer a method that tests the instrument over the full range of the beam currents I regularly utilize.

By the way, you mentioned that you do use high beam currents for minor and trace elements. I'm sure it has occurred to you that running a primary standard for a trace element, say Ti Ka using Ti metal or TiO2 (for maximum analytical precision) with the same beam current (100-200 nA?) as ones unknowns, will result in a very large dead time correction on the standard intensities, which although not the largest source of error for a trace element in an unknown (Donovan, 2011), is still worth correcting for accurately.  That is where the expanded dead time correction expression really comes into its own.

On the other hand if one chooses to run their primary standard at a lower beam current (say 30 nA) than ones unknowns, in order to avoid a large dead time correction, then the accuracy issue will be with the picoammeter calibration.  The accuracy issue cannot be avoided either way.

The nice thing about the constant k-ratio dead time constant calibration method is that when one acquires k-ratios using the same beam current for the primary and secondary standard (or unknown), one avoids any dependency on the picoammeter accuracy.  That's the beauty of a k-ratio measured at the same beam current!  And of course the whole point of the constant k-ratio method is that one should obtain the same k-ratio at *any* beam current.  One then simply adjusts the dead time constant until the k-ratios are as consistently constant as possible over the range of beam currents (which could be plotted in any order of beam current, even randomly!), of course using the high precision expanded dead time correction expression for best accuracy.

Not only that, but if one wants to then evaluate their picoammeter accuracy, then one can take their previously acquired constant k-ratio data and simply plot the secondary standard k-ratios against any *single* primary standard, as I showed in the post here:

https://probesoftware.com/smf/index.php?topic=1466.msg10972#msg10972

Try it, you'll like it!   

Once again, here is one of my objections to your approach, stated slightly differently:  You’ve roughly eliminated current as a variable in your calculation of dead time via collection of Si Ka k-ratio data at given current, yet, for the general case, you still use an expression for dead time that depends on accurate measurement of current.  This makes no sense to me, as your approach is inconsistent.

[Probeman]

I will give it a bit more thought!   

And the first thought that comes to my mind is that I'm pleased that you find the Heinrich ratio method to work for you.  I myself prefer a method that tests the instrument over the full range of the beam currents I regularly utilize.

By the way, you mentioned that you do use high beam currents for minor and trace elements. I'm sure it has occurred to you that running a primary standard for a trace element, say Ti Ka using Ti metal or TiO2 (for maximum analytical precision) with the same beam current (100-200 nA?) as ones unknowns, will result in a very large dead time correction on the standard intensities, which although not the largest source of error for a trace element in an unknown (Donovan, 2011), is still worth correcting for accurately.  That is where the expanded dead time correction expression really comes into its own.

On the other hand if one chooses to run their primary standard at a lower beam current (say 30 nA) than ones unknowns, in order to avoid a large dead time correction, then the accuracy issue will be with the picoammeter calibration.  The accuracy issue cannot be avoided either way.

The nice thing about the constant k-ratio dead time constant calibration method is that when one acquires k-ratios using the same beam current for the primary and secondary standard (or unknown), one avoids any dependency on the picoammeter accuracy.  That's the beauty of a k-ratio measured at the same beam current!  And of course the whole point of the constant k-ratio method is that one should obtain the same k-ratio at *any* beam current.  One then simply adjusts the dead time constant until the k-ratios are as consistently constant as possible over the range of beam currents (which could be plotted in any order of beam current, even randomly!), of course using the high precision expanded dead time correction expression for best accuracy.

Not only that, but if one wants to then evaluate their picoammeter accuracy, then one can take their previously acquired constant k-ratio data and simply plot the secondary standard k-ratios against any *single* primary standard, as I showed in the post here:

https://probesoftware.com/smf/index.php?topic=1466.msg10972#msg10972

Try it, you'll like it!   

Once again, here is one of my objections to your approach, stated slightly differently:  You’ve roughly eliminated current as a variable in your calculation of dead time via collection of Si Ka k-ratio data at given current, yet, for the general case, you still use an expression for dead time that depends on accurate measurement of current.  This makes no sense to me, as your approach is inconsistent.

That is incorrect.  No wonder it makes no sense to you!

The expanded expression for dead time correction is the same as the traditional (single term) expression, which also depends *only* on count rate.  The only difference is that instead of ignoring the infinite series of the Taylor expansion, it incorporates probability terms up to the 6th power for vastly improved precision at very high count rates.

Also by measuring the primary and secondary standard intensities using the *same* beam current, we have not merely "roughly eliminated  current as a variable in your calculation of dead time", but rather we have *completely* eliminated current as a variable in our calculation of dead time because we measure each k-ratio using the *same* beam current, at multiple beam currents, which simply acts as a proxy for different count rates.

You will note that like the traditional expression, beam current does not appear as a variable in our expanded dead time expression.  Only count rate and the dead time constant as shown here:

https://probesoftware.com/smf/index.php?topic=1466.msg10909#msg10909

[Brian Joy]

Hi John,

OK, I see what you mean.  I was mistaken.  Sorry about that.

To get the k-ratio, though, you must be measuring Si Ka, for instance, at different times on the primary and secondary standards.  Thus the current might not be exactly the same, and, obviously, more than one material must be analyzed.  One material might be carbon-coated more thickly than the other or might have contamination on the surface.  One material might be more easily beam-damaged than the other or prone to accumulation of static charge…  and so on.

The advantage of the ratio method of Heinrich et al. (1966) lies in the fact that measurements are performed simultaneously on a given material using different spectrometers such that all of the sources of error that I listed above are effectively eliminated completely.  The method also allows one to work exclusively with metals and semi-metals, and so conductivity and beam damage are not issues.  I believe it is the best approach, and I’ll continue to post results.

I’m genuinely not that interested in anything other than a reliable linear expression (on a plot of N’/N versus N’.)  If I want to analyze for very minor amounts of Co, Ni, and As in pyrite, then I calibrate on a pyrite standard (for Fe and S) at 50 or 100 nA and calibrate on Co, Ni, and As standards at 10 or 20 nA.  I try to put as little faith in the dead time correction as possible.  At the very least, the ratio method has demonstrated that my Xe counters show linear behavior up to at least 85 kcps.  On plots of N’/I versus N’, I’ve always had difficulty assessing this.

Brian

[Probeman]

Hi John,

OK, I see what you mean.  I was mistaken.  Sorry about that.

To get the k-ratio, though, you must be measuring Si Ka, for instance, at different times on the primary and secondary standards.  Thus the current might not be exactly the same, and, obviously, more than one material must be analyzed.  One material might be carbon-coated more thickly than the other or might have contamination on the surface.  One material might be more easily beam-damaged than the other or prone to accumulation of static charge…  and so on.

 

OK, no worries, glad it makes sense to you now. 

Yes, with the constant k-ratio method we are not measuring the intensities at exactly the same time, but that is also true of our quantitative measurements, so we do hope our beam currents are stable over the period of a few minutes.  Of course, for these k-ratios utilized in the constant k-ratio method we also measure the beam current and perform a beam drift correction, just as we do for normal quant measurements.  So the beam drift effects should be minimal for two materials acquired one after the other.

In fact, in Probe for EPMA we utilize exactly the same automation methods for this constant k-ratio dead time calibration method, as we do for normal quantitative measurements.  The only exception being that when using multiple (beam current) setups for quant analyses we would normally acquire all the multiple setups one after the other on each sample, but for the constant k-ratio method we need to acquire the multiple (beam current) setups one at a time on each sample (all samples at beam current N1, then all samples at beam current N2, then all samples at beam current N3, etc.).  That is why we added the "one at a time" checkbox in the Multiple Setups dialog in the Automate! window to automate this.

If we were instead automating multiple kilovolt setups (e.g., thin film analysis) it wouldn't have mattered because the software would automatically have figured out which primary standard is associated with each secondary standard at the different keVs, but of course we don't distinguish samples acquired with different beam currents in that manner for quantitative analysis.  So we need to use this new "one at a time" acquisition checkbox in PFE to ensure that the primary and secondary standards are acquired together in time, so the k-ratios are constructed using intensities acquired at the same beam current.

That's kind of the whole point of the constant k-ratio method using different beam currents as a proxy for count rate: the k-ratios at all beam currents should all be the same if our dead time calibrations are correct.   

As for different carbon coats on the materials, that's an easy one. Because we are using the same quant methods for this constant k-ratio method as we utilize for normal quantification, we already have the correction for coating material/thickness built into Probe for EPMA:

https://probesoftware.com/smf/index.php?topic=23.0

Though so far I have only utilized materials mounted on the same acrylic mount for these constant k-ratio calibrations:

https://probesoftware.com/smf/index.php?topic=172.msg8991#msg8991

The advantage of the ratio method of Heinrich et al. (1966) lies in the fact that measurements are performed simultaneously on a given material using different spectrometers such that all of the sources of error that I listed above are effectively eliminated completely.  The method also allows one to work exclusively with metals and semi-metals, and so conductivity and beam damage are not issues.  I believe it is the best approach, and I’ll continue to post results.

Sure, that is a nice aspect to the Heinrich method.  It's pretty clever actually, though I have to say, I'm not seeing beam damage effects on my constant k-ratio runs, but I am defocusing the beam somewhat for these beam sensitive materials. As you will see in my next post in the other dead time topic, I have acquired data on benitoite/SiO2 up to 400 nA and it yields consistent k-ratios, though I did defocus the beam to 15 nA for the test in an abundance of caution.

I would prefer to deal with beam damage effects separately using the TDI correction in PFE, which works great for these sorts of issues. In fact, the TDI correction could be applied to the acquisition of the constant k-ratio data because the process is exactly the same as for normal quant runs and can be completely automated.  In fact, I should turn on TDI acquisitions for my next constant k-ratio testing...  good idea!

The other thing that I think users will like is that they will find the constant k-ratio method very easy to use, because it's essentially just a normal probe run, so the process will be very familiar to them. It only takes a few minutes to set it up and then one can just let it acquire lots of data for 8 or 12 hours or more fully automated.

Once the constant k-ratio data is acquired and plotted as a function of beam current, it's very simple (and quite intuitive) to quantitatively evaluate the magnitude of the dead time calibration errors, and adjust the dead time constants accordingly.

I’m genuinely not that interested in anything other than a reliable linear expression (on a plot of N’/N versus N’.)  If I want to analyze for very minor amounts of Co, Ni, and As in pyrite, then I calibrate on a pyrite standard (for Fe and S) at 50 or 100 nA and calibrate on Co, Ni, and As standards at 10 or 20 nA.  I try to put as little faith in the dead time correction as possible.  At the very least, the ratio method has demonstrated that my Xe counters show linear behavior up to at least 85 kcps.  On plots of N’/I versus N’, I’ve always had difficulty assessing this.

Brian

Different beam currents for standards and unknowns is a reasonable strategy as I mentioned previously (though I have to say it surprised me when I first started in microanalysis, and found that the usual scientific precept of controlling for all variables was ignored in this way).

So instead of putting our faith in the dead time correction, we are rather putting our faith in the picoammeter linearity/accuracy.  Checking the accuracy of our picoammeters would be a reasonable next step, because we all utilize different beam currents in runs with major and minor elements. We are obtaining such a device (should arrive today in fact) for testing our picoammeter. I'll keep everyone posted on that process.

But I also think you really should consider the use of the expanded dead time expression as it would further extend your x-ray linearity.  The good news is that our dead time constants can now be accurately calibrated with two different methods without relying on the traditional beam current method. So this is real progress!  Next step: calibrating our picoammeters!   

[Brian Joy]

But I also think you really should consider the use of the expanded dead time expression as it would further extend your x-ray linearity.  The good news is that our dead time constants can now be accurately calibrated with two different methods without relying on the traditional beam current method. So this is real progress!  Next step: calibrating our picoammeters!   

Don’t forget about spectrometer alignment.  Something appears to be amiss with your channel 3.  On LPET, have you verified that you get the maximum count rate for Si Ka and Cr Ka when the stage is in optical focus (after doing a new peak search at each level of focus)?

[Probeman]

But I also think you really should consider the use of the expanded dead time expression as it would further extend your x-ray linearity.  The good news is that our dead time constants can now be accurately calibrated with two different methods without relying on the traditional beam current method. So this is real progress!  Next step: calibrating our picoammeters!   

Don’t forget about spectrometer alignment.  Something appears to be amiss with your channel 3.  On LPET, have you verified that you get the maximum count rate for Si Ka and Cr Ka when the stage is in optical focus (after doing a new peak search at each level of focus)?

Yes, I also think there's an issue specifically with spectro 3, but we just had Edgar Chavez out three weeks ago and he and our engineer went through this spectro alignment. So it could be an asymmetrical diffraction issue but then why would it show up for both large area crystals on that spectrometer?  Probably worth checking the alignment again...

And with this observation of spectro 3 k-ratios being an outlier compared to the other spectrometers, I feel compelled again to point out the flexibility of the constant k-ratio method. Specifically that it can be utilized for three separate calibration checks:

• For what it was intended for when John Fournelle and I first came up with it, and that is for calibrating ones dead time constants on each spectrometer by measuring k-ratios using multiple beam currents on both the primary standards and secondary standards. That is to say, these k-ratios, measured on each spectrometer, at a number of different beam currents, should all be the same (within statistics) if the dead time constant for that spectrometer is correct. This is independent of the picoammeter calibrations.
  
  
• Also for checking the picoammeter accuracy, by simply utilizing the same k-ratio data, but this time instead of using primary and secondary standards acquired at the same beam current, we calculate the k-ratios using a single primary standard (acquired at a single beam current) and plotting the k-ratios of the secondary standards to check that they are still consistent. This test will reveal problems with the picoammeter calibrations since one is extrapolating from a single beam current to other beam currents.
  
  
• And also, again using the same data set, by plotting the k-ratios of *all* the spectrometers using the same element and emission line, we obtain a "simultaneous k-ratio" test that compares the k-ratios from all spectrometers to see how well they agree with each other. Which may indicate a spectro alignment issue.
  

And finally I have to mention, by utilizing the expanded dead time expression for all three tests, as seen in the latest post here, we can obtain consistent k-ratios up to 250 nA (>400K cps on SiO2!) on a large TAP crystal on spectro 2:

https://probesoftware.com/smf/index.php?topic=1466.msg10982#msg10982

[Brian Joy]

Earlier this week, I collected data to determine dead time according to the ratio method of Heinrich et al. (1966) using the Ti Kα and Kβ lines.  As in previous cases, I used uncoated and recently polished Ti metal for the measurements and measured at the Kα or Kβ peak on three spectrometers simultaneously using LiF, LiFL, and LiFH.

When working with Ti using this method in conjunction with a sealed Xe counter, it is important to note that, while the energy of Ti Kα lies below that of the Xe L₃ absorption edge, the energy of Ti Kβ falls above it, and so electronic gain, anode bias, and baseline voltage must all be considered especially carefully.  When measuring Ti Kβ, the baseline needs to be set so that it always fully excludes the Xe escape peak when working at current ranging in my case from 5 nA to 540 nA in measurement set 1 and from 5 nA to 700 nA in measurement set 2.  During the first measurement set, I counted simultaneously at the Ti Kα peak position on channel 2/LIFL and at the Kβ position on channels 3/LiF and 5/LiFH.  During the second measurement set, I counted simultaneously at the Ti Kβ peak position on channel 2/LIFL and at the Kα position on channels 3/LiF and 5/LiFH. At I_PCD = 700 nA (V = 15 kV), the measured Ti Kα count rate on channel 3/LiF was ~60 kcps; on channel 5/LiFH, it was ~227 kcps.

The obtained values for τ are as follows:

Ti Kα/Kβ ratio method [μs]:
channel 2/LiFL:  1.43, 1.46
channel 3/LIF:  1.37
channel 5/LIFH:  1.42
Current-based method, Ti Kα [μs]:
channel 2/LiFL:  1.25
channel 3/LiF:  1.03
channel 5/LiFH:  1.20
Current-based method, Ti Kβ [μs]:
channel 2/LiFL:  0.81
channel 3/LiF:  -0.06
channel 5/LiFH:  1.04





Note that 1) the Ti dead time values calculated using the ratio method are essentially the same as those calculated for Cu and Fe and 2) use of the current-based method once again causes underestimation of the dead time, with channel 3/LiF showing the greatest departure.  Combining the results of the ratio method for Cu, Fe, and Ti, for channel 2 (six calculations total), the range of calculated dead time values is 1.43-1.50 μs, with an average of 1.46 μs.  The observed variation likely can be ascribed to counting error.  Dead time values for channel 3 (three calculations) are 1.45, 1.41, and 1.37 μs (Cu, Fe, and Ti, respectively), giving an average of 1.41 μs.  For channel 5 (three calculations), I’ve obtained 1.42, 1.38, and 1.41 μs, producing an average of 1.40 μs.

As an aside, assuming that linear behavior is maintained on a plot of N’/N versus N’ up to 80 kcps and assuming that τ = 1.45 μs, then N’/N = 0.884 (exactly) at that measured count rate, and so true count rate (N) is 90498 s^-1.  This represents a roughly 13% correction relative to the measured count rate.  Once again, I prefer to work with dead time corrections smaller than this, and so I’m happy to remain within the linear correction range.

Returning to the current-based method, an anomaly appears in all plots of N’/I versus N’ collected in both Ti measurement sets.  For instance, in the plot of the data for Ti Kα on channel 2/LiFL (below), the data take a trajectory in the form of an arc, such that initial data (lowest current) plot below the regression line, then subsequently well above it (around 35 kcps), and then, in the high end of the linear range (up to ~85 kcps), most fall slightly below it.  Above 85 kcps, significantly non-linear behavior truly is present, though assessing its onset on the plot is virtually impossible.  Whatever the nature/source of the problem (certainly related to the picoammeter), the anomaly appears to vary over time in magnitude and form.  Compare, for instance with my plots for Cu (anomaly pronounced but different) and Fe (anomaly much less pronounced).  Regardless, since it is present in each set of simultaneous measurements, the error cancels when calculating ratios.  Note that, had I chosen to use only apparent count rates falling below 35 kcps, my calculated dead time would have been 1.01 μs rather than 1.25 μs.



The same pattern is easily visible on channel 5/LiFH.  In calculation of dead time, had I considered only count rates below 35 kcps, I would have obtained 0.98 μs rather than 1.20 μs.



The following plot gives a more accurate depiction of the magnitude to which departure from linearity affects the dead time correction at high count rates.  I’ve plotted measured Ti Kβ count rate on channel 2/LiFL divided by the measured Ti Kα count rate on channel 5/LiFH versus the measured Ti Kβ count rate on channel 2/LiFL.  The Ti Kβ count rate on LiFL reaches only 33 kcps (at 700 nA), and so all obvious non-linearity should be due to Ti Kα on LiFH.  Above 85 kcps (for Ti Kα), all ratio values fall above the regression line, though only very slightly up to 113 kcps (I_PCD = 260 nA), and so I’ll take 85 kcps as my upper limit on channel 5; corresponding plots for Cu and Fe support this limit, which appears to be applicable to the other Xe counters as well.



My likely next step, when I get a chance, will be to determine dead time via the ratio method for Ti using my three PET crystals.  I’ll then move on to the spectrometers with gas-flow counters.