Duane-Hunt limit

[Nicholas Ritchie]

At the end of the day, reproducibility is more important than accuracy.  While beam energy is an important parameter in our matrix corrections, if the true beam energy is 10.25 keV when we've set it to 10 keV, the difference is probably buried within the uncertainty in the matrix correction so long as all data is collected at the 10.25 keV.

[Probeman]

At the end of the day, reproducibility is more important than accuracy.  While beam energy is an important parameter in our matrix corrections, if the true beam energy is 10.25 keV when we've set it to 10 keV, the difference is probably buried within the uncertainty in the matrix correction so long as all data is collected at the 10.25 keV.

Well, I for one sure would like to know if my high voltage was off by 250V!     

As I said previously the whole point of this topic is to determine the accuracy of our electron beam energy.  Yes, I agree that most of the time at sufficiently high over voltages the beam energy accuracy isn't too critical. But we at UofO often run at low over voltages in order to attain high spatial resolution analyses, so the question has concerned me enough to have started a thread to look into this.

And now we've made significant progress in learning about the convolution of the photon "cliff" at the Duane-Hunt limit... which no one had even suspected existed previously! I think this has been a fun and interesting exercise and appreciate all the help and support!   

The good news is that the over voltage curve method is also good for determining our beam energy precision/reproducibility...

[Probeman]

I made some very low over voltage measurements over the weekend on Ti Ka (~ 5 keV) and Ge Ka (~11 keV) and the results are very encouraging.

The Ti K edge energy is 4.964 (or 4.967 in another tabulation), so I started with a beam energy of 4.98 kV and went up from there.  At 4.98 kV, I obtained a count rate of 0.05 cps/nA with a variance of less than 0.0003!

I then increased the beam energy in increments of 0.04 KV, so the next measurement at 5.02 kV I obtained a count rate of 0.33 cps/nA +/- 0.02.

These were at 100 nA and counting 120 sec on-peak and 60 seconds on each off-peak.  I'm plotting them up now...

[Probeman]

This over voltage curve method is working better than I initially thought possible.

Unfortunately the instrument failed with a vacuum error about halfway through so the second data sets didn't get acquired, but here is the first attempt using 0.04 KV increments just about the TI K edge:



Looking at the counting scatter, this plot shows how reproducible this method is.  One can also conclude that the instrument is producing a fairly smooth curve in the high voltages... so perhaps we can conclude that the electron beam energy on my SX100 is very slightly lower than the nominal value at ~5 kV since the zero intercept is very slightly higher than the Ti K edge energy?

[Probeman]

And here is the dataset for Ge Ka:



This plot causes me to suspect that the electron beam energy on my Cameca SX100 at ~11 kV is slightly higher than the nominal value would indicate...  what do you all think?

Again, both the Ti and Ge pure metals were freshly polished and uncoated (though well grounded).

[Probeman]

Here's a zoom of of the D-H limit photon cliff in PENEPMA using continuum forcings:



These continuum forcings are seen here:

       >>>>>>>> Interaction forcing.
IFORCE 1 1 4 -100  0.9 1.0            [KB,KPAR,ICOL,FORCER,WLOW,WHIG]
IFORCE 1 1 5 -10   0.9 1.0            [KB,KPAR,ICOL,FORCER,WLOW,WHIG]

Interestingly if I run the same simulation but using characteristic forcings we see an even more prominent photon cliff:



The characteristic forcings used are here:

       >>>>>>>> Interaction forcing.
IFORCE 1 1 4 -10     0.9 1.0           [KB,KPAR,ICOL,FORCER,WLOW,WHIG]
IFORCE 1 1 5 -100    0.9 1.0           [KB,KPAR,ICOL,FORCER,WLOW,WHIG]

I do not have an explanation for this.

[Probeman]

Looking at the data in replies #63 and #64 above, it is interesting (to me anyway) how quickly the statistics improve at such low over voltages!

Here is the raw data from the Ti Ka measurements (K edge is 4.964 kV):

Ti Ka     kV      cps/100nA     STD DEV      %STD DEV
         4.98        4.80         0.30          7.23
         5.02       33.2          1.70          5.04
         5.06       67.8          3.90          5.80
         5.10      125.0          2.30          1.85
         5.14      190.4          2.90          1.54
         5.18      287.7          1.60          1.89
         5.22      370.6          9.00          2.42
         5.26      481.9          6.20          1.28
         5.30      624.6          9.20          1.47
         5.34      732.9         11.60          1.59

And here are the Ge Ka measurements (K edge is 11.103 kV):

Ge Ka     kV      cps/100nA     STD DEV      %STD DEV
        11.12       21.9          1.20          5.49
        11.16       49.9          2.30          4.63
        11.20       78.3          1.20          1.57
        11.24      115.3          1.40          1.17
        11.28      162.0          2.00          1.22
        11.32      229.7          1.00          0.44
        11.36      288.6          2.20          0.78
        11.40      354.8          1.80          0.51
        11.44      422.8          3.50          0.83
        11.48      530.2          3.60          0.69

We're at less than 1/2 kV above the edge energy and getting excellent statistics!

[Probeman]

I drafted these schematics in an attempt to explain how the photon "cliff" forms near the Duane Hunt limit (see previous post above). 

But the short answer is: if one is acquiring (or simulating) photons near the Duane-Hunt limit, one will accumulate photons below the electron beam energy, but one will NOT accumulate photons above the electron beam energy (aide from coincidence (continuum sum) photons of course).  So the appearance of  this so-called photon cliff should not surprise us.

Here is my schematic when accumulating photons into non-zero wide bins (assuming a detector with an energy resolution equal or better than the bin width:



The photons simply accumulate from zero. And here is the schematic for Monte Carlo simulations when the photon count is normalized to the number of incident electrons:



The difference being that the integer accumulation will show the photon cliff rising from a zero level with improving precision over time, while the simulated photon cliff stays at a constant value but also improves in precision over simulation time.

Of course since our EDS detector spectral resolutions are around 130 eV or more, we'd have to utilize 100 to 200 eV bins in order to see this photon cliff with actual spectra. 

Two questions: first, can anyone suggest any improvements for these schematics? We'll be presenting this at M&M this summer. And second, does anyone have access to a photon detector with at least an order of magnitude better spectral resolution (say, 10 eV or better), where we might be able to actually see this photon cliff using our typical 10 eV per channel EDS acquisition software?

[sem-geologist]

... So the appearance of  this so-called photon cliff should not surprise us.

... it had surprised me the first time I saw it from simulation and it still surprises me, even if it should not. I highly suspect it to be an artifact of modeling. I have doubt about the sharp cliff as WDS Duane-Hunt experiments show the region above the beam energy has some faint continuum and the level of continuum is clearly dependent from (electron) density of material. Cosmic radiation can be ruled out, and photon coincidence too as it is WDS not EDS, and if it would be for some weird reason coincidence or other orders - it could be cached with PHA, but it is not. If this sharp "cliff" could be proven to exist with higher spectral resolution detector (would that fancy calorimetric-EDS be any help here?) - then knowing that would give an advantage as such "cliff" position could be then resolvable by deconvolution and could provide the closest estimation of beam energy.

[Probeman]

... So the appearance of  this so-called photon cliff should not surprise us.

... it had surprised me the first time I saw it from simulation and it still surprises me, even if it should not. I highly suspect it to be an artifact of modeling.

We already know it is not an artifact of modeling, according to Llovet, Ducharme, Ritchie and Rohde  (it's actually a "binning" artifact if we assume non-zero width bins), and it should not surprise you because of what was posted previously:

Yeah, I'm always shocked!   And it is weird that an electron could come to a complete halt. Here is a figure that a colleague of mine in the Physics Department at the University of Oregon , Andrew Ducharme, wrote me recently in discussing the above photon "cliff":

The established physics is that bremsstrahlung cross-sections are nonzero for complete energy losses, or for production of photons with the beam energy E (see figure from Penelope manual), while the probability of producing a photon with energy E + 1 eV is 0. Wait long enough (assuming ideal detector, electron source) and you will count photons at the Duane-Hunt limit while never getting counts in the energy bin of E + 1 eV. The longer you wait, the more DH-energy photons, and the higher the cliff.

I think of it in the sense that the coulombic field of an atom is actually a pretty big target, so a few electrons are going to hit some atoms directly right at the surface...

Note that the cross sections at complete loss of energy by the electron (at 1.0 on the x axis) are non-zero. The implication of this is that it is possible to produce continuum photons up to and at the electron beam energy, but impossible to produce photons above the electron beam energy. That means that if we collect photons into bins of a non-zero width, the appearance of a photon cliff is inevitable, even if that photon cliff is obscured by the (limited) spectral convolution of our photon detector. This is the reason why we only observe the photon cliff in unconvolved simulations and when not using a continuous slowing down model (CSDA).  PENEPMA uses a discrete energy loss model so the photon cliff becomes apparent in simulations with sufficient electron trajectories.

However as you suggest, using a micro-calorimeter EDS could be a good idea as the authors claimed an ~10 eV spectral resolution:

D. A. Wollman, G. C. Hilton, K. D. Irwin, L. L. Dulcie, N. F. Bergren, Dale E. Newbury, Keung-Shan Woo, Benjamin Y. H. Liu, Alain C. Diebold, John M. Martinis; High-resolution microcalorimeter energy-dispersive spectrometer for x-ray microanalysis and particle analysis. AIP Conf. Proc. 24 November 1998; 449 (1): 799–804. https://doi.org/10.1063/1.56867

But I'm not sure who still has one up and running?  But if anyone does it would be an easy (and cool!) measurement, just use 10 eV wide energy channels and the photon cliff should appear given enough acquisition time.

[Probeman]

And here is the dataset for Ge Ka:



This plot causes me to suspect that the electron beam energy on my Cameca SX100 at ~11 kV is slightly higher than the nominal value would indicate...  what do you all think?

Again, both the Ti and Ge pure metals were freshly polished and uncoated (though well grounded).

Somebody asked about the counting times and beam current for these over voltage curve measurements and so I checked and they were 120 seconds on-peak and and 120 seconds off-peak at 100 nA.

Note that I used freshly polished pure metal standards for these tests, but in case ones samples are carbon coated, one can calculate the electron beam energy loss with a 20 nm carbon using the CalcZAF app and here is is for 5 keV:



So we're losing about 14 volts out of 5 KV. And here at 12 KV:



So only about 8 volts at 12 KV.  But I would still stick with freshly polished pure metals for these over voltage curve tests...