About NoiseWell, noise could have a big impact on the energy resolution, since each pulse must be processed as a separate event -- there is no way to average pulses to improve the SNR, because you don't know the photon energy beforehand. That's why I'm hoping that a least-squares fitting scheme will improve things. However, starting out with the best SNR possible can't hurt.
It is a serious limitation. For going after these (low) energy X-Rays directly from silicon, without a scintillator, the whole thing can fall flat on it's face without we pay attention to noise.
Incoming noise
Without full shielding, can't work in the face of the common electromagnetic racket in the average room, even without someone putting switching currents into the wall AC mains by using a welder. So OK, we know it needs screening, and by design, made to not respond to all the EMC threats, but we mean down to a new level way below what might be OK for average electronic kit. So good EMC immunity.
We are using the Am241 gamma to provoke a little racket of our own that we are interested in, and it must not be contaminated by "other stuff". I think it can only work if the gadget uses really good shielding, such that all of the path from the Am241 to the test material, and into the sensor sees only lead, and the test material. One mode uses lead sheet with the sample on it, and XRF à la HM gets plonked down over it. The other mode uses a smaller shield pushed onto the end, and it is pressed up against the test material. With a little care, it can still work even if the lead shield is not complete, say for example putting it up to a rod too long to be in a kind of "lead lined test box".
Amplifier device noise
With just my first forays into trying various op-amps in the circuit, and also exploring the circuits with discrete FET front-ends, I see that for "large" signals, given enough gain to get near 4V for the ADC, in the bad scenario noisy cases, we get enough "noise" of various kinds to have about 80dB SNR.
The lower noise circuits can get the racket down to below 160uVp-p at output, and I even have one at less that 90uV. That best one would have SNR at 93dB. I just threw in an example 80nA as the "biggest" signal, and I use 500pA to 5nA as the smallest signals. I do suspect the real signals might be as little as a third of that.
If we consider a "smallest" signal might from a 500pA signal, which pushes 440pA into the amp, and delivers 26mV at the output. Will that be seen above the noise racket? [The spice simulation does not show a transient analysis simulation of white noise added]!
The answer is "it might be", but perhaps at the cost of limiting the bandwidth, slowing it down, having a big phase delay. Also doing something about the shot noise, adding in noise filter stages. Ultimately, living with the "pulse" ending up as a much reduced, stretched thing with a loose approximation to the amplitude of the original, and an area under that has a very smudged relationship to the energy. This is where we get a threshold region below which we have the bumps from pulse that get going before the previous has finally died.
Here is where we come up against the noise figure of the amplifier. We have to deal with both voltage and current noise specs for the device (it seems there are two things accountable).
For most selection tables, "low noise" is anything <10nV/√Hz. Those "best" TIAs mostly have about 6nV/√Hz.
The (expensive) LTC6269 has 4.3nV/√Hz and 5.5fA/√Hz.
The "ultra low noise" FETs have specs a NF noise figure 1dB, which I would know as 75K
Low noise RF pHEMTs I used to work with have noise figures 0.4dB (28K). I would like that without 14dB of gain at 2GHz!
When we take a circuit that uses one of those low cost FETs, like about £0.54 each, or £4.30 for ten of them, we can have 0.9fV/√Hz, and lost a whole lot of shot noise.
Shot noise and the like.
I have not (yet) done much with the elaborate possible noise sources I posted earlier. The op-amps internal .MODEL have a noise contributor, as does all the resistors. Simply exploiting that we have no interest in slow signals down to DC, and AC coupling, loses noise. Then, if after the main sgain stage, we use a band-pass filter to start at (say) 100Hz, and drop off at (say) 200kHz, a huge amount of white noise is lost, as well as 50/60Hz remnants.
Real hardware
Everything about this so far is sloppy approximate - just about good enough to head for the main aim of deciding the basic amp with some gain, and hooking up a real circuit, and measuring some noise. I am not even using the right currents yet.
The integration of that 500pA current pulse over 10uS is 109.19pA x 10uS = 1.0919E15 Coulombs.
Dividing by the electron charge e = 1.602176634E-19 says this involved 6815 electrons.
That many, distributed over the 230pF would try to lift the voltage to 4.74uV at the input.
It would of course fail. Some would be lost in the 40Meg leak.
Pretty much all the rest would try to alter the state of the coupling capacitor, and fail again as the TIA zeros it out with the feedback, generating the voltage to make this happen at the amp stage output.
Where I get unglued is that the event that started this was a photon that could only have released one electron-hole pair when it smacked the sensor. I don't understand the physics process by which the thousands of electrons to make this current happen, came about. The pulse we get comes from a single photon arrival, not thousands in a bunch.
Maybe some experts here can set out what happens, but it may involve some extended lectures from Prof. M. R. Shenoy, Dept of Physics, Indian Institute of Technology, Delhi. About 46 lectures each 42min to 58 min long. The be-all and end all of photodiode semiconductor tech.
--> Photodiode Physics Course
Do I know what I am doing?
I think what actually happens is the incoming photon hits, and expends 4.15eV in liberating the electron, and the remaining energy is put into giving it KE kinetic energy. That speedy electron will ping around and (maybe) slam into atoms, trying to go right through, but in some way, liberating another electron, which ends up that bit slower. I don't know if the approach is correct, but I thought to find out how many work functions gets expended in this. Starting with (say) a 9.6keV photon from zinc, divided by 4.15eV silicon work functions, that might generate 2313 electrons.
2313 electrons is 3.7E-16 Coulombs.
If it delivered over 8uS, the average current would be 46pA.
Given the pulse shape we had, it's peak might be at 187pA.
Compared to my 500pA guess, the 26mV pulse might be nearer 11mV. It could be contaminated with 120uV of noise, making the numbers sampled jiggle about some unless we filter noise, or let computer averages do it for us. The very lowest energies might have us trying to sample 3mV pulses.
We are in the right ball-park!
Meantime, I do not have to understand the physics down to the last electron to make a better amp. So just working back from what the LSB from 16 bits can give, starting with a reasonable Vref, I get to the values I simulate with. Basically trying for minimum signals that sit up at least 10dB to 15dB above the noise at the input. We can do no better, and we have enough gain to see some volts at the output we can count, with still a dynamic range 80dB to 93dB. I expect it will work at least as well, and likely a good deal better than all those you and I have seen so far.
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