Yes - $100 - $68 = $32. You are saying the hassle of getting the perspex in, and making a polished taper exceeds $32, and on the face of it, I agree. I also get it why you went for LYSO. It's non hygroscopic! No extra waterproofs and X-ray transparent foils, etc.
A calculation
I was thinking about it entirely another way. Previously, @RJSakowski showed us that for the energies we want to capture, scintillator need only be 1 or 2mm thick to stop 98% of the X-rays.
The cost of a scintillator is very much about the amount of crystal, and the hassle of dealing with hygroscopic, but my considerations were all about not wasting the area, starting with the shower of X-Rays coming from the sample. I put a lot of value on the input area of the scintillator. I spent a while trawling the QST Photonics site looking for thin versions with area, unfortunately mostly among the hygroscopic ones.
For your scintillator.
For the size of the scintillator 4 x 4 x 22 mm, then using it against a 3x 3 photodiode gathers 56% of the flashes.
If the original area delivering the X-Rays was (reasonably) about 1" diameter, that was 506.2mm2. (Units!)
Those X-rays photons would be headed all directions, including deeper into the sample, and only half headed approximately in the hemisphere toward the scintillator (say about 25mm diameter, at about 40mm range), and only some of those aimed well enough to hit it.
In a crude volume estimate, what I am saying is that there is a huge waste of photons unless the scintillator area presents a bigger truncated cone capture region. If the end of the scintillator is 16mm2 compared to 506mm2, we get 56% of only 3.1%. I was hoping for better than this 1.8%.
If you cut your scintillator into 4 pieces with 3 cuts, using a metal disc and wet abrasive, losing (say) 1mm per cut, and then stacked them together to yield a 4.75mm thick scintillator of 64mm2 area, you get a massive increase in efficiency. If you are willing to use up both your scintillators like this, you get 128mm2.
I set this against getting hold of a chunk of this acryllic stuff.
Then turning a taper onto it, and rubbing up the surface with abrasive, finishing with toothpaste or whatever folk use to polish plastic. Even if the S-(PM) diode is only 1mm2, it captures the lot, compared to a horrifically expensive area array 314mm2.
The "bang for your buck" is much amplified by a 11 quid piece of plastic and the HM machining style effort.
I am sure there are many HM members who can offer good advice on how to make this thing.
The taper needs to be narrow enough that most rays starting from the scintillator end hit the inside at more than 42° to the normal.
Sometime around 1985 I had already abandoned any kind of 741 OP-Amp. To lose the DC drift, I would use chopper-type AC versions, or Instrumentation types. By around 2002, there was every kind of video speed op-amp including current feedback types. It is not too hard to find Op-Amps with >50MHz bandwidth. You can get a OPA2300AIDGSR from mouser.com for £2.03. It is a low noise 16-bit accurate 150MHz bandwidth (which would come down to about 10MHz actual operation with feedback). It's a very old device. LTC devices come with Spice models. We have nothing to fear here. It's a dual. Use one for a zero-loss peak detector.
This is one area I have not thought through, but I think there is massive potential in the PyMCA and all that free CERN software from the LHC, and the NRLXRF (Fortran) software to get somewhere.
Before even we go there, I think there is mileage in side-stepping a whole lot of it, and just do what you would do if you were just looking at it. Your original suggestion, similar to @RJSakowski 's switched capacitor, of a one-shot monostable timed capture-lockout-reset does a lot of the hard stuff in one hit.
Really fancy stuff, like capturing the whole scintillation, and/or parts thereof, subjecting to FFT analysis via octave, and all that clever stuff only has a point if there really is new information in the pulse to spot overlaps, etc. If the main information is the pulse amplitude, that is all we need, we can skip much of the rest.
The really clever sorting will be the analysis of accumulated counts of pulses meeting the criteria to be in the quantized energy "buckets". Assigning some kind of hash value, as a signature of the set of weighted buckets might allow a fast lookup on alloy probability, and display the conclusion on the plot. Maybe that is already done for us in PyMCA
Theremino
The biggest, nicest chunk of Theremino is the app display - a general purpose screen of graph that can be used on phones, etc. None of the XRF stuff that fed it need be replicated by us.
For me, the Raspberry Pi already comes with 2 full HDMI displays. Initially, I don't need a Theremino.
For folk who might want to use a smartphone, I think there are very likely apps that will do a USB link, or use Theremino. This is completely another branch to this job, something to be tacked on at the end.
Re: PCBs etc.
Most folk here would not have too much trouble hanging together evaluation boards. More difficult when it comes to soldering SOIC components. All one can fit on 1sq ft of PCB cost about £40 as a prototyping quantity, which would make about 10 lumps of electronics. I was hoping to use something like THIS
The "plug-in" prototyping breakout, ribbon cable extension, breadboard, and set of connector wires as seen in post #249 cost £8.49.
I will be trying out a S-(PM) like you are, and it will be one of the smallest cheapest. I am convinced that a light-pipe is a good idea. More problematic is a "big enough" scintillator, though it need not be very thick.
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. The python-based stuff you found may be helpful in the latter case. Just guessing here, but I suspect you have a decent 'scope to look at waveforms. I got spoiled at work so my tastes run on the expensive side there; but, really, a 1GSPS Hantek, Rigol or Siglent should be plenty good for stuff like this.