I recently learned about a really cool way to perform your own elemental analysis on a (relative) shoestring. Traditional approaches to elemental analysis have taken a number of different paths, starting with chemistry. Typically, analysis would involve dissolving the unknown(s) in an acid and then using various chemical reagents to determine the components. Some compounds are insoluble in water, so adding a solution with something that will react and cause some to become insoluble could be the first step in figuring out what you've got. Since there's a huge number of potential reactions (and possible confounding results), this approach requires a well-stocked chem lab and knowledge of many different reactions. Not for the hobbyist.
Another approach is spectroscopy. Based on the light emitted by excited elements, it's very powerful and sensitive. But there is the problem of heating the sample to a high enough temperature to get it to emit -- and then having the ability to separate the light into its individual components. This is more approachable because it's relatively easy to make or buy a visible-light spectrometer based on a diffraction grating (or blank DVD), plus a webcam to image the spectrum. The difficulties are the necessity to make something to ionize your sample long enough to acquire a spectrum; and the need to break your sample down in to some form to feed into your home-brew ionization device.
On the industrial side, you can buy analysis tools that look at the x-rays emitted by your sample. One approach that I'm pretty familiar with is to hit your sample with high-energy electrons (accelerated to 30KV or thereabouts), and look at the x-rays it emits. This requires an electron gun capable of accelerating your electrons. The x-rays are characteristic of the element(s), so an x-ray spectrometer has to be used in order to distinguish them. Two detection methods are used, EDX or WDX. EDX is "energy dispersive x-ray" and WDX is "wavelength dispersive x-ray" analysis. WDX is more straightforward, and uses a crystal lattice as a diffraction grating to produce a spectrum of the x-rays emitted. The crystal is rotated to direct the diffracted x-rays into a detector, so it is fairly slow because it must rotated in order to scan through the x-ray spectrum. In my experience, it also is less sensitive so scan times must be very slow in order to get decent signal to noise ratios. The detectors used for these analysis tools use relatively exotic gasses like argon + methane in a flowing tube.
On the other hand, EDX is much more sensitive and faster, because it uses a special kind of photo-detector. The detector outputs a pulse whose height is proportional to the energy of the incoming photon -- the shorter the x-ray wavelength, the more energy it has. So it is a kind of single-photon detector, but the energy of each photon is categorized and then entered into a "channel" of a multi-channel analyzer. So all detected photons are detected and characterized, which greatly increases the detection rate. The downside is that the detector has some losses -- x-rays can enter the detector (a type of semiconductor diode) but they may not deposit ALL their energy -- so they don't produce a signal that is exactly related to their original energy (determined by their wavelength and Planck's constant). These detectors also need to be operated at cryogenic temperatures, 77K (the temperature of liquid nitrogen) so they have to be inside a Dewar whenever they're are in use. Attempting to operate them at room temperature will destroy them, a VERY expensive proposition.
All this stuff is way out of the range for hobbyists, unless they buy something used and have the ability to get it to work. That could require a wide range of abilities, since these tools typically are attached to scanning electron microscopes. Not impossible, but a pretty high bar for most.
Another approach is XRF. It is a type of fluorescence, hence it's acronym -- "X Ray Fluorescence". You probably are familiar with ultraviolet fluorescence, from "black light" bulbs or tubes. When it occurs, the light emitted is characteristic of the materials involved. XRF is similar, but uses higher-energy X-ray photons to excite fluorescence at somewhat longer energy x-ray wavelengths. X=rays are emitted from inner-shell electrons so they are pretty much independent of the oxidation state of the elements -- so they are good for looking at individual elements. This sounds pretty exotic, but it actually has some advantages -- particularly from a hobbyist's point of view.
What are some of these XRF advantages? Well, for starters a lot of work for you as a DIY'er has already been done. Just Google "Theremino XRF" and you will see what I mean. It's not too expensive to buy scintillator crystals and photomultiplier tubes (and power supplies) from ebay to come up with something that can tell you (for instance) if the paint flakes you've got have lead in them or not. And it can be done with an x-ray source made from a few dead smoke detectors! Since this approach is all open-source based, you can take it as far as you want. Caveat: yep, you do need to have some experience with roll-your-own electronics, but it's not that high a lift.
Just to add to the attraction, the same setup can be used to look at materials to see if they're radioactive and provide some guidance on what the radionuclides might be. Not important, you say? well, Strontium-90 is a common contaminant from Nuke-reactor failures (think Fukushima, and apologies if I got the spelling wrong); and our bodies can't tell the difference between strontium and calcium. Living on the west coast, we've wondered just how much SR-90 we got from that, but authorities here have not been very helpful to resolve that concern, possibly due to economic issues. Yes, radioactive cesium also can be detected too.
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