OSHA, the U.S. Occupational Safety and Health Administration, has a method of using infrared radiation to determine it. The process is described in full here:
The technique described in the other answer’s reference is not to determine purity of quartz, but rather to determine quartz (silica) content of coal dust.
Generally various chemical analyses, from straight wet chemistry to microprobe analysis, would be used to determine the purity of something like quartz. It depends a lot on what you want to use it for, and what the nature of the material is. You’d do something different when seeking high-purity quartz sand for glass making versus pure quartz crystal for electronics.
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