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Both OSL and TLD testing release the trapped electrons which gives off radiation. That’s what they are measuring. Here’s a video explaining how OSL works. Skip to 9 minutes into the video to avoid her whole biography. That’s where she explains how OSL works.
You seem to be confused about what "trapped electron" means. It can mean different things in different contexts. When you're talking about solar flares, we are talking about protons and electrons expelled from the sun, which get trapped in the earth's magnetic field. Have you heard of the Van Allen radiation belt? The Van Allen belt was officially discovered in 1958; theorists had predicted it before then. Protons and electrons travel around the earth, guided by lines of the earth's magnetic field.
It is my understanding that Optically Stimulated Luminescence, OSL, works especially well with quarts crystals, although it also works with silicon dioxide crystals and other things. When sunlight strikes a quartz crystal, some of the outer electrons absorb a photon of light and go to a higher level, an excited state. These usually give off a photon and return to a normal state sometime soon. In a quartz crystal, anomalies in the crystal sometimes trap an electron in the excited state and it remains at a higher quantum level for long periods, even thousands of years.
There is no way that an electron "trapped" by the earth's magnetic field, whizzing around the earth in the Van Allen belt, can get into a quartz crystal buried underground in the Arabian desert. Think about it.
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