What's
claimed is that shooting electrons at the screen fast enough
both (a) generated X-rays, and (b) rendered that screen radioactive.
This has nothing to do with whether X-rays can make what they hit
radioactive.
Anyway, if an electron beam can cause something to be radioactive,
then X-rays can too, by using the photoelectric effect to generate an
electron beam within the material, no?
My guess would be "no".
Beta particles (electrons) don't cause radioactivity unless they meddle
with the nucleus. This requires surmounting a fairly hefty energy
barrier; for it to happen often enough to produce any significant
amount of radioactivity requires that the electrons be tootling along
at a pretty good clip. I suspect that by the time a photon can knock
an electron free of its atom with enough oomph left over to give it
that kind of energy, it's well above the X-ray region.
But I'm not physicist enough to be certain. Anyone?
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