The relativistic velocity of the electrons is important to the design of the accelerator.
GPS Satelites have to deal with the relativistic effects.
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Yeah, my initial reply there was poorly worded. My point was, the electronics are not at relativistic speed, so the electronics themselves don't have to deal with the effect of traveling at that speed.
Neither the linear accelerator itself nor the detection system itself is traveling. The xray (particle) is. The accelerator of course is designed with that in mind so it knows where the particle is as it accelerates the particle. But neither the accelerator nor detector themselves is traveling, the electronic components are stationary so they don't have to deal with relativistic effects on how the components work.
If the particle's energy level matters, then the particle's speed has to be factored in with energy calculation. Other than that, the application is rather like a lux-meter, absorb photon which is actually traveling at light speed and tell you how bright it is at the point, but the electronics themselves doesn't need to deal with any relativistic effect regarding itself traveling at relativistic speed because it isn't.
GPS: The electronics in the satellite are not traveling at relativistic speed, therefore the electronics doesn't have to deal with relativistic effect on its electronics due to traveling. The electronics themselves will work as well stationary or moving. However, the GPS application is one in timing, the timing calculation has to to deal with the time dilation.