Solid state detectors work similarly to proportional counter detectors, in that they work by an incoming X-ray freeing electrons which are then accelerated toward a positively-charged anode.
In the case of solid state detectors, however, the detector is not filled with gas, but with a semi-conducting solid material. When the X-ray enters the solid, it interacts with the material creating electron-hole pairs. (A “hole” is essentially an absence of an electron it doesn't really move like an electron. However, an electron in a neighboring atom may move into the hole, leaving a hole in the neighboring atom. In this way, the “hole” moves through the material.)
The signal at the anode is proportional to the energy of the X-ray that entered the detector, so just like proportional counters, solid state detectors allow scientists to measure the spectrum of X-ray light from objects in the sky.
Scintillation Detectors
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Charged Coupled Devices (CCD detectors)