The neutron temperature example is crucial: you just can't give the cross section of a target alone, the energy of the incoming beam also matters.
Figure 1.
Radium 226 source in a cloud chamber
. Source.
Video 1.
How to make a cloud chamber by Suzie Sheehy (2011)
Source.

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Most important application: produce X-rays for X-ray crystallography.
Note however that the big experiments at CERN, like the Large Hadron Collider, are also synchrotrons.
Predecessor to the synchrotron.

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Experimental particle physics is a branch of physics that investigates the fundamental constituents of matter and the forces that govern their interactions by conducting experiments. Unlike theoretical particle physics, which focuses on developing models and predictions based on mathematical frameworks, experimental particle physics involves the design, construction, and operation of experiments to test those theories and to discover new particles or phenomena.