The operating instrument in a Geiger counter is a Geiger-Muller tube, which is basically an electrified enclosure where the can is the negative electrode, and a rod down the middle is the positive electrode. The can is filled with gas at low pressure, and the electrodes are excited to a high voltage, but not quite high enough for a spontaneous arc of electricity between the core and the can. The gas ordinarily does not conduct electricity, but when a high energy particle or photon (ionizing radiation) passes into the assembly and collides with a gas molecule, it temporarily ionizes the gas, making it conductive. This provides a sufficiently conductive path for electricity to briefly arc between electrodes via the ion. Each such event represents a count, and often Geiger counters will have audible feedback of these electric arc events, which is the sound you are referring to.
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