How does a crystal oscillator work?
As a kid, having been given a 100,000 Hertz crystal of size 1&quot; by 1&quot; by 2&quot; from some WW2 military radio, I tried repeatedly to get that beast to oscillate, listening at the 5MHz WWV channel for some interference or beat note that would indicate successful oscillation.
I never could get oscillation.
The challenge with a crystal is the interface with outside circuitry.
One standard method is using PI_filtering mindset, where two external capacitors provide a closed_loop path for circulating currents.
Each end of the crystal is shunted to Ground; the size of the capacitors is proportional to the internal electrode_to_electrode capacitance; with thick quartz needed to resonate at 100,000 Hz, the spacing is high and the electrode_to_electrode capacitance is low, so you need only a few picoFarads lumped C at each end of the crystal.
As a kid, having had vector algebra and complex number math in school, I could have written the small_signal model of various  ARRL (HAM) oscillators, and perhaps extracted the conditions for successful oscillation. Maybe.
But my brain was not ready for small_signal modeling.
The 2_external_cap PI_filter interface to the quartz crystal has the useful behavior of INVERTING the voltage. This allows very simple one_transistor amplifiers, of common_emitter or common_source (of common_cathode) configuration, to be used.
As a kid, I did not have that understanding. Thus my 100,000 Hz circuits never oscillated.
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How to start the oscillation? how does one make a bell ring, or a crystal goblet ring? you lightly and briefly tap it.
And in electronics, there is random electron noise, as the electrons very quickly jump orbitals inside the atoms. That is the &quot;tap&quot; needed.
Probably bigger taps come from switching regulator spikes. Or just turning on the power. But the random and very-fast electron jumping around is always there.
HINT: a quick tap makes ANY bell or ANY goblet ring. That quick tap has a special name IMPULSE, meaning has some energy and very fast (ideally zero) time duration.
Thus the very_fast impulses make ANY QUARTZ CRYSTAL ring and ring and ring, as the transistor provides the boost in energy inside the quartz.
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Here is link to a previous answer, to explain the PI_filter usefulness. Scroll down, past the simulation, and examine the last two circuits.
How do crystal oscillators start?