The equivalent series resistance (ESR) is an electrical representation of the mechanical losses and at the series resonant frequency, fs, it is equal to R1 in the electrical model. The ESR is a very important parameter and can be found in the crystal data sheet. The ESR will usually be dependent on the crystal’s physical size and small crystals (especially small SMD crystals) typically have higher losses and ESR values than larger crystals.
Higher ESR values will load the inverting amplifier more and too high ESR may cause unstable oscillator operation. Unity gain will not be achieved and the Barkhausen criterion will not be fulfilled.