The mechanically vibrating system and the circuit shown above are "equivalent," because each can be described by the same differential equation.  The mass, spring and damping element (i.e., the dashpot) correspond to the inductor, capacitor and resistor.  The driving force corresponds to the voltage, the displacement of the mass to the charge on the capacitor, and the velocity to the current.
	A crystal resonator is a mechanically vibrating system that is linked, via the piezoelectric effect, to the electrical world.  In the (simplified) equivalent circuit (of one mode of vibration) of a resonator, on the next page, C0 is called the “shunt” capacitance.  It is the capacitance due to the electrodes on the crystal plate (plus the stray capacitances due to the crystal enclosure).  The R1, L1, C1 portion of the circuit is the "motional arm" which arises from the mechanical vibrations of the crystal.  The C0 to C1 ratio is a measure of the interconversion between electrical and mechanical energy stored in the crystal, i.e., of the piezoelectric coupling factor, k, and C1 is a measure of the crystal's "stiffness," i.e., its tunability - see the equation under the equivalent circuit on the next page.  When a dc voltage is applied to the electrodes of a resonator, the C0/C1 is also a measure of the ratio of electrical energy stored in the capacitor formed by the electrodes to the energy stored elastically in the crystal due to the lattice strains produced by the piezoelectric effect.  The C0/C1 is also a measure of the antiresonance-resonance frequency separation.  (Let r =  C0/C1, then fA - fRfR/2r, and 2r = (N/2k)2, where N = 1,3,5...is the overtone number.)
	Some of the numerous advantages of quartz crystal resonator over a tank circuit built from discrete R's, C's and L's are that the crystal is far stiffer and has a far higher Q than what could be built from normal discrete components.  For example, a 5 MHz fundamental mode AT-cut crystal may have C1 = 0.01 pF, L1 = 0.1 H, R1 = 5 , and Q = 106.  A 0.01pF capacitor is not available, since the leads attached to such a capacitor would alone probably contribute more than 0.01 pF.  Similarly, a 0.1 H inductor would be physically large, it would need to include a large number of turns, and would need to be superconducting in order to have a  5  resistance.
	E. Hafner, "Resonator and Device Measurements," in E. A. Gerber and A. Ballato, Precision Frequency Control, Vol. 2, pp.1-44,  Academic Press, 1985.
	V. E. Bottom, Introduction to Quartz Crystal Unit Design, Van Nostrand Reinhold Company, 1982.