Here is some of what the book "Crystal Oscillator Circuits" ISBN 0-471-87401-9 by Robert J. Matthys has to say about the Colpitts oscillator.
Physically, this parallel-resonant circuit is very simple, but analytically, it is very complex. The amplifier is an emitter follower with a gain of 1. The transistor conducts current over only a small portion of each cycle, usually about 15-20%. The transistor starts conducting a little before (about 10%) its base reaches the most positive peak of the sinusoid and stops conducting immediately after the positive peak. At the positive peak the transistor saturates and clamps the crystal to the power supply bus through the forward-biased base-collector junction. Positive peak saturation lasts about 5-10% of an oscillation cycle. The transistor shuts off and remains nonconducting over the rest (80-85%) of the cycle. Thus, three circuit conditions exist during each cycle: a short interval (10%) with the transistor conducting properly and acting as an emitter follower; a second short interval (5-10%) with the transistor saturated and shorting out the crystal; and a third long interval (80-85%) with the transistor shut off and nonconducting.
What values should be used for C1, C2, R1, and R2? The answer is that they should be what the crystal wants them to be, which the crystal indicates by maximizing the voltage across itself. The biggest factor is external shunt resistance across the crystal terminals, which is such a strong factor that it determines the lowest frequency the circuit will oscillate at. This shunt resistance is the parallel sum of R2 and the transistor's input resistance (R1 multiplied by the transistor gain). There is a minimum shunt resistance below which the crystal will not oscillate. C2 should be a fixed value of 40-70 pF. The time constant R1C1 should vary inversely with the oscillation frequency, R1C1=~0.5/f. Note that these values are for the basic transistor version. The Colpitts oscillator is also found in a frequency multiplier type circuit, a basic FET version, and an FET harmonic version.
