As shown in Fig. 2., the frequency vs. temperature characteristics of the crystal unit varies, depending on the cut of the crystal. For working frequencies of 3 MHZ and onwards, AT-cut crystal units are normally used. The temperature vs. frequency characteristics of the AT-cut form a 3rd order curve. The characteristics, compared with those of XY-cut tuning fork crystal units and longitudinal crystal units are frequently used.
The frequency f of a crystal unit can be represented generally by a cubic expression of temperature T:

Δf/ f0=α(T -T0)+β(T -T0)2+γ(T -T0)3 .........................................................(1)
Where, Δf = f - f0
T0 : Reference temperature (taken as 25℃ in most cases)
α: 1st order temperature coefficient at T0
β: 2nd order temperature coefficient at T0
γ: 3rd order temperature coefficient at T0

In the case of AT-cut crystal units, when the angle of the crystal piece is varied in 2' steps centering around a fixed angle, theα, β, andγvalues will also varies, thereby changing the form of the 3rd order curve; shown in Fig. 3. When the frequency tolerance is specified within the working range temperature, Fig. 3 can offer a measure for selecting the proper angle.
Meanwhile, for XY-cut tuning fork crystal units and longitudinal crystal units, whose frequency vs. temperature characteristics form a 2nd order curve, Equ. (1) may be written as.
Δf/ f0 =β(T- Tm)2.......................................................................................(2)