3-11
      
 Where fn = resonant frequency of n-th harmonic
             h = plate thickness
             r = density
            cij = elastic modulus associated with the elastic wave
                  being propagated          
where Tf is the linear temperature coefficient of frequency. The temperature
coefficient of cij is negative for most materials (i.e., “springs” become “softer”
as T increases). The coefficients for quartz can be +, - or zero (see next page).
Infinite Plate Thickness Shear Resonator
   The velocity of propagation, v, of a wave in the thickness direction of a thin plate, and the resonant frequency, fn, of an infinite plate vibrating in a thickness mode are
v=(cij/), and fn = n/2h(cij/)
respectively, where cij is the elastic stiffness associated with the wave being propagated,  is the density of the plate, and 2h is the thickness.  The frequency can also be expressed (taking the logarithm of both sides) as
log fn = log(n/2) - log h + 1/2 (log cij - log )
Differentiating this equation gives the above expression for Tf.
   Quartz expands when heated; dh/dT is positive along all directions, and d/dT is negative (see “Thermal Expansion Coefficients of Quartz” in chapter 4).  The temperature coefficient of cij also varies with direction.  It is fortunate that, in quartz, directions exist such that the temperature coefficient of cij balances out the effects of the thermal expansion coefficients, i.e., zero temperature coefficient cuts exist in quartz.


V. E. Bottom, Introduction to Quartz Crystal Unit Design, Van Nostrand Reinhold Company, 1982.