Measurement of the order parameter ofα-quartz by second-harmonic generation of light

We discuss the concept of an order parameter $\eta$ for the $\alpha -\beta$ transition of quartz following Landau's theory. Some physical properties, among them the second-harmonic-generation coefficient are found to vary as $\eta$, while others vary as ${\eta }^2$. A measurement of $d_{11}$ from room temperature to the transition is described. Assuming that $d_{11}=\eta$ one obtains the temperature variation of the order parameter; our data are in good agreement with Landau's theory. They also agree with the temperature variation of other properties, particularly birefringence, thermal dilatation, and some elastic constants. Our data can also be fitted to a formula $\eta ={\eta }^{*}+C{(t-t^{*})}^{\beta }$, with a critical exponent $\beta =\frac{1}{3}$ from room temperature to the transition, or with $\beta =\frac{1}{6}$ and ${\eta }^{*}=0\mathrm{}$ from 520 to 573.2°C the transition temperature. We show that this is not fortuitous but that one can always find these exponents in making a four- or three-parameter fit to Landau's formula. Thus, in general, there is considerable ambiguity when one tries to fit critical exponents to a first-order transition.