LEAST-SQUARES ESTIMATES OF THE NONLINEAR CONSTANTS OF PIEZOELECTRIC-CRYSTALS

The values of nonlinear constants of piezoelectric crystals are frequently calculated from overdetermined linear systems using the least-squares fit. In such systems, all nonlinear constants, whose values are known from earlier publications, are replaced with these values. The results of the least-squares fit, executed after the substitutions, are interpreted only mechanically, with no regard for the substitutions made. It is shown that this procedure frequently leads to a loss of valuable information and/or to unnoticed distortions and misinterpretations of the estimates of the nonlinear constants and of their standard errors. The reasons for these undesired phenomena and how to avoid them are explained. The calculation process should start with the least-squares fit with the substitutions to be made later, when their effect cannot be misunderstood. In a number of cases, however, the substitutions serve no useful purpose and should be avoided. A practical illustration is made for the nonlinear constants of quartz determined by the resonator method. The ideas and results of this paper apply generally to the determination of any material constants of crystals where the least-squares method is used.