INVERTING PIEZOELECTRIC MEASUREMENTS

The orientation distribution function (ODF) is a statistical descripti on of the orientation of the crystals which together make up a bulk sa mple. This paper describes an inversion of laboratory piezoelectric me asurements, made in different directions, to provide an estimate of th e smoothed, piezoelectrically-defined ODF for quartz crystals in a roc k. The inversion algorithm incorporates the properties of the piezoele ctric tensor so that the result is guaranteed to be consistent with th e piezoelectric properties of quartz. The case considered is that for which stress and electrical polarization are measured in the same dire ction. The procedure minimizes the RMS value of the ODF over the entir e sample, subject to the constraints provided by the measurements. The algorithm was tested by first generating synthetic data for a sample consisting of quartz crystals in a single known orientation, and simpl e aggregates of quartz crystals in different known orientations. The a greement between the ideal data and piezoelectric values calculated fr om the inversion of such synthetic samples was about twelve significan t figures, the limitation being due to the number of significant figur es output by the program generating the synthetic data. Adding random noise to the synthetic data had the result of worsening the agreement between the inversion results and the synthetic data by an amount appr oximately equal to the variance of the added noise. Thus, the inversio n can be used to evaluate an input data set. Data that are seriously i nconsistent with the piezoelectric tensor for quartz do not invert wel l in the sense that the result does not reproduce closely the input da ta. When used with a real data set comprised of 102 measurements of pi ezoelectric coefficients, residuals were significantly larger than we had expected. The size of the residuals can be accounted for by experi mental errors in the measurements or the inhomogeneity of the sample, or both. However, redundancy due to the extensive oversampling enabled us to produce an acceptable pole figure. An Appendix summarizes some of the important matrix relationships for piezoelectric materials and corrects some errors in published values.