Measurements in bulk materials

An excellent reference describing appropriate ways of measuring the piezoelectric coefficients of bulk materials is the IEEE Standard for Piezoelectricity [1], In brief, the method entails choosing a sample with a geometry such that the desired resonance mode can be excited, and there is little overlap between modes. Then, the sample is electrically excited with an alternating field, and the impedance (or admittance, etc.) is measured as a function of frequency. Extrema in the electrical responses are observed near the resonance and antiresonance frequencies. As an example, consider the length extensional mode of a vibrator. Here the elastic compliance under constant field can be measured from 

The ieee Standard on Piezoelectricity [1] describes these measurements in considerable detail, and gives the necessary sample geometries for determination of a number of the piezoelectric constants. For example, the relations that enable determination of the other common piezoelectric moduli in bulk ceramics are given in Equation (2.7). 

In most cases, resonance measurements are made at low excitation levels, so that small signal numbers are derived. Methods of extending resonance methods to higher powers are discussed in [11]. 

In such a measurement, the sample is clamped as lightly as possible, and the displacement of the surface in monitored. The amount of sample clamping is important, because the mechanical constraints can impact the ferroelastic response of the sample. That is, in samples where the mechanical coercive stress is low, it is possible to change the domain state of the material by improperly clamping it in the sample fixture. This is especially important in elastically soft piezoelectrics, such as many of the relaxor ferroelectric PbTiC 3 single crystals. 

Optical methods can also be utilized to track the displacement of samples surfaces. Various types of interferometers have been employed for measurements of this type, yielding accuracy in the surface displacement between 0.1 and 10-5 A [6,9,15]. Both single and double beam versions have been developed for measuring the d33 coefficient of bulk samples. Cubic samples can avoid some of the difficulties associated with sample flexure on actuation [9], In some cases, one of the probing beams can also be brought to the side of the sample, so that lateral and tangential displacements can be tracked simultaneously [15]. 

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