**4. Origin of piezoelectricity in single-crystal plates and ceramics**

From the viewpoints of elastic constants in single crystal and ceramics, the origin of piezo‐ electricity of both was proposed from viewpoints of elastic constants. In single crystal, low K accompanied with high Y and G and low σ is significant to realize high k because single crystal exhibits one grain (no grain boundary) with single domain (**Figure 17**). On the contrary, in ceramics high σ caused by high K accompanied with low Y and G is significant to realize high k because same charges introduced by domain alignment in each grain repulse each other after DC poling [11]. The repulsion of oriented domains in grains acts as increasing σ with increasing K and decreasing Y and G, as the results, high k can be achieve in ceramics with multi grains (practical single domain in one grain). Therefore, the origin of piezoelectricity in ceramics is said that the existence of grain boundaries; furthermore, the grain orientation accompanied with domain alignment between ceramic grains is the key. **Figure 19** is summarized the relationships between domain alignment, size dimensions, origin of piezoelectricity, and effect of DC poling on elastic constants in single crystal and ceramics through our investigation of the measurement on sound velocities. From this figure, it is clearly understand between single and oriented domains in single crystal, and the relationships between size dimensions, domain and grain alignments, and the effect of DC poling on elastic constants in piezoelectric materials.

**Figure 19.** Relationships between domain alignment, size dimensions, piezoelectricity, and effect of DC poling on elas‐ tic constants in single crystal and ceramics.
