Erratum to: Tensor Properties of Crystals

The online version of the original chapter can be found at http://dx.doi.org/10.1007/978-1-4419-7598-0_3

On page 72, Table 3.5 has the forms of 3rd rank tensors for all the crystallographic point groups. Five of these have factors of 2 in some of the components. These factors of 2 can be found in other published tables of 3rd rank tensors but they are only valid for a different type of notation that is not used in the book. To be consistent with the tensor notation used throughout this book, no factors of 2 should appear in these tensors. The Erratum provides a corrected Table 3.5 with the factors of 2 eliminated from all the tensor elements.

In addition, on page 73 an example is given in the second paragraph using one of the tensors from Table 3.5 that contains erroneous factors of 2 in some of its components. The Erratum provides a corrected paragraph without the factors of 2.

Table III.5 Form of third rank tensors for the crystallographic point groups.

Page 73, second paragraph

As a practical example, consider a quartz crystal that has D₃ symmetry at room temperature. The piezoelectric effect for this case is given by

$$ \left(\begin{array}{c}\hfill {P}_1\hfill \\ {}\hfill {P}_2\hfill \\ {}\hfill {P}_3\hfill \end{array}\right)=\left[\begin{array}{ccc}\hfill {d}_{111}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -{d}_{111}\hfill & \hfill d{}_{132}\hfill \\ {}\hfill 0\hfill & \hfill {d}_{132}\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill -{d}_{111}\hfill & \hfill -{d}_{132}\hfill \\ {}\hfill -{d}_{111}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill -{d}_{132}\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ {}\hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right]\left(\begin{array}{ccc}\hfill {\sigma}_{11}\hfill & \hfill {\sigma}_{12}\hfill & \hfill {\sigma}_{13}\hfill \\ {}\hfill {\sigma}_{21}\hfill & \hfill {\sigma}_{22}\hfill & \hfill {\sigma}_{23}\hfill \\ {}\hfill {\sigma}_{31}\hfill & \hfill {\sigma}_{32}\hfill & \hfill {\sigma}_{33}\hfill \end{array}\right) $$

.

so

• P ₁=d₁₁₁σ₁₁–d₁₁₁σ₂₂+d₁₃₂σ₃₂+d₁₃₂σ₂₃=(σ₁₁-σ₂₂)d₁₁₁+(σ₃₂+σ₂₃)d₁₃₂

• P ₂= -d₁₁₁σ₂₁–d₁₃₂σ₃₁-d₁₁₁σ₁₂-d₁₃₂σ₁₃= -d₁₁₁(σ₂₁+σ₁₂)-(σ₁₃+σ₃₁)d₁₃₂

• P ₃=0

If a uniaxial stress is applied in the σ₁₁ direction, P ₁=d₁₁₁σ₁₁ and P₂=0. The same tensile stress applied along σ₂₂ also produces a polarization along P ₁. The two-fold rotation axis P ₁ is the electric axis of quartz. Shear stress can produce polarization along P ₂ but no stress conditions can produce a polarization along P ₃.