# To derive the piezoelectric effect in crystals

• etotheipi
Finally, the piezoelectric coefficients are related to the moduli through the following relationships:$$e_{kij} = d_{ijk} - s_{ijkm} \epsilon_{m}$$and$$c_{ijkl} = s_{ijkl} + \frac{d_{ijmn} d_{klmn}}{\epsilon_0}$$Thus, we have derived the general constitutive equations for piezoelectric materials and derived relationships between the piezoelectric coefficients and the other moduli.
etotheipi
For a nice cubic non-centrosymmetric crystal like quartz/##\mathrm{SiO_2}## we can imagine that on application of stress the tetrahedral coordination polyhedra become distorted, and the central ion is displaced by a fraction ##\lambda## of the cell parameter ##a##. The total polarisation ##\mathbf{P} = \mathbf{P}_1 + \mathbf{P}_2## is the sum of two contributions: ##\mathbf{P}_1 \propto q (\lambda a)/a^3 \boldsymbol{e}_1## due to the 'frozen-in' displacement of the central ions in each unit cell, and ##\mathbf{P}_2 = \varepsilon_0 (\kappa - 1) \mathbf{E}## due to polarisation of the rest of the crystal (approximated to be linear and isotropic) due to the net field ##\mathbf{E}##. With the Gauss relation ##\nabla \cdot \mathbf{E} = \rho / \varepsilon_0## applied in integral form to a pillbox at the surface then we can show for a linear piezoelectric we get something like$$V = \frac{\sigma Ld}{\kappa \varepsilon_0}$$where ##\sigma## is the magnitude of stress, ##L## the width of the crystal and ##d## a piezoelectric coefficient. But for the general piezoelectric we have the constitutive equations\begin{align*} \sigma_{ij} &= c_{ijkl} S_{kl} - e_{kij} E_k \\ D_k &= e_{kij} S_{ij} + \epsilon_{ki} E_i\end{align*}with ##\sigma_{ij}## the stress tensor, ##S_{ij}## the strain tensor, ##\epsilon_{ij}## the dielectric tensor, ##c_{ijkl}## elastic constants, ##e_{jik}## piezoelectric constants and ##E_i## & ##D_i## as usual. Does anyone have reference to derive these equations? I don't know anything about elasticity theory. Thanks

Okay, I think I understand the argument now. I'll write up the main points in case anyone else is interested. First some notational housekeeping; in the more modern notation, we use ##\epsilon_{ij}## to refer to the strain tensor, ##\sigma_{ij}## the stress tensor, ##d_{ijk}## the piezoelectric moduli, and ##s_{ijkl}## the compliance tensor (the twin of the stiffness tensor ##c_{ijkl}##).

In the most general case, thermal, elastic and electrical quantities all depend on each other. We can choose a set of independent variables in which to describe the quantities ##\epsilon_{ij}## , ##D_i## and ##S## (entropy); we will here choose to express these as functions of the variables ##(\sigma_{ij}, E_i, T)##, i.e.$$d\epsilon_{ij} = \left(\frac{\partial \epsilon_{ij}}{\partial \sigma_{kl}}\right) d\sigma_{kl} + \left(\frac{\partial \epsilon_{ij}}{\partial E_k}\right) dE_k + \left(\frac{\partial \epsilon_{ij}}{\partial T}\right) dT$$ $$dD_i = \left(\frac{\partial D_i}{\partial \sigma_{jk}}\right) d\sigma_{jk} + \left(\frac{\partial D_i}{\partial E_j}\right) dE_j + \left(\frac{\partial D_i}{\partial T}\right) dT$$ $$dS = \left(\frac{\partial S}{\partial \sigma_{ij}}\right) d\sigma_{ij} + \left(\frac{\partial S}{\partial E_i}\right) dE_i + \left(\frac{\partial S}{\partial T}\right) dT$$Since all processes are assumed reversible,$$dU = \sigma_{ij} d\epsilon_{ij} + E_i dD_i + T dS$$Consider the function ##\Phi = \Phi(\sigma_{ij}, E_i, T)## such that$$\Phi = U - \sigma_{ij} d\epsilon_{ij} - E_i D_i -TS \implies d\Phi = - \epsilon_{ij} d\sigma_{ij} -D_i dE_i - SdT$$but we also have$$d\Phi = \left(\frac{\partial \Phi}{\partial \sigma_{ij}}\right) d\sigma_{ij} + \left(\frac{\partial \Phi}{\partial E_i}\right) dE_i + \left(\frac{\partial \Phi}{\partial T}\right) dT$$and hence we have the relationships ##\left(\frac{\partial \Phi}{\partial \sigma_{ij}}\right) = -\epsilon_{ij}##, then ##\left(\frac{\partial \Phi}{\partial E_i}\right) = -D_i##, and finally ##\left(\frac{\partial \Phi}{\partial T}\right) = -S##. But then, because partial derivatives commute, note that for instance$$\left(\frac{\partial \epsilon_{ij}}{\partial E_k}\right) = \left(\frac{\partial D_k}{\partial \sigma_{ij}}\right) \overset{\text{def}}{=} d_{kij}$$and a similar reasoning for the other combinations. This is interesting, because it means that the same components ##d_{ijk}## are the moduli for both the direct and converse piezoelectric effects! That is,\begin{align*} &\text{Direct effect:} \, P_i = d_{ijk} \sigma_{jk} \\ &\text{Converse effect:} \, \epsilon_{jk} = d_{ijk} E_i \end{align*}The general constitutive equations can be obtained by simply integrating the equations for ##\epsilon_{ij}## and ##D_i## written at the start of this post, and substituting in relational tensors like ##d_{ijk}##, etc. in place of the partial derivatives.

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Greg Bernhardt

## 1. What is the piezoelectric effect in crystals?

The piezoelectric effect is the phenomenon in which certain crystals can generate an electric charge when subjected to mechanical stress, such as pressure or vibration. This is due to the crystal's unique molecular structure, which allows it to convert mechanical energy into electrical energy.

## 2. How is the piezoelectric effect derived in crystals?

The piezoelectric effect is derived through the application of the laws of thermodynamics and electromagnetism. By studying the behavior of the crystal's molecules and their arrangement, scientists can mathematically determine the relationship between mechanical stress and electric charge.

## 3. What types of crystals exhibit the piezoelectric effect?

There are several types of crystals that exhibit the piezoelectric effect, including quartz, tourmaline, and topaz. These crystals have a symmetrical molecular structure, with a dipole moment that allows for the conversion of mechanical energy into electrical energy.

## 4. How is the piezoelectric effect used in technology?

The piezoelectric effect has a wide range of applications in technology, including in sensors, actuators, and transducers. It is commonly used in devices such as ultrasound machines, microphones, and accelerometers. It is also used in energy harvesting, where mechanical stress is converted into electrical energy.

## 5. What are the practical implications of understanding the piezoelectric effect in crystals?

Understanding the piezoelectric effect in crystals has led to advancements in technology and has opened up new possibilities for energy harvesting and sensing. It has also allowed scientists to better understand the behavior of materials at a molecular level, leading to further discoveries and developments in the field of materials science.

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