Question

Cane sugar, also called saccharose, has chemical composition \(\mathrm{C}_{12} \mathrm{H}_{22} \mathrm{O}_{11}\) and belongs to the crystal point group given by 2 in Hermann-Maguin notation [38]. Reference [38] lists values specified in cgse units for its piezoelectric constant as \(10.2 \cdot 10^{-8} \frac{\mathrm{esu}}{\mathrm{dyne}}\) and its pyroelectric coefficient as \(0.53 \frac{\mathrm{esu}}{\mathrm{cm}^{2,0} \mathrm{C}}\). Convert these values to the SI units of \(\frac{\mathrm{m}}{\mathrm{V}}\) and \(\frac{\mathrm{C}}{\mathrm{m}^{2} \cdot \mathrm{K}}\) respectively. Hint: The electrostatic unit or statcoulomb is a measure of charge [7] where $$ 1 \mathrm{esu}=1 \text { stat } \mathrm{C}=3.335641 \cdot 10^{-10} \mathrm{C} $$ and the dyne is a measure of force where 1 dyne \(=10^{-5} \mathrm{~N}\).

Short answer

The piezoelectric constant in SI units is \(3.403 \times 10^{-12} \frac{\text{m}}{\text{V}}\). The pyroelectric coefficient is \(1.77 \times 10^{-6} \frac{\text{C}}{\text{m}^2 \cdot \text{K}}\).

Step-by-step solution

Convert Piezoelectric Constant

The given piezoelectric constant is \(10.2 \cdot 10^{-8} \frac{\text{esu}}{\text{dyne}}\). First, convert esu (statC) to coulombs (C): \(1 \text{ esu} = 3.335641 \times 10^{-10} \text{ C}\). Then, convert dyne to newtons (N): \(1 \text{ dyne} = 10^{-5} \text{ N}\). Thus, \(10.2 \cdot 10^{-8} \frac{\text{esu}}{\text{dyne}} = 10.2 \cdot 10^{-8} \frac{3.335641 \times 10^{-10} \text{ C}}{10^{-5} \text{ N}}\). This simplifies to \(3.4029 \times 10^{-12} \frac{\text{C}}{\text{N}}\). In SI units, this is equivalent to \(3.4029 \times 10^{-12} \frac{\text{m}}{\text{V}}\) since \(1 \frac{\text{C}}{\text{N}} = 1 \frac{\text{m}}{\text{V}}\).

Convert Pyroelectric Coefficient

The given pyroelectric coefficient is \(0.53 \frac{\text{esu}}{\text{cm}^2 \cdot \text{C}}\). Convert esu to coulombs first: \(1 \text{ esu} = 3.335641 \times 10^{-10} \text{ C}\). Also convert cm^2 to m^2: \(1 \text{ cm}^2 = 10^{-4} \text{ m}^2\). Thus, \(0.53 \frac{\text{esu}}{\text{cm}^2} = 0.53 \cdot \frac{3.335641 \times 10^{-10} \text{ C}}{10^{-4} \text{ m}^2} = 1.76988493 \times 10^{-6} \frac{\text{C}}{\text{m}^2}\). In SI units, the mixed unit of \(\text{C}/\text{m}^2 / \text{C}\) is close to \(\text{C}/\text{m}^2 / \text{K}\) since they both involve incremental charge change with increment area, \(0.53 \frac{\text{C}}{\text{m}^2 \cdot K}\).

Key concepts

Piezoelectric Constants

Piezoelectric constants are values that describe the ability of certain materials to generate an electric charge in response to mechanical stress. This property is known as the piezoelectric effect, which is used in various applications like sensors and actuators. The piezoelectric constant is often denoted with \(d_{ij}\), where \i\ and \j\ denote specific directions within the crystal lattice. To convert a piezoelectric constant into SI units, we need to translate it from units of \(\frac{\text{esu}}{\text{dyne}}\) to \(\frac{\text{m}}{\text{V}}\). This involves converting electrostatic units (esu) to coulombs (C), knowing that \(1 \text{ esu} = 3.335641 \times 10^{-10} \text{ C}\). Additionally, a dyne must be converted to newtons (N), where \(1 \text{ dyne} = 10^{-5} \text{ N}\). Thus, the calculation for cane sugar's piezoelectric constant involves this unit conversion, resulting in a value of approximately \(3.4029 \times 10^{-12} \frac{\text{m}}{\text{V}}\). This means that for every newton of force applied per meter, cane sugar produces that amount of meters per volt in electrical response.

Pyroelectric Coefficients

Pyroelectric coefficients describe the ability of a material to generate temporary voltage as it is heated or cooled. This is termed the pyroelectric effect. Materials with significant pyroelectric coefficients are useful in thermal sensors and infrared detection. The pyroelectric coefficient is usually measured in \(\frac{\text{C}}{\text{m}^2 \cdot \text{K}}\) in SI units, representing the charge generated per unit temperature change over a unit area. In the original exercise, cane sugar's pyroelectric coefficient is given in \(\frac{\text{esu}}{\text{cm}^2 \cdot \text{C}}\), which needs conversion to \(\frac{\text{C}}{\text{m}^2 \cdot \text{K}}\). We again convert esu to coulombs, and \(\text{cm}^2\) to \(\text{m}^2\) (since \(1 \text{ cm}^2 = 10^{-4} \text{ m}^2\)). This results in cane sugar having a pyroelectric coefficient of about \(1.76988493 \times 10^{-6} \frac{\text{C}}{\text{m}^2 \cdot \text{K}}\), highlighting its efficiency in electric charge change when exposed to temperature variations.

SI Units

SI units, or the International System of Units, form the foundation for scientific measurements worldwide. This standardized system ensures that measurements are consistent and universally understood, facilitating collaboration and communication in the scientific community. The SI system is made up of seven base units which include meters (m) for length, kilograms (kg) for mass, seconds (s) for time, amperes (A) for electric current, kelvin (K) for temperature, mole (mol) for amount of substance, and candela (cd) for luminous intensity. In the context of conversion in physics, units like volts (V) and newtons (N) are derived from these base units, such as \(\frac{\text{kg} \cdot \text{m}}{\text{s}^2} = \text{N}\) for force, and \(\frac{\text{kg} \cdot \text{m}^2}{\text{s}^3 \cdot \text{A}} = \text{V}\) for voltage. Understanding and using SI units is crucial for accurately converting values like the piezoelectric constant and pyroelectric coefficient from other units like esu and dynes into their SI equivalents.

Crystal Point Groups

Crystal point groups form a way to categorize crystals based on their symmetry. These groups describe the symmetry properties of crystals, which are crucial in determining the behavior of piezoelectric and pyroelectric materials. Cane sugar, belonging to crystal point group 2, implies a certain degree of symmetry where the crystal structure is invariant under a particular set of symmetry operations, like rotations or reflections. The symmetry in point groups influences the material's ability to exhibit certain electric properties, such as piezoelectricity or pyroelectricity, meaning only crystals that lack a center of symmetry can be piezoelectric. Understanding crystal point groups helps predict which materials can have these electric properties, making it an essential concept in material science and engineering when working with and developing new materials for technological applications.