Question

Obtain an order of magnitude estimate for the spring constant \(\bar{k}\) associated with the interatomic forces in a silicate crystal such as forsterite by assuming \(\bar{k} \sim E b\), where \(E\) is Young's modulus and \(b\) is the average interatomic spacing. Young's modulus for forsterite is \(1.5 \times 10^{11}\) Pa. Obtain a value for \(b\) by assuming \(b^{3}\) is the mean atomic volume. The density of forsterite is \(3200 \mathrm{~kg} \mathrm{~m}^{-3}\). Estimate the maximum amplitude of vibration of an atom in a forsterite crystal at a temperature of \(300 \mathrm{~K}\). How does it compare with the mean interatomic spacing? What is the Einstein frequency at this temperature? The spring constant may also be estimated from the compressibility of forsterite using \(\bar{k} \sim 3 b / \beta,\) where the factor of 3 arises from the relation between fractional volume changes and fractional changes in length. How does this estimate of \(\bar{k}\) compare with the previous one? The compressibility of forsterite is \(0.8 \times 10^{-11} \mathrm{~Pa}^{-1}\).

Short answer

Forsterite's spring constant is around 6.24 N/m using Young's modulus, increasing to 15.6 N/m using compressibility. The thermal vibration amplitude is small compared to the interatomic spacing. Einstein frequency is about 5.15 × 10^13 rad/s.

Step-by-step solution

Calculate Interatomic Spacing (b)

The density of forsterite is given as \( \rho = 3200 \ \mathrm{kg} \ \mathrm{m}^{-3} \). To estimate the average interatomic spacing, we first determine the atomic mass (average for all constituent atoms). Assuming simple approximation, forsterite \( \mathrm{Mg}_2\mathrm{SiO}_4 \) has a molar mass \( M = 140.69 \ \mathrm{g/mol} \). The mean atomic volume is given by \( \frac{M}{N_A \cdot \rho} \), where \( N_A = 6.022 \times 10^{23} \ \mathrm{mol}^{-1} \). This gives us the mean atomic volume:\[ V_a = \frac{1.4069 \times 10^{-1} \ \mathrm{kg/mol}}{6.022 \times 10^{23} \ \mathrm{mol}^{-1} \times 3200 \ \mathrm{kg/m}^3} \approx 7.32 \times 10^{-29} \ \mathrm{m}^3. \]Assuming the volume \( b^3 \) yields \[ b = \sqrt[3]{V_a} \approx \sqrt[3]{7.32 \times 10^{-29}} \approx 4.16 \times 10^{-10} \ \mathrm{m}. \]

Calculate the Spring Constant Using Young's Modulus

Using \( \bar{k} \sim E \cdot b \) and the known Young's modulus \( E = 1.5 \times 10^{11} \ \mathrm{Pa} \):\[ \bar{k} \approx 1.5 \times 10^{11} \ \mathrm{Pa} \times 4.16 \times 10^{-10} \ \mathrm{m} = 6.24 \times 10^{1} \ \mathrm{N/m}. \]

Calculate Maximum Amplitude of Vibration

Using the formula for thermal vibration amplitude, \( A = \sqrt{\frac{k_B T}{\bar{k}}} \), where \( k_B \approx 1.38 \times 10^{-23} \ \mathrm{J/K} \) is Boltzmann's constant and \( T = 300 \ \mathrm{K} \):\[ A \approx \sqrt{\frac{1.38 \times 10^{-23} \times 300}{6.24 \times 10^{1}}} \approx 2.5 \times 10^{-11} \ \mathrm{m}. \]This amplitude is much smaller compared to the interatomic spacing \( b \approx 4.16 \times 10^{-10} \ \mathrm{m} \).

Calculate Einstein Frequency

The Einstein frequency \( \omega_E \) can be calculated using the relation \( \omega_E = \sqrt{\frac{\bar{k}}{m}} \), with \( m \) as the average atomic mass:\[ m = \frac{M}{N_A} = \frac{1.4069 \times 10^{-1}}{6.022 \times 10^{23}} \approx 2.34 \times 10^{-25} \ \mathrm{kg}. \]Then,\[ \omega_E \approx \sqrt{\frac{6.24 \times 10^1}{2.34 \times 10^{-25}}} \approx 5.15 \times 10^{13} \ \mathrm{rad/s}. \]

Calculate Spring Constant Using Compressibility

Given the compressibility \( \beta = 0.8 \times 10^{-11} \ \mathrm{Pa}^{-1} \), the alternative spring constant is estimated by \( \bar{k} \sim \frac{3b}{\beta} \):\[ \bar{k} \approx \frac{3 \times 4.16 \times 10^{-10}}{0.8 \times 10^{-11}} \approx 15.6 \times 10^{1} \ \mathrm{N/m}. \]

Compare Estimate of Spring Constants

In summary, the spring constant obtained using Young's modulus was \( 6.24 \ \mathrm{N/m} \). The compressibility method gave \( 15.6 \ \mathrm{N/m} \). These estimates show a discrepancy which can arise from approximations and assumptions in deriving \( b \) and using simple models to estimate \( \bar{k} \).

Key concepts

Young's modulus

Young's modulus is a fundamental property in material science that describes the stiffness of a material. It quantifies the amount of stretch or compression a material undergoes when a stress is applied. This modulus is essential when calculating the spring constant for interatomic forces. It provides a way to relate the macroscopic mechanical properties of a material to its microscopic characteristics.

For our exercise with forsterite, Young's modulus was given as \(1.5 \times 10^{11} \ \mathrm{Pa}\). This value is critical when estimating the spring constant of the interatomic forces within the crystal. The spring constant \(\bar{k}\) is approximated using the relation \(\bar{k} \sim E \cdot b\), where \(b\) is the interatomic spacing. By using Young's modulus, we're able to understand how much the material resists deformation, which directly influences the magnitude of \(\bar{k}\).

Understanding Young's modulus helps us comprehend how different materials can have vastly different elastic properties, affecting their behavior under stress and their applications.

atomic spacing

Atomic spacing, represented by \(b\), is the average distance between atoms in a solid structure. This spacing is fundamental in determining the physical properties of a crystal. It directly influences the material's density, thermal properties, and mechanical characteristics.

In the context of forsterite, calculating \(b\) involved analyzing the mean atomic volume and density of the material. Using the density \(\rho = 3200 \, \mathrm{kg/m^3}\) and the molar mass \(M\), the volume of an atom was determined, leading to \[ b = \sqrt[3]{V_a} \approx 4.16 \times 10^{-10} \, \mathrm{m}. \]

This value, \(b\), is crucial because it offers insight into how tightly packed the atoms are within the crystal lattice. It also helps us estimate the spring constant \(\bar{k}\) since a smaller atomic spacing usually signifies stronger interatomic forces.

compressibility

Compressibility is a measure that indicates how much a material can reduce in volume under pressure. It's a crucial parameter in understanding a material's response to external forces. In forsterite, compressibility gave an alternative method to estimate the spring constant \(\bar{k}\) of the interatomic forces in the crystal lattice.

Given the compressibility \( \beta = 0.8 \times 10^{-11} \, \mathrm{Pa}^{-1} \), the alternative spring constant was calculated using \( \bar{k} \sim \frac{3b}{\beta} \). This provides a view from a different angle compared to the approach using Young's modulus.

Compressibility can often highlight discrepancies in calculated results due to various assumptions and approximations related to atomic spacing and the nature of intermolecular interactions. Comparing the different methods to calculate \(\bar{k}\) helps validate the consistency and reliability of the calculated properties.

Einstein frequency

The Einstein frequency is a concept tied to the vibrational modes of atoms within a crystal at a given temperature. It describes the frequency at which each atom in the lattice oscillates.

For the forsterite crystal, the Einstein frequency \(\omega_E\) was determined using the relation \(\omega_E = \sqrt{\frac{\bar{k}}{m}}\), where \(m\) is the average atomic mass of the constituent atoms. With a calculated spring constant \(\bar{k} \) and known atomic mass, the frequency was estimated as \( 5.15 \times 10^{13} \, \mathrm{rad/s} \).

This frequency is pivotal in understanding the thermal and acoustic properties of materials. Higher Einstein frequencies typically indicate stronger interatomic bonds and a higher degree of order within the lattice. It provides insights into how energy propagates through the material, influencing properties like thermal conductivity and specific heat capacity.