In aluminium, \(v_{1}=6.32 \times 10^{3} \mathrm{~m} / \mathrm{s}\) and \(v_{\mathrm{t}}=3.1 \times 10^{3} \mathrm{~m} / \mathrm{s}\). The density of aluminium is \(2.7 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) and its atomic weight is \(26.97 .\) (a) Calculate the Debye cut off frequency for aluminium. (b) The Debye temperature for aluminium, as obtained from specific heat measurement, is \(375 \mathrm{~K}\). Find the cut off frequency and compare it with result obtained in (a). Ans. \(8.44 \times 10^{12} \mathrm{~Hz}, 7.8 \times 10^{12} \mathrm{~Hz}\). They are comparable.

In a solid, sound waves travel at different speeds depending on the mode of vibration: longitudinal or transverse. The average speed of sound, often denoted as \(v_{m}\), is a crucial parameter for calculating the Debye cut-off frequency. It combines both longitudinal \(v_{l}\) and transverse \(v_{t}\) speeds into a single effective speed. The formula to calculate \(v_{m}\) in a solid such as aluminum is: \[ v_{m} = \left( \frac{1}{3} \left( \frac{2}{v_{t}^{3}} + \frac{1}{v_{l}^{3}} \right) \right)^{-\frac{1}{3}} \] This equation ensures a balanced representation of the different wave speeds. For aluminum, using the given speeds \(v_{l} = 6.32 \times 10^{3} \text{ m/s}\) and \(v_{t} = 3.1 \times 10^{3} \text{ m/s}\), we find that \(v_{m} = 3.591 \times 10^{3} \text{ m/s}\). This average speed helps in determining many physical properties of the material.

The atomic volume is an essential concept that describes the volume occupied by a single atom in a material. It is calculated using the formula: \[ V_{a} = \frac{M}{\rho N_{A}} \] where \(M\) is the atomic weight, \(\rho\) is the density, and \(N_{A}\) is Avogadro's number. This equation connects the macroscopic density of a material with the microscopic properties of its atoms. For aluminum: substituting \(M = 26.97 \text{ kg/mol}\), \(\rho = 2.7 \times 10^{3} \text{ kg/m}^{3}\), and \(N_{A} = 6.022 \times 10^{23} \text{ atoms/mol}\), we get \(V_{a} = 1.659 \times 10^{-29} \text{ m}^{3} \text{ per atom}\). This minuscule volume highlights how densely packed aluminum atoms are in the solid state.

The Debye temperature, denoted as \(\Theta_{D}\), is a characteristic temperature of a material that correlates with the vibrational properties of its atoms. It is directly related to the highest frequency (Debye frequency) of the vibrational modes in a crystal. The formula connecting Debye temperature to the Debye cut-off frequency is: \[ \Omega_{D} = \frac{k_{B} \Theta_{D}}{\hbar} \] where \(k_{B}\) is Boltzmann constant and \(\hbar\) is the reduced Planck's constant. For aluminum with a Debye temperature \(\Theta_{D} = 375 \text{ K}\), this equation helps calculate the Debye frequency, giving a value of \(\Omega_{D} = 4.88 \times 10^{13} \text{ s}^{-1}\). This value can be converted to the cut-off frequency \(u_{D}\) as \(u_D = \frac{\Omega_D}{2\pi} = 7.8 \times 10^{12} \text{ Hz}\). This frequency indicates the upper limit on the vibrations' energy in the material.

Number density, symbolized as \(\frac{N}{V}\), denotes the number of atoms per unit volume in a material. It is a vital parameter for calculating the Debye cut-off frequency. Using the atomic volume \(V_{a}\), number density can be determined by: \[ \frac{N}{V} = \frac{1}{V_{a}} \] For aluminum, where \(V_{a} = 1.659 \times 10^{-29} \text{ m}^{3} \text{ per atom}\), we find \(\frac{N}{V} = 6.03 \times 10^{28} \text{ atoms/m}^{3}\). This high value of number density is typical of metals and reflects the tightly packed atomic structure. Knowing the number density helps us to understand the material's properties more deeply, such as how it will interact with sound waves and thermal energy at the atomic level.