Electrons in a piece of copper metal can be assumed to behave like an ideal Fermi gas. Copper metal in the solid state has a mass density of \(9 \mathrm{~g} / \mathrm{cm}^{3}\). Assume that each copper atom donates one electron to the Fermi gas. Assume the system is at \(T=0 \mathrm{~K}\). (a) Compute the Fermi energy, \(\varepsilon_{\mathrm{F}}\), of the electron gas. (b) Compute the Fermi "temperature", \(T_{\mathrm{F}}=\varepsilon_{\mathrm{P}} / \mathrm{k}_{\mathrm{B}}\).

The concept of Fermi energy is critical in understanding how electrons behave in a metal like copper. Fermi energy, denoted as \(\varepsilon_F\), represents the energy level at which the probability of finding an electron is 50% at absolute zero temperature (0 K). This energy level is fundamental because it determines the properties of the electrons, such as their speed and distribution within the material. The Fermi energy for a Fermi gas can be calculated using the equation: \[ \varepsilon_F = \frac{\hbar^2}{2m_e} (3 \pi^2 n)^{2/3} \] where \(\hbar\) is the reduced Planck constant, \(m_e\) is the electron mass, and \(n\) is the number density of electrons. By substituting these values into the equation, we find that the Fermi energy for copper is approximately 7.00 eV.

Number density is a measure of how many electrons are present in a given volume of the material. It is denoted as \(n\) and is crucial for calculating other properties such as Fermi energy and Fermi temperature. For copper, the number density can be calculated by using the mass density \(\rho\) and the molar mass \(M\): \[ n = \frac{N_A \cdot \rho}{M} \] Here, \(N_A\) is Avogadro's number. By performing the calculations step by step, we determine that the number density of electrons in copper is \(8.52 \times 10^{28} \:\text{electrons/m}^3\).

The Fermi temperature, denoted as \(T_F\), is another important concept derived from Fermi energy. It indicates the temperature at which the thermal energy \(k_B T\) is comparable to the Fermi energy. The Fermi temperature gives an idea of the temperature range in which quantum effects dominate the behavior of electrons in metals. It can be calculated using the equation: \[ T_F = \frac{\varepsilon_F}{k_B} \] Here, \(k_B\) is the Boltzmann constant. For copper, with a Fermi energy of 7.00 eV, the Fermi temperature is approximately \(8.12 \times 10^4 \:\text{K}\).

Statistical physics provides a framework to analyze the behavior of large groups of particles like electrons in a metal. It makes use of statistics and probability to explain macroscopic properties based on the microscopic behavior of particles. In the context of a Fermi gas, statistical physics helps us understand how electrons distribute themselves among energy states. Key concepts include:

• Fermi-Dirac statistics: governs the distribution of electrons in energy levels.
• Partition function: used to calculate thermodynamic properties.
• Ensemble theory: describes systems in terms of averages over large numbers of identical copies.

This theoretical approach is essential to derive properties like Fermi energy and number density.

Copper is a common material in which the concept of Fermi gas can be applied. It is highly conductive due to the free electrons that form a Fermi gas. These free electrons arise because each copper atom donates one electron. Some interesting points about copper include:

• Mass density: 9 g/cm³.
• Each atom contributes one electron to the electron gas.
• Copper's electronic properties can be explained using Fermi gas theory.

Understanding the electron behavior in copper helps in designing better electronic devices and materials.