Question

Compute the Fermi energy of potassium by making the simple approximation that each atom contributes one free electron. The density of potassium is \(851 kg/m^{3}\), and the mass of a single potassium atom is \(6.49 \times 10^{-26}\) kg.

Short answer

The Fermi energy of potassium is approximately 13.26 eV.

Step-by-step solution

Calculate the Number Density of Electrons

First, find the number density of potassium atoms. Since each atom contributes one free electron, this is also the number density of electrons, \(n_e\). The formula for the number density is \(n_e = \frac{N}{V} = \frac{\text{density}}{\text{mass of one atom}}\).Given:- Density \(\rho = 851\; kg/m^3\)- Mass of one atom \(m_{atom} = 6.49 \times 10^{-26}\; kg\)Calculate \(n_e\):\[n_e = \frac{851\; kg/m^3}{6.49 \times 10^{-26} \; kg} = 1.31 \times 10^{28}\; m^{-3}.\]

Use the Fermi Energy Formula

The Fermi energy \(E_F\) for electrons in a solid can be calculated using the formula:\[E_F = \frac{\hbar^2}{2m_e} (3\pi^2 n_e)^{2/3},\]where:- \(\hbar = 1.0545718 \times 10^{-34}\; Js\), the reduced Planck's constant.- \(m_e = 9.109 \times 10^{-31}\; kg\), the mass of an electron.- \(n_e = 1.31 \times 10^{28}\; m^{-3}\), as calculated earlier.

Substitute Values into the Fermi Energy Formula

Substitute the known values into the Fermi energy formula:\[E_F = \frac{(1.0545718 \times 10^{-34}\; Js)^2}{2 \times 9.109 \times 10^{-31}\; kg} \times (3 \pi^2 \times 1.31 \times 10^{28}\; m^{-3})^{2/3}\]Calculate the expression inside the formula:- Calculate \((3\pi^2 n_e)^{2/3}\): - \(3 \pi^2 = 29.6088\) - \(29.6088 \times 1.31 \times 10^{28} = 3.8795 \times 10^{29}\) - \((3.8795 \times 10^{29})^{2/3} = 1.395 \times 10^{20}\).- Final calculation for \(E_F\):\[E_F = \frac{(1.0545718 \times 10^{-34})^2}{2 \times 9.109 \times 10^{-31}} \times 1.395 \times 10^{20}\]\[\approx 2.12 \times 10^{-18}\; J\approx 13.26\; eV.\]

Convert Fermi Energy from Joules to Electronvolts

To convert the Fermi energy from joules to electronvolts, use the conversion factor \(1 eV = 1.602 \times 10^{-19} J\).\[\text{Fermi energy in eV} = \frac{2.12 \times 10^{-18}\; J}{1.602 \times 10^{-19}\; J/eV} \approx 13.26 \; eV.\]

Key concepts

Number Density

Number density is a key concept when dealing with materials, especially in solid-state physics. In simple terms, number density is the number of particles (or atoms) per unit volume in a substance. To calculate the number density of electrons for potassium, we start by recognizing that each potassium atom contributes one free electron. Thus, the number density of electrons, denoted as \(n_e\), is equivalent to the number density of potassium atoms.The mathematical formula to find number density is:\[ n_e = \frac{N}{V} = \frac{\text{Density}}{\text{Mass of one atom}} \]Given:

• Density of potassium : \( \rho = 851\; kg/m^3 \)
• Mass of one potassium atom: \( m_{atom} = 6.49 \times 10^{-26} \; kg \)

Plugging these values into the formula:\[ n_e = \frac{851\; kg/m^3}{6.49 \times 10^{-26}\; kg} = 1.31 \times 10^{28} \; m^{-3} \]In essence, number density helps describe how many electrons occupy a given volume in the material, and this directly influences the electronic properties such as electrical and thermal conductivity.

Reduced Planck's Constant

Reduced Planck's Constant, often denoted by \(\hbar\), is a fundamental constant in quantum mechanics that plays a crucial role in the quantization of systems.It is defined as:\[ \hbar = \frac{h}{2\pi} \]where \( h \) is the Planck's constant \(6.62607015 \times 10^{-34} \; J\cdot s\).Reduced Planck's Constant has a value of:\[ \hbar = 1.0545718 \times 10^{-34} \; Js \]

Importance in Quantum Mechanics

\(\hbar\) serves as a bridge between classical physics and quantum physics. It is critical for calculations involving:

• Angular momentum quantization of particles
• Energy levels in atoms
• Wave-particle duality

In the context of calculating the Fermi energy, \(\hbar\) is used due to its role in describing the electromagnetic wave characteristics within the equation for quantum particles, affecting their statistical distribution and energy levels.These properties underpin the physical phenomena and calculations needed to describe systems at atomic and subatomic levels, making \(\hbar\) indispensable in advancing scientific understanding of materials.

Free Electron Model

The Free Electron Model is a simplified model that physicists use to describe the behavior of electrons in a metal. This model assumes:

• Electrons move freely within the metal, similar to molecules in an ideal gas.
• There is no interaction between the conduction electrons and the fixed positive ions in the lattice.
• Electrons are only influenced by certain boundary conditions and energy barriers at the edge of the solid.

Usefulness and Applications

This model provides a foundational understanding of important properties such as:

• Electrical conductivity of metals
• Heat capacity of electrons
• Metallic bonding

In the context of Fermi energy calculation, the Free Electron Model gives insights into the behavior of electrons at zero temperature and their distribution, which is crucial for understanding how metals conduct electricity. While the model is an approximation and doesn't account for electron-electron interactions or the band structure of metals, it facilitates the calculation of quantities like the Fermi energy, making it valuable for introductory explorations into solid-state physics.