Problem 51 - University Physics with Modern Physics

Question

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

Step-by-Step Solution

Answer

The Fermi energy of potassium is approximately 5.59 eV.

Step 1: Calculate the number of atoms per cubic meter

First, find the number of potassium atoms in one cubic meter by dividing the density of potassium by the mass of a single potassium atom. The density \(\rho\) is 851 kg/m³, and the mass of one atom \(m_a\) is \(6.49 \times 10^{-26}\) kg. The number of atoms per cubic meter \(N\) is\[ N = \frac{\rho}{m_a} = \frac{851}{6.49 \times 10^{-26}} \approx 1.31 \times 10^{28} \text{ atoms/m}^3. \]

Step 2: Determine the number density of electrons

Assuming each atom contributes one free electron, the number density of electrons \(n\) is equal to the number of atoms per cubic meter, which we calculated previously as\[ n = 1.31 \times 10^{28} \text{ electrons/m}^3. \]

Step 3: Use the formula for Fermi energy

The Fermi energy \(E_F\) of a free electron gas can be calculated using the formula:\[ E_F = \frac{\hbar^2}{2m_e}(3\pi^2 n)^{2/3}, \]where \( \hbar \approx 1.0545718 \times 10^{-34} \text{ J} \cdot \text{s}\) is the reduced Planck's constant and \(m_e \approx 9.10938356 \times 10^{-31} \text{ kg}\) is the mass of an electron.

Step 4: Compute the Fermi energy

Substitute the values for \(\hbar\), \(m_e\), and \(n\) into the Fermi energy formula:\[E_F = \frac{(1.0545718 \times 10^{-34})^2}{2 \cdot 9.10938356 \times 10^{-31}}(3\pi^2 \cdot 1.31 \times 10^{28})^{2/3}. \]Perform the necessary calculations:\[E_F \approx 8.95 \times 10^{-19} \text{ J}. \]Convert Joules to electronvolts, knowing that 1 eV = \(1.60218 \times 10^{-19}\) J:\[E_F \approx \frac{8.95 \times 10^{-19}}{1.60218 \times 10^{-19}} \approx 5.59 \text{ eV}.\]

Key Concepts

Density of Potassium

When discussing the density of a substance, we are referring to the amount of mass packed into a given volume. For potassium, the density is measured at 851 kg/m³. This indicates how tightly the potassium atoms are packed in a cubic meter of space.
Density is calculated by the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). This value is crucial in calculations like determining how many atoms exist in a cubic meter.
By knowing the density and the mass of individual atoms, we can compute the number of atoms in a given volume, which becomes handy in advanced physics applications, such as calculating Fermi energy.

Free Electron Model

The Free Electron Model is an essential concept in understanding the behavior of metals. In this model, electrons within a metal are treated as free particles not bound to any atom. This means they can move freely within the entire volume of the metal, contributing to conductivity.
Potassium, like other alkali metals, is often assumed to contribute one free electron per atom. This assumption simplifies calculations, allowing physicists to model metals as a sea of electrons moving in a potential field without the complexities of atomic bonds.
Using this model helps us understand electrical properties and calculate the Fermi energy, a type of energy level crucial in quantum mechanics.

Quantum Mechanics

Quantum Mechanics is the branch of physics that describes the behavior of matter and energy on very small scales, such as atoms and subatomic particles. It introduces concepts that defy classical physics, such as wave-particle duality and uncertainty principles.
It also provides the framework to understand electron behaviors in materials, as it accounts for their wave-like characteristics and probabilistic nature.
In the context of the Fermi energy calculation, quantum mechanics allows us to understand how electrons fill energy levels within a material and how these electrons behave at absolute zero temperature, forming what's known as the 'Fermi sea.'

Planck's Constant

Planck's Constant is a fundamental quantity in quantum mechanics that relates the energy of a photon to its frequency. Its value is approximately \(6.62607015 \times 10^{-34} \text{ J} \cdot \text{s} \). In Fermi energy calculations, we use the reduced Planck's Constant, \( \hbar \approx 1.0545718 \times 10^{-34} \text{ J} \cdot \text{s} \), which accounts for a factor of \(2\pi\) in angular frequency.
This constant is pivotal in determining quantum scales and is integral to equations that describe how subatomic particles behave.
In our Fermi energy computation for potassium, Planck's Constant is used to relate the number density of electrons to their energy levels, providing insight into the electronic properties of the material.