How to Find the Number Density of Electrons?

The number density of electrons is found by determining the number of free electrons per unit volume of a material.

Essentially, it represents how many charge carriers (electrons) are available to conduct electricity within a specific material. The video reference states directly: "the number density is equal to your number of charge carriers" within a given volume. This highlights the direct relationship.

Here's a breakdown of how to calculate or determine electron number density, often denoted as 'n':

1. Define Number Density (n): Number density (n) is the number of free electrons (N) divided by the volume (V) they occupy. The formula is:

   n = N / V

2. Determine the Number of Free Electrons (N): This is the most crucial and potentially complex step, as it depends on the material:

   • For Metals: In metals, we often assume each atom contributes a certain number of free electrons (typically 1, 2, or 3, corresponding to the valence). Therefore:

     • Find the number of atoms per unit volume. This involves using the material's density (ρ), atomic mass (M), and Avogadro's number (N_A).
     • Calculate the number of atoms per unit volume: (ρ * N_A) / M
     • Multiply this by the number of free electrons per atom (e.g., 1 for copper, 2 for magnesium, 3 for aluminum).

     Therefore, N = (ρ N_A number of free electrons per atom) / M
     Combining with the original equation: n = (ρ N_A number of free electrons per atom) / (M * V), where V is the volume considered, often normalized to 1 m³ or 1 cm³.

   • For Semiconductors: In semiconductors, the number of free electrons is significantly lower than in metals and is highly dependent on temperature and doping. It can be determined through:

     • Hall Effect measurements: This experimental technique directly measures the carrier concentration.
     • Doping concentration: If the semiconductor is doped, the concentration of donor impurities will significantly affect the electron concentration (n ≈ N_D for n-type semiconductors).
     • Intrinsic carrier concentration (n_i): In intrinsic (undoped) semiconductors, the electron and hole concentrations are equal and temperature-dependent. The value of n_i can be found in material property tables at a given temperature.

3. Determine the Volume (V): Choose a convenient unit of volume (e.g., 1 m³, 1 cm³). When using the formulas involving density, atomic mass, and Avogadro's number, the resulting 'n' will represent the number of electrons per unit volume corresponding to the units used for density and atomic mass.

4. Units: The number density (n) is typically expressed in units of electrons per cubic meter (electrons/m³) or electrons per cubic centimeter (electrons/cm³).

Example (for Copper):

Let's calculate the electron number density for copper, assuming one free electron per atom:

• Density (ρ) = 8960 kg/m³
• Atomic mass (M) = 63.55 g/mol = 0.06355 kg/mol
• Avogadro's number (N_A) = 6.022 x 10²³ atoms/mol
• Number of free electrons per atom = 1

n = (ρ N_A number of free electrons per atom) / M
n = (8960 kg/m³ 6.022 x 10²³ atoms/mol 1) / 0.06355 kg/mol
n ≈ 8.49 x 10²⁸ electrons/m³

Therefore, the number density of electrons in copper is approximately 8.49 x 10²⁸ electrons per cubic meter.