For copper, take N/V =8.5×10^28m^-3, h=6.62×10^-34Js, and me =9.1×10^-31kg to calculate Fermi energy.
To calculate the Fermi energy for copper, we need to use the formula:
E_F = (h^2 / 2m_e) * (3π^2 * N / V)^(2/3)
where:
E_F is the Fermi energy,
h is the Planck's constant (6.62 × 10^-34 Js),
m_e is the mass of an electron (9.1 × 10^-31 kg),
N is the number of electrons per unit volume,
V is the volume of the material.
Given:
N/V = 8.5 × 10^28 m^-3,
h = 6.62 × 10^-34 Js,
m_e = 9.1 × 10^-31 kg.
First, let's substitute the given values into the formula:
E_F = (6.62 × 10^-34 Js)^2 / (2 * 9.1 × 10^-31 kg) * (3π^2 * 8.5 × 10^28 m^-3)^(2/3)
Calculating the expression inside the brackets gives:
(3π^2 * 8.5 × 10^28 m^-3)^(2/3) ≈ (3 * (3.14)^2 * 8.5 × 10^28 m^-3)^(2/3)
≈ (3 * 9.86 * 8.5 × 10^28 m^-3)^(2/3)
≈ (254.55 × 10^28 m^-3)^(2/3)
Now, let's find the cube root and square it using exponent properties:
≈ (254.55 × 10^28)^(2/3)
≈ 1.43 × 10^9
Now, substitute this value into the Fermi energy equation:
E_F ≈ (6.62 × 10^-34 Js)^2 / (2 * 9.1 × 10^-31 kg) * (1.43 × 10^9)≈ (4.37 × 10^-68 J^2s^2) / (1.82 × 10^-30 kg) * (1.43 × 10^9)
Simplifying the expression:
E_F ≈ 354.9 J ≈ 3.55 × 10^2 J
Therefore, the Fermi energy for copper is approximately 3.55 × 10^2 Joules.