Math Problem Statement

Solve this question as a solid state question

Calculate the density of states at the Fermi level at T = 0 K for 1 m³ of Na and for 1 mole of Na. Why are these values different? What fraction (in %) of 3s electrons has energy in the interval kT near the Fermi level at room temperature? (sodium density ρ = 0.975 * 10³ kg/m³, atomic weight A = 23 amu)

Solution

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Math Problem Analysis

Mathematical Concepts

Solid-state physics
Quantum mechanics
Density of states

Formulas

Density of states: g(E) = (8π√2/h³) * √(m³) * √(E)
Fermi energy: EF = (ħ²/2m) * (3π²n)^(2/3)
Fraction of electrons near Fermi energy: Δn/n = (kT/EF)

Theorems

Fermi-Dirac distribution
Relationship between electron density and Fermi energy

Suitable Grade Level

Undergraduate (Physics and Engineering)

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