Question

Metallic sodium crystal lizes in body-center cubic form, the lattice constant a is 4.25 A˚

. Assuming a free electron gas model, find the Fermi energy (in eV), Fermi velocity, and Fermi temperature for sodium. (Hint: electronic configuration for Na:[Ne3 s

Answer 1

Let's ,substitute the values and calculate:

`\[V = (4.25 \, \text{A})^3\]`

`\[N = \text{Number of electrons in sodium} = 11 \, \text{(atomic number)}\]`

Now, we can use these values in the Fermi energy and Fermi velocity equations to find
`\(E_F\)` and
`\(v_F\)`. Once we have
`\(E_F\),` we can calculate
`\(T_F\).`

Step-by-step explanation

To find the Fermi energy
`(\(E_F\))`, Fermi velocity
`(\(v_F\))`, and Fermi temperature
`(\(T_F\))` for sodium, we can use the following relations:

1. **Fermi Energy
`(\(E_F\))`:**

`\[E_F = (\hbar^2)/(2m) \left((3\pi^2N)/(V)\right)^(2/3)\]`

where:

-
`\(\hbar\)` is the reduced Planck constant
`(\(1.054 * 10^(-34) \, \text{J}\cdot\text{s}\)),`

-
`\(m\)` is the electron mass
`(\(9.109 * 10^(-31) \, \text{kg}\)),`

-
`\(N\)` is the total number of electrons,

-
`\(V\)` is the volume of the crystal.

2. **Fermi Velocity
`(\(v_F\))`:**

`\[v_F = (\hbar)/(m) \left((3\pi^2N)/(V)\right)^(1/3)\]`

3. **Fermi Temperature
`(\(T_F\))`:**

`\[T_F = (E_F)/(k_B)\]`

where:

-
`\(k_B\)` is the Boltzmann constant
`\(8.617 * 10^(-5) \, \text{eV/K}\)).`

Given the lattice constant
`\(a = 4.25 \, \text}\)` and assuming body-center cubic (BCC) structure, the volume
`(\(V\))` can be calculated as
`\(V = a^3\).`

Let's substitute the values and calculate:

`\[V = (4.25 \, \text{A})^3\]`

`\[N = \text{Number of electrons in sodium} = 11 \, \text{(atomic number)}\]`

Now, we can use these values in the Fermi energy and Fermi velocity equations to find
`\(E_F\)` and
`\(v_F\)`. Once we have
`\(E_F\)`, we can calculate
`\(T_F\).`