Solid state physics fermi surface

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Homework Statement

Some atoms in Cu crystal (Cu has a FCC lattice) are replaced by
Zn atoms. Taking into account that Zn is bivalent, while Cu is monovalent,
calculate the atomic ratio of Zn to Cu in ZnCu alloy at which the Fermi
surface touches the first Brillouin zone faces. Use the free-electron model and assume that lattice
constant does not change with Zn doping.

Homework Equations

N/A

The Attempt at a Solution

I would like some help getting started on this problem. Any help is much appreciated thank you very much!

 

[mark726]

Hello,

To solve this problem, we will use the free-electron model and assume that the lattice constant does not change with Zn doping. In this model, we treat the electrons in the crystal as a free electron gas. The Fermi surface is the boundary between the filled and empty states in the energy band diagram.

First, let's consider the FCC lattice of Cu. In this lattice, each Cu atom has 12 nearest neighbors. In order to replace one Cu atom with a Zn atom, we need to remove one electron from the system. This means that the total number of electrons in the system has decreased by one.

Now, let's look at the Brillouin zone. In the FCC lattice, the first Brillouin zone is a cube with each side having a length of 2π/a, where a is the lattice constant. Since we are assuming that the lattice constant does not change with Zn doping, the size of the Brillouin zone remains the same.

Next, we need to determine the number of Zn atoms in the system. Since Zn is bivalent, for every one Zn atom, we have two electrons. Therefore, the number of Zn atoms is equal to the number of electrons removed from the system.

Now, we can use the formula for the number of electrons in a Fermi sphere to determine the atomic ratio of Zn to Cu at which the Fermi surface touches the first Brillouin zone faces.

N = (2/3) * (π/a)^3 * (N/V)

Where N is the number of electrons, V is the volume of the system, and a is the lattice constant.

Since we are only considering the first Brillouin zone, the volume of the system is (2π/a)^3. Therefore, the number of electrons in the Fermi sphere is equal to:

N = (2/3) * (π/a)^3 * (N/(2π/a)^3)

Where N is the total number of electrons in the system.

Since we know that the number of electrons in the system has decreased by one, we can write:

N = (2/3) * (π/a)^3 * (N-1/(2π/a)^3)

Now, we can substitute this expression for N into our original equation and solve for the atomic ratio of Zn to Cu:

N = (2/3) * (π/a)^