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I know the density $\rho$ is $$ \rho=\frac{m}{V} \quad \bigg [\frac{\text{kg}}{\text{m}^3}\bigg ] \tag{1} $$ But in solid-state physics an another type of density if the free electron density $n$ $$ n=\frac{N}{V} \quad \bigg [\frac{\text{units?}}{\text{m}^3}\bigg ] \tag{2} $$ And in this link, $n$ written as $$ n=\frac{N_A}{M} \cdot \rho \quad \bigg [\frac{\text{units?}}{\text{m}^3}\bigg ] \tag{3} $$ where $N_A$ is Avogadro's number and M is the molar mass.

Questions:

What is the electron density $n$?

What is the relationship between the "regular" density $\rho=m/V$ and the electron density $n$?

And also, how can $n$ be written as both equation (2) and (3)?

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(1) The density n is the number of electrons per unit volume (here m^3).

(2) The mass density is related to the sum of atomic masses per unit volume. You would have to find the number of the atoms of different elements contained in the unit volume and multiply them with the respective number of electron in the electron shell of the atoms (atomic number). Thus you would get the total number of electron per unit volume, i.e., total electron density n, which is not the free electron density. To relate this to the mass density you have sum the atomic masses of different elements in the unit volume.

(3) The expression gives the number of atoms per volume in the material. According to the link this is also the number of free electron per unit volume because one free electron per Cu atom is assumed. The symbol n corresponds here to the free electron density in Cu. The total electron density would be different, see (2).

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