User:Wongzh/Free electron model

The free electron model simulates the quantum mechanical state of semiconductor or metal electrons and assumes that many electrons are considered as a set of electron systems, and this electronic state is assumed to be the simplest model of free electrons.

Introduction

 * For non-metallic elements, since they are electron-rich, they would have structures of lower coordination. In these structures, the electrons can occupy the bonding and non-bonding orbital, can left the antibonding empty.


 * However, when these structures apply to the metal, the situation will become contradictory. Structures used to explain non-metals cannot be used to explain structures of metals lacking electrons. The valence orbital of the metal is actually very diffuse. And the overlap with the nearby orbitals is strong compared to the non-metallic elements. Therefore, the orbitals are highly overlapped, and a wide conduction band is formed. In that case, the electrons can move freely in a constant potential. In 1927, Arnold Sommerfeld noticed this phenomenon and created the free electron model.


 * This free electron model can greatly explain the electron and bonding behaviors of the metals of the A sub-groups Ⅰ, Ⅱ, Ⅲ.

Application of the model to metals

 * The free electron model imagines the electrons as "gas". Compared to the atomic and molecular gas, the electrons are much lighter, and obey the exclusion principle. The density of the electrons is also much lighter a conventional gas, which leads to the unusual properties of the electron gas.


 * The model supposes the electrons are in a cube of metal, with the side length a. Therefore, the energy:

E=(nx2+ny2+nz2)h2/(8ma2)


 * The nx, ny, nz represent the number of half waves in the x, y, z directions in the cube. We can combine the quantum numbers (nx, ny, nz) as a point in a cubic lattice. Each point represents a state which has a distance R from the origin point:

R=nx2+ny2+nz2=8mE


 * In this imaginary lattice, the number of the electrons N is twice of the points. We can simply conclude that the number of the point is one-eighth the volume of the sphere. Therefore, the number of electrons N:

N=8Π/3(2mEmax/h2)3/2a3


 * In this way the Emax can be easily calculated as the function of density of electrons (ρ=N/a3):

Emax=h2/(2m)(3ρ/8Π)3/2


 * The Emax values calculated from this model are similar to the experimental values, which indicates that the free electron model could greatly explain the electron and bonding behaviors of the metal.
 * The density of state N(E) could also be calculated. By simple derivation, we can get the value of the N(E): N(E)=4Π(2m/h2)E1/2