Nearly Free Electron Theory PDF

Nearly free electron theory- Free electron model is assumed that electrons are weakly bonded to their atoms and move within the metal freely without being affected by the presence of atoms of mater and not interact with crystal ions. Free electron model was developed by Sommerfeld who combined clas...

Nearly free electron theory- Free electron model is assumed that electrons are weakly bonded to their atoms and move within the metal freely without being affected by the presence of atoms of mater and not interact with crystal ions. Free electron model was developed by Sommerfeld who combined classical Drude model with quantum mechanical Fermi Dirac statistics. Hence it is also known as Drude-Sommerfeld model. In free electron model, there is neither lattice nor electron-electron interaction but it given good result on electron specific heat, electric and thermal conductivities. In free electron model, energy of electron is Where, , where k is a wave number For 3D case, The allowed energy values are distributed continuously from zero to infinity. Failures of free electron model-  Heat capacities and Hall Effect of many metals are wrong.  Hall Effect can be positive  Does not explain why mean free path can be so long  Does not explain why some materials are metals, some insulators and some are semiconductors. For periodic boundary conditions over a cube of side ‘a’ Wave functions are in the form ̅ ̅ ⃗ The band structure of the crystal can be explained by nearly free electron model, for which bans electrons are treated as perturbed only by periodic potential of positive ions. These band gaps are of decisive significance in determining whether a solid is a insulators or conductors. Two standing waves pile up the electrons of different region so the two waves have different values of potential energy in the filed of ion core of the lattice. ( ) ( ) ( ) ( ) ( ) ( ) The parabolic relation between E as a function of wave number k. But in nearly free electron it is interrupted at value k. the energies corresponding to the k values are forbidden for electrons. Thus electrons energies are divided into forbidden and allowed bands. For the values of k,