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FREE ELECTRON FERMI GAS Outlines Free Electron Gas Electron Free
1. Energy levels in one dimension 2. Effect of temp on the Fermi-Dirac function 3. Free electron gas in 3D 4. Heat capacity of the electron gas 5. Thermal conductivity of metal
Physics UCF 2
In this unit, we study electrons in solid, in particular Inside a real metal, the electron in the conduction band electrons inside a metal. There are some overlap between this unit and unit#1, where we study the Drude model. 1. Will interact with other electron. (correlation, exchange) 3/20/2017 2. Will interact with the atomic potential (Coulomb force) 3/20/2017 In a simple metal, the atom at each lattice site usually has both core electrons and valence electrons. The core Gas Electron Free Gas Electron Free electrons are very similar to those electrons in a free When we ignore atom. However, the valence electrons are quite different from the outer electrons of the free atom. 1. >>>>>>>> Independent electron model 2. >>>>>>>> Free electron model
Of course, we can also treat electrons as classical particles or as quantum particles. In this unit, we will examine the success and the failure of this model, and 3 try to understand why. 4
Electron Energy Levels in one dimension One dimensional infinite square well For a particle in a 1D box, the boundary conditions are Let’s consider a free electron gas in one dimension. WE
3/20/2017 ; . The solution can be found 3/20/2017 take into account the fact that electrons are quantum particles satisfy the Schrodinger equation, Free Electron Gas Electron Free (3) Gas Electron Free (1) where is the wave-function of the electron, is where & Hamiltonian operator and is the eigenvalue of the Hamiltonian operator. Substitute (3) into (1) we obtain (2) (4) In this unit, we are dealing with free electrons, so the 5 6 potential is zero.
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So you can see from the figure below the first few energy For each of the energy levels associated with a quantum levels and the wavefunctions look like. number n, there can have only two electrons. For N electrons, the last filled level will be , and we define Since this is a one dimensional solid, so only one principle 3/20/2017 3/20/2017 quantum number n, but electron has an intrinsic Fermi energy as the energy of the topmost filled level in the quantum number called spin. So for 1D electron gas, ground state of the N electron system. there are two degree of freedom. Now electrons are Gas Electron Free Gas Electron Free fermions so they satisfy the Fermi-Dirac distribution. (5) For 6 electrons in a 1D infinite square well the ground state is given by: The Fermi-Dirac Distribution n Occupancy 1 ↑ 1 The Fermi-Dirac distribution gives the probability that an 1 ↓ 1 orbital at energy will be occupied in an ideal electron gas 2 ↑ 1 at thermal equilibrium 2 ↓ 1 3 ↑ 1 (6) 7 / 8 3 ↓ 1 4 ↑ 0
Free electron gas in 3D
The free electron in 3D satisfies the Schrodinger eq. 3/20/2017 3/20/2017 (7) Free Electron Gas Electron Free Gas Electron Free The solution is