Question

The Drude and Sommerfeld models ignore the Coulomb interactions between the conduc- tion electrons and seem also to neglect those between the conduction electrons and the ions They treat the conduction electrons as an ideal gas. This appears counter intuitive given that Coulomb interactions are a very strong force Later in band structure calculations, we will take into account the periodic potential pro- vided by the ions. The Coulomb repulsion between the electrons will be ignored throughout this course. What actually happens is that the ions shift slightly in their positions to screen those interactions between the conduction electrons. This justifies ignoring them at our level of detail To illustrate how strong the Coulomb forces actually are, let's estimate how far a conduction electron would be able to escape from the metal by crossing its surface. a. Drude reasoned (in 1902) that the conduction electrons in metals form a classical ideal gas. Recall that for such a gas Hm(v2)kBT. How fast do these electrons move around according to this at room temperature? b. Inside the metal, the ions form collectively a deep potential well. To test this, consider a single electron escaping from the metal surface perpendicularly. Assume it takes-off at a speed equal to the root mean square velocity in (a) at the moment it is already at a distance of one Bohr radius from the surface. Approximate the net charge it leaves behind as a mirror charge inside the metal at the same distance below the surface. Estimate how far it gets before being pulled back

Answer 1

Drude and sommerefold models ignored columb interactions they treated conduction electrons as an ideal gas Conduction electrons in metal forms ideal gas PV = n RT 1/2 m lang v^2 rang = 3/2 k_B T Therefore v^2 = 3 k_B T/m v = squareroot 3 K-B T M \r\n T = 293 K k_T = 1.38 times 10^-23 k-T = 1.38 times 10^-23 = > V = squareroot 3 times 1.38 times 10^-23 times 293/9.1 times 10^-31 V = squareroot 133.29 times 10^4 V = 11.54 times 10^4 V = 1.15 times 10^5 m/sec Inside metal ions forms collectively deep potential well Let electron takes of at velocity equal to rms velocity in a r = 1 bohr Net charge left behind as mirror charge inside metal at same same distance below surface. L^n = h^2 pi^2/2m n For n = 1 L6n = n pi^2 h^2/2m