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Download 1. The primitive translation vectors of the hexagonal space lattice
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Transcript
HW1 1. The primitive translation vectors of the hexagonal space lattice may be taken as 3 a 3 a axˆ ayˆ , a2 axˆ ayˆ , a 2 czˆ 2 2 2 2 (a) Derive the volume of the primitive cell. (b) Derive the primitive translations of the reciprocal lattice. Show that the lattice is its own reciprocal, but with a rotation of axes. (c) Sketch the first Brillouin zone of the hexagonal space lattice. a1 2. Show that the volume of the first Brillouin zone is 2 Vc 3 , where Vc is the volume of a crystal primitive cell. (Hint: The volume of a Brillouin zone is equal to the volume of the primitive cell parallelepiped in Fourier space. Recall the vector identity c a a b c a b a ) 3. For the crystal structure of diamond, the basis consists of 8 atoms if the cell is 1 1 1 , , , 4 4 4 1 1 3 3 1 1 1 1 3 3 1 1 3 1 3 , ,0 , , , , 0, , , , , , ,0, , , , . 2 2 4 4 4 2 2 4 4 4 2 2 4 4 4 (a) Find the structure factor S of this basis. (b) Discuss the allowed and forbidden reflections, i.e. the values of S in all cases. taken as the conventional cube. Carbon atoms locate at 0,0,0 , 4. NaCl crystallizes in a FCC lattice with a basis of Na and Cl ions separated by half the body diagonal of the cube. The atomic numbers of Na and Cl are 11 and 17, respectively. (a) Determine which X-ray reflections will be observed. Index them for the conventional cubic unit cell. (b) Of these which group will be strong and which group weak? HW2 1. Consider a (i) 3-D (crystal) (ii) 2-D (quantum well) (iii) 1-D (quantum wire) free electron gas in which the electrons are restricted to move freely within its boundary (3-D: L3, 2-D: L2, 1-D: L). (a) Show that the density of states, g E , for the 3-D, 2-D and 1-D systems are (i) E (ii) constant (iii) 1 E , respectively. (b) Draw the above result of g E versus E. 2. Metallic sodium crystallizes in body-centered cubic form, the length of the cube being 4.25 108 cm. (a) Find the concentration of conduction electrons. Assume one conduction electron. (b) What is the Fermi energy? 3. For metal, (a) Please show that the average kinetic energy of a conduction electron at 0 K is 3 given by E E f . 5 (b) The Fermi energy of copper at 0K is 7.05 eV. At what temperature would the average energy a molecule of an ideal gas equal the average energy of a conduction electron in copper at 0K? 4. Consider the general expression for the conductivity of metals in terms of density 1 of states g E f at E f given by e2v f 2 g E f . Show that within the free 3 ne 2 , the Drude expression. Explain what m* the main difference is between quantum and classical theory. electron theory, this reduces to