CondMat07FinalX

# CondMat07FinalX - 1 CONDENSED MATTER PHYSICS 171.405...

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Unformatted text preview: 1 CONDENSED MATTER PHYSICS 171.405 / 171.621 Instructor: N.P. Armitage Grader: Paula Mellado Fall 2007 Final Exam Due to NPA’s office by 5:00 PM, Tuesday Dec. 18th Use whatever reasonable resources you like (textbooks, internet, etc.). Explicit solutions to similar HW problems found elsewhere are not considered ‘reasonable’. All work must be done individually. ============================= Q 1: (a) Sodium transform from bcc to hcp at ≈ 23 K (the so-called ”martensitic” transition). Assuming that the density remains fixed across this transition, find the lattice constant a of the hexagonal phase, given that a = 4 . 23 ˚ A in the cubic phase and that the c/a ratio is indistinguishable from its ideal ratio. (b) The hcp lattice is the simple hexagonal Bravais lattice with a 2 atom basis. Show that the reciprocal lattice of the simple hexagonal lattice is also simple hexagonal, but with lattice vectors rotated with respect to the real space lattice. What are these vectors in terms of the original lattice constants? What angle are the reciprocal lattice vectors rotated with respect to the original real space lattice vectors? The real space primitive vectors for the hexagonal lattice are a 1 = a ˆ x , a 2 = a 2 ˆ x + √ 3 a 2 ˆ y , and a 3 = c ˆ z . (c) Calculate the structure factor of the hcp lattice. Q 2: Copper oxide planes are a common feature of high-temperature superconductors. In these planes, copper atoms lie at the vertices of a square lattice of side a . Oxygen atoms lie at the midpoints of the sides of the squares, i.e. halfway between nearest-neighbor copper atoms. For simplicity, lets ignore all other atoms and assume that there are simply stacked copper oxide layers with separation c between the layers. (a) What is the unit cell and what is the basis? (b) In La 2 CuO 4 , the oxygen atoms are actually displaced from the plane in an alternating fashion. In the figure, + atoms are above the plane by δc , while - atoms are below the plane by δc . What is the unit cell and lattice spacing? What is the basis? Describe (qualitatively) what happens in the X-ray diffraction pattern as the distortion decreases gradually to zero at a finite temperature structural phase transition....
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CondMat07FinalX - 1 CONDENSED MATTER PHYSICS 171.405...

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