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CondMat07PS4

# CondMat07PS4 - mixes substitutionally with copper its main...

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1 CONDENSED MATTER PHYSICS 171.405 / 171.621 Instructor: N.P. Armitage Grader: Paula Mellado Fall 2007 Problem Set 4 Due to Graders mail box by Nov. 14th ============================= Q 1: Kittel 9.4 ============================= Q 2: As a simple model of a metal consider a one-dimensional chain of 2N atoms at distances a 0 from each other. Let φ ( x - na 0 ) be the correct wavefunction for an electron with energy E i at the atom located at na 0 (eigenvalue for a single-isolated atom). Under what conditions can the one-electron wavefunction ψ ( x ) for an electron delocalized over the whole chain of atoms be approximated by a linear combination of atomic orbitals ψ ( x ) = N X n = - N c n φ i ( x - na 0 ) (1) Choose the coeﬃcients c n such that for N → ∞ the wavefunction ψ ( x ) is a Bloch wave. ============================= Q 3: Kittel 9.11 ============================= Q 4: Copper is a monovalent fcc nearly free electron metal. Zinc is a fcc divalent metal. In small quantities, zinc
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Unformatted text preview: mixes substitutionally with copper, its main eﬀect being to increase the electron concentration. (a) Assume that the zinc-copper alloy will undergo a phase transition to bcc as soon as the Fermi sphere ﬁrst touches some point of the edge of the Brillouin zone. Show that this transition occurs on the order of 36% zinc concentration. Remember that in a fcc conventional unit cell there are four ions. You should assume that the free electron approximation is valid. (b) As the concentration of zinc further increases, another phase transition occurs when the Fermi surface intersects the Brillouin zone of the bcc lattice. What is this second concentration of zinc? (c) Why do these transitions occur? ============================= For students of 621 only: Q 5: Kittel 9.8...
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