exam2 - Name: _ 2.57 Midterm Exam No. 2 Fall, 2004 Take...

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Name: ____________________ 2.57 Midterm Exam No. 2 Fall, 2004 Take Home Exam Distributed: Friday, November 19, 7:00 pm Due: November 24, 2:30 pm Four questions, each counts for 25 points Rules : (1) You are required to finish these problems independently, without consultation to other fellow students in the class or anyone else. Replies to specific questions will be shared with other students by em ail. You can check whatever references you can find. You are on your own honor. (2) State all the assumptions you made in solving each problem. 1. An fcc crystal has a lattice constant b and one atom at each lattice point. The phonon dispersions for transverse and longitudinal phonons are degenerate and are given by the following relation, ω = 2 K m sin ka 2 where a is the equivalent lattice constant of an isotropic crystal that has the same number of phonon modes as the fcc crystal, and 2 2 4 π k 2 = k x + k y + k 2 (k ,k ,k = ± 2 π , ± ,...) z x y z L L The phonon relaxation time τ in the crystal is a constant. Answer the following questions: (a) Relate the equivalent lattice constant “a” to the lattice constant of the fcc crystal “b”. (b) Derive an integral expression for the phonon specific heat (per unit volume) of the crystal
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exam2 - Name: _ 2.57 Midterm Exam No. 2 Fall, 2004 Take...

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