Problem_set_5 - Physics 545 Problem Set 5 Due l Vibrations...

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Unformatted text preview: Physics 545 Problem Set 5 Due l. Vibrations of square lattice. We consider transverse vibrations of a planar square lattice of rows and columns of identical atoms, and let all,” denote the displacement normal to the plane of the lattice of the atom in the lth column and mth row (Fig. 13). The mass of each atom is M . and C is the force constant for nearest neighbor atoms. (a) Show that the equation of motion is M(d2uim/dtz) = Cl:(ul+l,m + ul—l,m — 2“(111) + (ul,nl+l + ”Lm-l — 2“im)] - Figure 13 Square array oflattice constant a. The displacements considered are normal to the plane of the lattice. ‘ 0)) Assume solutions of the form uh" = “(0) expll‘iUKxa + mea “‘ (Him I where a is the spacing between nearest-neighbor atoms. Show that the equation of motion is satisfied if (02M = 2C(2 - cos lga - cos Kya) . This is the dispersion relation for the problem. (0) Show that the region of K space for which independent solutions exist may be taken as a square ofside 277/(1. This is the first Brillouin zone of the square lattice. Sketch a) versus K for K = K. with K,,.= 0,. ' and For K. = Ky. (d) For Ka < 1, show that l J .' w = '(Ca2/M)”2(K3 + K3)“ = (Caz/MWEK , so that in this limit the velocity is constant. ‘ 2. Kittel p. 102, #1 ' 3. 'Cons‘ider'a)‘q ys qfor a diatomic linear chain with in (light) and M (heavy) atoms in the unit cell. The alternate m’s and M’s are equidistant, the force constant between m and M being fl and spaced ‘a’ apatt. l 1 l l 2 4sin2ka 2 2 PD th 4 . ' a) 2 .._+_.._ i ___+____ _W I ‘m WWW“ q fl[m M] fl im Mi mM (i) Calculate expressions for a) q (+) and a) q (~) as a function of q, i.e., the optical and the acoustic branches, and display them in the lSt Brillouin Zone. (ii) Mark the a) (1’3 at the zone center and zone boundary in the reduced zone scheme. (iii) Calculate the dispersion curves form = M and show them on the above sketch. ...
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