homework5 - to two dimensions and the rods are connected by...

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Physics 481: Condensed Matter Physics - Homework 5 due date: Friday, Feb 18, 2011 Problem 1: Fibonacci chain (8 points) Determine the ratio between the numbers of A and B elements in Fibonacci chains of generation 2,3,4,5. Calculate the limiting value in an infinite chain. Use the inflation rule! Problem 2: Linear ionic crystal (12 points) Consider a one-dimensional chain of 2 N ions of alternating charge ± q ( N 1). In addition to the Coulomb interaction, there is a repulsive potential A/R n between nearest neighbors only. ( R is the distance between nearest neighbor ions.) a) Determine the equilibrium distance R 0 . b) Determine the cohesive energy E 0 for this distance and show that it can be written as E 0 = - N 2 ln 2 1 - 1 n q 2 R 0 . c) Determine the work necessary to compress the crystal such that R = R 0 (1 - δ ) to leading order in the small parameter δ 1 Problem 3: Polymer stiffness (Marder, problem 5.6, 20 points) Consider a polymer composed of a sequence of N rigid rods of length a . The polymer is confined to two dimensions, and the rods are connected by springs. If the angle between rod
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Unformatted text preview: to two dimensions, and the rods are connected by springs. If the angle between rod l and rod l + 1 is Θ l then the energy of this joint is κ Θ 2 l . (assume low temperatures such that κ/k B T ² 1). Show that long enough polymers behave as a random walks. To this end: a) Write down the probability of having a particular set of angles Θ 1 ,..., Θ N at temperature T (use canonical ensemble, i.e., Boltzmann distribution) b) Put one end of the polymer at the origin. Find the coordinates ( x N ,y N ) of the other end as a function of the angles Θ 1 ,..., Θ N . (Hint: It helps to formulate the problem in the complex plane!) c) Find the thermal average h x 2 N + y 2 N i . (Assume a sufficiently long polymer such that Nk B T ² κ. ) d) The result has the same form as expected for an ideal random walk, but the segment length a has to be replaced by by an effective length ˜ a . What is ˜ a ?...
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