MIT OpenCourseWare http ocw mit edu 8 512 Theory of Solids II Spring 2009 For information about citing these materials or our Terms of Use visit http ocw mit edu terms 8 512 Theory of Solids Problem Set 2 1 Due February 24 2004 Estimate the mean free path for plasmon production by a fast electron through a metal by the following steps a for small q it is a good approximation to assume that 1 q is dominated by the plasmon hole i e Im 1 A q pl q Determine the constant A q using the f sum rule b c 2 Using a write down an expression for the probability of scattering into a solid angle Q by emitting a plasmon Estimate the mean free path for plasmon emission Put in some typical numbers electron energy 100 KeV etc Dielectric constant of a semiconductor a In a periodic solid show that the dielectric response function is given within the random phase approximation by q 1 4 e q2 2 k G r r r rr r k q G e iq r k 2 f f k kr qr Gr kr i r r r k q G 1 r where G are the reciprocal lattice vectors b We will evaluate Eq 1 in an approximate way for a semiconductor in the limit 0 and q 0 We will work in the reduced zone scheme Argue that the energy denominator in Eq 1 may be replaced by the energy gap To estimate the numerator derive the following theorem r r 2 iq r a b a b e b h 2q 2 2m 2 where a and b are the eigenstate of a Hamiltonian H with a kinetic energy term h 2 2m This is a generalization of the f sum rule in atomic physics It is proven by evaluating the expectation value of r r iq r r r iq r H e e in the state a c By making the further approximation that the energy difference in Eq 2 may be replaced by the energy gap show that for a semiconductor h pl 2 q 0 0 1 where pl is the plasma frequency Estimate for Si and Ge and compare with experiment
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