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# ex06 - tial V local r = U Â Î´ r(2 shows a diï¬€erent...

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Solid State Theory Exercise 6 FS 2011 Prof. M. Sigrist Exercise 6.1 Lindhard function In the lecture it was shown how to derive the dynamical linear response function χ 0 ( q , ω ) which is also known as the Lindhard function: χ 0 ( q , ω ) = 1 Ω X k n F ( k + q ) - n F ( k ) k + q - k - ~ ω - i ~ η . (1) Calculate the static Lindhard function χ 0 ( q ) of free electrons for the 1 and 3 dimensional case at T = 0. Hint: We are only interested in the real part of χ 0 ( q , ω ). Therefore, use the equation lim η 0 ( z - ) - 1 = P (1 /z )+ iπδ ( z ). Furthermore, in 3 dimensions we can choose q = q e z to point in the z -direction due to the isotropy of a system of free electrons. Then change to cylindrical coordinates in order to calculate the integral. Exercise 6.2 Zero-sound excitations The dispersion relation of the plasmon excitation is finite for all q ’s. This appearance of a finite excitation energy is a consequence of the long range interaction of the Coulomb potential V Coulomb ( r ) = e 2 / | r | . A system consisting of fermions with a solely local poten-
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Unformatted text preview: tial V local ( r ) = U Â· Î´ ( r ) (2) shows a diï¬€erent behaviour at q = 0. In this exercise we basically follow the sections (3.2.1) and (3.2.2) of the lecture notes. a) As a warm-up, derive the relation between the particle distribution Î´n ( r ,t ) and its induced potential V ind ( r ,t ) in the ( k ,Ï‰ )-space. b) Find the imaginary part of the response function Ï‡ ( q ,Ï‰ ) for small q â€™s. What is the dispersion relation in the lowest order in q ? c) The upper boundary line of the particle-hole continuum is given by Ï‰ q, max = ~ 2 m ( q 2 + 2 k F q ) = ~ q 2 2 m + v F q, (3) where v F is the Fermi velocity and q = | q | . What is the condition on U for stable plasmon excitations (quasi-particles)? Oï¬ƒce hour: Monday, April 4th, 2011 - 9:00 to 11:00 am HIT K 31.3 Tama Ma...
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