Goethe-Universit¨ at Frankfurt Fachbereich Physik

Goethe-Universit¨ at Frankfurt Fachbereich Physik

Institut f¨ur Theoretische Physik Dr. Harald O. Jeschke

Dr. Yuzhong Zhang Dr. Hunpyo Lee

Frankfurt, Nov. 11, 2009

Theoretikum zur Einf¨ uhrung in die Theoretische Festk¨ orperphysik WS 2009/10

Exercise Set 4

(Due date: Tuesday, November. 17, 2009)

Exercise 9 (Vibrations in a linear chain) (20 points)

Consider a linear chain with two different masses M₁, M₂ per unit cell (lattice constant a) connected by springs (spring constant K).

M

M

a

a) Give the possible different force constants of the model and verify the symmetry properties of the matrix of force constants.

b) Show that the dynamical matrix is given by D =

2K

M1 −^√_M^2K

1M2 cos^qa₂

−^√_M^2K

1M₂ cos^qa_{2 M}^2K

2

!

and solve the eigenvalue problem.

c) Discuss the dispersion ω_s(^*q)close to the center and the boundary of the Bril- louin zone and visualize the corresponding motion of the masses. What happens for M₁ =M₂ =M?

Exercise 10 (Two-dimensinonal lattice dynamics) (15 points)

An ideal two-dimensional crystal consists of only one kind of atom (of mass M), and each atom has an equilibrium location at a point of a square lattice ^*R⁽rs⁰⁾ = a

r s

, where r,s = 1, 2, ...,N. The displacements from equilibrium are denoted by

u*_rs = u^x_rs

u^yrs

, i.e. R^*_rs =^*R⁽_rs⁰⁾+^*u_rs =

ra+u^x_rs sa+u^yrs

.

In the harmonic approximation the potential is given by V u^x_rs,u^y_rs

=X

r,s

K₁

u^x_(r+1)s−u^x_rs2

+ u^y_r(s+1)−u^y_rs2 +K₂

u^x_r(s+1)−u^x_rs2

+ u^y_(r+1)s−u^y_rs2 . For the case K₂ =0.1K₁:

a) Determine the general phonon dispersion relation ω_s(^*q) throughout the Bril- louin zone.

b) Sketch ω_s(^*q) as a function of ^*qfor

*q= ξ

0

, 06ξ6 π a^.