Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
(d) Making use of the approximation 1 q Vp2 – qx² : 2р for small x, determine the dispersion relation for the acoustic branch in the long-wavelength limit and thus find the group velocity of acoustic waves in the lattice. b a U2n-1 U2n U2n+1 U2n+2

(d) Making use of the approximation 1 q Vp2 – qx² : 2р for small x, determine the dispersion relation for the acoustic branch in the long-wavelength limit and thus find the group velocity of acoustic waves in the lattice. b a U2n-1 U2n U2n+1 U2n+2

icon
Related questions
Question

Good day, can you answer this question? Specifically part D please 

1. A linear lattice with lattice constant a has a basis of two identical atoms of mass m where the equilibrium spacing of the atoms within each unit cell is b (where b <). The displacements of the atoms from their equilibrium positions are given by u1, Uz, harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B1 and B2. (a) Develop: (i) The equation of motion for the 2nth atom in terms of forces exerted by the (2n – 1)th and (2n + 1)th atoms. (ii) The equation of motion for the (2n + 1)th atom in terms of forces exerted by the 2nth and (2n + 2)th atoms. (b) Using the equations of motion and assuming travelling wave solutions of the form ..., Uzn-1, U2n, U2n+1, .... The = Aei(wt-kna) and u2n+1 Beilwt-kna-kb). U2n derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations C11x + C12y = 0 C21X + C22y = 0 only has a non-zero solution for x and y when C11 С 12 = 0, %3D C21 C22 obtain an expression for w?. (d) Making use of the approximation 19 Vp2 – qx2 = p .2 X. - 2 р for small x, determine the dispersion relation for the acoustic branch in the long-wavelength limit and thus find the group velocity of acoustic waves in the lattice. B2 U2n-2 U2n-1 U2n Uzn+1 U2n+2
Transcribed Image Text:1. A linear lattice with lattice constant a has a basis of two identical atoms of mass m where the equilibrium spacing of the atoms within each unit cell is b (where b <). The displacements of the atoms from their equilibrium positions are given by u1, Uz, harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B1 and B2. (a) Develop: (i) The equation of motion for the 2nth atom in terms of forces exerted by the (2n – 1)th and (2n + 1)th atoms. (ii) The equation of motion for the (2n + 1)th atom in terms of forces exerted by the 2nth and (2n + 2)th atoms. (b) Using the equations of motion and assuming travelling wave solutions of the form ..., Uzn-1, U2n, U2n+1, .... The = Aei(wt-kna) and u2n+1 Beilwt-kna-kb). U2n derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations C11x + C12y = 0 C21X + C22y = 0 only has a non-zero solution for x and y when C11 С 12 = 0, %3D C21 C22 obtain an expression for w?. (d) Making use of the approximation 19 Vp2 – qx2 = p .2 X. - 2 р for small x, determine the dispersion relation for the acoustic branch in the long-wavelength limit and thus find the group velocity of acoustic waves in the lattice. B2 U2n-2 U2n-1 U2n Uzn+1 U2n+2
Expert Solution
steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer