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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 <Ə, The displacements of the atoms from their equilibrium positions are given by u,Uz, .--, Uzn-1, Uzn, Uzn+1 .... The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B, and B.. (a) Develop: (0 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 = Aeot-kna) and un, = Be(wt-kna-kb) 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 + C2,y = 0 only has a non-zero solution for x and y when C11 C12 =0, C22 C21 obtain an expression for w. (d) Making use of the approximation 1q x² 2p 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 2 || |

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 <Ə, The displacements of the atoms from their equilibrium positions are given by u,Uz, .--, Uzn-1, Uzn, Uzn+1 .... The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B, and B.. (a) Develop: (0 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 = Aeot-kna) and un, = Be(wt-kna-kb) 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 + C2,y = 0 only has a non-zero solution for x and y when C11 C12 =0, C22 C21 obtain an expression for w. (d) Making use of the approximation 1q x² 2p 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 2 || |

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Please show all working. 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 <5. The displacements of the atoms from their equilibrium positions are given by u,U2, ---, Uzn-1,Uzn, Uzn+1, -.. The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B, and B.. (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. (u) 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 Un = Aet-kna) and Um = Bei(@t-kna-kb) derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations C11x + C2y = 0 C2,x + Cy = 0 only has a non-zero solution for x and y when C12 =0. C22 obtain an expression for w. (d) Making use of the approximation 1q 2p 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.
Transcribed Image Text:Please show all working. 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 <5. The displacements of the atoms from their equilibrium positions are given by u,U2, ---, Uzn-1,Uzn, Uzn+1, -.. The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B, and B.. (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. (u) 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 Un = Aet-kna) and Um = Bei(@t-kna-kb) derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations C11x + C2y = 0 C2,x + Cy = 0 only has a non-zero solution for x and y when C12 =0. C22 obtain an expression for w. (d) Making use of the approximation 1q 2p 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.
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