Please show all working.1. A linear lattice with lattice constant a has a basis of two identical atoms of mass m where theequilibrium spacing of the atoms within each unit cell is b (where b <5. The displacements ofthe atoms from their equilibrium positions are given by u,Uz, ---, Uzn-1,Uzn, Uzn+1, -.. Theharmonic forces between nearest-neighbour atoms are characterised by the alternatinginteratomic 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)thand (2n + 1)*th atoms.(u) The equation of motion for the (2n + 1)th atom in terms of forces exerted by the 2nthand (2n + 2)th atoms.(b) Using the equations of motion and assuming travelling wave solutions of the formUn = 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 equationsC11x + C12y = 0C2,x + Cy = 0only has a non-zero solution for x and y whenC12=D0.C21obtain an expression for w.(d) Making use of the approximation1q2pfor small x, determine the dispersion relation for the acoustic branch in the long-wavelengthlimit and thus find the group velocity of acoustic waves in the lattice.B.

Answered: Image /qna-images/answer/36e5c877-64ae-438b-a592-e0fe224594ee.jpg

Answered: (b) Using the equations of motion and…

(b) Using the equations of motion and assuming travelling wave solutions of the form Uzn = Aer-Kna) and Uzn+1 = Beat-kna-kb) Uzn

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

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Concept explainers

Debye Theory of Solids

In solid-state physics, debye theory is used to estimate the phonons contributing to the specific heat capacity in a solid. It explains that the specific heat is a consequence of the vibrations of the atomic lattice of the solid, which is in contrast to the Einstein model. In the Einstein model, solids are treated as independent, non-interacting quantum harmonic oscillators. This model clearly explains the low-temperature dependence of the heat capacity, which is proportional to T3. This is known as Debye T3 law.

Question

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,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:
(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 + C12y = 0
C2,x + Cy = 0
only has a non-zero solution for x and y when
C12
=D0.
C21
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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