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, ... ,U2n-1, U2n, u2n+1, ... The 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 2nh 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 u2n = Ae(wt-kna) and uzn41 = 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 + c22y = 0 only has a non-zero solution for x and y when 11 С12 = 0, С21 С22 obtain an expression for w?.

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, ... ,U2n-1, U2n, u2n+1, ... The 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 2nh 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 u2n = Ae(wt-kna) and uzn41 = 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 + c22y = 0 only has a non-zero solution for x and y when 11 С12 = 0, С21 С22 obtain an expression for w?.

Name: Can all steps be included 1. A linear lattice with lattice constant a
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Description: Can all steps be included 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 <). The displacements ofthe atoms from their equilibrium

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ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

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Concept explainers

Simple harmonic motion

Simple harmonic motion is a type of periodic motion in which an object undergoes oscillatory motion. The restoring force exerted by the object exhibiting SHM is proportional to the displacement from the equilibrium position. The force is directed towards the mean position. We see many examples of SHM around us, common ones are the motion of a pendulum, spring and vibration of strings in musical instruments, and so on.

Simple Pendulum

A simple pendulum comprises a heavy mass (called bob) attached to one end of the weightless and flexible string.

Oscillation

In Physics, oscillation means a repetitive motion that happens in a variation with respect to time. There is usually a central value, where the object would be at rest. Additionally, there are two or more positions between which the repetitive motion takes place. In mathematics, oscillations can also be described as vibrations. The most common examples of oscillation that is seen in daily lives include the alternating current (AC) or the motion of a moving pendulum.

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Can all steps be included

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, ... ,U2n-1, U2n, u2n+1, ... The
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 2nh 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
u2n = Ae(wt-kna) and uzn41 = 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 + c22y = 0
only has a non-zero solution for x and y when
11
C12
= 0,
C21 C22
obtain an expression for w?.
(d) Making use of the approximation
14 „2
Vp? – qx² × p –x?
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.
a
U2n-2
U2n-1
U2n+1
U2n+2

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