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
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)...
Related questions
Question
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.
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
This is a popular solution!
Trending now
This is a popular solution!
Step by step
Solved in 2 steps with 2 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.Recommended textbooks for you
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
College Physics
Physics
ISBN:
9781305952300
Author:
Raymond A. Serway, Chris Vuille
Publisher:
Cengage Learning
University Physics (14th Edition)
Physics
ISBN:
9780133969290
Author:
Hugh D. Young, Roger A. Freedman
Publisher:
PEARSON
Introduction To Quantum Mechanics
Physics
ISBN:
9781107189638
Author:
Griffiths, David J., Schroeter, Darrell F.
Publisher:
Cambridge University Press
Physics for Scientists and Engineers
Physics
ISBN:
9781337553278
Author:
Raymond A. Serway, John W. Jewett
Publisher:
Cengage Learning
Lecture- Tutorials for Introductory Astronomy
Physics
ISBN:
9780321820464
Author:
Edward E. Prather, Tim P. Slater, Jeff P. Adams, Gina Brissenden
Publisher:
Addison-Wesley
College Physics: A Strategic Approach (4th Editio…
Physics
ISBN:
9780134609034
Author:
Randall D. Knight (Professor Emeritus), Brian Jones, Stuart Field
Publisher:
PEARSON