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, uzn+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 uzn = Aei(wt-kna) and uzn41 = Bei(wt-kna-kb) derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations
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, uzn+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 uzn = Aei(wt-kna) and uzn41 = Bei(wt-kna-kb) derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations
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can you explain/include all steps for part d. (in particular how/why sin^2 was removed and ka became k^2 a^2) please thank you. (is there a specific rule that allows this?)
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, ... ,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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