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, u2, ..., U2n-1, Uzn, Uzn+1, .... The harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B, and B2. (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. (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 u, = Aei(wt-kna) and u,n41 = Beilwt-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 = only has a non-zero solution for x and y when С11 С12 = 0, C21 C22 obtain an expression for w?. (d) Making use of the approximation 1 q Vp2 – qx2 : - -- 2 p 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 а U2n-1 U2n U2n+1 U2n+2

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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 <). The displacements of
the atoms from their equilibrium positions are given by u1, u2,
harmonic forces between nearest-neighbour atoms are characterised by the alternating
interatomic force constants B, and B2.
(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.
(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-1, U2n, Uzn+1, .... The
= Aei(wt-kna) and uzn+1
= Bei(wt-kna-kb)
U2n
derive two simultaneous equations for A and B.
(c) Making use of the fact that a homogeneous system of linear equations
C11x + C12y
C21x + C22y = 0
only has a non-zero solution for x and y when
y
С11
С12
= 0,
С21
С22
obtain an expression for w?.
(d) Making use of the approximation
Vp2 – qx?
- dz
2 р
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
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, u2, harmonic forces between nearest-neighbour atoms are characterised by the alternating interatomic force constants B, and B2. (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. (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-1, U2n, Uzn+1, .... The = Aei(wt-kna) and uzn+1 = Bei(wt-kna-kb) U2n derive two simultaneous equations for A and B. (c) Making use of the fact that a homogeneous system of linear equations C11x + C12y C21x + C22y = 0 only has a non-zero solution for x and y when y С11 С12 = 0, С21 С22 obtain an expression for w?. (d) Making use of the approximation Vp2 – qx? - dz 2 р 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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