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» Uznı Uzn+1, --- The harmonic forces between nearest-neighbour atoms are characterised by the altemating interatomic force constants B, and Bz. (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 Uzn = Aellat-kna) and uzn+1 = , Belwt-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 C11 C12 = 0, C21 C22 obtain an expression for w?. (d) Making use of the approximation 14 x Vp2 – qx² = p - 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. 2n-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<5. The displacements of the atoms from their equilibrium positions are given by u,, Uz, ... , Uzn-1» Uznı Uzn+1, --- The harmonic forces between nearest-neighbour atoms are characterised by the altemating interatomic force constants B, and Bz. (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 Uzn = Aellat-kna) and uzn+1 = , Belwt-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 C11 C12 = 0, C21 C22 obtain an expression for w?. (d) Making use of the approximation 14 x Vp2 – qx² = p - 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. 2n-2
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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<5. The displacements of
the atoms from their equilibrium positions are given by u,, Uz, ... , Uzn-1» Uznı Uzn+1, --- The
harmonic forces between nearest-neighbour atoms are characterised by the altemating
interatomic force constants B, and Bz.
(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
Uzn = Aellat-kna) and uzn+1
= ,
Belwt-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
C11
C12
= 0,
C21 C22
obtain an expression for w?.
(d) Making use of the approximation
14 x
Vp2 – qx² = p -
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.
2n-2
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