Show that the dispersion relation for the lattice vibrations of a chain of identical masses M, in which each is connected to its first and second nearest neighbours by springs of spring constants K and K, respectively, is Mo = 2K[1- cos(ka)]+2K,[1-cos(2ka)] where a is the equilibrium spacing. Show that: this dispersion relation reduces to that for sound waves in the long-wavelength limit (ensure that the velocity corresponds to that predicted by the elastic modulus of the crystal); the group velocity vanishes at k = t7/a; and o is periodic in k with period 2/a.
Show that the dispersion relation for the lattice vibrations of a chain of identical masses M, in which each is connected to its first and second nearest neighbours by springs of spring constants K and K, respectively, is Mo = 2K[1- cos(ka)]+2K,[1-cos(2ka)] where a is the equilibrium spacing. Show that: this dispersion relation reduces to that for sound waves in the long-wavelength limit (ensure that the velocity corresponds to that predicted by the elastic modulus of the crystal); the group velocity vanishes at k = t7/a; and o is periodic in k with period 2/a.
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![2. Show that the dispersion relation for the lattice vibrations of a chain of identical
masses M, in which each is connected to its first and second nearest neighbours by
springs of spring constants K and K, respectively, is
Mo = 2K[1- cos(ka)]+2K,[1- cos(2ka)]
where a is the equilibrium spacing.
Show that:
(a) this dispersion relation reduces to that for sound waves in the long-wavelength limit
(ensure that the velocity corresponds to that predicted by the elastic modulus of the
crystal);
(b) the group velocity vanishes at k = ±a/a; and
(c) o is periodic in k with period 27/a.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d7e5c83-44c9-4298-a22a-e4efb726c7ca%2F87c2daa6-bdf2-4de6-af59-e23d6e54e8c2%2Fkve8i8b_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2. Show that the dispersion relation for the lattice vibrations of a chain of identical
masses M, in which each is connected to its first and second nearest neighbours by
springs of spring constants K and K, respectively, is
Mo = 2K[1- cos(ka)]+2K,[1- cos(2ka)]
where a is the equilibrium spacing.
Show that:
(a) this dispersion relation reduces to that for sound waves in the long-wavelength limit
(ensure that the velocity corresponds to that predicted by the elastic modulus of the
crystal);
(b) the group velocity vanishes at k = ±a/a; and
(c) o is periodic in k with period 27/a.
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