Question A5 The angular frequency of vibrations in a one-dimensional monatomic crystal are given by 4K sin ka L(K) = √ (2). m where 2πT is the wavenumber of the vibration pattern in terms of its wavelength X, the spring constant between neighbouring atoms, and m the mass of each atom. (i) Show that the frequency does not change if one shifts 2π k → k + a (ii) Briefly explain the physical reason for this periodicity.
Question A5 The angular frequency of vibrations in a one-dimensional monatomic crystal are given by 4K sin ka L(K) = √ (2). m where 2πT is the wavenumber of the vibration pattern in terms of its wavelength X, the spring constant between neighbouring atoms, and m the mass of each atom. (i) Show that the frequency does not change if one shifts 2π k → k + a (ii) Briefly explain the physical reason for this periodicity.
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The angular frequency of vibrations in a one-dimensional monatomic crystal are given by
ω(k) = squareroot(4κ/m) l sin (ka/2) l,
where
k = 2π/λ
is the wavenumber of the vibration pattern in terms of its wavelength λ, κ the spring constant
between neighbouring atoms, and m the mass of each atom.
(i) Show that the frequency does not change if one shifts
k → k + 2π/a
(ii) Briefly explain the physical reason for this periodicity
Transcribed Image Text:Question A5
The angular frequency of vibrations in a one-dimensional monatomic crystal are given by
where
w(k)=
4K
m
n().
sin
2πT
is the wavenumber of the vibration pattern in terms of its wavelength X, the spring constant
between neighbouring atoms, and m the mass of each atom.
(i) Show that the frequency does not change if one shifts
2π
k → k +
a
(ii) Briefly explain the physical reason for this periodicity.
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