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LATTICE VIBRATION II, 1D crystal consisting of a chain of identical atoms…
LATTICE VIBRATION II
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Dispersion relation of the monatomic 1-D lattice 
w-K Relation This shown that minimum possible wavelength given by twice the equilibrium separation a between atoms
:check: wavelength, λ shorter than this can be mapped onto a wavelength longer than 2a
:check: only wavelengths longer than 2a are needed to represent the motion
:check: This means that K is limited by
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The maximum value of sin(1/2 ka) is 1 so that the maximum possible frequency of the waves is 2(K/M)^1/2
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Monoatomic Chain
- The chain is endless so that all the atoms have an identical environment
- Atoms separate by distance of “a”
- move in parallel to the chain.
- Only nearest neighbors interact (short-range forces).
- 1D chain of identical atoms with the same mass M connected by identical springs with the same spring constant K.
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1D crystal consisting of a chain of identical atoms (separated by atomic spacing, a)
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The cos term forces the group velocity down to zero at the edge of the first Brillouin zone (by substituting k=π/a in equation)