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Wave vector (Read 4070 times)
Gerrit-Jan Linker
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Wave vector
17.07.09 at 12:19:47
 
Wave vector
 
A wave vector k is a vector representation of a wave.  
 
Magnitude:
The wave vector k has magnitude indicating wavenumber.
    k = |k| = 2π/λ
 
Direction:
The direction of the wave vector k indicates the direction of wave propagation.
 
Common representation of a single wave:
At a single point in space: ψ(t) = A cos(φ + ωt )
 
For a wave that is travelling along an axis z, the phase at any point along the axis will be shifted from what it is at the origin. This is accounted for by an additional phase offset term:  
ψ(t,z) = A cos(φ + kz + ωt )
 
For a wave that is travelling in any particular direction:
ψ(t,r) = A cos(φ + k.r + ωt )
 
A is the amplitude
ω is the angular frequency
φ is the starting phase of the wave
t is time
 
Reference:
Wave vector
http://en.wikipedia.org/wiki/Wave_vector
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« Last Edit: 17.07.09 at 12:30:58 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
Linker IT Software
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Gerrit-Jan Linker
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Posts: 75
Wave vector in crystals
Reply #1 - 27.08.09 at 07:52:07
 
Wave vector in crystals
 
Wave vectors k represent directions in the reciprocal lattice in crystals. They are the symmetry labels of the electronic states in a band structure.  
 
In a crystal of size N1xN2xN3 where N1,N2,N3 are the number of unit cells in each direction there are N1xN2xN3 possible translations (symmetry operations) of the unit cell (when periodic boudary conditions are used). This leads to many possible symmetry labels and each direction in the crystal, that corresponds to a particular k vector, corresponds to such a symmetry label.
 
Note that k is also related to the crystal momentum.
 
k is an indication of the phase relationships of the band. When k=0 the monomer MO's are in phase and when k=π/a the monomer MO's are out of phase.
 
See also:
Reciprocal lattice
http://en.wikipedia.org/wiki/Reciprocal_lattice
Crystal momentum
http://en.wikipedia.org/wiki/Crystal_momentum
BLoch's theorem
http://www.oraxcel.com/cgi-bin/yabb2/YaBB.pl?num=1247827363
 
Source:
Modern Organic Physical Chemistry, Anslyn
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« Last Edit: 26.10.10 at 11:20:57 by Gerrit-Jan Linker »  

Gerrit-Jan Linker
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