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Faculty of Mathematics and Natural SciencesInstitute of Physics II Dep. of Physics

Phonon dispersion

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Options

 Mass M1 in parts of atomic units
 M1[u]:

 Mass M2 in parts of M1
 M2[M1]:

 Spring D1 between Mn,1 and Mn,2
 D1[u*THz2]:
 Spring D2 between Mn,2 and Mn+1,1
 D2[u*THz2]:

 Spring D3 between Mn,1 and Mn+1,1
 D3[u*THz2]:

 Spring D4 between Mn,2 and Mn+1,2
 D4[u*THz2]:
 Wave vector k in parts of the 1. Brillouin zone
 k[π/a]:
Transversal  Longitudinal Oscillator function
© II. Physikalisches Institut, Universität zu Köln / University of Cologne


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Instructions

1) What does the applet show?
 
The applet shows the characteristic lattice vibrations (in the language of quantum mechanics: phonons) of a 1-dimensional Bravais lattice with 2 atoms per cell. This chain is displayed in the lower part of the applet. The atoms with mass M1 and M2 are coupled by springs with coupling constants D1 and D2 according to the following scheme:
<-- M1 - D1 - M2 - D2 - M1 - D1 - M2 - D2 -->
You can also add springs which couple next-nearest neighbors, i.e., M1 & M1 or M2 & M2.
The phonon energy (or frequency) follows the dispersion relation omega(k) displayed in the upper left of the applet. In 1D with a 2-atom basis, there are two solutions omega(k) for each value of k, corresponding to the two branches plotted in the diagram. The branches are called optical and acoustic branch.
 
You may choose M1 = M2 and D1 = D2. This corresponds to another Bravais lattice with 1 atom per cell but half of the original lattice constant. In this case, there is only one branch omega (k), and the applet changes accordingly.  
  
 
2) How to use the applet?  

Below the dispersion relation, you may adjust the value of the crystal momentum k. All other adjustable values are located in the upper right section of the applet called 'Options'. You may
1) customize the masses of the atoms with sliders for M1 and M2/M1 (with M2<M1).
The values of M1 and M2/M1 can be varied over a certain interval. Note that a symmetric situation can only be achieved with M21/M1 = 1!
2) customize the coupling springs with sliders for D1 and D2.
Note that a symmetric situation can only be achieved with D1 = D2!
3) add next-nearest-neighbour coupling by adding springs D3 between M1&M1 and D4 between M2&M2. Note that a symmetric situation can only be achieved with D3 = D4.