skip to content

Phononen - Dispersion, optische/akustische

Anleitung

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 -->
Later 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 correpsonds 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?
 
All adjustable values are located in the upper right section of the applet called 'Options'. You may
1) adjust the value of the crystal momentum k
2) turn on / off the functions on which the atoms move
3) customize the masses of the atoms with sliders for m and delta_m:
M1 = m * (1 + delta_m) > m
M2 = m * (1 - delta_m) < m
The values of m and delta can be varied over a certain interval. Note that a symmetric situation can only be achieved with delta = 0!
4) customize the coupling springs with sliders for d and delta_d:
D1 = d * (1 + delta_d) > d
D2 = d * (1 - delta_d) < d
The values of d and delta_d again can be varied over a certain interval. Note that a symmetric situation can only be achieved with delta_d = 0!
5) add next-nearest-neighbour coupling by adding springs D3 between M1&M1 and D4 between M2&M2. Their values can also be adjusted by sliders either asymmetric or in the same way by using the checkbox 'D3 = D4' at the bottom of the panel. Note that a symmetric situation can only be achieved with D3 = D4, either achieved by using the checkbox or adjusting the values manually to the same value.