One Dimensional Lattice Dynamics
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Abstract
A one dimensional lattice can be viewed as a chain of atoms bonded together on a single axis. The dispersion relation, , for a one dimensional lattice structure can be obtained by exciting different wavelengths, (which have a corresponding frequency, ). Moreover, the dispersion can be described in terms of an Acoustic branch and Optic branch.
Introduction
The school of thought surrounding the “dispersion relation” comes from optics. A dispersion relation describes the different ways in which wave propagation changes with the wavelength or frequency of a wave. Further, it would be worthy to note that due to the wave nature present in all travelling objects, the transportation of energy and objects from one point to another can be studied through their dispersion relation. In this experiment, a one dimensional lattice is depicted by a row of sliders (carts) on an air track which are connected by springs. The springs correspond to the nearest neighbour bond between two of the adjacent atoms. The atoms are represented by the sliders and have varying mass. Depending on if the masses are equal there will be two types of branches that emerge as result of the dispersion relation. These two branches are known as the Acoustic Branch and the Optic Branch. The Acoustic branch represents low (acoustic) frequencies which depicts frequency as a function of wave number in crystal lattice vibrations. Similarly, the Optic branch represents high frequencies.
Theory
The dispersion relation is directly applied to the dependence of the frequency of the wave,4 on the wave vector, 5. Further, the dispersion relation for solids is given by:
where, M is the mass of slider 1, m is the mass of slider 2, C is the spring constant. It would be worthy to note that when the sine term for the dispersion relation vanishes.
Monatomic Chain
Behaviour of Dispersion Curve as
The behaviour of the dispersion curve near can be analysed. Using small angle approximation it can be seen that:
which makes the dispersion relation in the limit as
So,
Using a Taylor Series Expansion the dispersion relation becomes:
The negative root can be attributed to the acoustic branch and the positive root of equation 6 can be attributed to the optic branch.
Behaviour of Dispersion Curve at :
The sine function reaches a maximum when . This is known as the edge of the first Brillouin zone. At these values of the acoustic branch is at its maximum and the optic branch is at its minimum. Moreover, the sine term in equation goes to one for So the term in the curly brackets in equation reduces to:
Therefore the optical branch is given by:
The acoustic branch is given by:
Diatomic Chain
Behaviour of Dispersion Curve as
For the diatomic chain the dispersion relation for masses and is given by:
and
where and describe the relative amplitudes of the atoms of masses. At the zone center the acoustic branch has a dispersion relation of zero hence implying that the atoms will oscillate in phase and with the same amplitude. More over at the zone center for the optical branch the dispersion relation is:
Taking equation and substituting it into a relation between the relative amplitudes is obtained:
This means that the atoms of both the masses will oscillate out of phase with the mass moving less that the other mass.
Behaviour of Dispersion curve at :
At the zone edge of , the acoustic branch has a dispersion relation of:
and the optical branch has a dispersion relation of:
Procedure
The procedure for this experiment can be found in the Physics 360B lab handout, titled “Gamma Ray Spectroscopy”.
Results
Trial  Mass #1 (g) 
Mass #2 (g) 
1 
286.4 
136.1 
2 
284.8 
135.4 
3 
282.6 
135.2 
Mass #1 (M1) has a mass of 284.6g 0.05g which is 0.2846 kg 0.0001 kg and M2 has an average mass of 0.1356 kg 0.0001 kg. Moreover, the spring constant was determined by hanging different mass from the spring and noting its vertical displacement. This value was calculated to be 3.21 kg/s2 0.05 kg/s2.
Monatomic Chain (only Acoustic)
f1 
f2 
f4 

Voltage (V) 
0.0600 
0.1150 
0.1560 
Frequency (Hz) 
0.3278 
0.6284 
0.8525 
Period (s) 
3.050 
1.591 
1.173 
Measured Period (s) 
2.850 
1.555 
1.111 
Diatomic Chain – Acoustic
f1 
f2 
f3 

Voltage (V) 
0.7000 
0.1250 
0.1890 
Frequency (Hz) 
0.3825 
0.6831 
1.033 
Period (s) 
2.614 
1.464 
0.9683 
Measured Period (s) 
2.533 
1.441 
0.9524 
Diatomic Chain – Optical

f1 
f2 
Voltage (V) 
0.2500 
0.2150 
Frequency (Hz) 
1.366 
1.1750 
Period (s) 
0.7320 
0.8510 
Measured Period (s) 
0.7143 
0.8000 
Monatomic 1DLattice spacing: a = 0.2920 m 0.005 m
Diatomic 1DLattice spacing: a = 0.2430 m 0.005 m
Conclusion
Discussion
A graph of frequency versus wave number for the monatomic lattice is shown below in figure 1:
Figure 1: Frequency vs. Wave number for a Monatomic 1DLattice.
Moreover, a graph of frequency versus wave number for the diatomic lattice is shown below for the acoustic and optical case in figure 2:
Figure 2: Graph of Frequency versus Wave number for the 1D diatomic Acoustic & Optical Branch.
Band Gap
For the diatomic case the theoretical band gap or forbidden energy zone occurs at which implies as mentioned before that the sine term will disappear. So, equation for the optical case reduces to:
which further reduces to:
For the acoustic case the method of calculation is similar only with the dispersion relation having a minus sign in front of the second term. The dispersion relation reduces to:
In order to calculate the forbidden zone band gap energy a difference between both these value will be taken:
which yields Moreover, the experimental band gap was calculated to be The difference can be attributed to the fact that the one dimensional lattice on an air track is not a precise description of the atoms in a lattice. Also, there are other factors such as all the springs used might not have the same spring constants and also the masses are not completely equal.
Speed of Sound in Lattice
The speed of sound in a lattice (where bonds are depicted by springs) is given by the root of the spring constant divided by the density of the structure:
here is the mass per unit length since it is one dimensional. For the monatomic case this value is . For the diatomic case the value is .
Conclusion
In conclusion, the graphs of a frequency versus wave number for the monatomic and diatomic case were plotted and analyzed. The theoretical and experimental band gap energy were calculated and compared. There was a deviation from both the values but this is due to the fact that there are imperfections in the setup of the system.
References
“Experiment #23 One Dimensional Lattice Dynamics”, University of Waterloo, Physics 360B, 2009