One Dimensional Lattice Dynamics

Andy Chmilenko, Nick Kuzmin Instructor: Jeff Gardiner Section 1

(Dated: 1:30 pm Monday June 23, 2014)

I. ABSTRACT

II. INTRODUCTION

Studying Lattice dynamics, even if only in on the One Dimensional has many real world applications and some of the fundamental lessons learned in this lab can be carried forward to much more complicated situations. Through the use of the sliders and springs, we can study the propagation of phonons, which are quanta of sound travelling though a medium we can create a model and derive the dispersion relation related to atoms in a solid for both monoatomic lattices, and diatomic lattices. We will also be looking at the speed of sound though these mediums and observe the band gap between the acoustic and optical branches in the diatomic lattice dispersion relation. By studying phonons we can broaden our understanding and apply the concepts we learn to important fields of study such as thermal and electrical conductivity, as well as being an important part in Solid State physics.

III. THEORETICAL BACKGROUND

A. Vibrations in a Monoatomic Lattice

We can begin to derive the equations of motion for the monoatomic 1D lattice by assuming each atom in the lattice moves due the nearest-neighbour interaction, for example, in an array of couple atoms, atom n would be acted upon by the coupled atoms n − 1 and n + 1. Using Hooke’s law, F = Cu, where C is the spring constant and u is the displacement from equilibrium, we can describe the force acting on atom n by:

F_{n} = C(u_{n+1} − u_{n}) + C(u_{n−1} − u_{n})  (1)

We can then derive from (1), using the fact that F = ma = mü,

mü_{n} = m\frac{d^{2}u_{n}}{dt^{2}} = −C(2u_{n} − u_{n+1} − u_{n−1})  (2)

We can solve this differential equation by looking for solutions of the form

u_{n} = Ae^{i(kna−ωt)}  (3)

where k is the wave number, and a is the spacing between planes in the monoatomic lattice, then differentiating (3) twice w.r.t time we get,

ü_{n} = −ω^{2} · Ae^{i(kna−ωt)} = −ω^{2}u_{n}  (4)

By substituting (3) and (4) into (2), we get as follows:

mü_{n} = −C(2u_{n} − u_{n+1} − u_{n−1})
−mω^{2} ·[figure: Ae^{i(kna−ωt)}] = −C(2[figure: Ae^{i(kna−ωt)}] −[figure: Ae^{i(kna−ωt)}]e^{ika} −[figure: Ae^{i(kna−ωt)}]e^{−ika})
−mω^{2} = −C(2 − e^{ika} − e^{−ika})
−mω^{2} = −C(2 − (2 cos ka))

ω^{2} = 2\frac{C}{m}(1 − cos ka)  (5)

FIG. 1: The dispersion relation for a Monoatomic lattice, showing the Acoustic branch and ω_{max}.

and using the double angle formula for 1 − cos ka = 2 sin^{2} \frac{ka}{2}, from (5) we derive the Dispersion law as,

ω = ±2\sqrt{\frac{C}{m}}| sin \frac{ka}{2}|  (6)

Taking the continuum limit as k → 0, using the small angle approximation:

sin \frac{ka}{2} ≈ \frac{ka}{2}  (7)

substituting (7) into (6), we can get an expression for the speed of sound in the lattice v

\frac{ω}{k} = v = \sqrt{\frac{C}{m}}a  (8)

1. Behaviour of Dispersion Curve at k = ± \frac{π}{a}

When the wave number is a multiple of \frac{π}{a}, this is where there will be the maximum vibration frequency in the monoatomic lattice.

ω_{max} = 2\sqrt{\frac{C}{m}}  (9)

this is the edge of the Brillouin zones for this lattice, the first Brillouin zone spans −\frac{π}{a} ≤ k ≤ \frac{π}{a}, as seen in Fig. 1 between maxima.

B. Vibrations in a Diatomic Lattice

Similar to a monoatomic lattice, we can derive the equations for a diatomic lattice which, in one dimension, has two different atoms of different masses in a unit cell, repeating. We now have two coupled differential equations that need to be solved,

mü_{2n} = m\frac{d^{2}u_{2n}}{dt^{2}} = −C(2u_{n} − u_{2n+1} − u_{2n−1})  (10)

Mü_{2n+1} = M \frac{d^{2}u_{2n+1}}{dt^{2}} = −C(2u_{2n+1} − u_{2n+2} − u_{2n})  (11)

and similarly subbing in (3) into the two equations, we get the set of equations,

[\begin{matrix}u_{2n} \\ u_{2n+1} \end{matrix}] = [\begin{matrix}Ae^{ik2na} \\ Be^{ik(2n+1)a}e^{ika} \end{matrix}] e^{−iωt}  (12)

And substituting into (10) and (11) and cancelling out the factor of e^{−iωt}e^{ik2na}, we get:

[\begin{matrix}−mAω^{2} \\ −MBω^{2} \end{matrix}] = [\begin{matrix}−C(2A − B(e^{−ika} + e^{ika})) \\ −C(2B − A(e^{ika} + e^{−ika)}) \end{matrix}] = [\begin{matrix}−C(2A − 2B cos ka) \\ −C(2B − 2A cos ka) \end{matrix}]  (13)

then rearranging (10) grouping like terms A and B we get the system of equations:

[\begin{matrix}A(−mω^{2} + 2C) \\ −2AC cos ka \end{matrix}] = [\begin{matrix}2BC cos ka \\ B(−2C + Mω^{2}) \end{matrix}]  (14)

[\begin{matrix}(2C − mω^{2}) & −2C cos ka \\ −2C cos ka & (2C − Mω^{2}) \end{matrix}] [\begin{matrix}A \\ B \end{matrix}] = 0  (15)

and taking the determinant of the matrix in (15) we can solve for ω we get:

(2C − mω^{2})(2C − Mω^{2}) − 4C^{2} cos^{2} ka = 0
(2\frac{C}{m} − ω^{2})(2\frac{C}{M} − ω^{2}) − 4\frac{C^{2}}{mM} cos^{2} ka = 0
4\frac{C^{2}}{mM} − 2\frac{C}{m}ω^{2} − 2\frac{C}{M} ω^{2} + ω^{4} − 4\frac{C^{2}}{mM} cos^{2} ka = 0

(ω^{2})^{2} − (ω^{2})[2(\frac{C}{m} + \frac{C}{M})] + 4\frac{C^{2}}{mM} (1 − cos^{2} ka) = 0  (16)

Solving for the value of ω^{2} in (16) using the quadratic equation, we get a value of:

\begin{matrix}ω^{2} = (2(\frac{C}{m} + \frac{C}{M}) ± \sqrt{(2(\frac{C}{m} + \frac{C}{M}))^{2} − 4 · 4\frac{C^{2}}{mM} (1 − cos^{2} ka)}) · \frac{1}{2} \\ ω^{2} = (2C(\frac{1}{m} + \frac{1}{M}) ± 2C\sqrt{(\frac{1}{m} + \frac{1}{M})^{2} − 4\frac{1}{mM} (1 − cos^{2} ka)})[figure: ·\frac{1}{2}] \end{matrix}

ω^{2} = C(\frac{1}{m} + \frac{1}{M}) ± C\sqrt{(\frac{1}{m} + \frac{1}{M})^{2} − 4\frac{sin^{2} ka}{mM}}  (17)

Equation (17) is the dispersion relation, and by taking the continuum limit as k → 0, we can derive expressions for the acoustic and optic branch dispersion relations. We use the small angle approximation for sin, and we take the Taylor expansion of the root term in (17),

C\sqrt{(\frac{1}{m} + \frac{1}{M})^{2} − 4\frac{sin^{2} ka}{mM}}

⇒ C\sqrt{(\frac{1}{m} + \frac{1}{M})^{2}}[1 − \frac{4}{2} \frac{\frac{(ka)^{2}}{mM}}{( \frac{1}{m} + \frac{1}{M})^{2}} ]

⇒ C(\frac{1}{m} + \frac{1}{M})[1 − \frac{4}{2} \frac{\frac{(ka)^{2}}{mM}}{( \frac{m+M}{mM})^{2}} ]

⇒ C(\frac{m+M}{mM})[1 − 2\frac{(ka)^{2}mM}{(m+M)^{2}} ]

1. Behaviour of Dispersion Curve at k ≪ 1

Taking the positive branch of (17), for k ≪ 1 we get,

\begin{matrix}ω^{2} ≈ C(\frac{1}{m} + \frac{1}{M}) + C(\frac{1}{m} + \frac{1}{M})[1 − 2\frac{(ka)^{2}mM}{(m+M)^{2}} ] \\ ω^{2} ≈ 2C(\frac{1}{m} + \frac{1}{M}) + C[figure: m+M; mM] [2\frac{(ka)^{2}[figure: mM]}{(m+M)^{2}}] \end{matrix}

ω^{2} ≅ 2C(\frac{1}{m} + \frac{1}{M})  (18)

Equation (18) describes the root of the Optical branch of the dispersion relation. The Optical branch is when all atoms in the lattice vibrate out of phase by π. Alternatively, we can take the negative branch of (17) and derive the root for the Acoustic branch, which is when all the atoms in the lattice vibrate in phase with each other.

FIG. 2: The dispersion relation for a Diatomic lattice, showing the Acoustic and Optical branches with some key values.

\begin{matrix}ω^{2} ≈ C(\frac{1}{m} + \frac{1}{M}) − C(\frac{1}{m} + \frac{1}{M})[1 − 2\frac{(ka)^{2}mM}{(m+M)^{2}} ] \\ ω^{2} ≈ C([figure: m+M; mM])[2\frac{(ka)^{2}[figure: mM]}{(m+M)^{2}}] \end{matrix}

ω^{2} ≅ 2C \frac{(ka)^{2}}{m + M}  (19)

using (19), we derive the speed of sound though a Diatomic lattice as similarly to how we did for the monoatomic lattice.

\frac{ω}{k} = v = a\sqrt{\frac{2C}{m + M}}  (20)

2. Behaviour of Dispersion Curve at k = ± \frac{π}{2a}

We can explore what happens when the wave number k in (17) is a value of ±\frac{π}{2a}, this area is known as the edge of the first Brillouin Zone, which spans −\frac{π}{2a} ≤ k ≤ \frac{π}{2a}. At the edges the sin term in (17) has a value of 1, and the roots will allow us to find the band gap energy of the system.

ω^{2} = C(\frac{1}{m} + \frac{1}{M}) ± C\sqrt{(\frac{1}{m} + \frac{1}{M})^{2} − 4\frac{sin^{2} ka}{mM}}
ω^{2} = C(\frac{m+M}{mM}) ± C\sqrt{(\frac{m+M}{mM})^{2} − \frac{4}{mM}}
ω^{2} = C(\frac{m+M}{mM}) ± C\sqrt{(\frac{m^{2}+M^{2}+2mM}{(mM)^{2}} ) − \frac{4mM}{(mM)^{2}}}
ω^{2} = C(\frac{m+M}{mM}) ± C\sqrt{(\frac{m^{2}+M^{2}−2mM}{(mM)^{2}} )}
ω^{2} = C(\frac{m+M}{mM}) ± C\sqrt{(\frac{m−M}{mM})^{2}}
ω^{2} = C(\frac{m + M}{mM}) ± C(\frac{m − M}{mM})  (21)

Taking the positive branch of (21), we get the lowest frequency in the optical branch

ω = \sqrt{\frac{2C}{M}}  (22)

and similarly taking the negative branch of (21), we get the highest frequency of the optical branch,

ω = \sqrt{\frac{2C}{m}}  (23)

The difference between (22) and (23) will give us the band gap for m < M equal to,

ω_{gap} = \sqrt{\frac{2C(m − M)}{mM}}  (24)

IV. EXPERIMENTAL DESIGN AND PROCEDURE

Apparatus:

• Air-track with vacuum

• 6 Small Air-track sliders

• 3 Large Air-track sliders

• 7 identical springs • Eccentric cam motor and power supply • Oscilloscope

• Spring weights

• Electronic weight scale

First the spring constant was measured by applying weights sequentially and measuring the displacement of the spring using a ruler and recorded the values. We then measured the weight three small and three large air-track sliders individually using the electronic weight scale and recorded the values.

We first began with the monoatomic lattice set-up using the 6 small sliders each fastened together with the springs, with the end sliders attached to the mounting bracket and the eccentric cam motor respectively. Then, with the vacuum on, we waited for the system to stabilize and come to equilibrium, we then turned off the vacuum so the sliders wouldn’t move and measured the inter-atomic distance between sliders, by measuring the relative displacement on the air-track of two of the same ends of two adjacent sliders (both right edges for example). Then continuing with the experiment, the vacuum was turned on again and using the control knob on the eccentric cam motor power supply (with it first set to the lowest setting), the frequency of the motor was adjusted in small amounts, being sure to wait to give the system enough time to react, until we found a frequency at which the amplitude of the displacement of the sliders was comparatively large compared to previous frequencies (a normal mode harmonic). We then recorded the frequency from the oscilloscope readout. We did this until we found 4-6 harmonic frequencies of the system.

For the second part of the experiment, the diatomic lattice set-up, we replaced 3 of the small sliders on the track with large sliders, having them arranges in alternating sizes so the sliders are evenly distributed in the lattice, again connected with the springs. The vacuum was turned on and the system allowed to reach equilibrium before turning it off and measuring the relative displacement on the air-track of both edges of a small and both edges adjacent large slider, such that the center of masses and separation of center of masses could be calculated. Similar to the monoatomic lattice procedure, with the vacuum on again and with the eccentric motor power supply control knob in its lowest setting, the frequency of the motor was slowly increased until harmonic modes were achieved. The frequencies of motor at the excited modes was recorded from the oscilloscope readout. Once 3-4 acoustic modes and 2-3 optic modes were found and recorded, all the equipment was shut off and the station tidied.

V. ANALYSIS

VI. CONCLUSION

VII. REFERENCES

1. Jeff Gardiner. One Dimensional Lattice Dynamics. Waterloo, Ontario: University of Waterloo; c2014. 1p.

2. Charles Kittel. Introduction to Solid State Physics. Seventh Edition. New York: John Wiley & Sons Inc., 1996. 692p.