PHZ3400 Sound

Vibrational modes of a one-dimensional chain

Harmonic approximation: inter-atomic forces as springs

• If we put energy into a crystal the atoms will begin to vibrate.
• We are able to put energy into the crystal in two ways: Mechanical and Thermal.
  • Mechanical energy are sound waves
  • Thermal energy can be used to measure resistivity.
• Periodicity helps to simplify the problem of lattice vibrations.
• One atom moves with the frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega_0 = \sqrt{\frac{k}{m}}}
  • If we have many atoms moving together than they effectively have a larger mass, therefore they will have a smaller frequency.
• Broken Symmetry
• Collective Phenomenon - behavior changes when in a large group

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_2} Example

Consider an Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_2} molecule. Assume one is at rest while the other moves. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle ma = -\frac{dV(r)}{dt}}

,where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} is potential energy and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} is radius. This can be rewritten as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\frac{d^2r}{dt^2} = -\frac{dV(r)}{dt}}

This equation cannot be solved via conventional methods, so we must somehow simplify it. Let us only worry about very small oscillations. This reduces our problem to a harmonic oscillator. Small oscillations can be described simply since it is parabolic at the minimum energy.

Now we expand Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r)} in Taylor Series (note Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0} is the radius with minimum V)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V(r) = V(r_0) + V'(r_0)(r - r_0) + \frac{1}{2}K(r - r_0)^2}

,where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K = \frac{d^2V}{dt^2}} .Notice that the second term is the derivative of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle V} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r_0} , which is a minimum, therefore the derivative is zero and this term can be ignored. Now we have

$V(r)=V(r_{0})+{\frac {1}{2}}K(r-r_{0})^{2}$

Now let $u=(r-r_{0})$ and we have

$V(u)=V_{0}+{\frac {1}{2}}Ku^{2}$ and ${\frac {dV(u)}{du}}=Ku$

...

$\Rightarrow -\omega ^{2}u=-{\frac {k}{m}}u$

$\Rightarrow \omega ={\sqrt {\frac {k}{m}}}$

Two Atom Model

• Corresponds to putting $H_{2}$ in a potential well

• We can reduce this problem to particles on springs

• There is a spring with constant K holding the two atoms together. Also, we put them in a potential well such that we can imagine a spring connected each part individually to a wall, with spring constant Q

• Solve via eigenvalue analysis

• Two Modes
  • Both Particles move in concert
  • Both Particles move opposite to each other

• If one of these particles are displaced (ignore the effects of the second) it will act as a linear Harmonic Oscillator, so that it will act as a simple sine wave.
  • However, if you take into account the second particle, the problem will not be so simple since there will be a superposition of two since waves creating a beat pattern. If the displacement is described by one of the modes above then there will be a simple sine wave.

We can write the equations of motion for these two particles as

$m{\ddot {u}}_{1}=-Qu_{1}-Ku_{1}+Ku_{2}\!$

$m{\ddot {u}}_{2}=-Ku_{2}+Ku_{1}-Qu_{2}\!$

,where $K$ and $Q$ are the spring constants, $m$ is mass, and $u$ is displacement from the particles equilibrium position. This can be re-written as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ddot u_1 = u_1(Q + K) + u_2(-K)\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ddot u_2 = u_1(-K) + u_2(Q + K)\!}

We can create a matrix representation for this system of equations by letting

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \overrightarrow u = \begin{bmatrix} u_1 \\ u_2 \end{bmatrix}}

and choosing the interaction matrix, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbb{M}} to be

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} Q + K & -K \\ -K & Q + K \\ \end{bmatrix} }

Now we can rewrite our system of equations as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ddot{u} = \mathbb{M}\overrightarrow{u}}

One dimensional mono-atomic chain

For this example, we imagine a chain of identical atoms, with mass Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} connected by springs, with spring constant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K} . Since this is only in one dimension we assume the particles only move in a direction parallel to the chain.

There are two forces that contribute to the displacement of the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle nth} atom, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_n}

1)The force from the spring on the left Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K(u_n - u_{n-1})}

2)The force from the spring on the right Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle K(u_{n+1} - u_n)}

Combining these we can write the total force as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\ddot{u}_n = -K[2u_n - u_{n-1} - u_{n+1}]\!}

...

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -m\omega^2 = -K[2u_n - u_{n-1} - u_{n+1}]\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -m\omega^2 = -K[2 - e^{iKa} - e^{-iKa}]\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega(K) = 2\sqrt{\frac{K}{m}}|sin(Ka)|\!}