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 $\omega _{0}={\sqrt {\frac {k}{m}}}$
  • 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

• Consider an $H_{2}$ molecule. Assume one is at rest while the other moves.

$ma=-{\frac {dV(r)}{dt}}$

,where $V$ is potential energy and $r$ is radius. This can be rewritten as

$m{\frac {d^{2}r}{dt^{2}}}=-{\frac {dV(r)}{dt}}$

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 $V(r)$ in Taylor Series (note $r_{0}$ is the radius with minimum V)

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

,where $K={\frac {d^{2}V}{dt^{2}}}$.Notice that the second term is the derivative of $V$ at $r_{0}$, 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 ${\frac {dV(u)}{du}}=Ku$