PHZ3400 Sound: Difference between revisions

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* Collective Phenomenon - behavior changes when in a large group
* Collective Phenomenon - behavior changes when in a large group


* Consider an <math>H_2</math> molecule. Assume one is at rest while the other moves.
===<math>H_2</math> Example===
 
Consider an <math>H_2</math> molecule. Assume one is at rest while the other moves.
<math>ma = -\frac{dV(r)}{dt}</math>
<math>ma = -\frac{dV(r)}{dt}</math>



Revision as of 00:30, 12 February 2009

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
    • 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

Example

Consider an molecule. Assume one is at rest while the other moves.

,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)

,where .Notice that the second term is the derivative of at , which is a minimum, therefore the derivative is zero and this term can be ignored. Now we have

Now let and we have

and

...

One dimensional mono-atomic chain

Sound waves - acoustic modes

One dimensional diatomic chain: optical modes