PHZ3400 Sound: Difference between revisions
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** Thermal energy can be used to measure resistivity. | ** Thermal energy can be used to measure resistivity. | ||
* Periodicity helps to simplify the problem of lattice vibrations. | * Periodicity helps to simplify the problem of lattice vibrations. | ||
* One atom moves with the frequency <math>\omega_0 = \sqrt{\frac{k}{m}}</math> | |||
** 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 <math>H_2</math> molecule. Assume one is at rest while the other moves. | |||
<math>ma = -\frac{dV(r)}{dt}</math> | |||
,where <math>V</math> is potential energy and <math>r</math> is radius. This can be rewritten as | |||
<math>m\frac{d^2r}{dt^2} = -\frac{dV(r)}{dt}</math> | |||
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 <math>V(r)</math> in Taylor Series (note <math>r_0</math> is the radius with minimum V) | |||
<math>V(r) = V(r_0) + V'(r_0)(r - r_0) + \frac{1}{2}K(r - r_0)^2</math> | |||
,where <math>K = \frac{d^2V}{dt^2}</math>.Notice that the second term is the derivative of <math>V</math> at <math>r_0</math>, which is a minimum, therefore the derivative is zero and this term can be ignored. Now we have | |||
<math>V(r) = V(r_0) + \frac{1}{2}K(r - r_0)^2</math> | |||
Now let <math>u = (r - r_0)</math> and we have | |||
<math>V(u) = V_0 + \frac{1}{2}Ku^2</math> and <math>\frac{dV(u)}{du} = Ku</math> | |||
==One dimensional mono-atomic chain== | ==One dimensional mono-atomic chain== | ||
Revision as of 00:06, 11 February 2009
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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
- 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
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) + \frac{1}{2}K(r - r_0)^2}
Now let 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 = (r - r_0)} and we have
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(u) = V_0 + \frac{1}{2}Ku^2} 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 \frac{dV(u)}{du} = Ku}