]/(2
)/[(w-w')2+(/2)2].
Augustin-Louis Cauchy (1789-1857) of France contributed rigor to mathematics. His lectures and researches in analysis during the 1820's clarified the principles of calculus by developing it with limits and continuity. His theory of complex functions forms the basis of physics today. His theoretical work in optics provided a sound mathematical though physically unsatisfactory basis for the supposed pervasive ether thought to conduct light.
Hendrik Lorentz (1853-1928) of the Netherlands (jointly with Zeeman) won the 1902 Nobel Prize for Physics for his theory of electromagnetic radiation. His doctoral thesis of 1875 refined Maxwell's theory (1865) so as to better explain the reflection and refraction of light. Visible light is a narrow part of the broad electromagnetic spectrum which extends from long wavelength radio waves to short wavelength x-rays, and beyond--both ways. Electromagnetic radiation (a photon) is a precise oscillation of an electric and a magnetic field. Applied/external electric and magnetic fields have an effect on this oscillation and hence change the corresponding wavelength (frequency). The Lorentzian Transformation, with it's time dilation and length contraction superceded the work of Galileo/Newton and forms the basis of Einstein's 1905 work on Special Relativity.
The Lorentzian Distribution is thus often used to describe resonance behavior, things like swings (pendula) swinging, a bow on a violin string, a thin goblet shattering when the fat lady sings, that dreaded microphone feedback, or the rhythmic wind gusts which destroyed the Takoma Narrows Bridge. Soldiers learn early to break stride when crossing a bridge. A radio or TV receiver is tuned to resonate in response to a specific frequency, typically by changing the capacitance or inductance. Under resonance, energy flows rhythmically between the capacitor's electric field and the inductance's magnetic field, just like the interchange of potential (energy of position) and kinetic (energy of motion) energy in a swing. Like the Gaussian, the Lorentzian is symmetric, unimodal, and continuous. Under very general assumptions the following formula can be derived:
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[A]/(2)/[(w-w')2+(/2)2].
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By inspecting this equation closely we can see it is symmetric and
has a maximum when w=w'. w (the preferred symbol
is omega) is called the driving frequency
whereas w' is the resonance frequency.
A is the area under the curve and is
called the full width at half maximum or FWHM,
the parameter which characterizes the spread of the distribution.
FWHM is also commonly called the linewidth or halfwidth.
The Lorentzian distribution tends to be lower with fatter tails (often called wings) than a Gaussian distribution with equal FWHM. In fact, the wings are so extended that the standard deviation is (and higher moments are) not defined (the integrals are unbounded or there is no average distance from the mean)!
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The second is the duration of 9 192 631 770 periods of the
radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom. |
Leap seconds are now occasionally used to adjust between time as kept by the atomic cesium fountain clock and time as determined by the earth's motion on its axis and around the sun. In 1997 the International Committee on Weights and Measures confirmed that the definition of the second was at 0K, in other words, the coldest possible temperature. A common fallacy you will find in many popular writeups is that "all motion ceases" at 0K. This is quantum mechanically absurd.
Additional areas of application of the Lorentzian are: DNA microarray data used to measure gene expression; edges in MR brain images; and muon spin relaxation theory.
There is a large separation between the fine and hyperfine states. Whereas the D1 (3S1/2 to 3P1/2) and D2 (3S1/2 to 3P3/2) transitions are only 6 nm apart in sodium (witness the yellow of a high pressure sodium lamp), the D1 (6S1/2 to 6P1/2) and D2 (6S1/2 to 6P3/2) transitions are 42 nm apart in cesium. However, they are in the infrared (894 nm and 852 nm). The natural lifetime of these cesium states is about 30 ns hence the natural linewidth is about 5 Mhz. Although one would expect doppler broadening on the order of 400 Mhz for cesium, by forming a highly collimated thermal beam we are able to reduce that to about 5 Mhz. 894 nm corresponds to a frequency of about 335 Thz. My research involved precisely finding the peak of this Voigt profile to less than 3Khz. We achieved an accuracy of 7 parts per trillion (0.7 parts in 1011), or an improvement of an order of magnitude over H&#auml;nsch's 1999 result. Another way to say this is that our results were precise to twelve significant digits! We found the D1 centroid to be 335 116 048 748.1(2.4) kHz. This is clearly one reason for my insisting on at least 3, but no more than 5 significant digits, unless clearly indicated. Note this last notation is worth further discussion. The (2.4) given after the value is the one standard deviation error bar. We are thus 95% confident that the true value is within +/-4.7 kHz of the value given, where 1.96×2.4 = 4.7.
The new precision on the D1 frequency in cesium allowed an improved measurement of alpha, the fine structure constant or electromagnetic coupling constant which is a fundamentation constant of nature.
Shown below are typical scans of the F4 to F3 transistion taken on April 8, 2004. A 1 Gauss magnetic field has been applied in the z (vertical) direction to the bottom scan. Below that are fits of the 1 Gauss scan using a Gaussian (left) and Lorentzian (right). A better fit is found using a combination (Voigt profile). Due to good symmetry, however, any of these approaches finds the center to within about the same 1kHz.
This research took place in the very room at NIST in Boulder CO where the Krypton-86 wavelength was precisely measured. The Krypton-86 wavelength was used from 1960 until 1983 when this research lead to a redefinition of the meter in terms of the speed of light. We used the femtosecond laser frequency comb which is also being used to calibrate the Mercury and Calcium frequencies which might well become the new Thz time standards in the near future.
My dissertation can be located here.
The May 2004 Reader's Digest told the tale of a fellow who put into a cup one spoonful of many other chili's at a chili contest and ended up winning. My intent somewhat is for my web pages to do the same, blending the best features of some, while avoiding others.
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