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Lorentzian

Lorentzian (Area)

The Lorentzian peak with a_{0} as the peak area is defined as follows:

a_{0} = Area

a_{1} = Center (mode)

a_{2} = Width

Built in model: Lorentz

User-defined peaks and view functions: Lorentz(x,a_{0},a_{1},a_{2})

Lorentzian (Amplitude)

The Lorentzian peak with a_{0} as the peak amplitude is defined as follows:

a_{0} = Amplitude

a_{1} = Center (mode)

a_{2} = Width

Built in model: Lorentz[amp]

User-defined peaks and view functions: Lorentz[amp](x,a_{0},a_{1},a_{2})

Cauchy Distribution

The Lorentzian peak function is also known as the Cauchy distribution function. It is a symmetric function
whose mode is a_{1}, the center parameter. The tails of the Lorentzian are much wider than that
of a Gaussian. Moments do not exist.

The Lorentzian is encountered primarily in spectroscopy and is sometimes referred to as the “natural” shape of a spectral line.

Natural Line Broadening

This broadening is associated with the lifetimes of energy states and the Heisenberg principle where the energy uncertainty is inversely proportional to the uncertainty in time for the occupation of a particular energy state. There are upper and lower energy states associated with a simple transition where a photon is absorbed or emitted.

The probability per unit frequency of a transition yielding a frequency v is given by the following Lorentzian form (Jansson reference):

where g_{n} is the sum of the reciprocals
of the natural upper and lower state lifetimes, and n_{0}
is the center frequency of the emission.

Any spectral emission due to a transition between energy states would thus be expected to have some degree of Lorentzian broadening.

Collision Broadening

In the optical spectra of gases, one must also account for molecular interactions. Additional line broadening occurs from molecular collisions. When natural and collision broadening effects are combined, the resulting line shape is also a Lorentzian, but of greater width than that which would occur absent collisions (Jansson reference):

where g_{c}
is equal to twice the collision frequency. At low pressures, natural line broadening will dominate the
peak width. At high pressures, the peak width will be primarily due to collision broadening. A shift in
line frequency may also occur with high pressures.

Spectral Fitting Considerations

Note that the x in the Lorentzian above is in frequency units. When fitting spectral peaks, the x variable must be proportional to frequency, wave number, or energy. The y variable must be a quantitative measure. As such, you must convert wavelengths to wave number and transmission to absorption prior to fitting.

Most instrument response functions are Gaussian. This means that most spectral lines will have some measure of Gaussian character. Unless the Gaussian instrumental response broadening is nearly absent, the Voigt function is the theoretical line shape for most spectral peaks.

Central Limit Theorem

The central-limit theorem states that when a function f(x) is convolved with itself n times, in the limit n->infinity, the convolution product is Gaussian with variance n times the variance of f(x), provided that the area, mean, and variance of f(x) are finite.

As the equation for both natural and collision broadening suggests, this theorem does not hold for Lorentzians. When two Lorentzian distributions are convolved with one another, the result is also Lorentzian whose width is equal to the sum of the widths of the components.

When two Gaussians with equal half-maxima widths are convolved, the result is a Gaussian with sqrt(2) times the width. Convolving two Lorentzians with equal half maxima widths produces a Lorentzian with twice the width.

Lorentzian Areas

It is not particularly easy to envision a peak function without a variance or standard deviation, but this is true of the Lorentzian. Random samples of a Lorentzian distribution do not converge to a single mean and standard deviation as the size of the sample set increases.

Although the area of a Lorentzian is analytically defined and finite, it is not uncommon for some portion of a fitted Lorentzian’s area to lay outside the range of the spectrum. As such, analytical peak areas will be greater than the measured areas which report a numeric integration using the lower and upper x limits of the data.

Suggested References

Further information on spectral line broadening, and the measuring and deconvolution of Gaussian instrument response functions can be found in Peter A. Jansson, Deconvolution with Applications in Spectroscopy, Academic Press, 1984, ISBN 0-12-380220-2.

Mathematical properties of the Cauchy statistical distribution function can be found in Merran Evans, Nicholas Hastings, and Brian Peacock, Statistical Distributions, p.42-44, John Wiley and Sons, 1993, ISBN 0-471-55951-2.

Further information on the Cauchy distribution function is available in Norman L. Johnson and Samuel Kotz, Continuous Univariate Distributions, Vol 1, p.154-165, Houghton Mifflin, 1970.