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Encyclopedia > Voigt profile
(Centered) Voigt
Probability density function
Plot of the centered Voigt profile for four cases
Plot of the centered Voigt profile for four cases. Each case has a full width at half-maximum of very nearly 3.6. The black and red profiles are the limiting cases of the Doppler (γ =0) and the Lorentzian (σ =0) profiles respectively.
Cumulative distribution function
Centered Voigt CDF.
Parameters γ,σ > 0
Support
pdf
cdf
Mean (not defined)
Median 0
Mode 0
Variance (not defined)
Skewness (not defined)
Kurtosis (not defined)
Entropy
mgf (not defined)
Char. func.

In spectroscopy, the Voigt profile is a spectral line profile named after Woldemar Voigt and found in all branches of spectroscopy in which a spectral line is broadened by two types of mechanisms, one of which alone would produce a Doppler profile, and the other of which would produce a Lorentzian profile. Image File history File links Download high resolution version (1300x975, 161 KB) Probability density function for the Voigt profile File links The following pages link to this file: Voigt profile ... Image File history File links Download high resolution version (1300x975, 163 KB) Cumulative density function for the Voigt profile File links The following pages link to this file: Voigt profile ... In mathematics, the support of a numerical function f on a set X is sometimes defined as the subset of X on which f is nonzero. ... In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ... In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the variable X takes on a value less than or... In probability theory (and especially gambling), the expected value (or mathematical expectation) of a random variable is the sum of the probability of each possible outcome of the experiment multiplied by its payoff (value). Thus, it represents the average amount one expects to win per bet if bets with identical... In probability theory and statistics, the median is a number that separates the higher half of a sample, a population, or a probability distribution from the lower half. ... In statistics, the mode is the value that has the largest number of observations, namely the most frequent value or values. ... In probability theory and statistics, the variance of a random variable is a measure of its statistical dispersion, indicating how far from the expected value its values typically are. ... In probability theory and statistics, skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable. ... In probability theory and statistics, kurtosis is a measure of the peakedness of the probability distribution of a real-valued random variable. ... Entropy of a Bernoulli trial as a function of success probability. ... In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... Some mathematicians use the phrase characteristic function synonymously with indicator function. ... Spectrum of fluorescent lights showing prominent mercury peaks. ... Woldemar Voigt (September 2, 1850 - December 13, 1919) was a German physicist. ... A spectral line is a dark or bright line in an otherwise uniform and continuous spectrum, resulting from an excess or deficiency of photons in a narrow frequency range, compared with the nearby frequencies. ... The Doppler profile is a spectral line profile which results from the thermal motion of the emitting atom or molecule. ... ...


All normalized line profiles can be considered to be probability distributions. The Doppler profile is essentially a normal distribution and a Lorentzian profile is essentially a Cauchy distribution. Without loss of generality, we can consider only centered profiles which peak at zero. The Voigt profile is then the convolution of a Lorentzian profile and a Doppler profile: In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... This article is about the mathematical concept of convolution. ... ... The Doppler profile is a spectral line profile which results from the thermal motion of the emitting atom or molecule. ...

where x is frequency from line center, D(x |σ) is the centered Doppler profile:

and L(x |γ) is the centered Lorentzian profile:

The defining integral can be evaluated as:

where Re[w(z) ] is the real part of the complex error function of z  and In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, the error function (also called the Gauss error function) is a non-elementary function which occurs in probability, statistics and partial differential equations. ...

Contents


Properties

The Voigt profile is normalized:

since it is the convolution of normalized profiles. The Lorentzian profile has no moments (other than the zeroth) and so the moment-generating function for the Cauchy distribution is not defined. It follows that the Voigt profile will not have a moment-generating function either, but the characteristic function for the Cauchy distribution is well defined, as is the characteristic function for the normal distribution. The characteristic function for the (centered) Voigt profile will then be the product of the two: In probability theory and statistics, the moment-generating function of a random variable X is The moment-generating function generates the moments of the probability distribution, as follows: If X has a continuous probability density function f(x) then the moment generating function is given by where is the ith... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... Some mathematicians use the phrase characteristic function synonymously with indicator function. ... The Cauchy-Lorentz distribution, named after Augustin Cauchy, is a continuous probability distribution with probability density function where x0 is the location parameter, specifying the location of the peak of the distribution, and γ is the scale parameter which specifies the half-width at half-maximum (HWHM). ... The normal distribution, also called Gaussian distribution, is an extremely important probability distribution in many fields. ... Some mathematicians use the phrase characteristic function synonymously with indicator function. ...

The width of the Voigt profile

The full width at half-maximum (FWHM) of the Voigt profile can be found from the widths of the associated Gaussian and Lorentzian widths. The FWHM of the Gaussian profile is fG

The FWHM of the Lorentzian profile is just fL = 2γ. Define φ = fL / fG. Then the FWHM of the Voigt profile (fV ) can be estimated as:

where c0 = 2.0056 and c1 = 1.0593. This estimate will have a standard deviation of error of about 2.4 percent for values of φ between 0 and 10. Note that the above equation will have the proper behavior in the limit of φ = 0 and φ = ∞.


The uncentered Voigt profile

If the Gaussian profile is centered at μG and the Lorentzian profile is centered at μL, the convolution will be centered at μG + μL and the characteristic function will then be:

The mode and median will then both be located at μG + μL.


See also


  Results from FactBites:
 
www.myspace.com/deborahvoigt (1681 words)
Voigt has been one of the world's leading dramatic sopranos for some time, and it's a pleasure to report [on] that gleaming Cadillac of an instrument.
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Woldemar Voigt - Wikipedia, the free encyclopedia (1009 words)
Woldemar Voigt (September 2, 1850 - December 13, 1919) was a German physicist.
Voigt profile and Voigt notation are named after him.
Voigt's work was far ahead of its time and went apparently un-noticed by all those who contributed to the development of special relativity (George FitzGerald and Oliver Heaviside, Joseph Larmor, Hendrik Lorentz, Henri Poincaré and Albert Einstein)
  More results at FactBites »

 
 

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