VoigtFunc

VoigtFunc (X, Y)

VoigtFunc computes the Voigt function using an approximation that has an accuracy of, as described by the author, "typically at least 13 significant digits".

VoigtFunc returns values from a normalized Voigt peak centered at X=0 for the given value of X. The X input is a normalized distance from the peak center:

$$\displaystyle X=\sqrt{\ln (2)} \frac{v-v_{0}}{\gamma_{g}}$$

where

$$\displaystyle \gamma_{g}$$

is the Gaussian component half-width, and

$$\displaystyle v-v_{0}$$

is the distance from the peak center.

The parameter Y is the shape parameter. When Y is zero, the peak is pure Gaussian. When Y approaches infinity, the shape becomes pure Lorentzian. When Y is sqrt(ln(2)), the mix is half-and-half.

VoigtFunc was added in Igor Pro 7.00. The approximation used to compute it was changed Igor 8.00 for greater accuracy.

Details

The VoigtFunc function returns values from a normalized peak that can be used as the basis for user-defined fitting functions. The function is used as the basis for the built-in Voigt fitting function and the VoigtPeak function.

VoigtFunc Curve Fitting Example

Here is an example of a user-defined fitting function built on VoigtFunc:

Constant sqrtln2=0.832554611157698			// sqrt(ln(2))
Constant sqrtln2pi=0.469718639349826		// sqrt(ln(2)/pi)

Function MyVoigtFit(w,xx) : FitFunc
	Wave w
	Variable xx

	//CurveFitDialog/ These comments were created by the Curve Fitting dialog. Altering them will
	//CurveFitDialog/ make the function less convenient to work with in the Curve Fitting dialog.
	//CurveFitDialog/ Equation:
	//CurveFitDialog/ Variable ratio = sqrtln2/gw
	//CurveFitDialog/ Variable xprime = ratio*(xx-x0)
	//CurveFitDialog/ Variable voigtY = ratio*shape
	//CurveFitDialog/ f(xx) = y0 + area*sqrtln2pi*VoigtFunc(xprime, voigtY)
	//CurveFitDialog/ End of Equation
	//CurveFitDialog/ Independent Variables 1
	//CurveFitDialog/ xx
	//CurveFitDialog/ Coefficients 5
	//CurveFitDialog/ w[0] = y0
	//CurveFitDialog/ w[1] = area
	//CurveFitDialog/ w[2] = x0
	//CurveFitDialog/ w[3] = gw (FWHM)
	//CurveFitDialog/ w[4] = shape (Lw/Gw)

	Variable voigtX = 2*sqrtln2*(xx-w[2])/w[3]
	Variable voigtY = sqrtln2*w[4]
	return w[0] + (w[1]/w[3])*2*sqrtln2pi*VoigtFunc(voigtX, voigtY)
End

Parameter w[0] sets the vertical offset, w[1] sets the peak area, w[2] sets the location of the peak, w[3] gives the Gaussian component's full width at half max and w[4] is the ratio of the Lorenztian width to the Gaussian width.

After the fit, assuming you used a coefficient wave named voigtCoefs, you can calculate the width of the full Voigt peak as follows:

Variable/G wl = voigtCoefs[4]*voigtCoefs[3]
Variable/G wg = voigtCoefs[3]
Variable wv = wl/2 + sqrt( wl^2/4 + wg^2)

References

The code used to compute the VoigtFunc was written by Steven G. Johnson of MIT. You can learn more about it at http://ab-initio.mit.edu/Faddeeva.

See Also

VoigtPeak, VoigtFunc, Built-in Curve Fitting Functions