A detailed view of the Gaussian–Lorentzian sum and product functions and their comparison with the Voigt function

In this paper, we show that the use of methods of an operational nature, such as umbral calculus, allows achieving a double target: on one side, the study of the Voigt function, which plays a pivotal role in spectroscopic studies and in other applications, according to a new point of view, and on the other, the introduction of a Voigt transform and its possible use. Furthermore, by the same method, we point out that the Hermite and Laguerre functions, extension of the corresponding polynomials to negative and/or real indices, can be expressed through a definition in a straightforward and unified fashion. It is illustrated how the techniques that we are going to suggest provide an easy derivation of the relevant properties along with generalizations to higher order functions.

Download Full-text

Step size, scanning speed and shape of X-ray diffraction peak

Journal of Applied Crystallography ◽

10.1107/s002188989400186x ◽

1994 ◽

Vol 27 (5) ◽

pp. 716-722 ◽

Cited By ~ 7

Author(s):

H. Wang

Keyword(s):

Scanning Speed ◽

Peak Position ◽

Maximum Intensity ◽

X Ray Diffraction ◽

Step Size ◽

X Ray ◽

Scanning Time ◽

Integral Width ◽

Counting Statistics ◽

Voigt Function

The influences of step size and scanning speed on the shape of a single X-ray diffraction (XRD) peak are analyzed quantitatively. For this purpose, it is assumed that XRD peak shapes are a mixture of Cauchy and Gauss curves. Six equations are established for the calculation of position, maximum intensity and full width at half-maximum (FWHM) errors caused by step size and two for the FWHM errors caused by counting statistics. The ratio of step size to FWHM is proposed as the shape-perfect coefficient of the XRD peak. From these equations and the relationship between the FWHM and the integral width of a peak based on the pseudo-Voigt function or Voigt function, three basic elements of a single symmetric XRD peak (peak position, maximum intensity and FWHM) can be refined. The optimum step size and scanning time can also be set from them.

Download Full-text

The Use of the Pseudo-Voigt Function in the Variance Method of X-ray Line-Broadening Analysis

Journal of Applied Crystallography ◽

10.1107/s0021889896015464 ◽

1997 ◽

Vol 30 (4) ◽

pp. 427-430 ◽

Cited By ~ 101

Author(s):

F. Sánchez-Bajo ◽

F. L. Cumbrera

Keyword(s):

Crystallite Size ◽

Original Form ◽

Line Broadening ◽

X Ray Diffraction ◽

Stabilized Zirconia ◽

X Ray ◽

Variance Method ◽

The Mean ◽

Voigt Function

A modified application of the variance method, using the pseudo-Voigt function as a good approximation to the X-ray diffraction profiles, is proposed in order to obtain microstructural quantities such as the mean crystallite size and root-mean-square (r.m.s.) strain. Whereas the variance method in its original form is applicable only to well separated reflections, this technique can be employed in the cases where there is line-profile overlap. Determination of the mean crystallite size and r.m.s. strain for several crystallographic directions in a nanocrystalline cubic sample of 9-YSZ (yttria-stabilized zirconia) has been performed by means of this procedure.

Download Full-text

An efficient method for evaluation of the complex probability function: The Voigt function and its derivatives

Journal of Quantitative Spectroscopy and Radiative Transfer ◽

10.1016/0022-4073(79)90062-1 ◽

1979 ◽

Vol 21 (4) ◽

pp. 309-313 ◽

Cited By ~ 217

Author(s):

J. Humlíček

Keyword(s):

Efficient Method ◽

Probability Function ◽

Voigt Function

Download Full-text

Determination of the Coherency Strain of γ and γ′ Phases in Nickel-Base Superalloys at High Temperatures

Advances in X-ray Analysis ◽

10.1154/s0376030800019145 ◽

1992 ◽

Vol 36 ◽

pp. 515-526

Author(s):

Katsumi Ohno ◽

Tadaharu Yokokawa ◽

Toshihiro Yamagata ◽

Hiroshi Harada ◽

Michio Yamazaki ◽

...

Keyword(s):

Lattice Misfit ◽

Profile Fitting ◽

Overlapping Peaks ◽

Nickel Base Superalloys ◽

Nickel Base ◽

L12 Structure ◽

Fcc Structure ◽

Type Nickel ◽

Voigt Function

AbstractA method for using syncrotoron-radiation parallel-beam X-ray diffractometry for precise lattice parameters and strains of γ-γ′ type Nickel base superalloys at elevated temperature is described. The superalloys have γ′ precipitates which are an ordered L12 structure based on Ni3Al, in y-matrices having a disordered FCC structure. Lattice misfit between γ and γ′ phases was very small and peaks reflected from γ and γ′ phases made unresolved clusters of peaks.Profile fitting with a pseudo-Voigt function is used to resolve overlapping peaks. Instrumental broadening of the peak profile was removed using a deconvolution method. The standard errors of the calculated peak angle were less than 0.002°. The elastic strain of the γ′ precipitates in the alloys were smaller than those of γ-matrices.

Download Full-text

Relations of the Extended Voigt Function with Other Families of Polynomials and Numbers

Trends in Mathematics - Advances in Mathematical Inequalities and Applications ◽

10.1007/978-981-13-3013-1_6 ◽

2018 ◽

pp. 101-114

Author(s):

M. A. Pathan

Keyword(s):

Voigt Function

Download Full-text

Accurate Modeling of Size and Strain Broadening in the Rietveld Refinement: The “Double-Voigt” Approach

Advances in X-ray Analysis ◽

10.1154/s0376030800018048 ◽

1994 ◽

Vol 38 ◽

pp. 397-404 ◽

Cited By ~ 1

Author(s):

Davor Baizar ◽

Hassel Ledbetter

Keyword(s):

Rietveld Refinement ◽

Line Profile ◽

Fourier Coefficients ◽

Lattice Strain ◽

Harmonic Approximation ◽

Computer Code ◽

Isotropic Case ◽

Mean Square ◽

Line Broadening ◽

Voigt Function

In the “double-Voigt” approach, an exact Voigt function describes both size- and strainbroadened profiles. The lattice strain is defined in terms of physically credible mean-square strain averageid over a distance in the diffracting domains. Analysis of Fourier coefficients in a harmonic approximation for strain coefficients leads to the Warren-Averbach method for the separation of size and strain contributions to diffraction line broadening. The model is introduced in the Rietveld refinement program in the foliowing way: Line widths are modeled with only four parameters in the isotropic case. Varied parameters are both surface- and volumeweighted domain sizes and root-mean-square strains averaged over two distances. Refined parameters determine the physically broadened Voigt line profile. Instrumental Voigt line profile parameters are added to obtain the observed (Voigt) line profile. To speed computation, the corresponding pseudo-Voigt function is calculated and used as a fitting function in refinement. This approach allows for both fast computer code and accurate modeling in terms of physically identifiable parameters.

Download Full-text