The intensity and position of sidebands (satellites) on both sides of main diffraction peak in a great number of X-ray diffraction profiles of alloys always change with progress of aging. The sidebands position is det...The intensity and position of sidebands (satellites) on both sides of main diffraction peak in a great number of X-ray diffraction profiles of alloys always change with progress of aging. The sidebands position is determined by a newly optimized Voigt function in present investigation. Furthermore, for Cu-4 wt pet Ti alloy aged at 400℃ for 720 min and 1080 min, after introducing the weight factor of above two satellites intensity, the relative error between the fitting curves and X-ray diffraction profiles is less than 0.185%, which is more precise than the previously calculating result.展开更多
The Voigt function is the convolution of a Lorentzian and a Guaussian density. The computation of these functions is required in several problems arising in a variety of physicochemical subjects;such as nuclear reacto...The Voigt function is the convolution of a Lorentzian and a Guaussian density. The computation of these functions is required in several problems arising in a variety of physicochemical subjects;such as nuclear reactors, atmospheric transmittance and spectroscopy. In this work we suggest using a new formula for the calculation of the Voigt function. Our formula is a new integral representation for the Voigt function that gives the perfect results for the Voigt function calculation and is easily calculable. We give also a comparison between our results of calculation of Voigt function for the very small values of the parameter a, where the calculation is notoriously difficult, with those of the various algorithms of other authors.展开更多
The authors derive a set of unified representations of the Voigt functions in terms of familiar special functions of Mathematical Physics. Some deductions from these representations are also considered.
Previously we derived equations determining line broadening in ax-ray diffraction profile due to stacking faults. Here, we will consider line broadening due to particle size and strain which are the other factors affe...Previously we derived equations determining line broadening in ax-ray diffraction profile due to stacking faults. Here, we will consider line broadening due to particle size and strain which are the other factors affecting line broadening in a diffraction profile. When line broadening in a diffraction profile is due to particle size and strain, the theoretical model of the sample under study is either a Gaussian or a Cauchy function or a combination of these functions, e.g. Voigt and Pseudovoigt functions. Although the overall nature of these functions can be determined by Mitra’s R(x) test and the Pearson and Hartley x?test, details of a predicted model will be lacking. Development of a mathematical model to predict various parameters before embarking upon the actual experiment would enable correction of significant sources of error prior to calculations. Therefore, in this study, predictors of integral width, Fourier Transform, Second and Fourth Moment and Fourth Cumulant of samples represented by Gauss, Cauchy, Voigt and Pseudovoigt functions have been worked out. An additional parameter, the coefficient of excess, which is the ratio of the Fourth Moment to three times the square of the Second Moment, has been proposed. For a Gaussian profile the coefficient of excess is one, whereas for Cauchy distributions, it is a function of the lattice variable. This parameter can also be used for determining the type of distribution present in aggregates of distorted crystallites. Programs used to define the crystal structure of materials need to take this parameter into consideration.展开更多
基金supported by the Aeronautical Basic Science Foundation(No.00G53054)the National Natural Science Foundation of China(No.50171053).
文摘The intensity and position of sidebands (satellites) on both sides of main diffraction peak in a great number of X-ray diffraction profiles of alloys always change with progress of aging. The sidebands position is determined by a newly optimized Voigt function in present investigation. Furthermore, for Cu-4 wt pet Ti alloy aged at 400℃ for 720 min and 1080 min, after introducing the weight factor of above two satellites intensity, the relative error between the fitting curves and X-ray diffraction profiles is less than 0.185%, which is more precise than the previously calculating result.
文摘The Voigt function is the convolution of a Lorentzian and a Guaussian density. The computation of these functions is required in several problems arising in a variety of physicochemical subjects;such as nuclear reactors, atmospheric transmittance and spectroscopy. In this work we suggest using a new formula for the calculation of the Voigt function. Our formula is a new integral representation for the Voigt function that gives the perfect results for the Voigt function calculation and is easily calculable. We give also a comparison between our results of calculation of Voigt function for the very small values of the parameter a, where the calculation is notoriously difficult, with those of the various algorithms of other authors.
文摘The authors derive a set of unified representations of the Voigt functions in terms of familiar special functions of Mathematical Physics. Some deductions from these representations are also considered.
文摘Previously we derived equations determining line broadening in ax-ray diffraction profile due to stacking faults. Here, we will consider line broadening due to particle size and strain which are the other factors affecting line broadening in a diffraction profile. When line broadening in a diffraction profile is due to particle size and strain, the theoretical model of the sample under study is either a Gaussian or a Cauchy function or a combination of these functions, e.g. Voigt and Pseudovoigt functions. Although the overall nature of these functions can be determined by Mitra’s R(x) test and the Pearson and Hartley x?test, details of a predicted model will be lacking. Development of a mathematical model to predict various parameters before embarking upon the actual experiment would enable correction of significant sources of error prior to calculations. Therefore, in this study, predictors of integral width, Fourier Transform, Second and Fourth Moment and Fourth Cumulant of samples represented by Gauss, Cauchy, Voigt and Pseudovoigt functions have been worked out. An additional parameter, the coefficient of excess, which is the ratio of the Fourth Moment to three times the square of the Second Moment, has been proposed. For a Gaussian profile the coefficient of excess is one, whereas for Cauchy distributions, it is a function of the lattice variable. This parameter can also be used for determining the type of distribution present in aggregates of distorted crystallites. Programs used to define the crystal structure of materials need to take this parameter into consideration.
