Dc = 0.89λ /(B cos θ)(λ is the wavelength of X-ray, b is the full width at half maximum of diffraction peak, θ is the diffraction angle) Micro-stress in metal crystals is measured by Williams-Hall method.
The grain size is less than 0. 1μm, and the diffraction linewidth becomes wider when there is uneven strain. Quantitative calculation can be made by Sheryl equation or Hall method.
Reasons for broadening of 1 diffraction linewidth
The broadening of diffraction peak measured by diffractometer includes instrument broadening and broadening caused by sample itself. The broadening caused by the sample includes the effects of grain size, uneven strain (micro-strain) and stacking fault (causing long tail on the high angle side of diffraction peak). The latter two factors are caused by the incomplete crystal structure of the sample.
2 Sheryl equation
If it is assumed that there is no broadening caused by imperfect crystal structure in the sample, and the broadening of diffraction line is only caused by the size of crystal block, and the size of crystal block is uniform, Scheler equation can be obtained:
In the formula, Size represents the crystal block size (nm), k is a constant, generally K= 1, λ is the wavelength of X-ray (nm), FW(S) is the sample broadening (Rad), and θ is the diffraction angle (Rad).
When calculating the size of the crystal block, low-angle diffraction lines are generally used. If the size of the crystal block is large, it can be replaced by diffraction lines with higher diffraction angles. The applicable range of this formula is 1- 100nm.
3 micro-strain induced linear broadening
If there is micro-stress, the broadening of diffraction peak is expressed as:
In which, strain represents micro-strain, that is, the ratio of strain to surface spacing, expressed in percentage.
Hall method
FW(S) is measured at the full width at half maximum of more than two diffraction peaks. Because the size of the crystal block is related to the crystal plane index, the diffraction planes in the same direction should be selected, such as (11) and (222), or (200) and (400). Using the abscissa as the graph, the straight line is fitted by least square method. The slope of the straight line is twice the micro-strain, and the intercept of the straight line on the ordinate is the reciprocal of the size of the crystal block.
5 Full width at half maximum, sample widening and instrument widening
The diffraction peak broadening of the sample can be expressed by the full width at half maximum, and the full width at half maximum at FWHM of the sample is the convolution of the instrument broadening FW(I) and the sample properties (grain size refinement and micro-stress) broadening FW(S).
In order to obtain sample broadening FW(S), it is necessary to establish the relationship between instrument broadening FW(I) and diffraction angle θ, which is also called FWHM curve.
This curve can be obtained by measuring the diffraction spectrum of standard samples. The standard sample should be in the same crystalline state as the sample to be tested. Standard samples must be samples without stress and grain refinement, and the grain size is above 25μm, such as NISTA60Si and LaB6.