Application of function Pearson7 for $FW\frac{1}{5}/\frac{4}{5}M$ method

Rozk³ad wielko¶ci ziaren z dopasowania Pearson7

Presented $FW\frac{1}{5}/\frac{4}{5}M$ method of GSD determination is only as precise as measurements of both widths $FW\frac{1}{5}M$ and $FW\frac{4}{5}M$. This precision can be raised by fitting an analytical curve to the experimental data being evaluated, then measurement of widths of the analytical (instead of experimental) curve. One of possible choices could be popular function Pearson7:

$$P7(q,a_{0},a_{1},a_{2},a_{3})=\frac{a_{0}}{\left[1+4\left(\f... ...}{a_{2}}\right)^{2}\left(2^{1/a_{3}}-1\right)\right]^{a_{3}}},$$ (4)

where a₀ is line intensity, a₁ - line position, a₂ i a₃ are line widths. Putting a₀=1 and a₁=0 and comparing expression (4) to $h=\frac{1}{5}$ and $h=\frac{4}{5}$ we obtain equation for the width of Pearson7 curve at $\frac{1}{5}$ and $\frac{4}{5}$ of maximum:

$$\frac{1}{\left[1+4\left(\frac{\Delta q}{a_{2}}\right)^{2}\left(2^{1/a_{3}}-1\right)\right]^{a_{3}}}=h$$ (5)

Interesting solutions of above equation are:

$$FW\frac{1}{5}M(a_{2},a_{3}) 2\Delta q=2\frac{a_{2}\sqrt{-1+5^{1/a_{3}}}}{\sqrt{-4+2^{2+1/a_{3}}}}$$ = (6)

$$FW\frac{4}{5}M(a_{2},a_{3}) 2\Delta q=2\frac{a_{2}\sqrt{-1+\left(\frac{5}{4}\right)^{1/a_{3}}}}{\sqrt{-4+2^{2+1/a_{3}}}}$$ = (7)

Above expressions are functions of parameters a₂ and a₃, being immediate result of fitting in a crystallographic software (e.g. PeakFit). These values $FW\frac{1}{5}M(a_{2},a_{3})$ and $FW\frac{4}{5}M(a_{2},a_{3})$ can be placed in equations (3) and we obtain a recipe how to transform Pearson7 widths to the physical quantities of <R> and $\sigma$, defining Grain Size Distribution:

$$\begin{eqnarray*} A & = & arcctg\left[277069-105723\frac{FW\frac{1}{5}M(a_{2},a_... ...eft[0.002237-2101\cdot A\right]\\ C & = & -0.6515-463695\cdot A \end{eqnarray*}$$

$$\frac{2BC}{FW\frac{4}{5}M(a_{2},a_{3})}$$ <R> = (8)

$$\sigma \frac{2B\sqrt{C}}{FW\frac{4}{5}M(a_{2},a_{3})}$$ =