In analyzing residual stress by $x-$ray diffraction, the stress constant, $K1,$ is used, where

$K_1=\frac{Ecot}{2(1+v)si{n}^2}$

and where $E$ and $v$ are the elastic constants defined in this chapter, $$ is a Bragg angle, and $$ is an angle of rotation of the sample during the $x-$ray diffraction experiment $($generally $c=45).$ To maximize experimental accuracy, one prefers to use the largest possible Bragg angle, $.$ However, hardware configuration $($Figure $3\mathrm{.}34)$ prevents $u$ from being greater than $80.($a$)$ Calculate the maximum $$ for a $1040$ carbon steel using CrKa radiation $)=(0\mathrm{.}2291nm.($Note that $1040$ steel is nearly pure iron, which is a bec metal, and that the reflection rules for a bec metal are given in Table $3\mathrm{.}4.)($b$)$ Calculate the value of the stress constant for $1040$ steel.

TABLE $3\mathrm{.}4$

$\backslash$table$[[$Crystalstrueture$,$Diffraction does not oceur when,Diffraction occurs when$],[$Bedyeentered entic $($her$),,$II $+k+l=$ even number$],[$Fact$-$contierd cultic $($fce$),\backslash$table$[[,\backslash$table$[[$or all are ond numbers$)]],],[$Hexagonalclore picked $($hap$),($i$)+2k$ hn$,($oudd intanan integeri$),$Allothereases:$]],]]$