X$-$ray diffraction is a technique used to determine the structure of crystals. X rays are

light waves that have a wavelength of only a few nanometers or less. For wavelengths

this short, the parallel planes of atoms in a crystal, spaced less than a nanometer

apart, act very much like the slits of a diffraction grating. If x rays are sent through a

crystal, a careful analysis finds that the conditions for bright fringes of x$-$ray diffraction

is $2$d sin$($m$/2)=$ m$,$ where d is the spacing between the planes of the crystal$$called

the lattice spacing. This agrees with the diffraction$-$grating equation at small angles,

where sin$($m$/2)$ m$/2,$ but differs at slightly larger angles.

$($a$)$ If x rays with a wavelength of $0\mathrm{.}190$ nm are sent through a crystal of tungsten,

the first$-$order diffraction is observed at $34\mathrm{.}5.$ What is the lattice spacing of

tungsten?

$($b$)$ At what angle is second$-$order diffraction observed?