In their study of X-ray diffraction, William and Lawrence Bragg determined
that the relationship among the wavelength of the radiation \((\lambda)\), the
angle at which the radiation is diffracted \((\theta)\), and the distance
between planes of atoms in the crystal that cause the diffraction \((d)\) is
given by \(n \lambda=2 d \sin \theta\). \(\mathrm{X}\) rays from a copper
\(\mathrm{X}\)-ray tube that have a wavelength of \(1.54 \AA\) are diffracted at
an angle of \(14.22\) degrees by crystalline silicon. Using the Bragg equation,
calculate the distance between the planes of atoms responsible for diffraction
in this crystal, assuming \(n=1\) (first-order diffraction).