A narrow beam of `X`-rays impinges on the natural facet of a `NaCI` single srystal, whose density is `hro = 2.16 g//cm^(3)` at a galancing angle `alpha = 60.0^(@)`. The mirror reflection from this facet produces a maximum of seond order. Find the wavelength of radiation.

Explanation

The unit cell of `NaCl` is shown below. In an infinite crystal, there are four `Na^(+)` and four `Cl^(-)` inos per unit cell. (Each ion on the middle of the edge is shered by four unit cells, each ion on the face centre by two unit cells, the ion in the middle of the cell by one cell only and finally each ion on the corner by eigher unit cells.) Thus
`4(M)/(N_(A)) = rho.a^(3)`
where `M =` molecular weight of `NaCl` in
gms `= 58.5 gms`
`N_(A) =` Avogardo number `- 6.023 xx 10^(23)`
Thus `(1)/(2)a = sqrt((M)/(1N_(A) rho)) = 2.822 Å`
The natural facet of the crystal is one of the faces of the unit cell. The interplaner disatnce
`a = (1)/(2)a = 2.822 Å`
Thus `2d sin alpha = 2 lambda`
So `lambda = d sin alpha = 2.822Å xx (sqrt(3))/(2) = 244pm`.