X-rays of 173 pm wavelength are reflected by the (111) plane of a cubic primitive crystal at θ = 30°. The unit cell length (in pm) is closest to

Concept:

We can use Bragg's law to solve this problem:

nλ = 2d sinθ

where n is the order of the reflection (which is 1 in this case, since it is the first order reflection), λ is the wavelength of the X-ray, d is the distance between the crystal planes, and θ is the angle of incidence of the X-ray.

For a cubic primitive crystal, the distance between adjacent (111) planes is given by:

d = \(\frac{a}{\sqrt{3}}\)

where a is the length of the unit cell.

Explanation:

Substituting this into Bragg's law and solving for a, we get:

a =\(\frac{ d\times\sqrt{3}}{sin\theta }\)

a = \( \frac{\lambda}{2\times sin\theta }\times\sqrt{3}\)

a = \( \frac{173pm}{2\times sin30^{o}} \times\sqrt{3}\)

a ≈ 299.636 pm

Conclusion:
Therefore, the length of the unit cell is closest to 300 pm.