X-ray diffraction measurements on a newly discovered crystalline form of La revealed that La crystallizes in a body-centered cubic lattice in which the unit cell edge length is 426.3 picometers. Calculate the atomic radius of the La atoms in this crystal, based on the assumption that the atoms are tightly packed in the unit cell.Hint: Find the length of the cell body diagonal using the Pythagorean theorem.

Question

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The length of the cell body diagonal is:$$d = \sqrt{a^2 + a^2 + a^2} = \sqrt{3}a$$where a is the unit cell edge length.The atomic radius is half the length of the body diagonal:$$r = \frac{d}{2} = \frac{\sqrt{3}a}{2}$$Substituting the given value of a, we get:$$r = \frac{\sqrt{3} 	imes 426.3 	ext{ pm}}{2} = 184.6 	ext{ pm}$$

X-Ray Diffraction Measurements on a Newly Discovered Crystalline Form of La