Which of the following is the relation between the atomic radius, r, and the lattice parameter, a, for the BCC crystal?
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A.
r=a
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B.
r=a/2√2
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C.
r=a√3/4
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D.
r=a√2
Correct Answer:
C. r=a√3/4
Explanation:
In a Body-Centered Cubic (BCC) crystal structure, the atoms are in direct contact along the body diagonal of the unit cell. This diagonal passes through the center of the cube, connecting one corner to the opposite corner.
The length of this body diagonal can be expressed in two ways:1. In terms of the atomic radius (r): Since it spans two corner radii and the full diameter of the central atom, the diagonal length is 4r.
2. In terms of the lattice parameter (a): Using the Pythagorean theorem across the cube's dimensions, the body diagonal of a cube with side length a is equal to a√3.By equating these two expressions (4r = a√3) and solving for the radius, we find that r = a√3/4. Therefore, the correct relationship is represented by option C.
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