Define atomic radius and find its values for SC, BCC & FCC lattice

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Atomic radius in a crystal lattice is defined as half the distance between the centers of two nearest-neighbor atoms that are in contact. This fundamental concept helps us understand how atoms are arranged in different crystal structures. In a Simple Cubic lattice, atoms are located at the corners of the cube. The nearest-neighbor atoms touch along the edges of the cube. The distance between the centers of two touching atoms is equal to the edge length 'a'. Since this distance equals two atomic radii, we get the relationship r equals a divided by 2. In a Body-Centered Cubic lattice, atoms are located at the corners and one atom is at the center of the cube. The central atom touches the corner atoms along the body diagonal. The body diagonal length is square root of 3 times a. Along this diagonal, we have four atomic radii, so r equals square root of 3 times a divided by 4. In a Face-Centered Cubic lattice, atoms are located at the corners and at the center of each face. Nearest-neighbor atoms touch along the face diagonal of the cube. The face diagonal length is square root of 2 times a. Along this diagonal, we have four atomic radii, so r equals square root of 2 times a divided by 4. To summarize, we have derived the atomic radius formulas for three important crystal structures. For Simple Cubic, r equals a over 2. For Body-Centered Cubic, r equals square root of 3 times a over 4. For Face-Centered Cubic, r equals square root of 2 times a over 4. These formulas are fundamental for understanding crystal structures and calculating atomic dimensions from lattice parameters.