Fe (iron) has a BCC structure: The atomic radius for Fe atoms is 0.126 nm. What is the edge length (in the unit of nm) of the BCC cubic in Fe? Please Include 2 decimal places in your answer
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Response 1
Reasoned about BCC structure in iron for 7 seconds
An edge length of approximately 0.29 nm is obtained for the BCC cubic unit cell of iron.
In a body-centered cubic (BCC) crystal structure, atoms are located at the eight corners of the cube and one atom is positioned at the center. In this arrangement, atoms touch along the body diagonal, not along the edge. The geometric relationship in a cube shows that the length of the body diagonal is given by the product of the edge length of the cube (a) and the square root of three, expressed as a√3. Since the atoms along this body diagonal are in contact, the distance covered along the diagonal equals four times the atomic radius (4r), where the factor of four originates from the contribution of half the diameter of a corner atom, the full diameter of the center atom, and another half from the opposite corner atom. Equating these two expressions yields the equation: a√3 = 4r.
Rearranging the equation to solve for the edge length results in a = (4r)/√3. By substituting the provided atomic radius of iron, r = 0.126 nm, the computation proceeds as follows: a = (4 × 0.126 nm)/√3, which simplifies to a = 0.504 nm/1.732. The calculated value then evaluates to approximately 0.291 nm. Rounding this result to two decimal places gives an edge length of 0.29 nm.
This analysis illustrates how the spatial arrangement of atoms in a crystal lattice determines the dimensions of the unit cell. The geometric derivation not only confirms the importance of atomic radius in dictating cell parameters but also exemplifies how simple trigonometric relationships are harnessed to understand the structural properties of crystalline materials. The computation emphasizes the intrinsic connection between atomic geometry and macroscopic lattice dimensions, which is fundamental in materials science and solid-state physics. The derived unit cell parameter is critical for predicting various material properties such as density, packing efficiency, and mechanical behavior, underlying the importance of precise geometric determination in crystallography.
