(a) For BCC iron, calculate the diameter of the minimum space available in an octahedral site at the center of the (010) plane,

Question

29 days ago

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(a) For BCC iron, calculate the diameter of the minimum space available in an octahedral site at the center of the (010) plane,

and compare this to the diameter of a carbon atom. Assume that the iron atoms in the BCC structure are hard spheres that touch along the [111] direction.
(b) Compare the space available with the size of a carbon atom from Appendix C, and comment about the potential solubility of carbon in BCC iron in octahedral interstitial sites.

Engineering

1 answer:

choli [191] · Answer 1 · 2026-02-25T16:13:55+03:00

Response:

a) The diameter available is 0.0384 nm

b)This space is less than the size of a carbon atom, which has a radius of 0.077 nm, indicating that the carbon atom won't occupy these sites.

Clarification:

For BCC iron

According to the information in Appendix B, the lattice parameter (a) is determined to be 0.2866 nm

BCC iron encompasses 4 atomic radii, thus the body diagonal length = $a(3)^\frac{1}{2}$

which represents the atomic radius of BCC iron

4r = $a(3)^\frac{1}{2}$

Substituting the value of (a) from Appendix B, set as 0.2866 nm

4r = $0.2866 nm (3)^\frac{1}{2}$

leading to r = 0.4964 nm / 4 = 0.1241 nm

Refer back to Appendix C, where the atomic radius of BCC iron is stated as 0.1241 nm, assuming the atomic sizes for iron remain consistent.

Thus, the radius ratio = 0.62

According to Figure 3.2, the space necessary for an interstitial at the BCC site exists between atoms located at the FCC site, containing two atoms, each equal to a radius of 0.2482 nm

The diameter of the minimum space available

$d_{a} = a - r_{a}$

$r_{a} = atomic radii$ = 0.2482 nm

With a = 0.2666 nm

therefore

d $_{a}$ = 0.2866 nm - 0.2482 nm = 0.0384 nm

When comparing with the diameter of a carbon atom

This space is smaller than that of a carbon atom which has a radius of 0.077 nm, confirming that the carbon atom will not be able to occupy these positions.