Determine the radius of the largest atom that can be accommodated in the interstices of BCC iron (Fe) without stress. (Hint: the center of the largest site is at ½, ¼, 0; draw a unit cell - it helps.)
3.61*10^-9
To determine the radius of the largest atom that can fit in the interstices of the body-centered cubic (BCC) structure of iron (Fe) without stress, we need to follow a few steps.
Step 1: Understand the BCC structure of iron (Fe)
The BCC structure of iron consists of a cube with Fe atoms located at each of the eight corners and one Fe atom at the center of the cube. The lattice parameter, which represents the length of one side of the cube, is denoted as "a".
Step 2: Identify the interstitial site in BCC iron
In the BCC structure, there are interstitial sites located in the void spaces between the atoms. For BCC iron, the largest interstitial site is positioned at the coordinates (½, ¼, 0) relative to the lattice parameter "a".
Step 3: Calculate the radius of the largest accommodated atom
To determine the radius of the largest atom that can fit in the interstice, we need to calculate the distance between the interstitial site and the nearest neighboring Fe atom. This distance is equal to ¼ of the diagonal length of the cube.
Let's break down the calculations:
The diagonal length of the cube can be found using the Pythagorean theorem:
Diagonal length (d) = sqrt(a² + a² + a²) = sqrt(3a²)
The distance between the interstitial site and the nearest Fe atom can be found by multiplying the diagonal length (d) by a factor of ¼:
Distance = ¼ * sqrt(3a²)
Since the radius of the atom can be defined as half the distance between the interstitial site and the nearest atom, we obtain:
Radius = ⅛ * sqrt(3a²)
Therefore, the radius of the largest atom that can fit in the interstices of BCC iron without stress is given by ⅛ multiplied by the square root of 3 times the lattice parameter "a".