Question

Calculate the number of atoms per unit cell in each type of cubic unit cell. Enter three integers separated by commas (e.g., 1,2,3) in the following order: primitive cubic, body-centered cubic, and face-centered cubic.

Answer 1

Answer:

1, 2, 4

Explanation:

The primitive cubic unit cell has eight corners, and each corner is shared with 8 cubic unit cells. Therefore since we have one atom at each corner, we will have:

8 corner x  1 atom/8 corner = 1 atom

There is then the equivalent of one atom per unit cell in the primitive cubic cell.

For the body-centered cubic unit cell, we have 8 corners again shared each by 8 lattices as in the simple cubic plus we have one atom  in the center of the cubic lattice. Therefore, the number of atoms in the body -centerd unit cell is two:

8 corner x 1 atom/8 corner = 1 atom

                      +

1 atom in the center

= 2 atoms/ unit cel

For the face-centerd cubic again we have 8 atoms in the corners shared by 8 lattices, plus 1 atom in each of the faces shared by two unit cells:

8 corner x 1/8 atom/corner + 6 faces x 1 atom/face = 4 atoms/unit cell

Answer 2

In primitive cubic = 1 atom per unit cell, in body-centered cubic = 2 atoms per unit cell, and in face-centered cubic = 4 atoms per unit cell.  

• In primitive cubic or simple cubic unit cell, the existence of the atoms takes place only at the unit cell's corners. The total corners are 8, so there is contribution of 8×1/8 = 1 atom per unit cell.  

• In body-centered cubic cell, the existence of atoms takes place at the corners as well as at the center of the unit cell. Therefore, there is total contribution of 8×1/8 = 1 at the corners and 1 at the body center, which in total is 2 atoms per unit cell.  

• In face-centered cubic cell, the existence of atoms takes place at the corners as well as at the six faces. As calculated at corners it will contribute 1/8 while at faces it will contribute 1/2. So, in total it will be, 8×1/8 = 1 at the corners and 6×1/2 = 3 at the faces, which in total will be 4 atoms per unit cell.  

Thus, the number of atoms per unit cell in primitive cubic will be 1 atom per unit cell, in body-centered cubic will be 2 atoms per unit cell, and in face-centered cubic will be 4 atoms per unit cell, that is, (1,2,4).

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