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Body centred cubic and face centred cubi...

Body centred cubic and face centred cubic unit cells have `n_1 and n_2` effective number of atms. Which one the following `(n_1 , n_2)` combination is correct ?

A

(4,1)

B

(4,2)

C

(1,4)

D

(2,4)

Text Solution

AI Generated Solution

The correct Answer is:
To determine the effective number of atoms in body-centered cubic (BCC) and face-centered cubic (FCC) unit cells, we can follow these steps: ### Step 1: Calculate the effective number of atoms in a Body-Centered Cubic (BCC) unit cell. 1. **Identify the positions of atoms**: In a BCC unit cell, there are atoms at the corners and one atom at the center of the cube. 2. **Contribution from corner atoms**: There are 8 corner atoms, and each corner atom is shared by 8 unit cells. Therefore, the contribution from the corner atoms is: \[ \text{Contribution from corners} = 8 \times \frac{1}{8} = 1 \] 3. **Contribution from the body center atom**: The atom at the body center is not shared with any other unit cell, so its contribution is: \[ \text{Contribution from body center} = 1 \] 4. **Total effective number of atoms in BCC**: \[ n_1 = 1 + 1 = 2 \] ### Step 2: Calculate the effective number of atoms in a Face-Centered Cubic (FCC) unit cell. 1. **Identify the positions of atoms**: In an FCC unit cell, there are atoms at the corners and at the centers of each face of the cube. 2. **Contribution from corner atoms**: Similar to BCC, there are 8 corner atoms, each contributing: \[ \text{Contribution from corners} = 8 \times \frac{1}{8} = 1 \] 3. **Contribution from face center atoms**: There are 6 face-centered atoms, and each face atom is shared by 2 unit cells. Therefore, the contribution from the face-centered atoms is: \[ \text{Contribution from face centers} = 6 \times \frac{1}{2} = 3 \] 4. **Total effective number of atoms in FCC**: \[ n_2 = 1 + 3 = 4 \] ### Conclusion The effective number of atoms in the BCC unit cell is \( n_1 = 2 \) and in the FCC unit cell is \( n_2 = 4 \). Therefore, the correct combination of \( (n_1, n_2) \) is \( (2, 4) \). ### Final Answer The correct combination is \( (n_1, n_2) = (2, 4) \). ---
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The regular three dimensional arrangement of points in a crystal is known as crystal lattice and the smallest repeating pattern in the lattice is called unit cell. The unit cells are characterised by the edge lengths a, b, c and the angles between them alpha, beta and gamma respectively. Based on this, there are seven crystal systems. In a cubic unit cell: a=b=c and alpha = beta=gamma=90^(@) The number of points in simple, body centred and face centred cubic cells are 1,2 and 4 respectively In both the hcp and ccp of spheres, the number of tetrahedral voids per sphere is two while the octahedral voids is one. In a face centred cubic cell, an atom at the face contributes to the unit cell