3. A paramagnetic material has FCC structure with a cubic edge of 2.5 A°. If the saturation value of magnetization is 1.8 × 10º A m¯¹, Calculate the magnetization contributed per atom in Bohr magnetrons.

To solve this problem, we need to calculate the magnetization contributed per atom in terms of Bohr magnetons (\(\mu_B\)).

### Given:
- Cubic edge length of the unit cell: \( a = 2.5 \, \text{Å} = 2.5 \times 10^{-10} \, \text{m} \)
- Saturation magnetization: \( M_s = 1.8 \times 10^6 \, \text{A} \, \text{m}^{-1} \)
- The material has an FCC structure.

### Step 1: Calculate the number of atoms per unit cell.
For an FCC (Face-Centered Cubic) structure:
- The number of atoms per unit cell is 4.

### Step 2: Calculate the volume of the unit cell.
\[
V_{\text{cell}} = a^3 = \left(2.5 \times 10^{-10} \, \text{m}\right)^3 = 1.5625 \times 10^{-29} \, \text{m}^3
\]

### Step 3: Calculate the total magnetization per unit cell.
The total magnetization (\(M\)) of the unit cell can be calculated by multiplying the saturation magnetization by the volume of the unit cell:
\[
M_{\text{cell}} = M_s \times V_{\text{cell}} = \left(1.8 \times 10^6 \, \text{A} \, \text{m}^{-1}\right) \times \left(1.5625 \times 10^{-29} \, \text{m}^3\right) = 2.8125 \times 10^{-23} \, \text{A} \, \text{m}^2
\]

### Step 4: Calculate the magnetization per atom.
Since there are 4 atoms in the unit cell:
\[
M_{\text{atom}} = \frac{M_{\text{cell}}}{\text{Number of atoms per unit cell}} = \frac{2.8125 \times 10^{-23} \, \text{A} \, \text{m}^2}{4} = 7.03125 \times 10^{-24} \, \text{A} \, \text{m}^2
\]

### Step 5: Convert the magnetization per atom to Bohr magnetons.
The magnetization per atom can be expressed in Bohr magnetons (\(\mu_B\)), where:
\[
1 \, \mu_B = 9.274 \times 10^{-24} \, \text{A} \, \text{m}^2
\]
\[
\text{Magnetization per atom in } \mu_B = \frac{M_{\text{atom}}}{\mu_B} = \frac{7.03125 \times 10^{-24} \, \text{A} \, \text{m}^2}{9.274 \times 10^{-24} \, \text{A} \, \text{m}^2} \approx 0.758 \, \mu_B
\]

### Final Answer:
The magnetization contributed per atom in Bohr magnetons is approximately **0.758 \(\mu_B\)**.

Would you like more details or have any questions?

Here are five related questions to explore further:
1. How does the magnetization per atom change if the cubic edge length is doubled?
2. What is the significance of the Bohr magneton in quantum physics?
3. How does the FCC structure influence the magnetic properties of materials?
4. What other crystal structures exhibit paramagnetic behavior?
5. How does temperature affect the saturation magnetization of paramagnetic materials?

**Tip:** Remember, in crystal structures like FCC, the arrangement and number of atoms per unit cell directly impact material properties like magnetization.

Learn how to calculate the magnetization per atom in a face-centered cubic (FCC) structure with a cubic edge of 2.5 Å and a saturation magnetization of 1.8 × 10^6 A/m. Understand the formula for magnetization per atom and its conversion to Bohr magnetons. Suitable for advanced undergraduate students and researchers.

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