# Identifying the Type of Cubic Structure from XRD Pattern Let's solve the problem step by step. --- ## **Step 1: Understanding the Problem** - The XRD pattern for a new material is given. - The structure is a type of **cubic** (could be **simple cubic (SC)**, **body-centered cubic (BCC)**, or **face-centered cubic (FCC)**). - We need to identify **which type** of cubic structure it is. --- ## **Step 2: Bragg's Law and Cubic Diffraction** **Bragg's Law:** \[ n\lambda = 2d\sin\theta \] For cubic crystals: \[ d = \frac{a}{\sqrt{h^2 + k^2 + l^2}} \] So, \[ \sin^2\theta \propto (h^2 + k^2 + l^2) \] --- ## **Step 3: Allowed Reflections for Cubic Structures** - **Simple Cubic (SC):** All (hkl) allowed. - **Body-Centered Cubic (BCC):** Only (hkl) where **h+k+l is even**. - **Face-Centered Cubic (FCC):** Only (hkl) where **h, k, l are all odd or all even**. --- ## **Step 4: Extract Peak Positions from the XRD Pattern** Estimate peak positions (2θ) from the graph (approximate): | Peak | 2θ (°) | |------|--------| | 1 | ~32 | | 2 | ~38 | | 3 | ~55 | | 4 | ~65 | | 5 | ~78 | --- ## **Step 5: Calculate \(\sin^2\theta\) for Each Peak** Convert 2θ to θ (θ = 2θ/2 = θ). | Peak | 2θ (°) | θ (°) | sin θ | sin² θ | |------|--------|--------|-------|--------| | 1 | 32 | 16 | .276 | .076 | | 2 | 38 | 19 | .326 | .106 | | 3 | 55 | 27.5 | .462 | .214 | | 4 | 65 | 32.5 | .538 | .290 | | 5 | 78 | 39 | .629 | .396 | --- ## **Step 6: Find the Ratios of sin²θ** Take the first peak as reference: | Peak | sin² θ | Ratio to 1st Peak | |------|--------|-------------------| | 1 | .076 | 1.00 | | 2 | .106 | 1.39 | | 3 | .214 | 2.82 | | 4 | .290 | 3.82 | | 5 | .396 | 5.21 | --- ## **Step 7: Match Ratios to Allowed Reflections** ### **Theoretical Ratios for Cubic Structures:** #### **Simple Cubic (SC):** Allowed (hkl): (100), (110), (111), (200), (210), ... \[ (h^2+k^2+l^2): 1, 2, 3, 4, 5, ... \] #### **Body-Centered Cubic (BCC):** Allowed (hkl): (110), (200), (211), (220), (310), ... \[ (h^2+k^2+l^2): 2, 4, 6, 8, 10, ... \] Ratios: 2:4:6:8:10 → 1:2:3:4:5 #### **Face-Centered Cubic (FCC):** Allowed (hkl): (111), (200), (220), (311), (222), ... \[ (h^2+k^2+l^2): 3, 4, 8, 11, 12, ... \] Ratios: 3:4:8:11:12 → 1:1.33:2.67:3.67:4 ### **Compare with Experimental Ratios:** Experimental: 1.00, 1.39, 2.82, 3.82, 5.21 Closest theoretical: **FCC**: 1, 1.33, 2.67, 3.67, 4 --- ## **Step 8: Conclusion** **The experimental ratios closely match the FCC pattern.** --- ## **Step 9: Summary Table** | Peak | 2θ (°) | θ (°) | sin θ | sin² θ | Ratio | FCC (hkl) | (h²+k²+l²) | |------|--------|-------|-------|--------|-------|-----------|------------| | 1 | 32 | 16 | .276 | .076 | 1.00 | (111) | 3 | | 2 | 38 | 19 | .326 | .106 | 1.39 | (200) | 4 | | 3 | 55 | 27.5 | .462 | .214 | 2.82 | (220) | 8 | | 4 | 65 | 32.5 | .538 | .290 | 3.82 | (311) | 11 | | 5 | 78 | 39 | .629 | .396 | 5.21 | (222) | 12 | --- ## **Final Answer** ### **The material has a Face-Centered Cubic (FCC) structure.** If you need to see how to calculate the lattice parameter or further details, let me know!