Let's break this problem into **two steps**, as requested. We'll explain each part at the end of each step and provide the **final answer** at the end. --- ## **Step 1: Structure Factors and Reflection Conditions** ### **a) Structure-Factor Equations** The **structure factor** \( F_{hkl} \) for a unit cell is: \[ F_{hkl} = \sum_{j} f_j \exp\left[2\pi i (hx_j + ky_j + lz_j)\right] \] where \( f_j \) is the atomic form factor, and \( (x_j, y_j, z_j) \) are the fractional coordinates of the \( j \)-th atom. #### **Cell A: Base-centered Tetragonal** - Atoms at (,,) and (\(\frac{1}{2}, \frac{1}{2}, \)) \[ F_{hkl} = f \left[1 + \exp\left(2\pi i \left(\frac{h}{2} + \frac{k}{2}\right)\right)\right] \] \[ = f \left[1 + (-1)^{h+k}\right] \] - **Allowed reflections:** Only when \( h+k \) is even (\( F_{hkl} \neq \)), otherwise \( F_{hkl} = \). #### **Cell B: Simple Tetragonal** - Atom at (,,) \[ F_{hkl} = f \] - **All reflections allowed** (no systematic absences). --- ### **b) d-Spacings and Bragg Angles (2θ)** The general **d-spacing** for tetragonal cells: \[ \frac{1}{d_{hkl}^2} = \frac{h^2 + k^2}{a^2} + \frac{l^2}{c^2} \] For **Cu Kα** radiation, \( \lambda = 1.5406 \) Å. **Bragg's Law:** \( 2d \sin\theta = \lambda \) Thus, \( 2\theta = 2 \arcsin\left(\frac{\lambda}{2d}\right) \) #### **First 4 lines (lowest d, i.e., lowest \( h^2+k^2+l^2 \)):** Let’s enumerate for **Cell B** (all allowed): 1. (1 ): \( d_{100} = a = 2\,\text{Å} \) 2. ( 1): \( d_{001} = c = 3\,\text{Å} \) 3. (1 1 ): \( d_{110} = a/\sqrt{2} = 1.414\,\text{Å} \) 4. (1 1): \( \frac{1}{d^2} = \frac{1}{4} + \frac{1}{9} = \frac{13}{36} \implies d_{101} = 1.67\,\text{Å} \) For **Cell A**, only those where \( h+k \) is even: - (1 ): \( h+k = 1 \) (absent) - ( 1): \( h+k = \) (present) - (1 1 ): \( h+k = 2 \) (present) - (1 1): \( h+k = 1 \) (absent) - (2 ): \( h+k = 2 \) (present) - (1 1 1): \( h+k = 2 \) (present) So, the first four allowed for **Cell A** are: ( 1), (1 1 ), (2 ), (1 1 1) --- ### **End of Step 1:** - **Cell A**: Only reflections with even \( h+k \) (base-centered condition) - **Cell B**: All reflections allowed --- ## **Step 2: 2θ Calculations and Index Assignment** ### **a) Calculate 2θ for First Four Lines** #### **Cell B Allowed Planes & d-spacings:** | (hkl) | \( d_{hkl} \) (Å) | \( 2\theta \) (degrees) | |-------|-------------------|--------------------------| | ( 1) | 3.00 | \( 2\arcsin(1.5406/6.00) = 14.8^\circ \) | | (1 ) | 2.00 | \( 2\arcsin(1.5406/4.00) = 22.5^\circ \) | | (1 1 ) | 1.414 | \( 2\arcsin(1.5406/2.828) = 32.3^\circ \) | | (1 1) | 1.67 | \( 2\arcsin(1.5406/3.34) = 27.1^\circ \) | #### **Cell A Allowed Planes & d-spacings:** | (hkl) | \( d_{hkl} \) (Å) | \( 2\theta \) (degrees) | B-plane equivalent | |-------|-------------------|--------------------------|---------------------| | ( 1) | 3.00 | 14.8 | ( 1) | | (1 1 ) | 1.414 | 32.3 | (1 1 ) | | (2 ) | 1.00 | \( 2\arcsin(1.5406/2) = 49.9^\circ \) | (2 ) | | (1 1 1) | 1.154 | \( 2\arcsin(1.5406/2.308) = 40.^\circ \) | (1 1 1) | --- ### **b) Labeling and Index Matching** - The observed 2θ angles will be **identical** for both cells for planes representing the same atomic planes. - The **Miller indices** differ by the choice of cell, but **physical planes** (and therefore d-spacings and 2θ values) are the same. --- ### **c) Drawing the Cells** - **Cell A** is a base-centered tetragonal: has extra lattice points at (\(\frac{1}{2}, \frac{1}{2}, \)) - **Cell B** is a simple tetragonal: only at (,,) - Both cells describe the **same lattice**, just with different choices of unit cell. --- ## **Final Answer** ### **Summary Table** | 2θ (degrees) | Cell B (hkl) | Cell A (hkl) | Comments | |--------------|--------------|--------------|----------| | 14.8 | ( 1) | ( 1) | Same plane | | 22.5 | (1 ) | absent | extinct in A | | 27.1 | (1 1) | absent | extinct in A | | 32.3 | (1 1 ) | (1 1 ) | Same plane | | 40. | (1 1 1) | (1 1 1) | Same plane | | 49.9 | (2 ) | (2 ) | Same plane | ### **Key Points** - **Structure factor** for Cell A gives systematic absences for \( h+k = \text{odd} \). - **Observed lines (2θ values)** are the same for both cells for the same planes, though Miller indices differ. - **The diffraction pattern is independent of the cell choice**—it reflects the real atomic planes, not the cell description. --- ### **Explanations at Each Step** **After Step 1:** We determined the structure factors and systematic absences for both cells. This tells us which reflections are allowed for each unit cell type. **After Step 2:** We calculated d-spacings and Bragg angles for the first few reflections, showing that the same physical planes are observed, but indexed differently depending on the cell. The observed diffraction pattern is the same for both cells. --- **Final Answer:** The structure-factor equations and selection rules show that only certain reflections appear for each cell. The 2θ values for observed lines are the same for both cells, but the Miller indices differ. This demonstrates that the **diffraction pattern is independent of the unit cell choice**, even though the indices change. Drawing both cells shows how the same plane appears with different indices.