Let's solve the question step by step: --- ### 1(a) **Structure Factor and Diffraction Intensity of BCC** #### **Step 1: Structure Factor Formula** The general formula for the structure factor is: \[ F_{hkl} = \sum_{j} f_j \exp[2\pi i (h x_j + k y_j + l z_j)] \] where: - \( f_j \) = atomic scattering factor for atom \( j \) - \( (x_j, y_j, z_j) \) = fractional coordinates of atom \( j \) in the unit cell - \( (hkl) \) = Miller indices of the plane #### **Step 2: Atomic Positions in BCC** For a body-centered cubic (BCC) lattice: - Corner atoms at \( (0, 0, 0) \) - Body center atom at \( (\frac{1}{2}, \frac{1}{2}, \frac{1}{2}) \) Assume both atoms are the same type, so \( f_1 = f_2 = f \). #### **Step 3: Plug Positions into Structure Factor** \[ F_{hkl} = f \left[ e^{2\pi i (0)} + e^{2\pi i \left(h \frac{1}{2} + k \frac{1}{2} + l \frac{1}{2}\right)} \right] \] \[ = f \left[1 + e^{\pi i (h + k + l)} \right] \] Recall: - If \(h + k + l\) is even: \(e^{\pi i (h + k + l)} = 1\) - If \(h + k + l\) is odd: \(e^{\pi i (h + k + l)} = -1\) So: \[ F_{hkl} = \begin{cases} 2f & \text{if } (h + k + l) \text{ is even} \\ 0 & \text{if } (h + k + l) \text{ is odd} \end{cases} \] #### **Step 4: Diffraction Intensity** Intensity is proportional to the square of the structure factor: \[ I_{hkl} \propto |F_{hkl}|^2 \] So, \[ I_{hkl} = \begin{cases} 4f^2 & \text{if } (h + k + l) \text{ is even} \\ 0 & \text{if } (h + k + l) \text{ is odd} \end{cases} \] --- ### 1(b) **Diffraction Pattern Along Beam Direction [1̅11]** #### **Step 1: Understand the Zone Axis** The beam is along \([1\bar{1}1]\). The reciprocal lattice planes that satisfy the zone law: \[ ha + kb + lc = 0 \] For \([uvw]\) direction: \(hu + kv + lw = 0\) Plugging in for \([1\bar{1}1]\): \[ h(1) + k(-1) + l(1) = 0 \implies h - k + l = 0 \] So, only those (hkl) planes that satisfy \(h - k + l = 0\) will appear in the pattern. #### **Step 2: List Some Allowed (hkl) (Also Must Satisfy BCC Selection Rule)** Also, remember for BCC, \(h + k + l\) must be even. Let’s find some low-index points: - (0,1,1): \(0-1+1=0\), \(0+1+1=2\) ⇒ **Allowed** - (1,0,1): \(1-0+1=2\) ⇒ Not allowed by zone law - (1,1,0): \(1-1+0=0\), \(1+1+0=2\) ⇒ **Allowed** - (2,1,1): \(2-1+1=2\) ⇒ Not allowed by zone law - (2,0,2): \(2-0+2=4\), \(2+0+2=4\) ⇒ **Allowed** - (0,2,2): \(0-2+2=0\), \(0+2+2=4\) ⇒ **Allowed** - (2,2,0): \(2-2+0=0\), \(2+2+0=4\) ⇒ **Allowed** - (1,2,1): \(1-2+1=0\), \(1+2+1=4\) ⇒ **Allowed** #### **Step 3: Sketch the Pattern** - Place the direct beam at the center (labeled as 000). - Draw spots for the allowed reflections found above. - Index the points as (011), (110), (202), (022), (220), (121), etc. **[You would typically draw this as a 2D pattern with the central spot labeled 000, and the other spots symmetrically around it, with their respective indices.]** --- ### **Summary** #### **(a)** - **BCC Structure Factor:** \(F_{hkl} = 2f\) if \(h+k+l\) even, 0 if odd - **Intensity:** \(I_{hkl} = 4f^2\) if \(h+k+l\) even, 0 if odd #### **(b)** - **Diffraction pattern along \([1\bar{1}1]\):** - Only those (hkl) with \(h - k + l = 0\) and \(h+k+l\) even. - Example indices: (011), (110), (202), (022), (220), (121), etc. - Draw the pattern, center at (000), index the points. **If you need a hand-drawn sketch, let me know!**