Let's tackle each part step by step as requested. --- ### (a) **Bravais Lattice, Primitive Lattice Vectors, Unit Cell, Basis** **Step 1: Identifying Bravais Lattice and Primitive Lattice Vectors** - The CuO₂ layers are described as a square grid of copper atoms with oxygen atoms in between, stacked in 3D with spacing \(c\). - The simplest Bravais lattice that fits this is a **simple tetragonal lattice**: - **Lattice vectors:** \[ \vec{a}_1 = a\hat{x}, \quad \vec{a}_2 = a\hat{y}, \quad \vec{a}_3 = c\hat{z} \] - Here, **a** is the Cu–Cu spacing in-plane, and **c** is the interlayer spacing. - **Explanation:** The Cu atoms form a square grid in the plane, repeated in the stacking direction. This is a tetragonal lattice with primitive lattice vectors as above. --- **Step 2: Unit Cell and Basis** - **Unit cell:** The parallelepiped defined by the above primitive vectors (volume = \(a^2c\)). - **Basis:** - For the **CuO₂ layer**, each lattice point (Cu site) has nearby O atoms at positions (a/2, 0, 0) and (0, a/2, 0) relative to the Cu. - So, the **basis** is: - 1 Cu at (0, 0, 0) - 1 O at (a/2, 0, 0) - 1 O at (0, a/2, 0) - **Explanation:** The **unit cell** is the volume spanned by the primitive vectors. The **basis** is the set of atoms attached to each lattice point to complete the crystal structure. --- **(a) Final Answer:** - **Bravais lattice:** Simple tetragonal - **Primitive lattice vectors:** \[ \vec{a}_1 = a\hat{x}, \quad \vec{a}_2 = a\hat{y}, \quad \vec{a}_3 = c\hat{z} \] - **Unit cell:** Parallelepiped of size \(a \times a \times c\) - **Basis:** 1 Cu at (0,0,0), 1 O at (a/2, 0, 0), 1 O at (0, a/2, 0) --- --- ### (b) **Primitive Cell and Reciprocal Lattice for La₂CuO₄ with Alternating Oxygen Atom Displacements** **Step 1: Primitive Cell and Lattice Spacing** - With oxygen atoms alternately displaced out of plane (+ up, – down), the smallest repeating unit **doubles in one in-plane direction** (say, along x or y) because you need two Cu–O–Cu units to return to the same up/down pattern. - **Primitive cell:** - Lattice vectors become: \[ \vec{a}_1 = 2a\hat{x}, \quad \vec{a}_2 = a\hat{y}, \quad \vec{a}_3 = c\hat{z} \] (or doubled along y depending on direction of alternation) - **Lattice spacing:** Now \(2a\) along the direction of alternation (other directions unchanged). - **Explanation:** The alternation in O positions breaks the original square symmetry, **doubling the unit cell** along the direction of alternation. --- **Step 2: Reciprocal Lattice** - **Reciprocal lattice vectors** are given by: \[ \vec{b}_1 = \frac{2\pi}{2a}\hat{x} = \frac{\pi}{a}\hat{x} \\ \vec{b}_2 = \frac{2\pi}{a}\hat{y} \\ \vec{b}_3 = \frac{2\pi}{c}\hat{z} \] - So, the **reciprocal lattice is also tetragonal**, but with the in-plane spacing halved along the direction where the real-space cell doubled. - **Explanation:** Doubling the real-space lattice constant halves the corresponding reciprocal lattice vector, shrinking the Brillouin zone along that direction. --- **(b) Final Answer:** - **Primitive cell:** - Lattice vectors: \(\vec{a}_1 = 2a\hat{x}\), \(\vec{a}_2 = a\hat{y}\), \(\vec{a}_3 = c\hat{z}\) (assuming alternation along x) - Lattice spacing: \(2a\) along alternation, \(a\) perpendicular, \(c\) between layers - **Reciprocal lattice:** - Vectors: \(\vec{b}_1 = \frac{\pi}{a}\hat{x}\), \(\vec{b}_2 = \frac{2\pi}{a}\hat{y}\), \(\vec{b}_3 = \frac{2\pi}{c}\hat{z}\) - The Brillouin zone is halved along \(x\) compared to the undistorted case. --- **Summary Table:** | | Lattice vectors | Lattice spacing | Reciprocal lattice vectors | |---|------------------------|---------------------|------------------------------------| | (a) Flat CuO₂ | \(a, a, c\) | \(a, a, c\) | \(\frac{2\pi}{a}, \frac{2\pi}{a}, \frac{2\pi}{c}\) | | (b) Alternating O | \(2a, a, c\) (say, x) | \(2a, a, c\) | \(\frac{\pi}{a}, \frac{2\pi}{a}, \frac{2\pi}{c}\) | ---