Question

The 2 D unit cell of an element is shown. The two layers are placed one over the other and touching each other. Find the effective number of atoms in the unit cell:

• A. 1
• B. 2
• C. 3
• D. 6

Answer 1

Answer: (C)

3

Answer 2

The problem asks to find the effective number of atoms in a unit cell. The image shows a 2D arrangement of atoms: a central atom surrounded by six atoms in a hexagonal pattern. The question states that this is a "2D unit cell" and that "two layers are placed one over the other and touching each other".

Let's first determine the effective number of atoms in the 2D unit cell as shown in the figure.
The figure depicts a conventional hexagonal unit cell in 2D.

1. Central atom: There is one atom at the center of the hexagon. This atom is entirely within the unit cell. So, its contribution is 1.
2. Corner atoms: There are six atoms at the corners of the hexagon. In a 2D hexagonal lattice, each corner atom is shared by three adjacent hexagonal unit cells. Therefore, the contribution from each corner atom to this unit cell is $\frac{1}{3}$.
   Total contribution from corner atoms = $6 \times \frac{1}{3} = 2$.

The total effective number of atoms in this 2D hexagonal unit cell is $1 \text{ (center)} + 2 \text{ (corners)} = 3$.

Now, let's consider the phrase "The two layers are placed one over the other and touching each other." This implies the formation of a 3D structure by stacking these 2D hexagonal layers. The question then asks for the "effective number of atoms in the unit cell". This phrasing is slightly ambiguous. It could refer to the original 2D unit cell or a 3D unit cell formed by stacking.

However, if we are forming a 3D structure by stacking these 2D hexagonal layers, there are two common possibilities:

1. Simple Hexagonal (AA stacking): The layers are stacked directly on top of each other (atoms in the second layer are directly above atoms in the first layer). In this case, the 3D unit cell would be a simple hexagonal unit cell.
   For a simple hexagonal unit cell (P6/mmm space group), the effective number of atoms is calculated as:

   • 12 atoms at the corners of the hexagonal prisms, each shared by 6 unit cells: $12 \times \frac{1}{6} = 2$ atoms.
   • 2 atoms at the center of the top and bottom hexagonal faces, each shared by 2 unit cells: $2 \times \frac{1}{2} = 1$ atom.
     Total effective atoms in a simple hexagonal unit cell = $2 + 1 = 3$ atoms.

2. Hexagonal Close-Packed (HCP, ABAB stacking): The second layer (B) atoms sit in the depressions of the first layer (A). The third layer (A) is directly above the first.
   For an HCP unit cell, the effective number of atoms is:

   • 12 atoms at the corners: $12 \times \frac{1}{6} = 2$ atoms.
   • 2 atoms at the center of the top and bottom hexagonal faces: $2 \times \frac{1}{2} = 1$ atom.
   • 3 atoms in the body of the unit cell (between the two layers), fully inside: 3 atoms.
     Total effective atoms in an HCP unit cell = $2 + 1 + 3 = 6$ atoms.

Given that the question states "The 2 D unit cell of an element is shown" and then asks for "the effective number of atoms in the unit cell", it most likely refers to the 2D unit cell explicitly shown, or a 3D unit cell derived from it with simple stacking (AA type). If it were HCP, the image would typically show the 'A' and 'B' positions. Without further information on the stacking type, and given the explicit mention of "2D unit cell", the most direct interpretation is to calculate for the 2D unit cell shown.

If the question intends to imply a 3D structure, and since the 2D unit cell shown is a conventional hexagonal unit cell, the simple hexagonal (AA) stacking would be the most direct extension. In this case, the effective number of atoms in the 3D unit cell would also be 3. If it were HCP, the 2D unit cell shown would not be the direct repeating unit that forms the HCP structure in a simple stack.

Considering the common context of such problems, if a 2D unit cell is shown and then layers are mentioned, it often leads to a 3D structure. The most common 3D structure derived from a 2D hexagonal layer is HCP. However, the calculation for the 2D unit cell shown yields 3, and the calculation for a simple hexagonal 3D unit cell also yields 3. The HCP unit cell has 6 atoms.

Let's stick to the interpretation that the "2D unit cell" refers to the structure shown, and the question asks for the effective number of atoms within that unit cell. The mention of layers might be a distractor or context for a different question.

Effective atoms in the 2D hexagonal unit cell shown:

• Central atom: 1
• Corner atoms: $6 \times \frac{1}{3} = 2$
  Total = $1 + 2 = 3$.