What is the linear density of a body centered diagram (BCC) unit cell in the [0,-1,0] direction

To find the **linear density (LD)** of a body-centered cubic (BCC) unit cell in the \([0, -1, 0]\) direction, we need to follow these steps:

### 1. **Linear Density Formula:**
The linear density is defined as the number of atoms per unit length along a specific crystallographic direction, and is given by the formula:

\[
LD = \frac{\text{Number of atoms centered on the direction vector}}{\text{Length of the direction vector}}
\]

### 2. **BCC Structure:**
In a BCC structure:
- Atoms are located at each corner of the cube, and there is one atom in the center of the cube.
- Each corner atom is shared by 8 neighboring unit cells, so each corner contributes \(\frac{1}{8}\) of an atom to a single unit cell.
- The center atom belongs fully to the unit cell.

### 3. **Direction [0, -1, 0]:**
The [0, -1, 0] direction is equivalent to the [0, 1, 0] direction in terms of linear density, as only the magnitude of the y-component is considered, not the sign. This direction lies along one of the cube edges (along the y-axis), passing through two corner atoms.

### 4. **Length of the Direction Vector:**
The [0, 1, 0] direction runs from one corner of the unit cell to the opposite corner along the y-axis. The length of this vector is simply the edge length of the BCC unit cell, denoted by **a**.

\[
\text{Length of direction vector} = a
\]

### 5. **Atoms Along the [0, 1, 0] Direction:**
- Each corner atom contributes \(\frac{1}{8}\) of an atom.
- There are two corner atoms along this direction (one at each end), so the total contribution of atoms is:

\[
\text{Number of atoms} = 2 \times \frac{1}{8} = \frac{1}{4}
\]

### 6. **Linear Density Calculation:**
Now, we can plug the values into the linear density formula:

\[
LD = \frac{\frac{1}{4}}{a} = \frac{1}{4a}
\]

### Final Answer:
The linear density of the BCC unit cell in the \([0, -1, 0]\) direction is:

\[
LD = \frac{1}{4a}
\]

Would you like any further clarification or details?

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Here are 5 related questions:
1. What is the atomic packing factor (APF) for a BCC structure?
2. How do you calculate the planar density in a BCC unit cell?
3. What is the linear density for a BCC unit cell in the [1, 1, 1] direction?
4. How does linear density differ between BCC and FCC structures?
5. What is the coordination number of a BCC unit cell?

**Tip:** In BCC, atoms at the corners do not directly touch each other, but the central atom touches the corner atoms.

What is the linear density of a body centered cubic (BCC) unit cell in the [0,-1,0] direction?

Explore the linear density of a body-centered cubic (BCC) unit cell in the [0,-1,0] direction with detailed steps, including the linear density formula and key concepts from crystallography. Suitable for undergraduate students studying materials science or solid-state physics.

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