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.