# Nearest Neighbors in solid state, but with basis

In summary, the question is asking for the area of the primitive cell for a given lattice, with the nearest neighbor separation being "a." The solution depends on whether the nearest neighbor refers to any atom in the crystal or just lattice point-related neighbors. If it includes all atoms, the area would be 8a2, but if it only includes lattice point-related neighbors, the area would be a2. The sources consulted did not provide a clear answer for lattices with a basis.

## Homework Statement

What is the area of the primitive cell for the lattice shown below? The nearest neighbor separation is "a."

## Homework Equations

Here's the lattice we were given on our handout, and I have added the lines to indicate the square lattice (in red), the basis (in purple), and lattice points (circled in blue). Another potential basis is in green.

## The Attempt at a Solution

Is the "nearest neighbor" any atom in the crystal, or is it only referring to lattice point-related neighbors?

If any atom (not necessarily lattice points) counts as a nearest neighbor, then the primitive cell has area 8a2.

If it's only between lattice points, then the area would be a2.

Any text I refer to (I've looked at 4) only talk about nearest neighbors in terms of lattices without a basis. (For example, the SC lattice has 6 nearest neighbors, etc). But they never seem to indicate what the case might be when you have a basis...

Is the "nearest neighbor" any atom in the crystal, or is it only referring to lattice point-related neighbors?
Lattice-point-related neighbors only. For the overall crystal structure, the inner details of a cell do not matter.

## 1. What is the concept of "nearest neighbors" in solid state with basis?

The concept of "nearest neighbors" in solid state with basis refers to the closest atoms or molecules to a specific atom or molecule in a crystal lattice. In a solid state with basis, the crystal lattice is composed of a repeating unit cell with additional atoms or molecules (the basis) located at specific positions within the unit cell. The nearest neighbors are the atoms or molecules in the surrounding unit cells that are closest to the basis atom or molecule.

## 2. How are nearest neighbors determined in solid state with basis?

The determination of nearest neighbors in solid state with basis involves calculating the distances between the basis atom or molecule and each potential neighboring atom or molecule in the surrounding unit cells. The shortest distances are then identified as the nearest neighbors.

## 3. What is the significance of nearest neighbors in solid state with basis?

The concept of nearest neighbors is important in solid state with basis because it affects the properties and behavior of the material. The arrangement and number of nearest neighbors can impact properties such as conductivity, magnetism, and mechanical strength.

## 4. How does the number of nearest neighbors affect the properties of a material in solid state with basis?

The number of nearest neighbors can affect the properties of a material in solid state with basis in several ways. For example, a higher number of nearest neighbors can increase the strength and stability of the crystal lattice, leading to a stronger material. It can also impact the density and conductivity of the material.

## 5. Can the concept of nearest neighbors be applied to all types of solid state with basis?

Yes, the concept of nearest neighbors can be applied to all types of solid state with basis, including metals, semiconductors, and insulators. The specific number and arrangement of nearest neighbors may vary depending on the type of material, but the concept remains the same.

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