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The Kagome lattice is a two-dimensional lattice of corner sharing triangles. The...

Nov 14, 2024

The Kagome lattice is a two-dimensional lattice of corner sharing triangles. The distance between two neighbouring lattice points is given by the lattice constant a. a) How many atoms are in the unit cell of this lattice? b) Which Bravais lattice is formed by these unit cells? c) Calculate the packing fraction for edge contacted circular discs (with the centre of the discs located at the lattice sites).

Solution

The Kagome lattice consists of corner-sharing triangles, and in one unit cell, there are 3 atoms located at the vertices of the triangles. Therefore, the total number of atoms in the unit cell is

3

The Kagome lattice is a type of two-dimensional lattice that can be classified as a non-Bravais lattice due to the presence of two different types of atoms in the unit cell. However, it can be represented as a hexagonal Bravais lattice when considering the arrangement of the triangles

To calculate the packing fraction for edge-contacted circular discs, we first find the area occupied by the discs and the area of the unit cell. The area of one disc is

A_{disc} = \pi r^2

, where

r

is the radius of the disc. The area of the unit cell can be calculated as

A_{cell} = a^2

, where

a

is the lattice constant. The packing fraction is given by

\text{Packing Fraction} = \frac{N \cdot A_{disc}}{A_{cell}}

, where

N

is the number of discs in the unit cell. For the Kagome lattice,

N = 3

, thus the packing fraction becomes

\text{Packing Fraction} = \frac{3 \cdot \pi r^2}{a^2}

Answer

a: 3 atoms, b: Non-Bravais lattice (hexagonal), c:

\frac{3 \cdot \pi r^2}{a^2}

Key Concept

Lattice Structures and Packing Fraction: The packing fraction is a measure of how efficiently the space in a unit cell is filled with particles. It is calculated as the ratio of the area occupied by the particles to the total area of the unit cell. The Kagome lattice is a specific arrangement of particles that can be analyzed using these concepts.

Explanation

The answers provided detail the number of atoms in the unit cell, the classification of the lattice, and the formula for calculating the packing fraction, which is essential for understanding the efficiency of space utilization in crystal structures.