Crystalline silicon has the same structure as diamond, with...

Crystalline silicon has the same structure as diamond, with a unit cell edge length of $5.430 \AA$ A. (a) What is the $\mathrm{Si}-\mathrm{Si}$ distance in this crystal? (b) Calculate the density of crystalline silicon.

Key Concepts

Diamond Cubic Structure

This concept refers to a type of crystal structure where atoms are arranged in a cubic lattice with a basis, similar to that of diamond. It is characterized by atoms placed at the corners and face centers of the cube as well as additional atoms located within the cube, arranged in a tetrahedral configuration. This structure determines the spatial relationships between atoms, affecting both the closest interatomic distances and the overall packing density.

Unit Cell and Lattice Parameters

A unit cell is the minimal repeating unit which, when translated through space, fully recreates the crystal lattice. Lattice parameters, such as the unit cell edge length, are fundamental in determining the geometrical and physical properties of a crystal. In this context, the edge length is used to derive interatomic distances and to compute the unit cell’s volume, which is essential for calculating the material's density.

Nearest Neighbor Distance Calculation

In crystals with a diamond cubic structure, the bond length between nearest neighbor atoms is not equal to the unit cell edge length but is related through the geometry of the tetrahedral arrangement. Specifically, the nearest neighbor distance can be expressed as a multiple of the edge length, often given by (?3/4) times the unit cell edge length. This relation is crucial for assessing the bond length in crystalline materials and understanding their bonding characteristics.

Density Calculation in Crystals

Density in crystalline materials can be calculated by dividing the mass of atoms contained within a unit cell by the volume of that cell. This requires knowing the number of atoms per unit cell, the atomic weight of the element (appropriately converted to mass per atom using Avogadro’s number), and the unit cell volume derived from the lattice parameter. This calculation integrates concepts from crystallography, atomic mass, and geometric volume to yield a material density consistent with macroscopic measurements.

Transcript

Please give Ace some feedback