Question

Calculate the percentage of occupied space in bcc and hcp arrangements.

Short answer

After calculating, the packing efficiency in bcc is found to be about 68%, and for hcp, it is approximately 74%. So, in bcc 68% of the space is occupied by atoms, and in hcp 74% is occupied.

Step-by-step solution

Calculations for BCC

The body-centered cubic (bcc) arrangement consists of an atom at each of the eight corners of a cube and one atom at the center. The atoms at the corners contribute (1/8)*8 = 1 atom and the one in the center contributes 1 atom, so total 2 atoms. The atomic radius \( r \) relates to the length of the cube edge \( a \) by \( a = 4r / \sqrt{3} \). Therefore, the volume of the cube \( V_c = a^3 = (4r / \sqrt{3})^3. \) The volume of the atoms \( V_a = 2*(4/3)*πr^3 \). Hence, the packing efficiency, which is the ratio of the volume of the atoms to the volume of the cube and then multiplied by 100, can be calculated.

Calculations for HCP

The hexagonal close-packed (hcp) arrangement consists of three layers of atoms. The top and bottom layers are identical and consist of six atoms that form a regular hexagon and a seventh atom that sits in the middle of the hexagon. The middle layer has three atoms nestled into the triangular grooves of the top and bottom layers. Thus, the repeating pattern includes 6 atoms. The volume of the unit cell for this arrangement is \( V_c = \sqrt{3}/2 * a^2 * c \) where \( a \) is the side length and \( c \) is the height, and relates to the atomic radius \( r \) by \( a = 2r \) and \( c = \sqrt{8/3}*a \). The volume of the atoms \( V_a = 6*(4/3)*πr^3 \). Hence, the packing efficiency can be calculated, similar to the bcc case.

Summary of Results

Having calculated the packing efficiency for both bcc and hcp, it can be concluded about the percentage of occupied space in both these arrangements. This will give an understanding about the relative densities of materials having these structures.

Key concepts

Body-Centered Cubic (BCC) Structure

The body-centered cubic (BCC) structure is one of the ways in which atoms can arrange themselves in a crystal lattice. In a BCC lattice, each unit cell consists of one atom at each of the eight corners of a cube and one atom at the very center. This arrangement leads to a distinctive geometric pattern that is found in various metal crystals.

In terms of packing efficiency, BCC structures do not pack atoms as closely as some other structures, but they are still quite dense. To calculate the packing efficiency of a BCC structure, the relationship between the atomic radius and the cube edge is crucially considered. Specifically, the cube edge \( a \) is mathematically related to the atomic radius \( r \) through the equation \( a = \frac{4r}{\sqrt{3}} \). By determining the atomic radius, the volume of the atoms within the unit cell can be calculated and compared to the total volume of the cell, revealing the packed space percentage within BCC crystals.

Hexagonal Close-Packed (HCP) Structure

The hexagonal close-packed (HCP) structure is another common type of crystal lattice arrangement. Each unit cell in an HCP structure is formed by two layers of atoms where each layer consists of six atoms forming a hexagon, with another atom at the center of the hexagon. Above and below this pair of layers, there are half-layers that fit into the grooves of the hexagonal pattern, creating a three-dimensional stacking sequence.

To measure the packing efficiency of an HCP structure, one must understand the geometric relationship between the atomic radius \( r \) and the dimensions of the hexagonal unit cell. Specifically, the edge length \( a \) is equal to twice the atomic radius \( a = 2r \) and the cell height \( c \) is \( c = \sqrt{\frac{8}{3}}a \). With these relations, the volume occupied by atoms within the unit cell can be computed. HCP structures are known for their high packing efficiency, rendering them highly dense with minimal empty space between atoms.

Crystal Lattice Arrangements

Crystal lattice arrangements refer to the regular, repeating patterns in which atoms, ions, or molecules are arranged in crystalline solids. The arrangement of particles within the lattice affects the material's properties, such as density, melting point, and mechanical strength. There are several types of crystal lattice structures, each with its geometric pattern and coordination number— the number of immediate neighboring particles surrounding a particle.

The BCC and HCP structures discussed are both examples of such arrangements, but there are others, like face-centered cubic (FCC) and simple cubic (SC). Variations in packing efficiency, determined by how tightly the particles fit together, distinguish these structures from one another and contribute to the diversity of material characteristics observed in nature.

Atomic Radius

The atomic radius is a measure of the size of an atom. It is defined as half the distance between the nuclei of two identical atoms that are bonded together. The atomic radius plays a significant role in determining the spacing between atoms in a crystal lattice and therefore has a direct impact on the crystal's packing efficiency and overall structure.

Understanding how to relate atomic radius to the dimensions of the unit cell in various crystal lattices is fundamental when calculating packing efficiency. It is essential to account for the atomic radius to predict the density and other physical properties of the material. For instance, in BCC and HCP structures, knowing the atomic radius allows us to establish the unit cell dimensions and thereby calculate the extent to which space within the crystal is filled by atoms.