Question

What type of lattice-primitive cubic, body-centered cubic, or face-centered cubic-does each of the following structure types possess: (a) \(\mathrm{CsCl}\), (b) \(\mathrm{Au}\), (c) \(\mathrm{NaCl}\), (d) \(\mathrm{Po}\), (e) \(\mathrm{ZnS}\) ?

Short answer

(a) The CsCl structure has a Primitive Cubic lattice type. (b) Gold (Au) has a Face-Centered Cubic (fcc) lattice type. (c) The Sodium Chloride (NaCl) structure has a Face-Centered Cubic (fcc) lattice type. (d) Polonium (Po) has a Primitive Cubic lattice type. (e) Zinc Sulfide (ZnS) in the Sphalerite form has a Face-Centered Cubic (fcc) lattice type.

Step-by-step solution

(a) CsCl Lattice Type

The CsCl structure consists of two interpenetrating simple cubic lattices. Cesium (Cs+) ions occupy the corners of a cube, and a chloride (Cl-) ion is at the center of the cube. The lattice type for CsCl is called Primitive Cubic.

(b) Au Lattice Type

Gold (Au) has a cubic close-packed structure, also known as face-centered cubic (fcc) lattice. In this arrangement, every face of the unit cell has atoms at their center and their corners, resulting in a total of 14 lattice points per unit cell.

(c) NaCl Lattice Type

The Sodium Chloride (NaCl) structure consists of two interpenetrating face-centered cubic lattices. It can be visualized as a cubic close-packed array of chloride (Cl-) ions with sodium (Na+) ions occupying the octahedral holes. This formation results in each ion being surrounded by six other ions of the opposite charge. The lattice type for NaCl is Face-Centered Cubic (fcc).

(d) Po Lattice Type

Polonium (Po) has a simple cubic lattice. It means that the unit cell consists of an atom at each vertex, forming a cube. This arrangement results in one lattice point per unit cell, and it is called Primitive Cubic.

(e) ZnS Lattice Type

Zinc Sulfide (ZnS) forms two types of structure: Wurtzite and Sphalerite. The Wurtzite structure is a hexagonal close-packed lattice, while the Sphalerite structure is a cubic close-packed lattice. However, Sphalerite is more commonly known and has a face-centered cubic (fcc) lattice similar to NaCl. In this structure, both Zn and S ions occupy half of the tetrahedral holes of one another fcc lattice. The lattice type for ZnS (Sphalerite) is Face-Centered Cubic (fcc).

Key concepts

Primitive Cubic Lattice

Imagine a simple grid made from equally spaced dots that signify atoms. A primitive cubic lattice is just like that grid made 3D. Each atom sits on a corner of a cube. This configuration is the simplest of the crystal lattice structures and is also called 'simple cubic' or 'P-lattice'.

In a primitive cubic unit cell, we have an atom at each of the eight corners of a cube but each atom is shared between eight adjacent unit cells. This means that for the calculation of the number of molecules per unit cell in a primitive cubic structure, we consider just 1/8th of each atom at the corners. In essence, it offers one atom per unit cell.

Body-Centered Cubic Lattice

A body-centered cubic lattice (BCC) builds on the concept of a primitive cubic lattice with an extra twist: there's an additional atom smack dab in the center of the cube besides the atoms at the corners. This makes it more tightly packed than the primitive cubic lattice.

While still a simple shape, a body-centered cubic cell provides a different arrangement which results in two atoms per unit cell (one from the center, and the contributions from the corner atoms).

Face-Centered Cubic Lattice

Compared to others, the face-centered cubic lattice (FCC) is a bit like the social butterfly of crystal structures - it has atoms on the corners of the cube and one in the middle of each face. This increases the density of the lattice.

The face-centered cubic lattice turns out to be quite the dense package with four atoms per unit cell. Since the atoms at the faces are shared between two cells and the corner atoms are shared among eight, it results in a highly efficient packing arrangement. FCC lattices are known for being 'cubic close-packed' structures because they maximize the number of atoms in a given volume.

Unit Cell

Consider the unit cell as a repeating building block, like a Lego block, of crystal lattices. We use it to study the structural arrangement of a crystal. Each lattice type has a characteristic unit cell.

For all types of cubic lattices, the unit cell is, well, a cube. However, how atoms are placed within that cell is what distinguishes one type of cubic lattice from another - primitive has them at the corners, body-centered adds one in the middle, and face-centered features atoms smack in the center of each face too.

Interpenetrating Lattices

Interpenetrating lattices may sound complicated, but really they are just lattices that crisscross each other without actually disrupting the pattern. They share the same space.

It's like two overlapping fishnets. Each net retains its structure without being intertwined with the other. In crystallography, it’s like having two separate lattices that intersect but don’t share atoms, giving rise to complex structures like that of sodium chloride (NaCl), where two face-centered cubic lattices penetrate one another.

Octahedral Holes

Imagine a die, the kind with dots that represent numbers. Now, picture an atom snugly fit at the center of it - that's essentially an octahedral hole. It is a space within the lattice that is surrounded by, and touching, six atoms arranged at the corners of an octahedron.

In a face-centered cubic lattice like NaCl, these nooks and crannies host ions or atoms, and are crucial for the assembly of more involved crystal structures. Each FCC lattice has eight octahedral holes per unit cell, helping form more packed and stable structures.

Cubic Close-Packed

Cubic close-packed (ccp) structure is really just another name for a face-centered cubic lattice. It's the arrangement of atoms within the lattice in a certain sequential layering pattern where each layer is offset and nests comfortably into the lower layer's gaps.

These stacked layers 'close-pack' the atoms so that they take up the maximum amount of available space, leading to a very dense, tightly packed structure. This packing is found in some of the most common elements like gold (Au) and also crystals such as Zinc Sulfide (ZnS) in its Sphalerite form.