The metal polonium (which was named by Marie Curie after her homeland, Poland) crystallizes in a primitive cubic structure, with an atom at each corner of a cubic unit cell. The atomic radius of polonium is \(167 \mathrm{pm}\). Sketch the unit cell and determine (a) the number of atoms per unit cell; (b) the coordination number of an atom of polonium; (c) the length of the side of the unit cell.

Understanding the atomic radius is critical when studying crystal structures, especially for elements like polonium that crystallize in a primitive cubic structure. In simple terms, the atomic radius refers to the size of an atom, specifically the distance from the center of the nucleus to the boundary of the surrounding cloud of electrons. In a primitive cubic lattice, the atoms are arranged in such a way that each atom's radius reaches halfway across the unit cell edge to meet the radius of a neighboring atom without overlapping.

For polonium, with its given atomic radius of 167 picometers (pm), we know that the entire length of the edge of the cube can be calculated as twice the radius, as this is the distance from the edge of one atom to the edge of the atom on the opposite side of the unit cell. This relationship is vital for not only visualizing the structure but also for determining other properties of the crystal, such as its density and packing efficiency.

When you picture the structure in three dimensions, imagine small spheres (atoms) with their centers fixed at the corners of a cube, just touching each other along the edges.

The coordination number is a term used to described the number of immediate neighboring atoms surrounding a specific atom within a crystal lattice. For a primitive cubic structure, such as that of polonium, each atom is located at a corner of a cubic unit cell.

As one might visualize using the cube sketch from the exercise, an atom at one corner of a cube has only one direct neighbor along each of the three axes - left/right, up/down, and front/back. Because the structure is cubic and symmetrical along all three dimensions, the coordination number is 6 - one for each face of the cube.

This number is a reflection of how atoms are 'connected' in a metal and is important in determining the properties of the material. For example, a higher coordination number often correlates with higher packing density and can affect the mechanical strength and thermal and electrical conductivity of the metal.

The concept of a unit cell is fundamental to crystallography and materials science. It is the smallest repeating unit that contains the entire pattern of a crystal lattice, essentially the 'building block' of the larger structure. In a primitive cubic structure, the unit cell is a cube with atoms at its corners.

Each atom at a corner is shared by eight adjacent unit cells, which means that any calculations involving the number of atoms in a single unit cell must take this sharing into account. As in the polonium example from our exercise, this results in there being effectively one atom per unit cell.

Furthermore, the dimensions of the unit cell, like the length of its sides, are directly linked to the atomic radius of the element forming the crystal. In our case, knowing the atomic radius of polonium enables us to calculate the size of the unit cell and, from there, derive other important characteristics of the crystalline structure, such as its volume, density, and the spacing between atoms.