How to pack atoms in a crystal?

A crystal system enumerates or categorizes different ways in which the space can be filled without voids using unit cells, which are the periodically repeating blocks of a crystal. In other words, a crystalline solid exhibits a regular, long-range atomic arrangement, distinguishing it from amorphous solids that lack such order. Nature has already been thinking about symmetry, order, or even long-range order through the evolution. The concept of symmetry and order has also intrigued humans over the past centuries. Among the early thinkers, Robert Hooke, in the 1600s, was notable for contemplating the atomic arrangement of salt crystals.

Let’s explore the concept of “unit cell” in more detail. As we mentioned before, the unit cell is the fundamental building blocks of a crystal, which enumerates space. It can be defined using lattice vectors. These vectors, labeled as “a” and “b” below in 2D space, act like stamps that, when translated along the x and y axes, generate a periodic tiling of space. From another perspective, the periodic repetition of unit cells forms the crystalline lattice. A key question arises once the lattice vectors are established: What should be placed at each lattice point? Suppose we place circles at lattice points to mimic atomic representation. Another important question follows: What is the maximum possible packing density for this arrangement? If we increase the radius (r) of these circles that are positioned at each lattice point, they eventually touch each other, demonstrating the simplest form of maximum packing. In two dimensions, the maximum packing fraction can be quantified by dividing the maximum area occupied by the circles in the unit cell by the total area of the unit cell. For a simple square lattice, this fraction is found to be approximately 78% in 2D space.

In 3D space, there are seven distinct crystal systems, determined by the principles of group theory. To simplify the discussion, we will focus on only the cubic crystal system, which is commonly adopted by many elements in the periodic table. The cubic crystal system is one of the most symmetric and simplest systems, characterized by three mutually orthogonal lattice vectors of equal length (a=b=c) and angles of 90° between them. When considering how to pack atoms within a cubic system, we refer to the concept of Bravais lattices. Bravais lattices enumerate the packing within the unit cell rather than the whole space. Auguste Bravais identified 14 unique ways to arrange unit cells in 3D space across the seven crystal systems. Within the cubic system, three distinct Bravais lattice types exist:

1. Simple Cubic (SC)
2. Body-Centered Cubic (BCC)
3. Face-Centered Cubic (FCC)

In the simple cubic (SC) arrangement, atoms are located only at the corners of the cube. Each atom is shared among 8 adjacent unit cells, so the unit cell contains 8 x 1/8 = 1 atom. Coordination number in this arrangement is 6 meaning that each atom has six nearest neighbors (NNs), forming a relatively loose atomic packing. The maximum packing fraction can be calculated by determining the volume occupied by atoms within the unit cell. Given a unit cell with a side length of “a,” the atomic radius will be “a/2.” The volume of a single atom (assumed to be a sphere) and the volume of the unit cell can be calculated accordingly as shown below. Since a simple cubic arrangement contains only one atom per unit cell, the atomic packing fraction (APF) is calculated by dividing the volume of atoms in the cubic unit cell by the volume of the unit cell. Substituting the values, the APF for the SC lattice is found approximately 52%. This indicates inefficient packing, leaving almost half of the space empty within the structure. Due to its low packing efficiency, the simple cubic structure is rare in nature. In fact, only one naturally occurring element in the periodic table, polonium (Po), adopts this structure. Polonium is a heavy element (atomic number 84), and in such elements, relativistic effects become significant. This effect is due to the inner electrons (particularly the 1s electrons) move at speeds close to the speed of light, causing an increase in their effective mass. Therefore, the inner electrons are pulled closer to the nucleus, which reduces the shielding effect on the outer electrons. This phenomenon is known as contraction of the 6s orbital. On the other hand, the outer electrons experience less effective nuclear charge, causing these orbitals to expand, which refers to expansion of 6p orbital in this case. A weaker binding of the 6p electrons makes them more available for bonding, further stabilizing the simple cubic lattice arrangement.

In the body-centered cubic (BCC) arrangement, atoms are located at the corners of the cube and one atom at the center of the cube. The unit cell contains 8 x 1/8 = 1+ 1 (for the central) = 2 atoms. Coordination number is 8 for each atom and APF is found to be approximately 68%. This indicates relatively stronger packing compared to the SC arrangement, with increased contact capacity for each atom. A common example of this category is iron (Fe).

In the face-centered cubic (FCC) arrangement, atoms are located at the corners of the cube and at the centers of each side face. The unit cell contains (8 x 1/8) + (1/2 x 6) = 4? atoms. Coordination number is 12 and APF is found to be approximately 74%. Aluminum (Al) and Copper (Cu) are well known examples of this category.

High packing efficiency often results in dense, strong materials with high atomic coordination and often high ductility, whereas lower APF leads to loosely packed structures with more empty space, making them less stable and less common in nature. The concept of maximum packing is critically important in the context of protein crystallization, as it directly influences the stability, quality, and utility of protein crystals.