## Crystal Structure: Packing Spheres

Consider the arrangement of spheres within a lattice to form a view of the structure and complexity of crystalline materials.

### Learning Objectives

Illustrate how atoms or molecules pack in crystalline materials.

### Key Takeaways

#### Key Points

• The coordination number of a given atom in a crystalline substance identifies the number of atoms that directly neighbor the given atom.
• The atomic spheres of crystalline substances pack into unit cells, which are the fundamental building blocks of crystal lattices.
• An understanding of how spheres are packed within a given crystalline material contributes to researchers’ understanding of the properties of the given material.

#### Key Terms

• unit cell: In a crystal, the smallest repeating structure (parallelepiped) of atoms from which the structure of the complete crystal can be inferred.
• coordination number: The total number of atoms that directly neighbor a central atom in a molecule or ion.
• crystal structure: The unique three-dimensional arrangement of atoms or molecules in a crystalline solid.

### Three-Dimensional Lattice

The unique arrangement of atoms or molecules within a crystalline solid is referred to as the crystal structure of that material. A crystal structure reflects the periodic pattern of the atoms which compose a crystalline substance. Crystalline materials are so highly ordered that a crystal lattice arises from repetitions along all three spatial dimensions of the same pattern. The crystal lattice represents the three-dimensional structure of the crystal’s atomic/molecular components.

### The Unit Cell

The structure seen within the crystalline lattice of a material can be described in a number of ways. The most common way to describe a crystal structure is to refer to the size and shape of the material’s characteristic unit cell, which is the simplest repeating unit within the crystal. In principle, one can reconstruct the structure of an entire crystal by repeating the unit cell so as to create a three-dimensional lattice.

Unit Cell of Fluorite: This shows the simplest repeating unit within a crystal of the molecule fluorite, or calcium fluoride (CaF2). Notice that this unit cell contains several atoms. Calcium ions are depicted as gray spheres, and fluorine ions are yellow.

### Packing of Atoms Within a Unit Cell

Within a crystalline material, each atom can be thought of as a sphere. These spheres are packed into unit cells. Each sphere that participates in a crystal structure has a coordination number, which corresponds to the number of spheres within the crystalline structure that touch the sphere that is being evaluated. For a sphere in the interior of a crystal lattice, the number of spheres contacting the sphere that is being evaluated is known as the bulk coordination number. For a sphere at the surface of a crystal, the number of spheres contacting the sphere being evaluated is known as the surface coordination number.

Crystalline Lattice: Sodium chloride crystal lattice

By considering how atomic spheres are arranged relative to one another, their coordination numbers, and the dimensions of the unit cell, it is possible to form a general view of the structure and complexity of particular crystal structures.

## Crystal Structure: Closest Packing

Closest packing refers to the most efficient way to arrange atoms in a crystalline unit cells.

### Learning Objectives

Discuss the two ways in which atoms/molecules pack in the most efficient way in crystals.

### Key Takeaways

#### Key Points

• The most efficient conformation atomic spheres can take within a unit cell is known as the closest packing configuration.
• Densely packed atomic spheres exist in two modes: hexagonal closest packing (HCP) and cubic closest packing (CCP).
• The packing conformation of spheres into a unit cell can effect the physical, chemical, electrical, and mechanical properties of a given crystalline material.

#### Key Terms

• closest packing: A phenomenon resulting in the crystal structure of atoms/molecules having their component parts as near to each other as possible.
• lattice: A regular spacing or arrangement of geometric points.

A crystalline material’s structure is typically visualized as being composed of unit cells. These cells are periodically arranged to give rise to a crystal’s lattice structure. This section considers how the packing of atoms within unit cells contributes to a crystalline solid ‘s lattice structure.

Interactive: Molecular View of a Solid: Explore the structure of a solid at the molecular level.

### Two Types of Atom Packing in a Crystal

The three dimensional structure of a solid crystalline material is established through the periodic patterning of the atoms that make up the crystal. The most efficient conformation of atomic spheres within a unit cell is known as the closest packing formation. In a three dimensional representation of this hypothetical unit cell—with the spheres packed as efficiently as possible—there are two methods to densely pack the cell.

Imagine a single layer of spheres packed into the bottom of a unit cell.

1. In the first method, each successive layer of spheres covers gaps in the previous layer. Three neighboring spheres in the first layer will form a hollow space where they meet. Spheres in one layer align to fit in the hollows formed in the previous layer. The third layer aligns directly above the first layer. Because the third layer is aligned the same way as the first, this configuration is referred to as “ABA” and results in hexagonal closest packing (HCP).
2. Alternatively, the gaps in the first layer are covered by the second layer. But the third layer is offset relative to the intersphere gaps of the first layer. The third layer of spheres does not align with the first layer. This configuration is referred to as “ABC” and results in cubic closest packing (CCP).

Two ways to stack closest packed spheres: Two methods of packing spheres into a unit cell yield the most common closest packing conformations: CCP and HCP.

A CCP arrangement has a total of 4 spheres per unit cell and an HCP arrangement has 8 spheres per unit cell. However, both configurations have a coordination number of 12.

The packing efficiency is the fraction of volume in a crystal structure that is occupied by constituent particles, rather than empty space. In order to find this, the volume of the spheres needs to be divided by the total volume (including empty spaces) occupied by the packed spheres. For both HCP and CCP, the packing efficiency is 74.05 %.

### The Importance of Packing

The arrangement of the atoms in a crystalline solid affects atomic coordination numbers, interatomic distances, and the types and strengths of bonding that occur within the solid. An understanding of atomic packing in a unit cell and crystal lattice can give insight to the physical, chemical, electrical, and mechanical properties of a given crystalline material.

## Determining Atomic Structures by X-Ray Crystallography

X-ray crystallography is a method of determining the arrangement of atoms within molecules.

### Learning Objectives

Describe the method of x-ray crystallography as it is used for determining the structure of molecules.

### Key Takeaways

#### Key Points

• To obtain an x-ray diffraction measurement, three components are necessary: a crystal sample, a source of x-ray beams, and a detector.
• Data from an X-ray crystallography experiment is used to generate a three-dimensional model of the molecules comprising the crystal. Scientists’ knowledge of molecular shapes, bond angles and lengths, are all based on results from such experiments.
• X-ray crystallography is a powerful tool that has broad applications in the determination of the structures of both organic and inorganic compounds.

#### Key Terms

• reflections: Diffraction of X-rays by the layers of atoms within a crystal produces spots, or reflections recorded by a detector. The presence and intensity of reflections is the raw data of an X-ray crystallographic experiment.
• x-ray crystallography: X-ray crystallography is a method of determining the three-dimensional arrangement of atoms within a molecule.

### X-ray Crystallography

X-ray crystallography is a method for determining the arrangement of atoms within a crystal structure. Substances including inorganic salts and minerals, semiconductors, and organic and biological compounds can form crystals under suitable and specific conditions. The method is useful in determining the structure of molecules, which allows researchers to characterize and understand their behavior and function. This method of structure determination has provided the most reliable evidence scientists have about the way molecules are shaped and what their bonds angles and lengths are.

### The Diffraction Experiment

To obtain x-ray diffraction measurements, three components are necessary:

• a crystal sample
• a source of x-ray beams
• a detector

The best x-ray crystallographic structures are derived from the purest crystal samples, meaning samples that contain only molecules of one type and as few impurities as possible. Crystal samples contaminated with impurities, samples that are too small, and samples that are not uniform may result in the formation of imperfect crystals, whose defects affect the quality of the data that can be obtained. Once a crystal has been deemed of sufficient quality, it is generally mounted on special instruments and “shot” with an intense beam of X-rays.

X-Ray Crystallography: When bombarded with x-ray radiation, crystals exhibit a characteristic diffraction pattern.

This process reveals the geometry of the atoms within the molecules. The x-ray beams are diffracted in a characteristic pattern that gives rise to reflections, dark spots on the detector which represent places where constructive interference of the diffracted light has occurred. The detector records the reflections on a two-dimensional surface. The crystal is typically rotated with respect to different axes and shot again with X-rays, so that diffraction patterns from all angles of the X-rays hitting the crystal are recorded.

X-Ray Diffraction Pattern of a Protein: An X-ray diffraction pattern of a crystallized protein molecule. The two dimensional reflection pattern can be used to determine the atomic structure of the protein.

Then mathematical algorithms are applied in order to decode the information contained within the recorded reflections. A map is constructed to describe the electron density of the molecules in the crystal. Atomic models of the molecules are also created; these can explain the experimentally observed electron density.

Workflow for Solving a Molecular Structure Using X-ray Crystallography: The steps to the process of determining the three dimensional structure of a molecule are outlined in this figure.

The final result is the three-dimensional structure of the molecules in the crystal. This is the most direct method that exists for “seeing” what molecules look like. Details such as atomic radii, bond angles and lengths, and molecular geometry are revealed through this method.

### Historical Significance

X-ray crystallography is a powerful tool that has broad applications in the determination of the structures of both organic and inorganic compounds. Throughout the history of chemistry and biochemistry, x-ray crystallography has been one of the most important methods in helping scientists understand the atomic structure and bonding. X-ray diffraction data have proven useful in identifying the structures of protein parts of viruses, such as HIV, which was instrumental in the design of drugs that can specifically target the virus’ needed machinery for its life cycle.