# 2. Metal Packing: Layers

Metals are composed of atoms in ordered layers. These atoms form a three-dimensional, crystalline structure. That means that the individual units within the solid -- the atoms, in this case -- are organized in a regularly repeating pattern, like the stars on the American flag. Three-dimensional things can be difficult to think about. It's much easier to start in two dimensions. On this page, we're going to look at a couple of ways that metal atoms could arrange themselves in a single layer. We'll build up from there later on. In the meantime, we'll try to develop familiarity with a number of terms that are used to describe crystalline structures.

### Simple Squares

We live in a Cartesian society. Our houses and buildings are mostly square, our rooms are square, our streets are straight and meet at right angles, unless we live in a meandering place on the water, like Boston or Venice. It's probably easiest to imagine layers of atoms that form nice squares. Suppose a group of atoms forms a nice, straight line. Another group forms a second line, and each atom in the second line sits directly behind an atom in the first line. A third row forms in the same way. The resulting layer has neat rows and columns of atoms. The pattern of atoms regularly repeats as you look from right to left, and as you look from front to back.

Figure ME2.1. A simple square layer of atoms.

We will call this type of layer a simple square layer.

• In a square layer, atoms are arranged in rows and columns.

### Hexagonal Layers

Some metals atoms may be arranged in simple square layers. Others adopt a slightly different arrangement. Start with the same, neat line of atoms in the first row. This time, when the second row of atoms forms, we won't put them directly behind the atoms in the first row. We'll put them just in between the atoms in front. Each atom in the second row sits behind two atoms in the first row, as if it were peeking between them, like kids in a kindergarten class photo (before they all squirm around).

The third row will also be slightly offset from the row in front of it. If you look carefully, you can see that puts each of the third row atoms directly behind an atom in the first row, but an extra row farther away. The fourth row will be slightly offset from the third, and so on.

Figure ME2.2. A hexagonal layer of atoms.

We still have a crystalline structure. There is a regularly repeating pattern as we look from left to right. There is also a repeating pattern as we move from front to back, but now that pattern repeats every other row, instead of every row.

This type of layer is called a hexagonal layer. Sometimes, it is called a close-packed layer.

• In a hexagonal layer, the rows are offset from each other by half an atom.
• Each atom in a hexagonal layer is surrounded by six other atoms.

### Holes

Unlike some other geometric shapes, circles do not pack together with no space in between them. The spaces in between atoms within a layer are called holes. Note that the use of the word "holes" here has nothing to do with holes in conductivity-speak. On this page, holes are just spaces between atoms. In conductivity, holes are places where there is positive charge that could capture an electron.

Sometimes, the spaces between the atoms are described in other ways, such as "interstitial sites". Later on, these interstitial sites or holes will be important in building compounds from mixtures of elements. You will see that the empty spaces in an array of atoms leave room for a second kind of atom to pack into the same structure.

### Unit Cells

The simplest repeating unit in a pattern is sometimes called the unit cell. In a layer, the unit cell can be traced again and again over the layer, so that every part of the layer is accounted for (but depending where you start tracing, you might run out of room at the edges).

Figure ME2.3. A simple row of atoms. The unit cell is a repeating unit along the row.

Because each layer is composed of individual atoms, we might call a single atom the unit cell. However, we also use the unit cell to illustrate the spatial relationship between the atoms in the layer. For that reason, we often shift the unit cell to show multiple atoms.

For example, in a row of atoms, a unit cell might be a square. The ends of the square would cut through two neighboring atoms. The unit cell contains two half-atoms, for a total of one atom. The square could be repeated in a straight line to reproduce the row of atoms.

• The unit cell is the smallest repeating unit in the structure.
• The unit cell shows the pattern that is found in the structure.

In a layer of atoms, the unit cell might once again be thought of as a single atom. However, we can more clearly show the relationships between these atoms by choosing a slightly different cell. The unit cell in a simple square layer is just a square that contains the corners of four neighboring atoms. The unit cell in a hexagonal layer is a rhombus that contains the corners of four neighboring atoms.

Figure ME2.4. The unit cell in a simple square layer (left) and in a hexagonal layer (right).

### Packing Efficiency

Packing efficiency is one of the questions people wonder about in crystal structures. It deals with how tightly the atoms are packed together in a structure. Often, the more tightly packed the structure, the better (but it doesn't always happen that way).

Figure ME2.5. Packing efficiency refers to how tightly atoms are packed together.

Why might tighter packing of metal atoms be favored? It has to do with the fact that the metal atoms are interacting with each other. The closer they are together, the more strongly they can interact.

There are a couple of ways we can think about that interaction. In the simplest idea of the electron-sea model, metal atoms have become metal cations, and are surrounded by electrons. The closer the metal atoms can get to the electrons, the more strongly they will interact. In general, we think of the metal atoms as sharing their electrons; the closer the metal atoms are to each other, the more efficiently the electrons can be shared.

Let's just work with that idea of electrostatic attraction. Imagine a group of atoms in a simple square layer. All of the atoms are touching their neighbors on either side. Let's think about the distance between a nucleus and an individual electron in the electron sea. We know that the nucleus is in the center of the atom. We don't know exactly where a free electron is, but let's assume it is in the center of the nearest hole. Remember, the closer the electron is to the nucleus, the lower the energy.

Figure ME2.6. Packing efficiency has an influence on the strength of interactions between metal nuclei and the electrons in the "electron sea."

For comparison, maybe there is another set of atoms, also in a simple square layer. Suppose they are well-separated from each other; maybe they are far enough apart that you could fit an extra atom between each pair if you wanted to. If the free electron is in the same place -- the middle of the nearest hole -- you can see that it is much farther from the nucleus in this case. The force of attraction is much lower in this case, and the overall energy is not as low.

• Most metals pack very efficiently together to form a solid.
• Efficient packing leads to stronger bonding interactions.

That first case, with atoms packed more tightly together, may be preferable, because of the stronger interaction between the metal nucleus and the free electron. For reasons like this, understanding the packing efficiency in a crystal can be very important.

### Coordination Number

There are a number of terms used in describing the relationships between atoms and the atoms to which they are directly bonded. For example, we use the term "coordination number" to describe the number of other atoms that a specific atom is touching in a regular layer.

### Coordination Geometry

Take a look at all of the atoms surrounding one specific atom in a layer. If you were to draw a line from the center of one of these atoms to the next, and go all the way around until you got back to the beginning, you would get a specific shape. That shape is described as the coordination geometry of the central atom. It's closely related to the coordination number. In two dimensions, two atoms with the same coordination number always have the same coordination geometry. Occasionally in three dimensions, two atoms with the same coordination number can have different coordination geometries; the atoms that they are touching may be arranged in slightly different ways.

### Lattice

Sometimes, the arrangement of atoms in a crystal is referred to as "an array" or "a lattice." At this point, an array or lattice is just the two-dimensional network of atoms in a layer. Later, we will see three-dimensional arrays of atoms. Metallic solids are usually three-dimensional, not two-dimensional. However, as we start to build up to a third dimension, you will see that most of the ideas on this page have analogies in 3D.