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.
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.
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.
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.