Chapter 3. The structures of simple solids
Structures of Solids
Crystalline solids
The atoms, molecules or ions pack together in an ordered arrangement
Such solids typically have flat surfaces, with unique angles between faces and unique
3-dimensional shape
Examples of crystalline solids include diamonds, and quartz crystals
Amorphous solids

No ordered structure to the particles of the solid
No well defined faces, angles or shapes
Often are mixtures of molecules which do not stack together well, or large flexible molecules
Examples would include glass and rubber
The packing of spheres
2.1 Unit cells and the description of crystal structure
Metals and ionic compounds can be treated as 3-d arrays of hard spheres and in many cases satisfactory results for energy calculations can be obtained by completely neglecting the covalent character which is always present but often small.
In simple cases it is possible to understand and even predict crystal structures based on the notion that the cations and anions pack together as would hard spheres with certain constraints, e.g. cations (often smaller) tend to be surrounded by anions.When atoms, molecules or ions pack in a regular arrangment which can be repeated "infinitely" in three dimensions a crystal is formed. Acrystalline solid, therefore, possesses a long-range order; its atoms, molecules or ions occupy regular positions which repeat in three dimensions. The network of atoms, molecules or ions is known as a crystal latticeor simply as a lattice.
Any location in a crystal lattice is known as a lattice point. Since the crystal lattice repeats in three dimensions, there will be an entire set of lattice points which are identical. That means that if you were able to make yourself small enough and stand at any such lattice point in the crystal lattice, you would not be able to tell which lattice point of the set you were at - they would all appear identical. Of course, you could move to a different site which would look different. This would constitute a different lattice point. For example, when we examine the sodium chloride lattice later, you will notice that the environment of each sodium ion is identical. If you were to stand at any sodium ion and look around, you would see the same thing. If you stood at a chloride ion, you would see a different environment but that environment would be the same at every chloride ion. Thus, the sodium ion locations form one set of lattice points and the chloride ion locations form another set.

Because the atoms, molecules or ions pack in a regular array, it is possible, to use x-ray diffraction to determine the location of the atoms in crystal lattice. When such an experiment is carried out we say that we have determined the crystal structureof the substance. The study of crystal structures is known as crystallography and it is one of the most powerful techniques used today to characterize new compounds. You will discuss the principles behind x-ray diffraction in the lecture part of this course.

The forces which stabilize the crystal may be ionic (electrostatic) forces, covalent bonds, metallic bonds, van der Waals forces, hydrogen bonds, or a combination of these. The properties of the crystal will change depending upon what types of bonding is involved in holding the atoms, molecules or ions in the lattice.

The fundamental types of crystals based upon the types of forces that hold them together are:

metallic / metal cations held together by a sea of electrons
ionic / cations and anions held together by predominantly electrostatic attractions
network / atoms bonded together covalently throughout the solid (also known as covalent crystal or covalent network).
molecular / collections of individual molecules; each lattice point in the crystal is a molecule

Often crystals held together by more than one type of force and thus may have intermediate properties.
Since the crystal lattice is made up of a regular arrangement which repeats in three dimensions, we can save ourselves a great deal of work by considering the simple repeat unit rather than the entire crystal lattice. The basic repeat unit is known as the The Unit Cell
Crystalline solids often have flat, well-defined faces that make definite angles with their neighbors and break cleanly when struck. These faces lie along well-defined directions in the unit cell.The unit cell is the smallest, most symmetrical repeat unit that, when translated in three dimensions, will generate the entire crystal lattice.

  • Since the crystal is made up of an arrangement of identical unit cells, then an identical point on each unit cell represents an identical environment within the crystal
  • The array of these identical points is termed the crystal lattice
  • The unit cells shown are cubic
  • All sides are equal length
  • All angles are 90°
  • The unit cell need not be cubic
  • The unit cell lengths along the x,y, and z coordinate axes are termed the a, b and c unit cell dimensions
  • The unit cell angles are defined as:
  • , the angle formed by theb and c cell edges
  • , the angle formed by the a and c cell edges
  • , the angle formed by the a and b cell edges

Cubic Cells
There are many types of fundamental unit cells, one of which is the cubic cell. In turn, there are three subclasses of cubic:
Simple cubic (sc) cell
Body-centered cubic (bcc) cell
Face-centered cubic (fcc) cell .

Several aspects of each cell type:

  • the number of atoms per unit cell
  • the efficiency of the packing of atoms in the volume of each unit cell

the number of nearest neighbors (coordination number) for each type of atom

The unit cell is normally selected to be the simplest of the possible repeating units.

  • The basic unit cell in three dimensions is a parallelipedwith side langths and angles as defined below.
  • The angles and lengths used to define the size of the unit cell are known as the unit cell parameters.

This unit cell has no symmetry in that the cell parameters and and angles may take any values. An increasing level of symmetry produces relationships between the various cell parameters and leads to the seven crystal classes.

Crystal Class / Restrictions on Unit Cell Parameters / Highest Type of Symmetry Element Required
Triclinic / a is not equal tob is not equal toc is not equal to  is not equal to. / no symmetry is required, an inversion center may be present
Monoclinic / ais not equal to bis not equal to c =is not equal to. / highest symmetry element allowed is a C2axis or a mirror plane
Orthorhombic / ais not equal to bis not equal to c == / has three mutually perpendicular mirror planes and/or C2 axes
Tetragonal / a =bis not equal to c == / has one C4 axis
Cubic / a =b =c== / has C3and C4 axes
Hexagonal, Trigonal / a =bis not equal to c ==120 / C6 axis (hexagonal); C3 axis (trigonal)
Rhombohedral* / a =b =c ==is not equal to / C3 axis (trigonal)

It is possible to have a number of different choices for the unit cell. By convention, the unit cell that reflects the highest symmetry of the lattice is the one that is chosen. A unit cell may be thought of as being like a brick which is used to build a building. Many bricks are stacked together to create the entire structure.

Because the unit cell must translate in three dimensions, there are certain geometrical constraints placed upon its shape. The main criterion is that the opposite faces of the unit cell must be parallel. Because of this restriction there are only six parameters that we need to define in order to define the shape of the unit cell. These include three edge lengths a, b and c and three angles ,  and . Once these are defined all other distances and angles in the unit cell are set. As a result of symmetry, some of these angles and edge lengths may be the same. There are only seven different shapes for unit cells possible. These are given in the chart above.

*There is some discussion about whether the rhombohedral unit cell is a different group or is really a subset of the trigonal/hexagonal types of unit cell.
In contrastamorphous solids do not possess any long-range order, so they do not typically form well-defined faces nor do they fracture evenly. Glass is an example of an amorphous solid. Amorphous solids have very interesting properties in their own right that differ from those of crystalline materials. We will not consider their structures in this laboratory exercise.

2.2 The close-packing of spheres
Closest Packing (Close-packing) of Spheres (Section 2.2)
A crystal is a repeating array. In describing this structure we must distinguish between the pattern of repetition (the lattice type) and what is repeated (the unit cell) described below.
Packing and Geometry
Imagine a stack of balls in a box. This was actually how structures were first visualised before computers (BC). The balls will pack together to fill up all the space. This is called close packing - you can see how it works if you look at a pile of oranges in the supermarket. Notice how the oranges form a pattern. Each orange labelled A will be surrounded by six other oranges within one layer. Notice the holes labelled B and C. We can place a second layer of close-packed oranges on either the B-sites or theC-sites (but not both). In this way we can build up a 3D structure.
The closest packing of identical spheres forms the basis for the structures of a number of metals, and the arrangement of the largest ions (often the anions) in a number of simple ionic structures. Close-packing of spheres is one example of an arrangement of objects that forms such an extended structure. Extended close-packing of spheres results in 74% of the available space being occupied by spheres (or atoms), with the remainder attributed to the empty space between the spheres. This is the highest space-filling efficiency of any sphere-packing arrangement. The nature of extended structures as well as close-packing, which occurs in two forms called hexagonal closest packing (hcp) and cubic closest packing (ccp), will be explored in this lab activity.
Sixty-eight of the ninety naturally occurring elements are metallic elements. Forty of these metals have three-dimensional submicroscopic structures that can be described in terms of close-packing of spheres. Another sixteen of the sixty-eight naturally occurring metallic elements can be described in terms of a different type of extended structure that is not as efficient at space-filling. This structure occupies only 68% of the available space. This second largest subgroup exhibits a sphere packing arrangement called body-centered cubic (bcc).

There are two packing methods:

  • Hexagonal close-packed (hcp) Type ABABAB... (Figs. 2.2a and 2.3a)

Actually there is another common form of close-packing, corresponding to layers with stacking AB.AB... or AC.AC... (these are equivalent). This is called hexagonal close-packing HCP, and the competition between CCP and HCP is determined by longer range forces between the atoms. This is the structure of sodium at low temperatures. No, we can't transform sodium to gold by stacking the atoms differently ! For such simple materials, the different properties are mainly due to the differences between the sodium and gold atoms themselves.

  • Cubic close-packed (ccp) or face-centred cubic (fcc) Type ABCABCABC... (Figs. 2.2b and 2.3b)

Does this structure correspond to anything in nature (apart from oranges in supermarkets) ? Of course ! A stack of layers of types ABC.ABC... represents the cubic close-packed CCP atomic structure of gold as determined by X-rays. Atoms lie on the corners of a cube, with additional atoms at the centers of each cube face: for that reason it is often called face centered cubic or FCC. There is one host atom at each corner, one host atom in each face, and the host atoms touch along the face diagonal (a = 2.8284r, Z = 4).Many simple metals have this FCC structure, whose symmetry is described as Fm-3m where F means Face-centered,m signifies a mirror-plane (there are two) and -3 tells us that there is a 3-fold symmetry axis (along the body diagonal) as well as inversion symmetry. where A, B and C are hexagonally packed layers of spheres. Both fill 74% of the space in an extended structure.

Simple Structures which are not Close-packed (Section 2.5)
There are two other simple packing types which are found:

  • Body-centered cubic - 68% filled (Fig 2.6)

The third common metallic structure is called body-centered cubic BCC, and consists of a unit cube with atoms at its corners and center. The BCC structure is slightly less closely packed than FCC or HCP and is often the high temperature form of metals that are close-packed at lower temperatures. For example sodium changes from HCP to BCC above -237 degrees C ! The structure of iron (Fe) can be either CCP or BCC depending on its heat treatment, while metals such as chromium are always BCC. Metals which are BCC are, like chromium, usually harder and less malleable than close-packed metals such as gold. When the metal is deformed, the planes of atoms must slip over each other, and this is more difficult in the BCC structure. Note that there are other important mechanisms for hardening metals, and these involve introducing impurities or defects which also block slipping. (The body-centred cubic type is relatively common, but only -Po is know to adopt the primitive cubic structure.) There is one host atom at each corner of the cubic unit cell and one atom in the cell center. Each atom touches eight other host atoms along the body diagonal of the cube (a = 2.3094r, Z = 2).

  • Primitive (or simple) cubic - 52% filled(Fig 2.7)

Simple Cubic (SC)- There is one host atom ("lattice point") at each corner of a cubic unit cell. The unit cell is described by three edge lengths a = b = c = 2r (r is the host atom radius), and the angles between the edges, alpha = beta = gamma = 90 degrees. There is one atom wholly inside the cube (Z = 1). Unit cells in which there are host atoms (or lattice points) only at the eight corners are called primitive.

You should be able to calculate the % of void space using simple geometry.

When spherical objects of equal size are packed in some type of arrangement, the number of nearest neighbors to any given sphere is dependent upon the efficiency of space filling. The number of nearest neighbors is called the coordination number and abbreviated as CN. The sphere packing schemes with the highest space-filling efficiency will have the highest CN. Coordination number will be explored in this lab activity. A useful way to describe extended structures, which, in principle, can be infinitely large, is to conceive of a three-dimensional parallelepiped, which is a six-sided solid having parallel faces. This represents a unit cell which can be moved in three directions to duplicate the entire structure of the crystal; for a cubic unit cell, the cell is shifted in the three perpendicular X, Y and Z directions. The unit cell is the repeating three-dimensional pattern for extended structures. A unit cell has a pattern for the objects as well as for the void spaces. Unit cells will be explored in this lab activity.

The remaining unoccupied space in any sphere packing scheme is found as void space. This void space occurs between the spheres and gives rise to so-called interstitial sites.
2.3 Holes in close-packed structures

Holes ("Interstices") in Closest Packed Arrays

Tetrahedral Hole - Consider any two successive planes in a closest packed lattice. One atom in the A layer nestles in the triangular groove formed by three adjacent atoms in the B layer, and the four atoms touch along the edges (of length 2r) of a regular tetrahedron; the center of the tetrahedron is a cavity called the Tetrahedral (or Td) hole; a guest sphere will just fill this cavity (and touch the four host spheres) if its radius is 0.2247r.

Octahedral Hole - Adjacent to the Td hole, three atoms in the B layer touch three atoms in the A layer such that a trigonal antiprismatic polyhedron (a regular octahedron) is formed; the center of the octahedron is a cavity called the Octahedral (or Oh) hole. A guest sphere will just fill this cavity (and touch the six host spheres) if its radius is 0.4142r. It can be shown that there are twice as many Td as Oh holes in any closest packed bilayer.
coordination number
The coordination number is the number of particles surrounding a particle in the crystal structure.

  • In each packing arrangement (simple cube), a particle in the crystal has a coordination number of 6
  • In each packing arrangement (body centerd cube), a particle in the crystal has a coordination number of 8

In each packing arrangement (hexagonal close pack, cubic close pack), a particle in the crystal has a coordination number of 12