Physical Chemistry


arrangement of atoms in solids


Atoms in solids generally occupy set positions.  Those positions and their chemical bonds maintaining those positions determine almost all the important physical properties of solids.  So there is value in studying the most common of these lattice arrangements.

First, we should recall that some solids which we call glass, are amorphous, meaning without form.  When broken these solids typically fracture forming curved surfaces.  They generally lack precise melting temperature, gradually softening over a temperature range suggesting that the strength of forces between atoms varies considerably as distances between atoms varies due to irregular packing arrangements.  For the present we will turn attention to those solids which have a repeating structural pattern, called crystalline.  These generally have precise and characteristic melting temperatures, and if brittle, cleave such that the new flat surfaces which form indicating their lattice layering.

We should note that metals also use lattice arrangements.  But these typically shiny, often silver colored solids which are good conductors of heat and electricity, allow slippage between their layers such that they are malleable and ductile, deforming rather than cleaving under stress.  We shall find that all these properties can be understood and predicted in terms of their lattice.

Obviously real atoms are too small to see their lattice arrangement.  Yet crystals containing enormous numbers of atoms, when grown free from hinderance, exhibit a shape characteristic of the lattice.  Since early 20th Century, it has been possible to confirm lattice of crystals by X-ray diffraction and more recent techniques such as scanning tunneling microscopy.


It is often valuable to create models to help understand the behavior of complicated systems such as collections containing large numbers of atoms and molecules.  Sometimes those models are mathematical.  But visual models, either virtual on computers, or physical models, can improve understanding.  Of course models are never exactly like the things being modeled.  But value occurs when the model shares and illustrates many features with the real thing.  In this experiment paper templates can be folded to represent atoms which join together in patterns characteristic of four common lattice.


  1. Print several pages of each paper template by either clicking each image to obtain full page gif images or by opening PDF files for lattices called
    1. Simple Cubic,
    2. Hexagonal Close Packed,
    3. Cubic Close Packed, and
    4. Body Centered Cubic.
    Construct a minimum of atoms to make each unit cell described in the table below.  (8 additional Cubic Close Packed atoms will be needed if you wish to construct Face Centered Cubic lattice.

  2. Cut out each atom along the outside of each figure, and fold backwards along each line.  Assemble each atom, gluing the shaded tabs under the neighboring side.

  3. While nearly everyone can imagine the packing of simple cubes, it may be worth constructing a layer so that at least one atom is surrounded by other atoms.  Stack (but do NOT now glue) atoms below and above so all sides of the chosen central atom have adjoining atoms.  Count the number of atoms that have adjoining sides to the central atom.  This is called the coordination number, a property useful when considering which substances use which lattice.
  4. Match and glue together trapezoid sides to form layers for a Hexagonal Close Packed lattice.  Assemble layers of atoms for a unit cell described in the table below but do NOT now glue layers together.  The larger layer should have a central atom with the others around.

  5. Similarly assemble layers to eventually make unit cells or Cubic Close Packed lattice and Body Centered Cubic lattice.  Match and glue parallelogram sides to make layers for Cubic Close Packed lattice.  Unlike in the close packed lattice where atoms adjoin their neighbors with a layer, the atoms in a layer of Body Centered Cubic lattice do not touch!  To assemble the Body Centered Cubic lattice, glue the triangular corners of the central atom to upside down triangular corners of the other atoms.  That is, if a corner of one triangle is up, the attached triangle should not match, but have its corner down, with only the centers of the triangles aligned.  (Yeah, kind of strange!  You might want to ponder why this needs to be.)
Lattice Unit Cell layering Glide
Features Examples
Simple Cubic 4 + 4 = 8
Hexagonal Close Packed 3 + 7 + 3 = 13
Cubic Close Packed 3 + 7 + 3 = 13
Body Centered Cubic 4 + 1 + 4 = 9
Face Centered Cubic 1 + 6 + 6 + 1 = 14

  1. Notice that the layers of atoms in the two close packed lattices nestle into the adjoining layers.  With the layers still not glued together gently try to slide the layers from side to side.  Compare the difficulty doing so with sliding layers in the Simple Cubic lattice.  This is called glide difficulty, and is one of the factors on how easy a solid, such as a metal, deforms.  If enough Body Centered Cubic atoms are constructed to conduct a similar test, you would find that its atoms are even more deeply nestled and so are even harder to glide.

  2. Just as you did with the Simple Cubic lattice, for each of the other lattices count the coordination number of neighbors sharing sides with a surrounded atom.  This too is a factor determining hardness: The more neighbors restraining an atom, the more difficult the motion.  The number of valence electrons per atom is a third factor.

  3. Note the hexagonal shape of the two largest layer pieces in Hexagonal Close Packed and Cubic Close Packed.  Count the dimples on the tops and bottoms of both.  Place one of the appropriate smaller layers under each and note how many of the dimples are occupied by the peaks of atoms.  Why aren't the other three used?  Now also place the remaining appropriate small layers on top of the large layers.  Note whether the top atoms occupy the same positions as the bottom layer, or the alternate positions.  Do the position of atoms alternate every other layer, or every third layer?  (...finally differences between the two kinds of lattice!)

  4. To construct a unit cell of Face Centered Cubic lattice, retrieve the small 3 atom pieces of Cubic Close Packed layers.  Add three additional atoms to one side of each so that there are rows of 1, 2, and 3 atoms long forming a larger triangular piece of layer.  Nestle the two pieces together so the triangles point opposite directions.  Add single atoms to the exposed face of both triangular pieces.  Note that you have constructed a cube!  But unlike the simple cube (where there are only corner atoms), or the Body Centered Cube (where an atom is in the middle of the body holding the corners apart), here there is an atom in the middle of each of the six faces of the cube (thus it is called Face Centered Cubic).  But note also that the layers and its atoms are still Cubic Close Packed as well.  So these two names describe the same lattice, Cubic Close PackedFace Centered Cubic, as seen rotated so that their unit cells are chosen from different viewing perspectives.

You might reflect on the difficulty envisioning the three dimensional geometry.  This may be a skill area than many people haven't well developed?

Communicating technical information such as observations and findings is a skill used by scientists but useful for most others.  If you need course credit, use your observations in your journal to construct a formal report.

Related information

Metalsperiodic chart of metals

Recall that there is an energy advantage (universal preference for minimum stored energy; e.g., balls roll down hill!) for atoms to have fully occupied electron energy levels (or equivalently, empty energy levels).  So chemicals react so their atoms with partially filled energy levels either empty or fill those levels.  In some cases this is achieved by taking or giving electrons (resulting in ionic bonding), but other substances share electrons to fill energy levels (resulting in covalent bonding).

The substances, classified as metals since long before electrons were known, occupy the left side of typical periodic charts, indicating they have few valence electrons.  Having few valence electrons, metals often relinquish those valence electrons to non-metals, forming ionic salts.  Since metals have less than four valence electrons, in the absence of nonmetals pure metals cannot reach minimum stored energy by either traditional covalent bonding or ionic bonding.  Metals solve that apparent dilemma by using a variation of covalent bonding.  Instead of sharing pairs of electrons between two atoms, valence electrons are communally shared with all neighboring metal atoms.

This sea of communally shared electrons which metal atoms share with neighbor atoms is responsible for the well known metallic properties: Because metal atoms have few valence electrons, metallic elements prefer lattices with large coordination numbers.  These lattice provide more neighboring atoms and thus more valence electrons for sharing.  (Why are the lattice used by many 7th period metals ...the white boxes at bottom of chart... still undetermined?)

Ionic SolidsNaCl crystals

Ionic Solids are the result of a non-metal with high electronegativity essentially pulling one or more electrons away from a metal with low electronegativity resulting in ions with opposite electrical charges.  Since opposite electrical charges attract, the ions pack together in lattices that allow opposite charged ions to be closer together than identically charged ions.  If you consider arranging ions in either of the close packed lattices, you will find it impossible to do so without having adjoining identical ions.  Thus ionic solids seem to prefer the less compact forms of lattice rather than a close packed lattice.  Can you guess the type of lattice in the NaCl (table salt) shown at right?

If you are not familiar with cleavage, you might view a short video of a NaCl crystal being cleaved.

Covalent Solids

For covalently bonded materials, the shape of the particular molecule is most important.  (The non-metals on the chart are white because at room conditions they form molecules rather than pack in lattice.)  Typically the bonds within each molecule are strong while bonds between molecules are much weaker.  So if the molecules are relatively small, the substances are often gases or liquid at room temperature.  But larger molecules and network solids (held together by a network of covalent bonds) may use lattice structures.  And smaller molecules form solids at higher pressures or lower temperatures.  Can you guess the kind of lattice in the SiO2 (quartz) shown at right?

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created 1/17/2006
revised 1/27/2006
by D Trapp
Mac made