The simplest structures are those of elemental solids like metals.  The atoms are usually viewed as spheres, and the structures are defined by showing their unit cell.  Note a useful website is http://www.chem.lsu.edu/htdocs/people/sfwatkins/MERLOT/lattice/01intro.html.  This shows pictures of many of the lattice structures, as well as other information about them.

unit cell--a component of a crystal that reproduces the crystal structure when stacked together repeatedly. Therefore all of the structural information needed to describe the solid is contained in the unit cell.  Some sample unit cells (transparency) show how portions of atoms can be included in the unit cell (face 1/2, edge 1/4, corner 1/8).

primitive cubic (simple cubic) -- one atom on each corner of cubic unit cell.

The coordination number of an atom--is the number of nearest neighbors it has in the lattice.

body-centered cubic -- primitive cubic plus an atom in the middle.  

face-centered cubic -- primitive cubic plus atoms on each face.  (this is a closest-packed structure...)

Close-packed structures

Close-packed structures--are structures that, treating the atoms as spheres, have the minimum amount of unused space and the maximum coordination number for each atom.

• hexagonal close-packed--where the spheres of a third layer of atoms lie directly above the spheres of the first.  (ABA)

• cubic close-packed, or face-centered cubic--where the spheres of the third layer of atoms lie above the gaps in the first layer.  (ABC)

When talking about crystal structures of solids, a hole is an unoccupied space.

• octahedral hole--lies between 2 oppositely directed planar triangles of spheres in adjoining layers. It can accomodate a sphere of radius <= 0.414r (r = radius of lattice atom). There are N octahedral holes in a crystal of N atoms.

• tetrahedral hole--made from a capped triangle of spheres, these can accomodate spheres of radius <= 0.225r. There are 2N tetrahedral holes in a crystal of N atoms.

Diamond

Very covalent bonding, tetrahedral coordination about each carbon.

Properties of metals:  

metallic conductor--a substance with an electric conductivity that decreases with increasing temperature.

dislocations -- imperfections in metallic crystals, allowing deformation to happen more easily

Generally larger atoms (or ions) are closest packed, and the smaller ones go into either the tetrahedral or octahedral holes.  The structure depends on 1. radius ratios, and 2. formula.  (e.g. N octahedral holes in a crystal of N atoms is consistent with a 1:1 stoichiometry)

Radius Ratios:

a.  Limiting radius ratios are calculated from simple geometric considerations, for example bcc is shown on transparency. 

b.  The lattice type of a particular ionic compound is affected by the relative ionic sizes; (transparency, table 7-1), e.g. for NaCl r+/r- = 116pm/167pm = 0.69.  Table 7-1 shows that a ratio of 0.414 would allow a C.N. of 4, so this cation is too big for that.  Therefore it would go into the  C.N. = 6 category, octahedral. 

Note that these are rough predictions, and there are many exceptions.

Structure Types:

Named for the most common compound with that structure.  See figures.

• Sodium Chloride--fcc of Na and fcc of Cl, filling each other's spaces
• Cesium Chloride -- Cs and Cl atoms form simple cubic structures, again in each other's spaces
• Zinc Blende (ZnS) -- like diamond but alternating Zn and S
• Wurtzite (ZnS) -- Zn and S form hexagonal close packed structures, in each other's spaces
• Fluorite (CaF₂) -- Ca atoms are ccp, with F ions in the tetrahedral holes
• NiAs -- As atoms are in AAA close-packed layers, with Ni ions in octahedral holes
• Rutile (TiO₂) -- TiO₂ octahedra with shared edges

e.g. Na⁺_(g) + Cl^-_(g) --> NaCl_(s) DH = -U (exothermic)

Lattice energy is defined as the energy released when ions come together from infinite separation to form a crystal.

a. calculated by theoretical models such as :

U =(MN_Ae²Z_AZ_B)(4pe₀r₀)^-1 (1-30/r₀)

where M = Madelung constant (depends on lattice type), e = charge on electron, N_A = Avogadro's number, Z_N = charge on atom N, e₀ = vacuum permittivity, and r₀ = separation between nearest neighbors in pm.  Note that you do NOT need to memorize this equation, but you should be able to use it.

This generally predicts U ~3-5% lower than experimental values (see transparency). You must know the lattice type to obtain M. The agreement between this equation and the experimental value of U is often used to estimate the solid's ionic character.

b. U can be experimentally determined by a Born-Haber cycle, e.g.:

DH_f = DH_lat + IE + DH_EA + DH_AM(DH_s) + DH_AX(DH_d)

(all quantities except DH_lat can be experimentally determined)

(DH_s = sublimation energy, and DH_d = standard enthalpy of dissociation, DH_f = heat of formation, DH_lat = heat of lattice formation.)

Practice for NaCl:  From Table B-2, IE for Na is 495.8 kJ/mol.  From Table B-3, EA for Cl is 349.0 kJ/mol.  Bond dissociation energy for Cl₂ is 239.7 kJ/mol.  Heat of sublimation for Na is 108.4 and NaCl heat of formation is -410.9 kJ/mol.  What is the calculated U? 

*Be careful about signs when doing Born-Haber cycles!   EA = -DH_EA , DH_lat is usually negative,  DH_f is often negative...

Predicting Stable Lattices:  U negative,   stable oxidation states...

Solubility

Experimentally, it is found that large cation/large anion combinations form relatively stable lattices--they don't dissolve as easily as big/small combinations, and don't decompose as easily.  This is most simply explained by hard/soft arguments.  Hard-hard bonding tends to be favored over hard/soft, and in general hardness trends follow size trends.  If there is a big size mismatch there is likely to be a hard/soft mismatch as well, which would destabilize the lattice.  

You can imagine a solid as an indefinitely large molecule. Each constituent atom will have atomic orbitals that must be combined with the atomic orbitals of the other atoms to make molecular orbitals. But if there is an infinite number of atoms, there will also be an infinite number of orbitals. In fact both of these numbers are finite, but they are very large.

You can imagine mixing atomic orbitals one by one to make more and more MO's until there is what appears to be a continuum of available orbital energies, known as a band. The band formed by mixing all of the s orbitals is called the s band, the one formed from the p orbitals the p band, and so on. Just like in discrete molecules, if the s and p orbital energies are close they can interact with each other; in the metal this is evidenced by overlap of the s and p bands. If the energy difference is large enough, there will be a gap between the different bands in the solid.    

You can fill the solid molecular orbitals using the same building up principle as before, and you can end up with a HOMO. But in an extended solid, where we have defined bands rather than specific orbital energies, the highest occupied orbital energy level is described not as the HOMO but rather as the Fermi level (at 0 K). The energy of the Fermi level is called the Fermi energy, E_F.   The highest energy band with electrons is the valence band, while the lowest band with holes (electron vacancies) is called the conduction band.  

As you increase the temperature of the solid, you give enough energy to it to excite electrons out of the lowest energy levels (the energy difference is very small) so that you end up with a distribution of electronic energies. The resulting population of the energy levels is called the Fermi-Dirac distribution.  At room temperature the Fermi level is the energy at which an orbital is equally likely to be filled or empty (essentially the centerpoint of the Fermi-Dirac distribution).  

Density of states--refers to how many orbital energy levels there are in a given increment of energy. There is a higher density of states where the orbitals are closely spaced in energy, and lower where they are spaced far apart.

Semimetals--have a full band and an empty band that are right next to each other in energy but have a zero density of states right at the intersection.

Metalloid--is a solid with character intermediate between that of a metal and that of a nonmetal--it is not necessarily the same as a semimetal.

X-ray emission and absorption--are techniques analogous to photoelectron spectroscopy, used on solids to help determine their band structure.

Types of Solids:

metallic conductor--a substance with an electric conductivity that decreases with increasing temperature.

semiconductor--a substance with an electric conductivity that increases with increasing temperature.

insulator--is a substance with a very low electrical conductivity.

Conductivity requires either: a partially filled band, or overlap of filled and empty bands.

In an electric conductor, electrons are able to move relatively easily through the solid. The electron movement can be viewed as changing molecular orbitals; and if there are a number of available orbitals that are closely spaced in energy then it is easy for the electrons to change orbitals with only slight perturbations. An electric field is one perturbation that will slightly shift the energies of certain orbitals relative to each other, causing electron movement (to the now-lower-energy orbitals).  The reason that increasing temperature decreases conduction in a metallic conductor is carrier scattering: vibrations of atoms in the metallic lattice that disrupt the smooth flow of electrons from one side to the other of the molecule.

Insulators are solids that have a full lower band, and a significant energy gap between it and the next empty band. Thus the electrons do not have readily accessible empty energy levels to move into.

Semiconductors are solids that have a band gap energy intermediate between that of a conductor and that of an insulator. They can be either intrinsic or extrinsic.

• An intrinsic semiconductor is one that has a small enough band gap to allow some electrons to populate the upper band , leaving some electrons in the upper band, and some "holes" in the lower band to allow conductivity.

• An extrinsic semiconductor becomes semiconducting when "doped."

• A dopant either adds or subtracts electrons to give a conduction band:

• p-type (positive holes)--e.g. Si doped with B

• n-type (negative charges)--e.g. Si doped with P

These doped semiconductors can be viewed either with a "localized" or "delocalized" model; see handouts.  The color of a semiconductor is related to its band gap energy E_g; and the band gap (and therefore color) can be tuned by the dopant (see handout).  Remember that E = hn and ln = c (c = speed of light in a vacuum, 2.998 x 10⁸ m/s), so the absorbed photon has wavelength < hc/E_g.  

Diodes are formed by p-n junctions; a depletion zone is formed between the two semiconductors (see handouts).  The p-n junction can be drawn with band diagrams.  

If you put the diode in a circuit, you can either 

1. reverse bias--the depletion zone increases, it becomes more insulating
2. forward bias--remove the depletion layer, current can flow
3. reverse bias + photons--despite depletion zone, photons promote electrons into the conduction band, and current can flow (light-sensitive switches)
4. no bias + photons--photovoltaic cells; electrons promoted and current flows

At the junction, as current flows, electrons are dropping energy and going into empty orbitals, releasing energy in the process.  Will the energy be released as heat, or light, or something else?  If you prefer light (to form an LED), you will need a "direct band gap" material such as GaAs.  If the transition includes a change in momentum or vibrational energy it will create phonons, thermal energy.  Direct band gap materials will have radiative transitions, and release photons.  Finally, a third layer can be used to make a laser.  

superconductors--are substances with zero electrical resistance.

Meissner effect--the property of expelling all magnetic flux when cooled below the critical temperature T_c, at magnetic fields below the critical magnetic field value H_c.  Type I superconductors exhibit this effect.  

Type II superconductors have three regions; 1. exclude all magnetic field, 2. exclude part of the magnetic field, 3. normal conductance.

Used in NMR spectrometers, trains, and other applications.  

Cooper Pairs--theory of superconductivity that electrons pair in solid to form "Cooper pairs" that travel easily through the solid.

Ionic model of binding--treats the atoms in the solid as ions interacting by simple coulombic forces (electrostatic attraction and repulsion).  This is used for lattice energy calculations.

Properties:

• conduct electricity in solution and in molten state, but not in solid state.

• high mp, bp (breaking of strong bonds must occur)

• hard and brittle

• soluble in high dielectric constant solvents (why?).

Mⁿ⁺/X^n- where M = group 1,2, or 13(IIIA) metal (except boron) or most transition metals

X = group 17 (VIIA) or 16 (VIA), sometimes 15 (e.g. P^3-)