Cohesion of Metals Cohesion of Metals, Hydrogen Bonding, Molecular Solids Metals � Metals are characterized by a high electrical conductivity (inverse of resistivity), and a large number of electrons[.]
Cohesion of Metals, Hydrogen Bonding, Molecular Solids Metals • Metals are characterized by a high electrical conductivity (inverse of resistivity), and a large number of electrons which are mobile throughout the lattice Conductivity (units of 1/ohm) Current density J=δE Electrical field • These electrons are called conduction electrons • Most pictures of metals are of the postively charged cores, with a sea of mobile conduction electrons • Metals tend to be shiny and are malleable because of this sea of electrons (weakly bound positive charges) They can also form alloys easily (weak, non-directional bonding) • Metals also interact with visible light (why?) Metallic Gold (Au) Sea of valence electrons Metals • • • • The cohesive energy for elemental metals (ie Li, Na, K) is typically around 1-4 eV/atom This tends to be less than ionic solids (eg ~8 eV/atom for NaCl) This is, in general, because the ions in a metal are only weakly held together in the sea of electrons The cohesive energy is higher for d-electron elemental metals (ie W) + + + + sea of electrons + + + + This configuration lowers the free energy of the system for some materials However, as you might imagine, the cohesive energy is usually much less than say for the ionic solids Cohesive Energies Transition metals – strongly bound Alkali metals – Intermediate values Inert gases – weakly bound Metals • • • • One of the reason why metals form is because of the uncertainty principle : ∆x∆px ~ h Delocalizing the electrons increases ∆x, which lowers the momentum spread of the particles (and therefore lowers the energy of the electrons) However, the kinetic energy of the electrons is so low that it leads to relatively weak binding on average (compared to say ionic solids) Metals tend to have a variety of melting points from low (Hg melts at -39 C) to high (W melts at 3410 C) This illsutrates that the forces holding metals together are complicated For the most part, the transition metals (d-electrons) have higher melting points, and therefore higher cohesive energies, because of the large overlap of the extended d-electron orbitals, which are larger than s and pshells This provides more of an extended overlaping orbital picture instead of a completely delocalized picture (a snooker ball floating in a pool of Hg) Metal Melting pt (C) Silver 961 Copper 1083 Mercury -39 Iron 1535 Tungsten 2410 Sodium 98 d-electron overlap • • • So, some materials (even oxides) show metallic behaviour (ie easily delocalized electrons) if they just have large, extended dorbitals Think of the electrons as moving in rivers of overlapping d-orbitals – the binding forces are stronger than that of completely delocalized electrons For the most part, though, metals are elements, or alloys of different elements, and they have FCC, HCP, or BCC structures Iron oxide, Fe3O4, forms from a complicated network of overlapping p and d orbitals This basically delocalizes the electons, so they can hop around from site to site and the material is a metal (and is a ferromagnet) Cohesion in free electron metals • The cohesive energy can be written as: U = Ucoul + Ukin + Uexchange • • • Where we have terms for the coulomb energy, the kinetic energy of the conduction electrons, and the exchange energy (which is a quantum mechanical property) The coulomb energy is similar to the Madelung energy, which in this case is the total electrostatic attraction between the lattice of positive charge, and a mostly homogeneous sea of negative charge (difficult to calculate!) The kinetic energy of the electrons is determined by the Pauli Exclusion Principle (they can’t all have the same energy, so they end up forming a “band” of energies up to a maximum energy called the Fermi Energy We will see more of this in Chapter Six) The exchange energy is an attractive energy term (!) driven by the fact that electrons are fermions Exchange Energy • • • • • • The exchange energy for electrons in a metal is actually negative, meaning that it lowers the energy of the system What is this mysterious term? Let’s look at two electrons moving in a lattice Each of them has a wave function that looks like Φ(r) ~ eikr The total wavefunction for this system is a product of the two functions However, this product has to flip sign if we exchange two electrons, because they are fermions (the wavefunctions have to be antisymmetric when you exchange them This is a consequence of the Pauli Exclusion Principle) Therefore, our total wavefunction ψ ~ Φa(r1)Φb(r2) - Φb(r1)Φa(r2) (for particles and that can be in states a and b) The exchange energy varies with 1/(distance between the particles) So, we have to calculate = Uex Exchange Energy • So, we have (using 1/r12 for 1/|(r1-r2)|): < φa (r1 )φb (r2 ) − φb (r1 )φa (r2 ) φa (r1 )φb (r2 ) − φb (r1 )φa (r2 ) > r12 =< φa (r1 )φb (r2 ) 1 φa (r1 )φb (r2 ) > + < φb (r1 )φa (r2 ) φb (r1 )φa (r2 ) > r12 r12 − < φb (r1 )φa (r2 ) 1 φa (r1 )φb (r2 ) > − < φa (r1 )φb (r2 ) φb (r1 )φa (r2 ) > (Exchange energy) r12 r12 • • (Direct energy) Note that the 2nd set of terms, where we exchange the coordinates of the electrons, is always negative So, this lowers the energy of the system (neat, huh?) and it is larger than the first terms This only works because electrons are fermions If they were bosons, then the particles would have a symmetric wavefunction (with a positive sign) and this energy would be positive Hydrogen Bonding • • • • The hydrogen bond is an exceptionally strong van der Waals type force that exists between covalently bounded hydrogen in one polar molecule, and covalently bounded oxygen, flourine, nitrogen, or chlorine in another (very electronegative atoms) Water molecules are the best example of this The structure of ice is due to the fact that hydrogen bonding keeps the H2O units together The cohesive energy of these materials is higher than the inert gas solids because of the unusual strength of the hydrogen bonds Ice has some peculiar properties due to hydrogen bonding (ie It is less dense than the liquid form!) (Hexagonal Symmetry Of Ice Crystal Growth) Hydrogen Bonding Stronger H-O bonds Molecular Crystals I2 units held together by covalent bonds Molecular I2 • • • • • Molecular solids, of which H2O is an example, compose the largest class of solids They are composed of molecular units on the lattice sites, typically FCC or HCP Examples: Solid I2, CO2 (dry ice), solid ethanol, biological molecules These are held together by hydrogen bonding, or weak van der Waals forces (induced dipoles) Ice is among the strongest, with cohesive energies on the order of ~ 10% of the average covalent bond (~10 eV/atom) Gases tend to form these as well, such as solid O2 and N2 (very low temperatures) (weak van der Waals forces between I2 units) Solid Melting Temp H 2O 0C NH3 -78 C O2 -218 C C2H5OH -117 C C6H6 5.5 C H 2S -85.5 C Molecular solids Inert gases (~ 1-2 eV/atom) (~ 0.02-0.2 eV/atom) Covalent bonds (~5-8 eV) Summary Alkali metals (~ eV/atom) d-electron Transition metals (~ 5-8 eV/atom) Last topic: Atomic Radii • • • • We can determine bond distances by x-ray diffraction How can we determine absolute sizes of the atoms? This becomes even more difficult for covalent bonds rather than for ionic (where does one atom end and the other begin?) Let’s see if this can first be done for the ionic case, where the atoms are more like isolated spheres We can often this by taking the neutral atoms (ie Na and F for the NaF structure), and adding/subtracting a tiny amount to account for the size of the ions (Na+ and F- in this case) Eg NaF, which has the NaCl structure How large are these atoms? Atomic Radii to Crystal Radii • • • • • • • • Table (Chap 3) of Kittel shows a list of radii for different ions (ie Na+, Li+, O2-, Cl-, etc.) You can often just use these for the atom sizes in ionic solids So, for NaF, this would be : 0.97 (Na+ value) + 1.36 (F- value) = 2.33 Å, which compares very well to the x-ray diffraction value (2.32 Å) For covalent structures, this is a bit trickier But, if the structure is similar, you can use the same atomic radii from structure to structure Example: the interatomic bond distance in diamond is 1.54 Å, so the radius of the carbon atom is 0.77 Å for this structure In Si, which has the same structure, the radius is 1.17 Å What is the bond distance of SiC (which has the ZnS structure)? D = 1.17 + 0.77 = 1.89 Å, which agrees pretty well with experiment (1.94 Å) (same as SiC) Atomic Radii to Crystal Radii • • The examples we have shown are very simple For structures with more or less nearestneighbours, the atomic sizes are distorted For ionic solids, the interatomic distance, D, can be represented as: DN = RC + RA + ∆N Radii of cation, anion • Correction term This depends upon the coordination (no of nearest neighbours) about each ion (if this is small, the distance will be smaller – atoms will be held tighter) N ∆N (Å) N ∆N (Å) -0.50 0.04 -0.31 0.08 -0.19 0.11 -0.11 10 0.14 -0.05 11 0.17 12 0.19 (determined by Zachariasen, table 10 of Kittel) An example: BaTiO3 (perovskite) • • • • • So, for something like BaTiO3, perovskite structure, we can try to determine the crystal radii (the atomic radii of Ba, Ti, and O for this structure) The Ba2+O2- units are 12-fold coordinated (12 O n.n for every Ba unit) The Ti4+O2- units are 6-fold coordinated Using the above table, which assumes ionic bonding, we can get: for BaO: ∆12 = 1.35 + 1.40 + 0.19 = 2.94 Å For TiO: ∆6 = 0.68 + 1.40 + = 2.08 Å Both of these results give a = 4.16 Å, which is close to experiment More sophisticated methods exist: (eg Valence Bond Theory How we know that Ti is 4+? It could be 3+, 2+, etc? Valence Bond Theory tells us the coordination number from bond distances – larger the bond distance, the less electrons are on the atom, and therefore the higher the oxidation state) Ti O O Ti Ti O O Ti O O Sr O O Ti O Ti O O Ti O Ti a (SrTiO3, which has the same structure as BaTiO3)