Crystals of Inert Gases Crystals of Inert Gases Van der Waals interaction � Last time, we showed that for 2 coupled harmonic osciallators, there were 2 modes of vibration These correspond to symmetric[.]
Crystals of Inert Gases Van der Waals interaction • Last time, we showed that for coupled harmonic osciallators, there were modes of vibration These correspond to symmetric and antisymmetric vibrations: • Symmetric vibrations: Antisymmetric vibrations: 2e 2e ω s = ωO 1 − − + CR CR + - + - 2e 2e ωa = ωO 1 + − + CR CR + - + - Van der Waals interaction • For these coupled oscillators, the zero point energy is defined as being the sum of the lowest energies of the harmonic oscillator: ε n = (n + / 2)hω = / 2hω ( n = 0) • So, for these two modes of vibration, the lowest energy is: U= 2 2e 2e 1 1 1 ) hω s + hωa = hωO + h(− h ω h ( ) + + − O CR CR 2 2 2 2 2e A = − → ∆U = −hωO • CR R6 • Where in this case, ∆U is the change from uncoupled oscillators which would have energies of ½ ħωO each Van der Waals Interaction • So, the net results is that the induced dipole-dipole interaction gives rise to a reduction in energy that goes like 1/R6 • This is entirely a quantum effect: test this by letting ħ→0 In this case, ∆U →0, so this effect wouldn’t occur for classical systems This interaction is what keeps inert gas crystals together (and also many organic molecules) It does not depend upon the overlap of charge densities of the two atoms “A” is a constant, which is usually determined experimentally (but it can also now be calculated from first principles) What happens when the atoms get close together – like charges tend to repel one another (or they? What about chemical bonding? Isn’t this overlap of electronic orbitals? How they not repel one another to form bonds?) • • • • Repulsive Interaction • • • • • As two atoms are brought close together, their charge distributions gradually overlap At close separations, the overlap provides a Inert gas atoms –filled shells repulsive force (and therefore, increases the total energy of the system) This is largely due to the Pauli exclusion principle: electrons cannot have all their quantum numbers equal So, if both spins are pointing upwards, for example, they will be in Ne Ne the same quantum state if their orbitals overlap – they will repel one another The Pauli exclusion principle is what creates the periodic table – the electrons cannot all be in the These atoms repel one another lowest ground state, so they stack up in different energy states and create atomic structures For closed shell atoms (like inert gases), in order Unfilled shells – bonding can occur to get a new quantum number for overlap, the electrons would have to be excited to a higher energy state (there is no room in the orbitals to accommodate other electrons) So, this does not happen – the atoms just repel one another However, if the shells aren’t filled, they can occupy other orbital states – electrons can occupy each orbital (ie in s orbitals, in p, but these are grouped into px, py, and pz) This is why you can have bonding in these materials – electronic overlap occurs, but the spins are paired up (usually one up and one down) and therefore not in the same quantum state Pauli Exclusion Principle • To illustrate this effect, consider fig from Kittel In this case, the H atoms are pushed close enough to overlap + H 1s H 1s + H 1s Total electron energy = -78.98 eV = He 1s2 = H 1s Total electron energy = -59.38 eV He 1s2s (can’t be in same orbital) Therefore, the Pauli exclusion principle has increased the energy by 19.60 eV (neglecting the proton repulsions) This is huge effect – we never see bonding in inert gas crystals because of this The Repulsion Term • • The raising of energy from the repulsive term is difficult to calculate Experimentally, we can measure this, and it is known to be empirically: B/R12 (B is a constant) So, the total potential energy for two atoms at separation R is: This is called the Lennard-Jones Potential • The force between atoms is given by –dU/dR Lennard-Jones Potential Minimum – equilibrium position The parameters increase as the size of the atoms increase Equilibrium Lattice Constants • • • • • • So, our atoms will tend to sit at equilibrium positions, attracted by the dipolar interactions, but repelled if they get too close For N atoms, we can some up the potential energy terms: Where pIJR is the distance between atom I and atom J, expressed in terms of the nearest neighbour distance R The factor of ½ occurs with the N to make sure we don’t count pairs twice You can calculate these terms For the FCC structure, we find: Interesting: there are 12 n.n for the FCC lattice This means that the nearest neigbours contribute mostly to the interaction energy of the FCC crystal HCP lattice: we get 12.13228 and 14.45489 (almost identical Would you expect this?) Equilibrium Lattice Constants • If we take this equation to be the total energy of the crystal, we can find the equilibrium value for these atoms in the FCC structure by setting the derivative to be zero (ie All the forces are balanced, or the energy is minimized for this configuration) • And therefore, solving we find: • This is the same for all elements with an FCC structure (neat, huh?) The observed values for the inert gas elements are: Ro/δ • Ne 1.14 Ar 1.11 Kr 1.10 Xe 1.09 (a remarkable result) The departure from 1.09 for light atoms is due to zero-point motion (lighter atoms – more motion, and for He, we don’t get a crystal at all) Cohesive Energy • The cohesive energy of inert gas crystals at absolute zero and at zero pressure can be obtained by substituting the values for an FCC gas into our energy equation: • • This is the same for all inert gases Remember, this is for atoms which are at rest in their equilibrium positions The atoms are actually in motion (due to quantum effects), so this reduces the binding energy (the equation above) by 28, 10, 6, and percent for Ne, Ar, Kr, and Xe (again, notice the trend – this effect is largest for smaller atoms) Neat tidbit: the lattice constant of a Ne20 crystal (lighter isotope) is slightly larger (4.4644 Å) than Ne22 (4.4559 Å) at the same temperature This is again, because the higher quantum kinetic energy expands the lattice slightly • Ionic Crystals • • • • These are made up of positive and negative atoms Typical example: Na+ClThe ionic bond arises from the electrostatic interaction of oppositely charged ions Cs+Cl- is another example The electronic configuration of all ions of a simple ionic crystal corresponds to closed shells In this case, however, the atoms have residual charge, and this creates an attractive interaction which holds the lattice together NaCl structure CsCl structure Ionic Crystals • • • • • Example of electronic configurations: LiF Li: 1s22s, F:1s22s22p5 Li+:1s2, F-:1s22s22p6 These atoms are stable because they have closed shells They are spherically symmetric (x-ray diffraction shows this as well – see figure in Chapt of Kittel) • As mentioned before, when forming an ionic crystal, the cohesive energy is not just determined by the ionization potential, I (the energy needed to form a positive ion), and the electron affinity, E (energy gained in forming a negative ion) There is also the electrostatic attraction which has to be taken into effect An example: NaCl • • • NaCl can be thought of as forming in steps: Na + 5.14 eV (Ionization Energy) → Na+ + ee- + Cl → Cl- + 3.61 eV (Electron Affinity) Na+ + Cl- → Na+Cl- + 7.9 eV (Cohesive Energy) So, the total energy, per molecule unit of NaCl of a crystal is: (7.9 – 5.1 + 3.6) = 6.4 eV lower than the energy of separated neutral atoms Where does the cohesive energy come from? A large part of it is just from the electrostatic energy of the Na+Cl- atoms We can, for example, calculate this by taking atoms which are 2.81 Å apart (separation distance for NaCl atoms in the crystal), and find what the attractive potential is: R QQ (1.602 x10 −19 C ) • (−1.602 x10 −19 C ) (SI) V= Na+ • Cl- 4πε o R = 4πε o (2.81x10 −10 m) V = −5.1eV So, a large part of this energy is just from the electrostatic attraction (it is close to the experimentally found value of 7.9 eV) We can a better job though (this only takes into account pairs Actual solids have many nearest neighbours) The Electrostatic, or Madelung Energy • • • • • What forces hold ionic crystals together? The van der Waals force is still here, but it only accounts for 1-2 % of the cohesive energy (which, remember, is larger for ionic crystals than inert gas crystals) The electrostatic energy is the prime culprit We have to think of a way of calculating the electrostatic energy for a whole lattice, which has a huge number of atoms, with attractive (for unlike charges) or repulsive terms (for like charges) that have energies +/e2/R This seems like a very complicated problem, but it helps that the forces die off with increased distance The energy associated with the electrostatic attraction of all the atoms in the lattice is called the Madelung energy + - + - + + - + - - + + - + - + + - + - + + - + How we calculate the total electrostatic energy of a lattice of N atoms? The Madelung Energy • If UIJ is the interaction energy between atoms I and J, we define a sum UI which contains all interactions involving the ion I: UI = Σ’ UIJ J • • • Where the summation includes all ions except J=I (the same atom) Let’s assume that the only forces acting on the ions are due to (1) an electrostatic force (which give rise to an attractive or repulsive force) and (2) a repulsive force due to atomic overlap (remember, these Na+ and Cl- atoms have filled shells, so they don’t want to get close to each other), which we will use the form λexp(-λρ) (this has the similar R dependance as the B/R12 repulsive term, but it is easier to deal with) Note that λ and ρ are empircal parameters Therefore, for each atom I, we have terms for the interaction with atoms J which will depend upon the distance rIJ between atom I and atom J: UIJ = λ exp(-rIJ/ρ) +/- q2/rIJ (+ for like charges, - for negative) (CGS units) The Madelung Energy • • Now, the constants λ and ρ can be found experimentally (usually by measuring the compressibility) We can now take this sum However, life will be easier if we realize that the repulsion term (from electronic overlap) will only be significant for nearest neighbours Also, we will introduce rIJ = pIJR (R = nearestneighbour separation) UIJ = • { q2 λ exp(− R / ρ ) − R q2 ± p IJ R (nearest-neigbours) (otherwise) So for 2N ions, where we have N bonds, the total energy is: Utot = NUI= N(zλe-R/ρ –αq2/R) where z is the number of nearest neighbours, and α is the Madelung constant The Madelung Constant • The Madelung constant, α, is defined to be: (±) α = ∑' pIJ J • • The pIJ values are constants, which are different for each neighbour J The signs change too, depending if the neighbour is a like charge (+) or unlike charge (-) The rest of this section will discuss how to calculate these for different crystals Some examples: NaCl CsCl ZnS α 1.7475 1.7626 1.6381 Cohesive Energy, in terms of the Madelung Constant • Note that for equilibrium separtion, dUtot/dR = 0, so dU λ α Nz N q i Utot = NUI= N(zλe-R/ρ –αq2/R) N =− exp(− R / ρ ) + =0 ρ dR R R=RO • Or, by rearranging: RO exp(− RO / ρ ) = ραq / zλ • This determines the equilibrium separation, Ro We can then write the total lattice energy in terms of this U tot Nα q ρ =− (1 − ) R RO • The Madelung Energy is –Nαq2/Ro We will solve for ρ in the next section, which will tell us how short-ranged the repulsive force is (it is usually ~ 0.1 Ro)