Solid State Physics - Philip Hofmann

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described here holds mainly for simple metals with only s and p electrons in the outer shells. For these, metallic bonding is not as strong as covalent or ionic bonding, but the cohesive energy can still amount to almost-equals1–2 eV per atom. Stronger bonding is found in transition metals, i.e. metals that have partially filled d shells in addition to the s and p electrons (an overview of the cohesive energies is provided in Figure 4.16). The explanation for this is that a mixed bonding situation arises in these cases. The s and p electrons turn into delocalized metallic conduction electrons, whereas the d electrons create more localized, covalent‐type bonds.

      Hydrogen atoms possess only one electron and can form only one covalent bond. If this bond is formed with a very electronegative atom (like F or O), the electron is mostly located close to that atom and the hydrogen nucleus represents more or less an isolated positive (partial) charge. This can lead to a considerable charge density because of its small size, and it can therefore attract negative (partial) charges in other molecules to form an electrostatic bond. This type of bonding is called hydrogen bonding. It is usually quite weak, but in some cases, the cohesive energy can be up to several hundred meV per atom. It is responsible for the intermolecular attraction in (liquid and solid) water and for the bonding of the double helix strands in DNA.

      Several of the bonding types discussed here are identical to those relevant for the formation of molecules (with the exception of metallic bonding). They are therefore discussed in great depth in the literature for chemistry and molecular physics. A good overview on bonding in solids is given in:

       Ibach, H. and Lüth, H. (2009). Solid State Physics, 4th edn. Springer.

       Kittel, C. (2005). Introduction to Solid State Physics, 8th edn. John Wiley & Sons.

      Discussion

      1 2.1 Why is a typical interatomic potential, such as the one in Figure 2.1, asymmetric?

      2 2.2 Which elements are likely to form crystals through ionic bonding?

      3 2.3 What types of forces are important for ionic bonding?

      4 2.4 How does the lattice energy in an ionic crystal depend on the interatomic distance?

      5 2.5 Explain the difference between cohesive energy and lattice energy.

      6 2.6 Which elements are likely to form metals?

      7 2.7 Where does the energy gain in metallic bonding come from?

      8 2.8 What is the difference between a simple metal and a transition metal (definition and typical physical properties)?

      9 2.9 Why is van der Waals bonding much weaker than most other bonding types?

      Basic Concepts

      1 2.1 Interatomic potential: Consider the model interatomic potential as a function of interatomic distance given by the equation(2.15) where and are parameters of the model.What is the physical origin of the strongly repulsive part of the potential?The Coulomb repulsion between the two nuclei.The increase of the electron energies due to the overlap of the electron clouds.What is the equilibrium distance between two atoms if their interaction is described by the potential in Eq. (2.15)?....What is the minimum value of ?

      2 2.2 Cohesive energy: In an ionic solid, which energy is the largest?The cohesive energy.The lattice energy.Depends on the particular ionic solid.

      3 2.3 Bonding and cohesive energy: When ordering the following elemental solids according to their cohesive energy (smallest to largest), which is the correct order?Pb, Ar, W.Ar, Pb, W.W, Pb, Ar.Ar, W, Pb.

      4 2.4 Bonding type and structure: Which bonding types do typically lead to close‐packed crystal structures?Covalent bonding.Metallic bonding.Van der Waals bonding.A. and B.B. and C.

      Problems

      1 2.1 Ionic bonding: Calculate the potential energy for an ion in a sodium chloride crystal (the interatomic distance is 2.81 Å) in units of electron volts and joules. Neglect the influence of the repulsive potential. From this, calculate the lattice energy of sodium chloride and compare the result to the experimental value of 776 kJ . Also, calculate the cohesive energy in the same units.

      2 2.2 Ionic bonding: The Madelung constant for a three‐dimensional NaCl crystal was given in Section 2.2. (a) Derive the Madelung constant analytically for a one‐dimensional NaCl chain as the one shown in Figure 2.5. (b)* Calculate the Madelung constant numerically for a one‐dimensional, two‐dimensional, and three‐dimensional NaCl lattice and plot the result as a function of the number of neighbor “shells” included in the computation. Compare the result for the one‐dimensional case to the analytical value from part (a).Figure 2.5 One‐dimensional chain of ions.

      3 2.3 Covalent bonding: (a) Given the linear system of equations (2.7) and the solution in Eq. (2.8), show that for the bonding “” solution and for the antibonding “” solution . (b) We could thus choose the resulting wave functions to be , but these are not normalized. Calculate the proper normalization factor, assuming that the atomic wave functions and are normalized.

      4 2.4 Metallic bonding: The most important contribution to the stability gained by metallic bonding is the lowering of kinetic energy. To verify this, consider an electron in a one‐dimensional box, inside of which the potential is zero and outside of which the potential is infinite. Consider first a box with a length corresponding to the size of an atom – say, twice the Bohr radius – and calculate the lowest energy eigenvalue (you can consider any standard text in quantum mechanics for help with this problem). Give the result in electron volts. Clearly, this energy is only kinetic energy. By how much is the kinetic energy lowered when you increase the size of the box by a factor of 10, so that it is roughly the size of the interatomic spacing in a crystal?

      5 2.5

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