1. Objectives By the end of this section you should: understand the concept of the radius of an atom or ion know how the Lennard-Jones [12,6] potential describes the interaction between atoms be able to calculate the van der Waals radius, the distance between the atoms that minimises their energy (PX only) know the trends in ionic radius with coordination number, oxidation state, group know about the radius ratio and be able to calculate this for octahedral and 8-fold coordination

2. Each atom exludes other from the space it occupies Attraction? Electrons are moving so that, at some instant, distribution is uneven Positively (electron-deficient) and negatively (electron- rich) charged regions  electrical dipole Dipole induces an opposing dipole in neighbouring atom  attraction Close Packing

3. Attractive force is known as: van der Waals interaction London interaction induced dipole-induced dipole interaction

4. The total potential energy for two atoms a distance, r, apart can be written as: This is called the Lennard-Jones (12,6) potential function First term is repulsive, second term is attractive. We want to find a minimum - so differentiate w.r.t. r

5. This is the van der Waals radius, the distance between the atoms that minimises their energy

6. Substituting back in to the (12-6) potential gives the minimum energy:

7. Energy has a minimum value of -  at the van der Waals radius

8. Ionic radii and bond distances Ionic radii cannot be “measured” - estimated from trends in known structures (reference - Shannon, Acta Cryst. (1976) A32 751) Oxide ion: r 0 taken as 1.26 Å

9. Refs: Krug et al. Zeit. Phys Chem. Frankfurt 4 36 (1955) Krebs, Fundamentals of Inorganic Crystal Chemistry, (1968)

11. Notes:  Ion radii for given element increase with CN

12. Notes:  Ion radii for given element increase with CN  Ion radii for given element decrease with increasing oxidation state/positive charge

13. Notes:  Ion radii for given element increase with CN  Ion radii for given element decrease with increasing oxidation state/positive charge  Radii increase going down a group  Anions often bigger than cations

14. Radius ratio rules Rationalisation for octahedral coordination: R= radius of large ion, r=radius of small ion

15. If r/R < 0.414, the cation is too small and can “rattle” inside the octahedral site If r/R > 0.414, the anions are pushed apart If r/R > 0.414, coordination changes: A simple prediction tool, but beware - it doesn’t always work! Coordination Minimum r/R Linear, 2- Trigonal, 30.155 Tetrahedral, 40.225 Octahedral, 60.414 Cubic, 80.732 Close packed, 121.000

16. Radius ratio rules Rationalisation for 8-fold coordination: Radius ratio rules Rationalisation for 8-fold coordination: Unit cell edge a = 2R Atoms touch along diagonal (if small ion fits perfectly into space) so a  3 = 2(R+r) Divide:  3 = (R+r)/R Multiply out  3R = R+r R(  3 -1) = r r/R =  3 -1 = 0.732

17. Other ways of classifying structures 1) Structure Field Maps e.g. for A x B y O z compounds, plot radius of A against radius of B and note trends of structure as r A and r B change. 2) Mooser-Pearson plots Focuses on the covalent character of bonds. Plot of difference in electronegativity versus average principal quantum number of atoms involved.

20. Summary  the attraction/repulsion between two atoms of size, r, can be adequately described by the Lennard-Jones [12,6] potential  the point of minimum energy in the LJ potential is the van der Waals radius  Ion radii for given element increase with increasing CN and with decreasing oxidation state  Ionic radii increase going down a group  It is possible to calculate the radius ratios which give an indication of the likely coordination of a given ion