Set of supplementary notes.

3.2. COMPLEX MATTER 37
the other ions is proportional to a −1 ∝ n 1/3 . Thus the higher the density, the larger the kinetic
energy relative to the potential energy, and the more itinerant the electrons. 3 By having a high
coordination number, one can have relatively large distances between neigbours - minimising
the kinetic energy cost - in comparison to a loose-packed structure of the same density.
Screening. Early schooling teaches one that a metal is an equipotential (i.e. no electric
fields). We shall see later that this physics in fact extends down to scales of the screening length
λ ≈ 0.1nm, i.e. about the atomic spacing (though it depends on density) - so that the effective
interaction energy between two atoms in a metal is not Z 2 /R (Z the charge, R the separation),
but Z 2 e −R/λ /R and the cohesion is weak.
Trends across the periodic table. As an s-p shell is filled (e.g. Na,Mg,Al,Si) the
ion core potential seen by the electrons grows. This makes the density of the metal tend to
increase. Eventually, the preference on the right-hand side of the periodic table is for covalent
semiconductor (Si, S) or insulating molecular (P, Cl) structures because the energy is lowered
by making tightly bound directed bonds.
Transition metals. Transition metals and their compounds involve both the outer s-p
electrons as well as inner d-electrons in the binding. The d-electrons are more localised and
often are spin-polarised in the 3d shell when they have a strong atomic character (magnetism
will be discussed later in the course). For 4d and 5d transition metals, the d-orbitals are more
strongly overlapping from atom to atom and this produces the high binding energy of metals
like W (melting point 3700 K) in comparison to alkali metals like Cs (melting point 300 K).
3.2 Complex matter
Simple metals, semiconductors, and insulators formed of the elements or binary compounds
like GaAs are only the beginning of the study of materials. Periodic solids include limitless
possibilities of chemical arrangements of atoms in compounds. Materials per se, are not perhaps
so interesting to the physicist, but the remarkable feature of condensed matter is the wealth of
physical properties that can be explored through novel arrangements of atoms.
Many new materials, often with special physical properties, are discovered each year. Even
for the element carbon, surely a familiar one, the fullerenes (e.g. C 60 ) and nanotubes (rolled up
graphitic sheets) are recent discoveries. Transition metal oxides have been another rich source
of discoveries (e.g. high temperature superconductors based on La 2 CuO 4 , and ferromagnetic
metals based on LaMnO 3 ). f−shell electron metals sometimes produce remarkable electronic
properties, with the electrons within them behaving as if their mass is 1000 times larger than
the free electron mass. Such quantum fluid ground states (metals, exotic superconductors, and
superfluids) are now a rich source of research activity. The study of artificial meta-materials
becins in one sense with doped semiconductors (and especially layered heterostructures grown
by molecular beam epitaxy or MBE), but this subject is expanding rapidly due to an influx of
new tools in nanomanipulation and biological materials.
Many materials are of course not crystalline and therefore not periodic. The physical description
of complex and soft matter requires a separate course.
3 Note the contrast to classical matter, where solids are stabilised at higher density, and gases/liquids at
lower density.