Fig. 2.9 of the notes contains schematic drawing of the energy bands in sodium
(valence I) as function of the internuclear separation. Assume that roughly similar
diagram pertains to Mg (valence 2) and Al (valence 3). which are the next two elements
in the same row of the periodic table. [This only true in rough approximation Mg
and Al do not even have the same crystal structure as Na. However, the three elements
do have the common property that the bands arising from the 3s and 3p states overlap at
separations near the equilibrium separation, ro.]
Fig. 2.1: Energy bands as function of internuclear distance in Na
Now suppose that you are given a means for controlling the internuclear
separation precisely; for example, you may employ a well -disciplined troop of
Maxwell demons who push the atoms apart or pull them together on command Show
that at critical value of the equilibrium separation Mg changes from metal to a
semiconductor or insulator. but Na and Al remain metallic until the atoms are pushed so
far apart that they behave as independent atoms.
(a) Materials that are covalently bonded are ordinarily either semiconductors or
(b) Materials that are ionically bonded are ordinarily either insulators or ionic
(c) In the "chemist's" model, what type of bonding leads to high conductivity?
A crystal consists of building block (an atom or group of atoms) that is repeated
periodically through space
(a) Describe the difference between the "crystal lattice", the arrangement of lattice
points in space, and the "crystal structure' the arrangement of atoms in space.
(b) Explain how every crystal lattice has unit cell that has the shape of a
parallelopiped and is "primitive" in the sense that contains only one lattice point.
(c) Given (b), why do we often choose unit cell that contains more than one
lattice point (for example, the face -centered cubic and body-centered cubic cells)?
As two-dimensional example of the principles adduced in problem 3:
(a) Consider a two-dimensional crystal made up of spherical atoms that are
packed together as tightly as possible in the plane. Show that the unit cell can be drawn
as hexagon with an atom in the center and atoms at each of the six corners. How many
atoms are there per cell?
(b) Show that is also possible to draw the unit cell as parallelogram that has an
empty center and atoms at each of its corners. How many atoms does this cell contain?
How many unit cells of type (b) are contained within single hexagonal cell of type (a)?
(c) Consider a two-dimensional crystal that has hexagonal unit cell with an
empty center and atoms located the corners of the hexagor (the structure of plane of
carbon atoms in graphite) Draw the atom positions generated by several neighboring
cells. Show that the unit cell used in part (b) is nota correct unit cell for this structure
and draw proper unit cell. [Note that the effective "building block" of this structure is
an atom pair.]
(a) Draw the following crystallographic directions in cubic unit cell: ,
(b) Draw all directions in the family <111 1> in a cubic crystal.
(c) Draw the following planes in cubic unit cell (010),(112), (123).
(d) Draw all the planes in the family (111) in 8 cubic crystal.
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In the band model there are 2 situations that lead to metal conduction. A band necessarily contains an even number of electron states. Elements that have odd valences like Na and Al are metallic unless the atoms are gathered into molecular groupings that have an even number of valence electrons. The 2nd kind of metal has an even number of valence electrons: The elements from groups II A and II B like Mg are metals because the band of excited states ( called conduction band) overlaps...