Transcript Close-packed structure

The structures of simple solids
 The majority of inorganic compounds exist as solids and comprise ordered
arrays of atoms, ions, or molecules.
 Some of the simplest solids are the metals, the structures of which can be
described in terms of regular, space-filling arrangements of the metal
atoms.
These metal centres interact through metallic bonding
The description of the structures of solids
The arrangement of atoms or ions in simple solid structures can often be
represented by different arrangements of hard spheres.
3.1 Unit cells and the description of crystal structures
 A crystal of an element or compound can be regarded as constructed from
regularly repeating structural elements, which may be atoms, molecules, or
ions.
The ‘crystal lattice’ is the pattern formed by the points and used to represent the
positions of these repeating structural elements.
(a) Lattices and unit cells
A lattice is a three-dimensional, infinite array of points, the lattice points, each
of which is surrounded in an identical way by neighbouring points, and which
defines the basic repeating structure of the crystal.
The crystal structure itself is obtained by associating one or more identical
structural units (such as molecules or ions) with each lattice point.
A unit cell of the crystal is an imaginary parallel-sided region (a ‘parallelepiped’)
from which the entire crystal can be built up by purely translational
displacements
Unit cells may be chosen in a variety of ways but it is generally preferable to
choose the smallest cell that exhibits the greatest symmetry
Two possible choices of
repeating unit are shown but
(b) would be preferred to (a)
because it is smaller.
All ordered structures adopted by compounds belong to one of the following
seven crystal systems.
The angles (, β, ) and lengths (a, b, c) used to define the size and shape
of a unit cell are the unit cell parameters (the ‘lattice parameters’)
A primitive unit cell (denoted by the symbol P) has just one lattice point in the
unit cell, and the translational symmetry present is just that on the repeating
unit cell.
Lattice points describing the translational
symmetry of a primitive cubic unit cell.
body-centred (I, from the German word innenzentriet, referring to the lattice
point at the unit cell centre) with two lattice points in each unit cell, and
additional translational symmetry beyond that of the unit cell
Lattice points describing the translational
symmetry of a body-centred cubic unit cell.
face-centred (F) with four lattice points in each unit cell, and additional
translational symmetry beyond that ofthe unit cell
Lattice points describing the translational
symmetry of a face-centred cubic unit cell.
We use the following rules to work out the number of lattice points in a
three-dimensional unit cell.
The same process can be used to count the number of atoms, ions, or
molecules that the unit cell contains
1. A lattice point in the body of, that is fully inside, a cell belongs entirely to
that cell and counts as 1.
2. A lattice point on a face is shared by two cells and contributes 1/2 to the cell.
3. A lattice point on an edge is shared by four cells and hence contributes 1/4 .
4. A lattice point at a corner is shared by eight cells that share the corner, and
so contributes 1/8 .
Thus, for the face-centred cubic lattice depicted in Fig. the total
number of lattice points in the unit cell is (8×1/8 ) +(6× 1/2) = 4.
For the body-centred cubic lattice depicted in Fig. , the number of lattice points
is (1×1) + (8×1/8 ) = 2.
The close packing of identical spheres can result in a variety of polytypes
cubic closepacked (ccp)
hexagonally close-packed (hcp)
In both the (a) ABA and (b) ABC close-packed arrangements,
the coordination number of each atom is 12.
Dr. Said M. El-Kurdi
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The close packing of spheres
Many metallic and ionic solids can be regarded as constructed from entities,
such as atoms and ions, represented as hard spheres.
Close-packed structure, a structure in which there is least unfilled space.
The coordination number (CN) of a sphere in a close-packed arrangement
(the ‘number of nearest neighbours’) is 12, the greatest number that
geometry allows
A close-packed layer of hard spheres
Interstitial holes: hexagonal and cubic
close-packing
Close-packed structures contain octahedral and tetrahedral
holes (or sites).
There is one octahedral hole per sphere, and there are
twice as many tetrahedral as octahedral holes in a closepacked array
Tetrahedral hole can accommodate a sphere of radius  0.23
times that of the close-packed spheres
Octahedral hole can accommodate a sphere of radius 0.41
times that of the close-packed spheres
3.5 Nonclose-packed structures
Not all elemental metals have structure based on close-packing and some
other packing patterns use space nearly as efficiently.
Even metals that are close-packed may undergo a phase transition to a
less closely packed structure when they are heated and their atoms
undergo large-amplitude vibrations.
Non-close-packing: simple cubic and body centred cubic arrays
Unit cells of (a) a simple cubic lattice and (b) a
body-centred cubic lattice.
The least common metallic structure is the primitive cubic (cubic-P) structure ,
in which spheres are located at the lattice points of a primitive cubic lattice,
taken as the corners of the cube. The coordination number of a cubic-P
structure is 6.
One form of polonium (-Po) is the only
example of this structure among the
elements under normal conditions.
Body-centred cubic structure (cubic-I
or bcc) in which a sphere is at the
centre of a cube with spheres at each
corner
Metals with this structure have
a coordination number of 8
Although a bcc structure is less closely packed
than the ccp and hcp structures (for which the
coordination number is 12),
6.4 Polymorphism in metals
Polymorphism: phase changes in the solid state
If a substance exists in more than one crystalline form, it
is polymorphic.
under different conditions of pressure and temperature
The polymorphs of metals are generally labelled , β,
,...with increasing temperature.
Solid mercury (-Hg), however, has a
closely related structure: it is obtained
from the cubic-P arrangement by
stretching the cube along one of its body
diagonals
A second form of solid mercury (β-Hg)
has a structure based on the bcc
arrangement but compressed along
one cell direction
Phase diagrams
A pressure–temperature phase diagram for iron
6.5 Metallic radii
The metallic radius is half of the distance between the
nearest neighbor atoms in a solid state metal lattice, and is
dependent upon coordination number.
6.7 Alloys and intermetallic compounds
compound of two or more metals, or metals and non-metals;
alloying changes the physical properties and resistance to
corrosion, heat etc. of the material.
Alloys are manufactured by combining the component
elements in the molten state followed by cooling.
Substitutional alloys
In a substitutional alloy, atoms of the solute occupy sites in
the lattice of the solvent metal
similar size
same coordination environment
sterling silver
which contains 92.5% Ag and 7.5% Cu
Interstitial alloys
In an interstitial solid solution, additional small atoms occupy
holes within the lattice of the original metal structure.
Interstitial solid solutions are often formed between metals and small
atoms (such as boron, carbon, and nitrogen)
One important class of materials of this
type consists of carbon steels in which C
atoms occupy some of the octahedral
holes in the Fe bcc lattice.
Intermetallic compounds
When melts of some metal mixtures solidify, the alloy formed
may possess a definite structure type that is different from
those of the pure metals.
e.g. b-brass, CuZn. At 298 K, Cu has a ccp lattice and Zn has a
structure related to an hcp array, but b-brass adopts a bcc
structure.
The structures of metals and alloys
Many metallic elements have close-packed structures, One consequence of this
close-packing is that metals often have high densities because the most mass is
packed into the smallest volume.
 Osmium has the highest density of all the elements at 22.61 g cm−3 and the
density of tungsten, 19.25 g cm−3, which is almost twice that of lead (11.3 g
cm−3)
Calculate the density of gold, with a cubic close-packed array of atoms of molar
mass M=196.97 g mol−1 and a cubic lattice parameter a = 409 pm.
Gold (Au) crystallizes in a cubic close-packed structure (the face-centered cube)
and has a density of 19.3 g/cm3. Calculate the atomic radius of gold.
The unoccupied space in a close-packed structure amounts to 26 per cent of the
total volume. However, this unoccupied space is not empty in a real solid
because electron density of an atom does not end as abruptly as the hardsphere model suggests.
Calculating the unoccupied space in a close-packed array
Calculate the percentage of unoccupied space in a close-packed arrangement
of identical spheres.
6.8 Bonding in metals and semiconductors
Electrical conductivity and resistivity
An electrical conductor offers a low resistance (measured in
ohms, ) to the flow of an electrical current (measured in
amperes, A).
The electrical conductivity of a metal decreases with
temperature; that of a semiconductor increases with
temperature.
Band theory of metals and insulators
A band is a group of MOs, the energy differences between
which are so small that the system behaves as if a continuous,
non-quantized variation of energy within the band is possible.
The relative energies
of occupied and empty
bands in
(a) an insulator,
(b) a metal in which
the lower band is only
partially occupied,
(c) a metal in which
the occupied and
empty bands overlap,
and
(d) a semiconductor.
A band gap occurs when there is a significant energy
difference between two bands.
6.9 Semiconductors
For C, Si, Ge and -Sn, the band gaps are 5.39, 1.10, 0.66
and 0.08 eV respectively.
C being an insulator
Each of Si, Ge and -Sn is classed as an intrinsic semiconductor
 Electrons present in the upper conduction band act as
charge carriers and result in the semiconductor being
able to conduct electricity.
 removal of electrons from the lower valence band
creates positive holes into which electrons can move,
again leading to the ability to conduct charge.
A charge carrier in a semiconductor is either a positive
hole or an electron that is able to conduct electricity.
Extrinsic (n- and p-type) semiconductors
Extrinsic semiconductors contain dopants; a dopant is an
impurity introduced into a semiconductor in minute
amounts to enhance its electrical conductivity.
In Ga-doped Si, the substitution of a Ga (group 13) for a Si
(group 14) atom in the bulk solid produces an electron
deficient site.
In As-doped Si, replacing an Si (group 14) by an As (group
15) atom introduces an electron-rich site.
(a) In a p-type semiconductor (e.g. Ga-doped Si), electrical
conductivity arises from thermal population of an acceptor level
which leaves vacancies (positive holes) in the lower band. (b) In an
n-type semiconductor (e.g. As-doped Si), a donor level is close in
energy to the conduction band.
6.10 Sizes of ions
Ionic radii
Values of the ionic radius (rion) may be derived from X-ray
diffraction data.
internuclear distance >>> we generally take this to be the sum of the
ionic radii of the cation and anion
Ionic solids
Characteristic structures of ionic solids
Many of the structures can be regarded as derived from arrays in which
the larger of the ions, usually the anions, stack together in ccp or hcp patterns
and the smaller counter-ions (usually the cations) occupy the octahedral or
tetrahedral holes in the lattice
The rock salt (NaCl) structure type
In salts of formula MX, the coordination numbers of M and X must be equal.
Na+ and Cl- ion is 6-coordinate in the crystal
lattice
The number of formula units present in the unit cell is commonly denoted Z
Show that the structure of the unit cell for sodium chloride (Figure) is consistent
with the formula NaCl.
The caesium chloride (CsCl) structure type
cubic unit cell with each corner occupied by an anion and a cation occupying the
‘cubic hole’ at the cell centre (or vice versa); as a result, Z =1.
The coordination number of both types of
ion is 8, so the structure is described as
having (8,8)-coordination.
The fluorite (CaF2) structure type
Ca ions are shown in red and
the F ions in green
In salts of formula MX2, the coordination number of X must be half that of M.
The antifluorite lattice
The antifluorite structure is the inverse of the fluorite structure in the sense
that the locations of cations and anions are reversed.
The latter structure is shown by some alkali metal oxides, including Li2O.
In it, the cations (which are twice as numerous as the anions) occupy all the
tetrahedral holes of a ccp array of anions.
The coordination is (4,8) rather than the (8,4) of fluorite itself.
The sphalerite structure, which is also known as the
zinc-blende structure, it is based on an expanded
ccp anion arrangement but now the cations occupy
one type of tetrahedral hole, one half the
tetrahedral holes present in a close-packed
structure.
Each ion is surrounded by four neighbours
and so the structure has (4,4)coordination and Z= 4.
The wurtzite structure
polymorph of zinc sulfide
This structure, which has (4,4)-coordination, is
adopted by ZnO, AgI, and one polymorph of
SiC, as well as several other compounds
The rutile structure, a mineral form of titanium(IV) oxide, TiO2. The structure can
also be considered an example of hole filling in an hcp anion arrangement, the
cations occupy only half the octahedral holes.
Each Ti atom is surrounded by six O
atoms and each O atom is surrounded by
three Ti ions; hence the rutile structure
has (6,3)-coordination.
6.13 Lattice energy: estimates from an electrostatic
model
The lattice energy, U(0 K), of an ionic compound is the change in
internal energy that accompanies the formation of one mole of the
solid from its constituent gas-phase ions at 0 K
Coulombic attraction within an isolated
ion-pair
For an isolated ion-pair:
"0 permittivity of a vacuum = 8.854 × 1012Fm1
Born forces
The ions have finite size, and electron– electron and
nucleus–nucleus repulsions also arise; these are Born forces.
The Born-Lande´ equation
r0 Equilibrium separation
L Avogadro number
A Madelung constant (no units)
n Born exponent
The Madelung constant reflects the effect of the geometry
of the lattice on the strength of the net Coulombic
interaction.
6.14 Lattice energy: the Born-Haber cycle
Let us consider a general metal halide MXn
by application of Hess’s law of constant heat summation
Rearranging this expression and introducing the approximation
that the lattice energy U(0 K)  latticeH(298K)
First, construct an appropriate thermochemical cycle
6.17 Defects in solid state lattices:
an introduction
Schottky defect
A Schottky defect consists of an atom or ion
vacancy in a crystal lattice, but the stoichiometry of
a compound (and thus electrical neutrality) must be
retained.
(a) Part of one face of an ideal NaCl structure;compare
this with Figure 6.15. (b) A Schottky defect involves
vacant cation and anion sites
Frenkel defect
In a Frenkel defect, an atom or ion occupies a normally
vacant site, leaving its ‘own’ lattice site vacant.
A Frenkel defect in AgBr involves the migration of Ag+ ions into
tetrahedral holes