Download 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
11
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.
Calculate the maximum radius of a sphere that may be accommodated in
an octahedral hole in a closepacked solid composed of spheres of radius r.
0.414r
3.8 Alloys
An alloy is a blend of metallic elements prepared by mixing the molten
components and then cooling the mixture to produce a metallic solid.
Alloys typically form from two electropositive metals
(a) Substitutional solid solutions
 Involves the replacement of one type of metal atom in a structure by
another.
Substitutional solid solutions are generally formed if three criteria are
fulfilled:
1. The atomic radii of the elements are within about 15 per cent of each other.
2. The crystal structures of the two pure metals are the same.
3. The electropositive characters of the two components are similar.
Sodium and potassium are chemically similar
and have bcc structures,
the atomic radius of Na (191 pm) is 19 per cent smaller than that of K (235 pm)
and the two metals do not form a solid solution.
Copper and nickel, have similar electropositive character,
similar crystal structures (both ccp),
and similar atomic radii (Ni 125 pm, Cu 128 pm, only 2.3 per cent different),
and form a continuous series of solid solutions, ranging from pure nickel to pure
copper.
(c) Intermetallic compounds
Intermetallic compounds are alloys in which the structure adopted is different
from the structures of either component metal.
when some liquid mixtures of metals are cooled, they form phases with definite
structures that are often unrelated to the parent structure. These phases are
called intermetallic compounds.
They include β-brass (CuZn) and compounds of composition MgZn2, Cu3Au,
NaTl, and Na5Zn21.
Composition, lattice type and unit cell content of iron and its alloys
What are the lattice types and unit cell contents of (a) iron metal (Fig. a) and (b)
the iron/chromium alloy, FeCr
The structure type is the bcc
there are two Fe atoms in the unit cell
the lattice type is primitive, P.
There is one Cr atom and 1 Fe atom in the
unit cell
Ionic solids
3.9 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 relation of structure to the filling of holes
(a) Binary phases, AXn
The simplest ionic compounds contain just one type of
cation (A) and one type of anion (X) present in various
ratios covering compositions such as AX and AX2.
Several different structures may exist for each of these
compositions, depending on the relative sizes of the
cations and anions and which holes are filled and to
what degree in the close-packed array
The rock-salt structure is based on a ccp array of bulky
anions with cations in all the octahedral holes.
Because each ion is surrounded by an octahedron of
six counter-ions, the coordination number of each
type of ion is 6 and the structure is said to have (6,6)coordination.
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.
many 1:1 compounds in which the ions are complex units such
as [Co(NH3)6][TlCl6].
The structure of this compound can be considered as an array of closepacked
octahedral [TlCl6]3− ions with [Co(NH3)6]3+ ions in all the octahedral holes.
Similarly, compounds such as CaC2, CsO2, KCN, and FeS2 all adopt structures
closely related to the rock-salt structure with alternating cations and complex
anions
The structure of CaC2 is based on
the rock-salt structure but is
elongated in the direction parallel
to the axes of the C22− ions.
caesium-chloride structure
which is possessed by CsCl, CsBr, and CsI, as well as some other compounds
formed of ions of similar radii to these.
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 structure of ammonium chloride, NH4Cl, reflects the ability of the
tetrahedral NH4+ ion to form hydrogen bonds to the tetrahedral array of Cl−
ions around it.
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
It derived from an expanded hcp anion array
rather than a ccp array
This structure, which has (4,4)-coordination, is
adopted by ZnO, AgI, and one polymorph of
SiC, as well as several other compounds
The fluorite (CaF2) lattice
Each cation is 8-coordinate and each anion 4coordinate; six of the Ca2+ ions are shared
between two unit cells and the 8-coordinate
environment can be appreciated by envisaging
two adjacent unit cells.
The unit cell of CaF2; the Ca2+ ions
are shown in red and the F− ions in
green.
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 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 Ti4 atom is surrounded by six O
atoms and each O atom is surrounded by
three Ti4 ions; hence the rutile structure
has (6,3)-coordination.
(b) Ternary phases AaBbXn
it is difficult to predict the most likely structure
type based on the ion sizes and preferred
coordination numbers.
The mineral perovskite, CaTiO3, is the
structural prototype of many ABX3 solids
The perovskite structure is cubic with each A
cation surrounded by 12 X anions and each B
cation surrounded by six X anions
the coordination number of the Ti4+ ion in the
perovskite CaTiO3 is 6