Lecture 3: Description crystal
structures / Defects
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Coordination
Close packed structures
– Cubic close packing
– Hexagonal close packing
– Metallic structures
– Ionic structures with interstitial sites
Important structure types
Defects
– Point defects
– Solid solutions
– Extended defects
– Dislocations
Figures: AJK
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Coordination (1)
Ref: Müller p. 5
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Coordination (2)
(dodecahedral = 12 faces. The
polyhedron shown is actually a
snub disphenoid)
Ref: Müller p. 5
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Close Packing (1)
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Many metallic, ionic, covalent and molecular crystal structures can be described
using the concept of close packing (cp)
The structures are usually arranged to have the maximum density and can be
understood by considering the most efficient way of packing equal-sized spheres
The most efficient way to pack spheres in two dimensions is shown below
Each sphere, e.g. A, is in contact with six others -> six nearest neighbours and the
coordination number, CN = 6 (the largest possible for a planar arrangement)
(a) a cp layer of equal-sized spheres;
(b) a non-cp layer with coordination
number 4;
Within a cp layer, three close packed
directions xx’, yy’, and zz’ occur
Ref: West p. 19
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Close Packing (2)
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The most efficient way to pack spheres in three dimensions is to stack cp layers on
top of each other
There are two simple ways to do this, resulting in hexagonal close packed and
cubic close packed structures
The most efficient way for two cp layers A and B to be in contact is for each sphere
of one layer to rest in a hollow between three spheres in the other layer (P or R)
Addition of a third cp layer can also be done in two ways:
– Hexagonal close packing (hcp): Third layer at S, layer sequence …ABABAB…
– Cubic close packing (ccp): Third layer at T, layer sequence …ABCABC…
(c, d) alternative positions P and R for a second cp layer
Ref: West p. 20
Two cp layers A and B. The B layer occupies
the P positions
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Close Packing (3)
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The simplest layer stacking sequences hcp and ccp are the most important ones
More complex sequences with larger repeat units, e.g. ABCACB or ABAC can occur
and some of these give rise to the phenomenon of polytypism.
In a 3D cp structure, each sphere is in contact with 12 others
74.05% of the total volume is occupied by spheres (maximum density possible in
structures constructed of spheres of only one size)
Ref: West p. 21
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ccp = fcc
The unit cell of a ccp arrangement is in fact face
centered cubic (fcc)
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Body diagonal
A
B
C
A
F (fcc)
Figure: Wikipedia
Cu (fcc metal)
Figures: AJK
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hcp
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hcp structure of Zn metal (space group P63/mmc)
– The structure is slightly distorted, with 6 neighbors at 2.66 Å and 6 at 2.91 Å
A
B
A
B
Figure: AJK
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Structures of common metals
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Most metals crystallise in one of the three arrangements, ccp (fcc) hcp or bcc
– bcc is not a close packed structure!
It is still not well understood why particular metals prefer one structure type to
another
Calculations reveal that the lattice energies of hcp and ccp metal structures are
comparable and, therefore, the structure observed in a particular case probably
depends on fine details of the bonding requirements or band structures of the metal
Ref: West p. 25
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Periodic table of crystal structures
Legend:
.. / .. = mixed structure
[...] = predicted
structure
Figures: Wikipedia
P
I (bcc)
F (fcc)
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Close packing in ionic structures
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The structures of materials such as NaCl, Al2O3, Na2O and ZnO, in which the anion
is larger than the cation, are built of cp layers of anions with the cations placed in
interstitial sites
Many structures are possible in which the variables are the anion stacking
sequence, either hcp or ccp, and the number and type of interstitial sites occupied
by cations.
The cations are, however, often too large for the prescribed interstitial sites and
the structure can accommodate them only by expanding the anion array
Consequently, the anion arrangement is the same as in cp, but the anions may not
be in contact (the term eutactic has been suggested for such structures)
Furthermore, the rigid sphere model is an oversimplification of reality since, in
ionic structures, it can be difficult to specify ion sizes exactly.
Two types of interstitial site, tetrahedral and octahedral, are present in cp
structures
Ref: West p. 26
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T+ ,
T– ,
and O
sites
Ref: West p. 27
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Examples of interstitial sites
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It is rare that all the interstitial sites in a cp structure are occupied; often one set is
full or partly occupied and the remaining sets are empty
Ref: West p. 28
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Octahedral interstitials in
NaCl structure
Na
Cl
Figures: AJK
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Cation sites in an fcc anion array
Ref: West p. 29
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Close packing in covalent
structures
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Some materials such as diamond and silicon carbide, which have very strong,
directional, covalent bonds, can also be described as cp structures or eutactic
structures
Diamond can be regarded as a sphalerite structure as ionic compounds in which
half of the C atoms form a ccp array and the other half occupy T+ sites
– The two “types” are equivalent since there are only carbon atoms!
Classification of diamond as a eutactic structure is useful since in diamond all
atoms are of the same size and it is unrealistic to distinguish between packing
atoms and interstitial atoms
F (fcc)
Ref: West p. 31
Diamond structure,
T+ sites highlighted
Figure: AJK
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Cubic structure types with fcc
anion array
Figures: AJK
Zn
Na
Na
Cl
S
NaCl (rocksalt)
O sites occupied
ZnS (zincblende),
T+ (or T-) sites occupied
(anion and cation sites
are interchangeable)
O
Na2O (antifluorite),
T+ and T- sites occupied
F (fcc)
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NaCl/ZnS -type structures
Ref: West p. 39-40
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Fluorite/antifluorite structures
Figures: AJK
Na
F
O
Ca
CaF2 (fluorite)
Ref: West p. 41
Na2O (antifluorite)
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CsCl structure
Primitive cubic, not body
centered cubic since there
are different ions at corner
and body center positions!
The anion and cation sites
are interchangeable
Figures: AJK
Ref: West p. 48
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Hexagonal structure types with hcp
anion array
Figures: AJK
Ni
Zn
As
O
ZnO (wurtzite - ZnS),
T+ (or T-) sites occupied
NiAs (NiAs): O sites occupied.
Coordination: Ni octahedral (6);
As trigonal prism (6)
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Wurtzite/NiAs –type structures
Ref: West p. 46-47
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Interatomic distances in
some simple structures
Ref: West p. 42
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Defects
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In a perfect crystal, all the atoms are at rest on their correct lattice positions
Such a perfect crystal can be obtained, hypothetically (or computationally), only at
absolute zero; at all real temperatures, crystals are imperfect
Atoms vibrate, which may be regarded as a form of defect, but also a number of
atoms are inevitably misplaced
In some crystals, the number of defects may be very small, <<1%, as in, e.g., highpurity silicon, diamond, or quartz
In highly defective crystals, the question often arises as to whether the defects
themselves should be regarded as forming a fundamental part of the structure
rather than as some imperfection in an otherwise ideal structure.
Pure diamonds, before and after irradiation and annealing.
Clockwise from left bottom:
1) Initial (2×2 mm)
2–4) Irradiated by different doses of 2-MeV electrons
5–6) Irradiated by different doses and annealed at 800 °C.
Ref: West p. 83
Figure: Wikipedia
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Why do crystals have defects?
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Crystals are invariably imperfect because the presence of defects up to a certain
concentration leads to a reduction of free energy (Figure 2.1)
Let’s consider the effect on the free energy of a perfect crystal of creating a single
defect (e.g. a vacant cation site)
– This requires a certain amount of energy, ΔH
– It also causes a considerable increase in entropy, S, because of the large
number of positions which this defect can occupy (configurational entropy)
At some point, the energy required to create more defects will be larger than any
subsequent gain in entropy
ΔG = ΔH - TΔS
S = kB ln W
W = number of possible
configurations
1023 for one mol of cations!
Ref: West p. 84
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Defect types (1)
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For a given crystal, curves such as shown in Fig. 2.1 can be drawn for every
possible type of defect and the main difference between them is the position of
their free energy minimum
The defect that predominates is the one which is easiest to form, i.e. with the
smallest ΔH and for which the free energy minimum occurs at the highest defect
concentration
In Table 2.1, the defects which predominate in a variety of inorganic solids are
summarized
Ref: West p. 84
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Defect types (2)
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One classification of defect types:
– Stoichiometric defects, in which the crystal composition is unchanged on
introducing the defects (also called intrinsic defects)
– Non-stoichiometric defects, which are a consequence of a change in
composition (also called extrinsic defects)
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Another classification based on the size of the defect:
– point defects involve only one atom or site, e.g. vacancies or interstitials,
although the atoms surrounding the defect are also somewhat perturbed
– line defects, i.e. dislocations, are effectively point defects in two dimensions
but in the third dimension the defect is very extensive or infinite;
– in plane defects, whole layers in a crystal structure can be defective
– Sometimes the name extended defects is used to include all those which are
not point defects
Ref: West p. 84
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Schottky defect
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In ionic solids such as halides or oxides, the Schottky defect, a stoichiometric defect, is a
pair of vacant sites, an anion vacancy and a cation vacancy
To compensate for the vacancies, there should be two extra atoms at the surface of the
crystal for each Schottky defect
The Schottky defect is the principal point defect in the alkali halides
Equal numbers of anion and cation vacancies required to preserve local electroneutrality
Vacancies tend to associate because they attract each other
For NaCl at room temperature, typically one in 1015 of the sites is vacant
In other terms, 1 mg of NaCl (~1019 atoms) contains ~104 Schottky defects
Schottky defects are responsible for the optical and electrical properties of NaCl
Ref: West p. 85
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Frenkel defect
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An atom displaced off its lattice site into an interstitial site that is normally empty
AgCl (with the NaCl crystal structure) has predominantly this defect (interstitial Ag)
There is probably some covalent interaction between the interstitial Ag+ ion and its
four Cl− neighbours which acts to stabilise the defect and give Frenkel defects, in
preference to Schottky defects, in AgCl
Na+ with its “harder”, more cationic character, would not find much comfort in a
site which was tetrahedrally surrounded by four other Na+ ions. Frenkel defects
therefore do not occur to any significant extent in NaCl
Calcium fluoride, CaF2, has predominantly anion Frenkel defects in which F−
occupies interstitial sites
As with Schottky defects, the vacancy and interstitial are oppositely charged and
may attract each other to form a pair
Ref: West p. 85
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Color centers (1)
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The best known example of a color center is the F-center (Farbzentrum), which is
an electron trapped on an anion vacancy (Figure)
F-centres can be prepared by heating an alkali halide in vapour of an alkali metal
NaCl heated in Na vapour becomes slightly non-stoichiometric due to the uptake
of Na to give Na1+δCl (δ << 1), which has a greenish yellow colour
The trapped electron provides a classic example of an “electron in a box”.
A series of energy levels are available for the electron within this box and the
energy required to transfer from one level to another falls in the visible part of the
electromagnetic spectrum
Ref: West p. 91
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Color centers (2)
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Many other colour centres have been characterised in alkali halide crystals
H- and V-center containing the chloride molecule ion Cl2– are shown in the Figure
Other defect centers which have been identified in the alkali halides include:
– the F’-center, which is two electrons trapped on an anion vacancy
– the FA-center, which is an F-center, one of whose six cationic neighbors is a
foreign monovalent cation, e.g. K+ in NaCl
– the M-center, which is a pair of nearest neighbour F-centres
– the R-center, which is 3 nearest neighbour F-centres located on a (111) plane
– ionised or charged cluster centres, such as M+, R+ and R−
Ref: West p. 91
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Solid solutions
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In extrinsic defects associated with dopants or impurities, the dopants either
occupy interstitial sites or substitute for atoms or ions in the parent lattice
As the dopant concentration rises, above ∼0.1–1%, it is common practice to refer
to the materials as solid solutions rather than as doped materials, but these two
terminologies are interchangeable
A solid solution is basically a crystalline phase that can have variable composition.
As with doped crystals, simple solid solutions are one of two types:
– in substitutional solid solutions, the atom or ion that is being introduced
directly replaces an atom or ion of the same charge in the parent structure
– in interstitial solid solutions, the introduced species occupies a site that is
normally empty and no ions or atoms are left out
Starting with these two basic types, a considerable variety of more complex solid
solution mechanisms may be derived
Ref: West p. 96
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Some extended defects
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Stacking faults are plane defects that are common in materials with layered
structures, especially those that exhibit polytypism
– For example: Co metal exhibits both polytypism and stacking faults. It occurs
in two main polytypes, either ccp (ABC) or hcp (AB).
– Stacking disorder occurs when the normal stacking sequence is interrupted by
the presence of ‘wrong’ layers, e.g. . . . ABABABABCABABA
Subgrain boundaries and antiphase domains (boundaries)
– Within the domains, typically ∼10 000 Å in size, the structure is relatively
perfect, but at the interface between domains there is a structural mismatch
– Another type of boundary, an antiphase boundary, involves a relative lateral
displacement of two parts of the same crystal
Ref: West p. 110
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Dislocations
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Dislocations are an extremely important class of crystal defect
– They are responsible for the relative weakness of pure metals and in certain
cases (after work-hardening) for just the opposite effect of extra hardness
– The mechanism of crystal growth from either solution or vapour appears to
involve dislocations
– Reactions of solids often occur at active surface sites where dislocations
emerge from the crystal
Dislocations are stoichiometric line defects
Dislocations can be one of two extreme types, edge or screw dislocations, or can
have any degree of intermediate character
Ref: West p. 112
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Edge dislocation
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A simple edge dislocation is an extra half-plane of atoms, i.e. a plane of atoms that
goes only part of the way through a crystal structure
To understand the effect of dislocations on the mechanical properties, consider the
effect of applying a shearing stress to a crystal that contains an edge dislocation
Ref: West p. 112
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