Crystal Structure

Solid State Chemistry, a sub-discipline of Chemistry, primarily

involves the study of extended solids.
Solid state chemistry deals with the study of synthesis, structure,
bonding, reactivity, and physical properties of solids.
➢ Except for helium, all substances form a solid if sufficiently
cooled at 1 atm.
➢ The vast majority of solids form one or more crystalline phases –
where the atoms, molecules, or ions form a regular repeating
array.
➢ The properties of solids are related to its structure and bonding.
➢ Classification of solids can be based upon atomic arrangement,
binding energy, physical and chemical properties or the
geometrical aspects of the crystalline structure.
 2

Classification of Solids
(Shechtman)
 3

Crystalline Solids
The solids in which atoms, ions or molecules are arranged in a
definite pattern, constantly repeated, giving a definite geometrical
shape, characteristic of the substance and independent of the
sources from which they have been obtained are called crystals.
A CRYSTAL is any solid material in which the component atoms are
arranged in a definite patter and whose surface regularity reflects its
internal symmetry.
 NaCl crystal CsCl crystal
A crystalline solid is an aggregate of minute crystals, packed
together in a well-defined order.
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Properties of crystals
i. In a crystal, there is perfect and well-ordered arrangement of
molecules throughout the entire body, each molecule is
surrounded by a set of other molecules in a definite symmetrical
way.
ii. A crystal when melted expands only about 10% in volume or
about 3% in inter-molecular spacing.
iii. The crystals are bounded by surfaces which are planar and
arranged on a definite plane.
iv. The crystals have a rigorous geometrical order. Thermal motions
cause disorder. It is clear that if a small region of disorder is
introduced into a crystal, it would cause disturbance in the long
range and destroys the crystalline arrangement. This explains the
reason of sharp melting points of crystalline substances.
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v. When subjected to a mechanical stress, a crystal tends to

fracture along a perfectly definite direction.
vi. The important feature of crystal is the periodicity of
arrangement along with regularity.
vii. A crystalline substance is anisotropic, i.e., its physical
properties like mechanical, electrical and optical properties are
different in different directions. For example, the velocity of
 light passing through a crystal changes with the direction in
which it is measured. Moreover, in silver iodide crystal, the
coefficient of thermal expansion is positive in one direction and
negative in another direction. Anisotropy offers a strong
evidence for the presence of well ordered molecular
arrangement in crystals.
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6 Crystallography is the experimental science of determining the

arrangement of atoms in the crystalline solids.
Classification of Crystalline Solids based on Different Binding Forces
 7

A molecular solid is a solid that consists of atoms or molecules

held together by intermolecular forces (van der Waals forces
and/or hydrogen bonding). Many solids are of this type. Examples
include solid SO2, solid neon, solid water (ice), and solid carbon
dioxide (dry ice) or "cardice" (chiefly by British chemists).
Sulfur
8
A metallic solid is a solid that consists of positive cores of atoms
held together by a surrounding “sea” of electrons (metallic
bonding). In this kind of bonding, positively charged atomic cores
are surrounded by delocalized electrons. Examples include iron,
copper, silver etc.

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 Fe Cu Au
An ionic solid is a solid that consists of cations and anions held
together by the electrical attraction of opposite charges (ionic
bonds). Examples are cesium chloride, sodium chloride, and zinc
sulfide etc .

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 CsCl ZnS
A covalent network solid is a solid that consists of atoms held
together in large networks or chains by covalent bonds. Diamond
is an example of a three-dimensional network solid. Every carbon
atom in diamond is covalently bonded to four others, so an entire
crystal might be considered an immense molecule.
Examples are diamond (three-dimensional network); graphite
(sheets); asbestos (chains); boron nitride etc.

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 Diamond Graphite
Amorphous Solids
The solids which do not have any definite geometrical shape are
called amorphous solids, e.g., glass, rubber, plastic, starch, and
proteins.

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Physical properties of amorphous solids:
i. Amorphous solids are considered to be super-cooled liquids in
which the force of attraction holding the molecules together is

15
 so great that the substance is rigid and there is no regularity of
structure.
ii. Amorphous solids do not have sharp melting points. They
gradually soften on heating. Absence of sharp melting point
suggests the absence of long-range order in amorphous solids.
On increasing the temperature, the viscosity of amorphous
substances decreases and gradually changes into the liquid
state.
iii. In amorphous solids, there is non-periodicity of the
arrangement along with no regularity.

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iv. Amorphous solids are isotropic, i.e., their physical properties like
mechanical, thermal, and electrical properties are the same in all
directions. In amorphous substances, the particles are randomly
arranged and disordered. Due to this, all directions are equivalent,
so all the properties remain the same.
Micro-crystalline or Poly-crystalline Solids
A crystal structure with an aggregate of many small crystals (or) grains is
separated by well-defined grain boundaries. These crystals will have a
sharp melting point.
In many solids, we may not clearly see the shape of the crystals, because
several small or micro-sized crystals are tightly packed together without
any specific order.
Therefore, a substance, which in fact is crystalline, but is superfine to be
seen as a crystal, is called a micro-crystalline or poly-crystalline solid,
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e.g., Diamond, Copper, Platinum, Silver, Polonium, Gold, Aluminum,
Nickel, Cadmium, Iron etc..
A microcrystalline material is a crystallized substance or rock that
contains small crystals visible only through microscopic examination.

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 Cuprite: Copper (I) oxide
While most copper compounds are blue
or green, one is red.

Microcrystalline
cellulose

Single Crystal:
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A single-crystal, or monocrystalline, solid is a material in which the
crystal lattice of the entire sample is continuous and unbroken to
the edges of the sample, with no grain boundaries.
The absence of the defects associated with grain boundaries can
give monocrystals unique properties, particularly mechanical,
optical and electrical, which can also be anisotropic, depending on
the type of crystallographic structure.

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Polymorphism or Allotropy
Many elements or compounds exist in more than one crystalline
form under different conditions of temperature and pressure. This
phenomenon is termed polymorphism and if the material is an
elemental solid is called allotropy.
Polymorphism is where the same chemical compound occurs in two
or more atomic structures.
Some examples are:
– diamond and graphite, fullerene, graphene (C)
– calcite and aragonite (CaCO3)
– andalusite, kyanite and sillimanite (Al2SiO5)
➢ Polymorphs have different physical properties

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➢ Polymorphs form under different physical conditions to each
other

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Aragonite - CaCO3 Calcite - CaCO3
Orthorhombic Trigonal
 Diamond Structure

Crystal System Cubic

Crystal form Octahedron
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Hardness Hardest substance
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Graphite Structure
Crystal System Hexagonal
Crystal form Hexagonal flakes
Hardness One of softest known

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 Graphene

C60 (Fullerene)

Carbon nanotubes
(Fullerene)
 21

Iron (Fe: Z = 26)

Liquid above 1539oC.
-iron (BCC) between 1394 and 1539oC.
-iron (FCC) between 912 and 1394oC. α-
iron (BCC) between -273 and 912oC.

912oC 1400oC 1539oC

α iron  iron  iron liquid iron
BCC FCC BCC

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Isomorphism
Isomorphism is where two minerals canhave different
compositions but the same structure.
Isomorphs have the same proportions of ions that are
approximately the same size
Examples are the Olivine minerals
– Forsterite Mg2SiO4
– Fayalite Fe2SiO4
– General formula for Olivine group is (Mg,Fe)2 SiO4
Isomorphism occurs by substitution of ions
➢ In Olivine, Fe2+ can substitute for Mg2+ because:
– the ionic radii are very similar: Fe = 0.66, Mg = 0.74 Å
– the charges are the same
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 Also called “Solid-Solution”
– analogous to liquid solution
➢ Ionic substitution is common in minerals ➢ Can
occur wherever:
– different ions have similar ionic size
– charge balance in the lattice can be maintained

Olivine

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 (Fe,Mg)2SiO4
Isotropy and Anisotropy
Amorphous substances are said to be isotropic because they
exhibit the same value of any property in all directions. Thus
refractive index, thermal and electrical conductivities, coefficient
of thermal expansion in amorphous solids are independent of the
direction along which they are measured.
In amorphous substances, as in liquids, the arrangement of
particles is random and disordered. Therefore all directions are
equivalent and properties are independent of direction.
Crystalline substances, on the other hand, are anisotropic and the
magnitude of a physical property varies with directions, i.e.,
mechanical, electrical and optical properties are different in
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different directions. Anisotropic behaviour of crystalline
substances can be understood form figure.

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 Figure: Anisotropy in crystals is due to different arrangements of
particles in different directions.
Crystallization
Crystallization is the (natural or artificial) process of formation of
solid crystals precipitating from a solution, melt or more rarely
deposited directly from a gas .
Crystallization is also a chemical solid–liquid separation technique,
in which mass transfer of a solute from the liquid solution to a
pure solid crystalline phase occurs.

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35
36
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Figure: (a) Pictorial diagram of single-crystal growth. (b) and (c) Images of
the MAPbI3 and MAPbBr3 single crystals, respectively, with scale.

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 Crystal Growth
Crystal growth is the process where a pre-existing crystal becomes
larger as more growth units (e.g. molecules, ions) add in their
positions in the crystal lattice or a solution is developed into a
crystal and further growth is processed.
Nucleation and Growth are the main factors of crystal growth. If
nucleation rates are slow and growth is rapid, large crystals will
result. On the other hand, if nucleation is rapid, relative to growth,
small crystals or even polycrystalline samples will result.
Condition of Crystal Growth
❖ Achievement of super saturation or super cooling
❖ Formation of crystal nucleus of microscopic size
❖ Successive growth of crystals to yield distinct faces
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Basic growth methods available for crystal growth

The basic growth methods available for crystal growth are broadly
1. Growth from a melt.
2. Growth from vapor .
3. Growth from solution.
4. Growth from solid.
Crystal Growth Techniques
❖ Bridgmann method
❖ Czochralski method
❖ Vernuil method
 ❖ Zone melting method ❖ Kyropoulos technique ❖ Skull
melting.
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Czochralski Crystal Growth Process

❖ The Czochralski method or Czochralski process, is a method of
crystal growth used to obtain single crystals of semiconductors
(e.g. silicon, germanium, and gallium arsenide), metals (e.g.
palladium, platinum, silver, gold), salts and synthetic gemstones.
❖ It is also known as Pulling Technique
❖ This method is widely used for growing semi-conducting material
crystals. The shape of the crystal is free from constraint due to
the shape of the crucible.

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❖ In this method the charge is melted and maintained at a
temperature slightly above the melting point. The pulling rod is
lowered to just touch the melt. Since the rod is at a lower
temperature, crystallization occurs at the pointed tip of the
pulling rod. The crystal is pulled slowly.
Czochralski Crystal Growth Process

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44
Czochralski Process

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❖ The rate of pulling depend upon various factors like thermal
conductivity, latent heat of fusion of charge and rate of cooling
of the pulling rod. The seed is rotated to keep the grow crystal
uniform and cylindrical.
❖ A seed crystal is attached to a rod, which is rotated slowly.
❖ The seed crystal is dipped into a melt held at a temperature
slightly above the melting point.
❖ A temperature gradient is set up by cooling the rod and slowly
withdrawing it from the melt (the surrounding atmosphere is
cooler than the melt)
❖ Decreasing the speed with which the crystal is pulled from the
melt, increases the quality of the crystals (fewer defects) but
decreases the growth rate.

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 Application
The most important application of the Czochralski Process may be
the growth of large cylindrical ingots, or boules, of single-crystal
silicon used in the electronics industry to make semiconductor
devices like integrated circuits. Other semiconductors, such as
gallium arsenide can also be grown by this method.
Monocrystalline silicon (mono-Si) grown by the Czochralski method
is often called monocrystalline Czochralski silicon (Cz-Si). It is the
basic material in producing integrated circuits used in computers,
TVs, mobile phones, and all electronic equipment and semiconductor
devices. Monocrystalline silicon is also used in large quantities by the
photovoltaic industry to produce conventional mono-Si solar cells.
The almost perfect crystal structure yields the highest light-
toelectricity conversion efficiency for silicon.
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Advantages
❖ This method is used to grow large single crystals. Thus it is used
extensively in the semiconductor industry.
❖ There is no direct contact between the crucible walls and the
crystal which helps to produce unstressed single crystal.
Disadvantages
❖ In general this method is not suitable for incongruently melting
compounds and of course the need for a seed crystal of the
same composition limits is used as tool for exploratory synthetic
research.
Space lattice (or) crystal lattice
A three dimensional collection of points in space are called space
lattice (or) crystal lattice.

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A geometrical representation of the crystal structure in terms of
lattice points are called space lattice, provided that the environment
about every point is identical to that of every other
point.
Space lattice can be
described as an infinite
three-dimensional
array of points.

Figure-1: Space lattice of ideal crystalline solid.

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 Unit Cell
The basic repeating structural unit of a crystalline solid is a unit cell.
The size and shape of the unit cell can be described by three lattice
vectors a, b, and c, originating from one corner of the unit cell
(Figure-b).
The axial lengths a, b, and c and the inter axial angles α, β, and γ are
the lattice constants of the unit cell.
( a) (b)

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 Figure-2: Unit cell showing lattice constants (b).
Figure-2a shows a unit cell and its extension in three dimensions.
Each sphere represents an atom, an ion, or a molecule and is called a
lattice point.
A Lattice point is the position in the unit cell or in a crystal where the
probability of finding an atom or an ion is the highest.
In other words, the atoms or ions occupy the lattice points in a
crystalline solid.

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Unit cell parameter or lattice parameters
The three intercept quantities a, b and c are called
the fundamental translational vectors (or) axial
lengths (or) intercepts (a, b and c) and three inter
facial angles are called unit cell parameter (or)
lattice parameters.

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Primitive cell: A primitive cell is the simplest type of unit cell which
contains only one lattice point per unit cell (contains lattice points at its
corner only) Simple Cubic (SC)
Non-Primitive cell: If there are more than one lattice point in an unit cell,
it is called a non-primitive cell (BCC and FCC).
Co-ordination number: Co-ordination number is the number of nearest
neighboring atoms to a particular atom. (Or)
Co-ordination numberis the number of nearest neighbors directly
surrounding a given atom.
Crystal Systems and Bravais Lattices
By assigning specific values for axial lengths and interaxial angles,
unit cells of different types can be constructed.
Crystallographers have shown that only seven different types of unit
cells are necessary to create all point lattices.
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The 14 possible ways of arranging points in space lattice such that,
all lattice points have exactly the same surroundings. These 14
lattice in seven crystal structure are called Brava is lattice.
These crystal systems are listed in Table 1.
There are four basic types of unit cells:
➢ Simple
➢ Body-centered
➢ Face-centered
➢ Base-centered

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Table-1: Classification of Space Lattices by Crystal System
(Bravis Lattices)

55
In the cubic system there are three types of unit cells:

Simple cubic Face-centered cubic Body-centered cubic In the

orthorhombic system all four types are represented.

56
 Simple Base-centered Face- Body-centered orthorhombic orthorhombic
centered orthorhombic
orthorhombic
In the tetragonal system there are only two:

Simple tetragonal Body-centered tetragonal

The monoclinic system has simple and base-centered unit cells:

57
 Simple monoclinic Base-centered monoclinic
The rhombohedral, hexagonal, and triclinic systems have only one
simple type of unit cell.

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Simple rhombohedral

59
 Simple hexagonal Simple triclinic

Simple cubic Body-centered cubic Face-centered cubic

Figure-3: Unit cells in the cubic crystal system: In the line-and-ball
drawings in the top row, only the centers of spheres (atoms) are shown

60
at their respective positions in the unit cells. The space-filling models in
the bottom row show contacts between spheres (atoms).

Figure-4: How spheres are shared between or among unit cells. For a
sphere in the middle of the unit cell, there is no sharing; on a face 1/2 of
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 the sphere is in the unit cell; at an edge only 1/4 of the sphere is in the
unit cell; and in a corner only 1/8 is contained within the unit cell.
Effective number of atoms in a cubic unit cell:

Body-centered Face- Simple cubic cubic centered

BCC lattice: cubic
In the BCC unit cell effective number of atoms = 8 corner atoms ×
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(1/8) (each atom is shared by 8 unit cells) + 1 center atom = 2
FCC lattice:
In the FCC unit cell effective number of atoms = 8 corner atoms ×
(1/8) (each atom is shared by 8 unit cells) + 6 face-centered atoms
× 1/2 (each shared by two unit cells) = 4
SC lattice:
In the SC unit cell effective number of atoms = 8 corner atoms ×
(1/8) (each atom is shared by 8 unit cells) = 1

Relationship between the edge length of the unit cell, a, and the
radius of the atoms, r:
Because the contact between atoms is different for each of the
three types, the relationship between the edge length and the atom
radius is also different for each.
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Simple cubic cell
In the simple cubic cell, where the atoms meet along the edge of the
cell, the edge of each cell is twice the radius of the atom: a = 2r.

Simple cubic cell : a = 2r

Show that for BCC and FCC crystal structure, the lattice constants
𝟒𝒓 𝟒𝒓 are given by a =and a =respectively
where r is the atomic
64
 𝟑𝟐
radius.

65
In the face‐centered cell, the contact is along the face diagonal. A
face diagonal passes through the diameter of the atom in its face
(a distance of 2r) and half way through each of two corner atoms
for a distance of r from each. The total distance along the face
diagonal is therefore 4r and each edge has a length of a. From the
Pythagorean theorem, a2 + a2 = (face diagonal)2 or face diagonal =
a = 2 𝟐r.

b= 4r b2 = a2 +
a2
2a2 = 16r2
𝟐a = 4r a =
2 𝟐r
 53

In the body‐centered cell, the atoms meet along the body

diagonal. The diagonal passes through the diameter of the atom in
the center of the cell, but also passes half way through each of the
corner atoms. The contribution from each of the two corner atoms
is r, and the contribution from the center atom is 2r, so that the

an edge (a) and a face diagonal (a 𝟐 ), outlined in red in the figure

below, to give a2 + (a 𝟐 )2 = (4r)2 or a 𝟑 = 4r.
b2 = a2 + a2 = 2a2
c2 = a2 + b2

c = 𝟑a =4r
c = 3a2
b
𝟒𝒓
𝟑
a= 67
entire body diagonal has a length of 4r. In this case the right
triangle for the calculation is composed of the body diagonal (4r),

68
Atomic packing factor
Atomic packing factor (APF) or packing efficiency indicates how
closely atoms are packed in a unit cell and is given by the ratio of
volume of atoms in the unit cell and volume of the unit cell.
Volume of atoms in unit cell*
APF =
Volume of unit cell
*assume hard spheres

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Simple Cubic Structure : � 3
Volume of a atom (sphere) =�πr
volume
atoms atom
4
unit cell 1 p (0.5a) 3
3
APF = = 0.52
a3
a = 2r
𝟑
Only Polonium (Po) has this structure)
Body Centered Cubic Structure (BCC)
Atoms touch each other along cube diagonals within a unit cell.
Examples: Cr, W, Fe (α), Tantalum, Molybdenum

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 𝟒𝒓
2 2
b = a + a = 2a ; b = 𝟐a c = a + b = 3a ; c = 𝟑a =4r; a =
2 2 2 2

atoms volume
4
unit cell 2 p ( 3a/4 ) 3 atom
3
APF =
3 volume
a
unit cell
= 0.68

𝟑
2 2

Face Centered Cubic Structure (FCC)

Atoms touch each other along face diagonals.
Examples: Al, Cu, Au, Pb, Ni, Pt, Ag

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 𝟐 a = 4r; r = 𝟐�/ 𝟒
b= 4r; b2 = a2 + a2; 2a = 16r ;

atoms volume
4
unit cell 4 p ( 2a/4 ) 3 atom
3
APF =
3 volume
a
= 0.74 unit cell
The maximum achievable APF! 22

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 Hexagonal and cubic close-packing
The following figures show the different ways of packing atoms.
Figures (b) and (c) show the most efficient way in which atom can
arrange themself. Such an arrangement is close-packed, and
spheres that are not on the edges of the assembly are in contact
with six other spheres within the layer (c).

Square packed Close-packed Hexagonal motifs

 (a) (b) (c)
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The two close-packed arrangements are distinguished in that one

contains two repeating layers, ABABAB. . . , while the second
 Layer A
Layer B
Layer C

Layer A
Layer A Layer A Layer B
Layer B Layer A

contains three repeating layers, ABCABC. . .

ABABAB. . . Stacking Sequence: hexagonal close-packing (hcp)
ABCABC. . . Stacking Sequence: cubic close-packing (ccp)
 59

Close-Packed Structure
In chemistry, crystallography, and materials science the
coordination number of a central atom in a molecule or crystal is
the number of its near neighbors.

HCP CCP
ABA and ABC close-packed arrangements, the coordination number of
each atom = 3+6+3 = 12.
Examples (HCP): Cd, Mg, Ti, Zn, Re, Co
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 Number of atoms in HCP unit cell

▪ Three full atoms within the volume of per unit cell.

▪ The atoms at the center of the top face and base are shared by
only two unit cells.
▪ Each of the 12 atom at the corner of the top face and the base
are shared by 6 HCP unit cells.
3  1 = 3 center atoms
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 2  (½ ) = 1 face atom = 6 atoms
12  (1/6) = 2 corner atoms
Show that the atomic packing factor for HCP is 0.74.
The APF is the ratio of the total sphere volume, Vs (volume of atoms
in unit cell) to the unit cell volume, Vc.

APF = 𝐕𝐕𝐬𝐜
For HCP, there are the equivalent of six
𝟒
spheres per unit cell, and thus
Vs = 6  𝟑 πr3 =
Now, the unit cell volume 8pr3 is the product of the
base area times the cell height, c. The
following figure shows an HCP unit cell and the basal plane

79
 The area of equilateral triangle, OAB = ½  AB  OP
= ½  AB  AO Sin60o
= ½  a  a Sin60o
𝟑
2

=a 𝟑 𝟑
𝟒
2

𝟑
Thus, the area of the basal plane = 6 a2 = a.
𝟒 𝟐
Further, as can be seen from the figure of
basal plane , a = 2r.
𝟑𝟑
Therefore, the base area = (2r)2 = 6r2 𝟑
𝟐
The relation between the unit cell height, c and the basal plane
length, a is given as: c/a = 1.63
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 Thus, c = 1.63a = 1.63  (2r) = 3.26r
The unit cell volume can now be calculated as:
Vc = c  base area = 3.26r  10.392r2
Vc = 33.878r3
𝐕𝐬 𝟖𝛑𝐫𝟑

Thus, APF = 𝐕𝐜= 𝟑𝟑.𝟖𝟕𝟖𝐫𝟑 = 0.74

Mass of Atoms inUnitCell

Density =  =
Total Volume of UnitCell n
A
 = V C NA

81
 where n = number of atoms/unit cell
A = atomic weight
VC = Volume of unit cell = a3 for cubic
NA = Avogadro’s number
= 6.023 x 1023 atoms/mol
Example: Cr (BCC) A = 𝟒𝒓
52.00 g/mol r = For BCC, a =
0.125 nm n = 2 𝟑

actual = 7.19 g/cm3

atoms
g
unit cell 2 52.00
mol
=
a3 6.023 x 1023
volume atoms
unit cell mol 82