Lecture 16 - Lattices and Unit Cells - Solid State Chemistry Notes | EduRev

: Lecture 16 - Lattices and Unit Cells - Solid State Chemistry Notes | EduRev

 Page 1


 Module 4 : Solid State Chemistry
Lecture 16 : Lattices and Unit Cells
 
 
  
Objectives
 
In this Lecture you will learn the following
Definition of a lattice and a unit cell
Classification and characterization of different lattices
Some examples of different lattices.
Estimates of lattice energies.
16.1 Introduction
 
Solids may be classified into different types such as crystalline, amorphous, glassy and so on. At low
temperatures and high pressures, most substances condense into a solid state. The formation of a solid is a
consequence of a variety of intermolecular forces such as ionic, covalent as well as non-covalent (such as van
der Waals forces). In this lecture we will classify crystalline solids into various types of lattice structures and
give examples of each type. We will also estimate lattice energies of ionic crystals.
  
 
The variety and beauty of patterns in crystals is due to the presence of repeating units. The repeating units
extend periodically in three dimensional space giving a space filling structure. The structurally repeating unit
may be a group of one or more atoms, molecules or ions. If each one of these units is represented by a point,
a space filling pattern can be obtained by regularly repeating this unit in three dimensions. This space filling
pattern is called a space lattice or a crystal lattice or a Bravais lattice. Bravais showed in 1875 that there can
be only 14 distinct lattices in three dimensions. Each point of a Bravais lattice can be associated with a unit
cell, which is an imaginary parallelopiped (i.e., a figure with parallel sides) that contains one unit of the
translationally repeating pattern.
 
The fourteen Bravais lattices can be grouped into seven crystal systems by using the symmetry properties of
unit cells, as well as by the relations between the sides and angles of the unit cell. Consider two unit cells as
shown below.
Page 2


 Module 4 : Solid State Chemistry
Lecture 16 : Lattices and Unit Cells
 
 
  
Objectives
 
In this Lecture you will learn the following
Definition of a lattice and a unit cell
Classification and characterization of different lattices
Some examples of different lattices.
Estimates of lattice energies.
16.1 Introduction
 
Solids may be classified into different types such as crystalline, amorphous, glassy and so on. At low
temperatures and high pressures, most substances condense into a solid state. The formation of a solid is a
consequence of a variety of intermolecular forces such as ionic, covalent as well as non-covalent (such as van
der Waals forces). In this lecture we will classify crystalline solids into various types of lattice structures and
give examples of each type. We will also estimate lattice energies of ionic crystals.
  
 
The variety and beauty of patterns in crystals is due to the presence of repeating units. The repeating units
extend periodically in three dimensional space giving a space filling structure. The structurally repeating unit
may be a group of one or more atoms, molecules or ions. If each one of these units is represented by a point,
a space filling pattern can be obtained by regularly repeating this unit in three dimensions. This space filling
pattern is called a space lattice or a crystal lattice or a Bravais lattice. Bravais showed in 1875 that there can
be only 14 distinct lattices in three dimensions. Each point of a Bravais lattice can be associated with a unit
cell, which is an imaginary parallelopiped (i.e., a figure with parallel sides) that contains one unit of the
translationally repeating pattern.
 
The fourteen Bravais lattices can be grouped into seven crystal systems by using the symmetry properties of
unit cells, as well as by the relations between the sides and angles of the unit cell. Consider two unit cells as
shown below.
 
Figure 16.1 Unit cells. Sides are a, b and c and the angles are  (between b and c in the bc plane),  (in the
ac plane, between a and c) and  (between a and b in the ab plane. (a) cubic unit cell, (b) non-cubic unit cell
 
. 
In a cubic crystal system (formed from cubic unit cells placed at the lattice points) there are four C
3
 axes
placed in a tetrahedral arrangement. In Fig 16.1(a) the line joining the points 3 and 5, for example is a C
3
axis. What this means is that if the unit cell/crystal is rotated by 120
o
, 240
o
 and 360
o
 (three angles, multiples
of 120
o
), we get an arrangement which is indistinguishable from the original arrangement. Having only a C
1
axis is as good as having no symmetry at all because every object has a C
1
 axis of symmetry, i.e., if you
rotate it with respect to any axis by 360
o
, you will recover the original arrangement. A triclinic crystal has no
symmetry or has only a C
1
 symmetry axis
 
The symmetry elements of the seven crystal systems are given in Table 16.1
 
 Table 16.1 Essential symmetries of the seven crystal systems.
 
Sr no System Symmetries
   
1 Cubic four C
3
 axes tetrahedrally arranged
   
2 Hexagonal one C
6
 axes
   
3 Tetragonal one C
4
 axes
   
4 Rhombohedral one C
3
 axes
   
5 Orthorhombic three perpendicular C
2
 axes
   
6 Monoclinic one C
2
 axes
   
7 Triclinic none ! or only C
1
 axes
  
 
The seven cubic systems can also be classified in terms of the relations between their unit cell parameters a,
b, c and , and . These relations are shown in Table 16.2
  
 The parameters characterizing the seven crystal systems are given in Table 16.2
  
Page 3


 Module 4 : Solid State Chemistry
Lecture 16 : Lattices and Unit Cells
 
 
  
Objectives
 
In this Lecture you will learn the following
Definition of a lattice and a unit cell
Classification and characterization of different lattices
Some examples of different lattices.
Estimates of lattice energies.
16.1 Introduction
 
Solids may be classified into different types such as crystalline, amorphous, glassy and so on. At low
temperatures and high pressures, most substances condense into a solid state. The formation of a solid is a
consequence of a variety of intermolecular forces such as ionic, covalent as well as non-covalent (such as van
der Waals forces). In this lecture we will classify crystalline solids into various types of lattice structures and
give examples of each type. We will also estimate lattice energies of ionic crystals.
  
 
The variety and beauty of patterns in crystals is due to the presence of repeating units. The repeating units
extend periodically in three dimensional space giving a space filling structure. The structurally repeating unit
may be a group of one or more atoms, molecules or ions. If each one of these units is represented by a point,
a space filling pattern can be obtained by regularly repeating this unit in three dimensions. This space filling
pattern is called a space lattice or a crystal lattice or a Bravais lattice. Bravais showed in 1875 that there can
be only 14 distinct lattices in three dimensions. Each point of a Bravais lattice can be associated with a unit
cell, which is an imaginary parallelopiped (i.e., a figure with parallel sides) that contains one unit of the
translationally repeating pattern.
 
The fourteen Bravais lattices can be grouped into seven crystal systems by using the symmetry properties of
unit cells, as well as by the relations between the sides and angles of the unit cell. Consider two unit cells as
shown below.
 
Figure 16.1 Unit cells. Sides are a, b and c and the angles are  (between b and c in the bc plane),  (in the
ac plane, between a and c) and  (between a and b in the ab plane. (a) cubic unit cell, (b) non-cubic unit cell
 
. 
In a cubic crystal system (formed from cubic unit cells placed at the lattice points) there are four C
3
 axes
placed in a tetrahedral arrangement. In Fig 16.1(a) the line joining the points 3 and 5, for example is a C
3
axis. What this means is that if the unit cell/crystal is rotated by 120
o
, 240
o
 and 360
o
 (three angles, multiples
of 120
o
), we get an arrangement which is indistinguishable from the original arrangement. Having only a C
1
axis is as good as having no symmetry at all because every object has a C
1
 axis of symmetry, i.e., if you
rotate it with respect to any axis by 360
o
, you will recover the original arrangement. A triclinic crystal has no
symmetry or has only a C
1
 symmetry axis
 
The symmetry elements of the seven crystal systems are given in Table 16.1
 
 Table 16.1 Essential symmetries of the seven crystal systems.
 
Sr no System Symmetries
   
1 Cubic four C
3
 axes tetrahedrally arranged
   
2 Hexagonal one C
6
 axes
   
3 Tetragonal one C
4
 axes
   
4 Rhombohedral one C
3
 axes
   
5 Orthorhombic three perpendicular C
2
 axes
   
6 Monoclinic one C
2
 axes
   
7 Triclinic none ! or only C
1
 axes
  
 
The seven cubic systems can also be classified in terms of the relations between their unit cell parameters a,
b, c and , and . These relations are shown in Table 16.2
  
 The parameters characterizing the seven crystal systems are given in Table 16.2
  
 Table 16.2 Seven crystal systems or Bravais unit cells
 
Sr no Type/category cubic Edge lengths Internal angles
    
1 Cubic a = b= c
 =  =  = 90
o
    
2 Hexagonal a = b c
 =  = 90
o
,  = 120
o
    
3 Tetragonal a = b c
 =  =  = 90
o
    
4 Rhombohedral a = b = c
 = =     90
o
    
5 Orthorhombic a  b c
 =  =  = 90
o
    
6 Monoclinic a  b c
 = = 90
o
,   
    
7 Triclinic a  b c
            90
o
  
 
The cubic unit cell can be further categorized as simple cubic, face centered cubic and body centered cubic.
The fourteen Bravais lattices are shown in fig 16.2
  
Page 4


 Module 4 : Solid State Chemistry
Lecture 16 : Lattices and Unit Cells
 
 
  
Objectives
 
In this Lecture you will learn the following
Definition of a lattice and a unit cell
Classification and characterization of different lattices
Some examples of different lattices.
Estimates of lattice energies.
16.1 Introduction
 
Solids may be classified into different types such as crystalline, amorphous, glassy and so on. At low
temperatures and high pressures, most substances condense into a solid state. The formation of a solid is a
consequence of a variety of intermolecular forces such as ionic, covalent as well as non-covalent (such as van
der Waals forces). In this lecture we will classify crystalline solids into various types of lattice structures and
give examples of each type. We will also estimate lattice energies of ionic crystals.
  
 
The variety and beauty of patterns in crystals is due to the presence of repeating units. The repeating units
extend periodically in three dimensional space giving a space filling structure. The structurally repeating unit
may be a group of one or more atoms, molecules or ions. If each one of these units is represented by a point,
a space filling pattern can be obtained by regularly repeating this unit in three dimensions. This space filling
pattern is called a space lattice or a crystal lattice or a Bravais lattice. Bravais showed in 1875 that there can
be only 14 distinct lattices in three dimensions. Each point of a Bravais lattice can be associated with a unit
cell, which is an imaginary parallelopiped (i.e., a figure with parallel sides) that contains one unit of the
translationally repeating pattern.
 
The fourteen Bravais lattices can be grouped into seven crystal systems by using the symmetry properties of
unit cells, as well as by the relations between the sides and angles of the unit cell. Consider two unit cells as
shown below.
 
Figure 16.1 Unit cells. Sides are a, b and c and the angles are  (between b and c in the bc plane),  (in the
ac plane, between a and c) and  (between a and b in the ab plane. (a) cubic unit cell, (b) non-cubic unit cell
 
. 
In a cubic crystal system (formed from cubic unit cells placed at the lattice points) there are four C
3
 axes
placed in a tetrahedral arrangement. In Fig 16.1(a) the line joining the points 3 and 5, for example is a C
3
axis. What this means is that if the unit cell/crystal is rotated by 120
o
, 240
o
 and 360
o
 (three angles, multiples
of 120
o
), we get an arrangement which is indistinguishable from the original arrangement. Having only a C
1
axis is as good as having no symmetry at all because every object has a C
1
 axis of symmetry, i.e., if you
rotate it with respect to any axis by 360
o
, you will recover the original arrangement. A triclinic crystal has no
symmetry or has only a C
1
 symmetry axis
 
The symmetry elements of the seven crystal systems are given in Table 16.1
 
 Table 16.1 Essential symmetries of the seven crystal systems.
 
Sr no System Symmetries
   
1 Cubic four C
3
 axes tetrahedrally arranged
   
2 Hexagonal one C
6
 axes
   
3 Tetragonal one C
4
 axes
   
4 Rhombohedral one C
3
 axes
   
5 Orthorhombic three perpendicular C
2
 axes
   
6 Monoclinic one C
2
 axes
   
7 Triclinic none ! or only C
1
 axes
  
 
The seven cubic systems can also be classified in terms of the relations between their unit cell parameters a,
b, c and , and . These relations are shown in Table 16.2
  
 The parameters characterizing the seven crystal systems are given in Table 16.2
  
 Table 16.2 Seven crystal systems or Bravais unit cells
 
Sr no Type/category cubic Edge lengths Internal angles
    
1 Cubic a = b= c
 =  =  = 90
o
    
2 Hexagonal a = b c
 =  = 90
o
,  = 120
o
    
3 Tetragonal a = b c
 =  =  = 90
o
    
4 Rhombohedral a = b = c
 = =     90
o
    
5 Orthorhombic a  b c
 =  =  = 90
o
    
6 Monoclinic a  b c
 = = 90
o
,   
    
7 Triclinic a  b c
            90
o
  
 
The cubic unit cell can be further categorized as simple cubic, face centered cubic and body centered cubic.
The fourteen Bravais lattices are shown in fig 16.2
  
 
  
 
                                                    Figure 16.2 The 14 Bravais lattices
  
 
In the following table (16.3), we list some common substances which are found in the 14 Bravais lattices. How
a given substance chooses to be found in a given lattice type is determined by the sizes of the constituent
particles and the detailed interactions between them.
  
 Table 16.3 Common substances in seven crystal systems.
1     Cubic Metal like Ni, Ag, Au, Cu, Al; Na
+
 ions in NaCl lattice [FCC] one form of
Fe, V, Cr, Mo, W; CsCl, taking Cs
+
 and Cl
 -
 together [BCC], -
Page 5


 Module 4 : Solid State Chemistry
Lecture 16 : Lattices and Unit Cells
 
 
  
Objectives
 
In this Lecture you will learn the following
Definition of a lattice and a unit cell
Classification and characterization of different lattices
Some examples of different lattices.
Estimates of lattice energies.
16.1 Introduction
 
Solids may be classified into different types such as crystalline, amorphous, glassy and so on. At low
temperatures and high pressures, most substances condense into a solid state. The formation of a solid is a
consequence of a variety of intermolecular forces such as ionic, covalent as well as non-covalent (such as van
der Waals forces). In this lecture we will classify crystalline solids into various types of lattice structures and
give examples of each type. We will also estimate lattice energies of ionic crystals.
  
 
The variety and beauty of patterns in crystals is due to the presence of repeating units. The repeating units
extend periodically in three dimensional space giving a space filling structure. The structurally repeating unit
may be a group of one or more atoms, molecules or ions. If each one of these units is represented by a point,
a space filling pattern can be obtained by regularly repeating this unit in three dimensions. This space filling
pattern is called a space lattice or a crystal lattice or a Bravais lattice. Bravais showed in 1875 that there can
be only 14 distinct lattices in three dimensions. Each point of a Bravais lattice can be associated with a unit
cell, which is an imaginary parallelopiped (i.e., a figure with parallel sides) that contains one unit of the
translationally repeating pattern.
 
The fourteen Bravais lattices can be grouped into seven crystal systems by using the symmetry properties of
unit cells, as well as by the relations between the sides and angles of the unit cell. Consider two unit cells as
shown below.
 
Figure 16.1 Unit cells. Sides are a, b and c and the angles are  (between b and c in the bc plane),  (in the
ac plane, between a and c) and  (between a and b in the ab plane. (a) cubic unit cell, (b) non-cubic unit cell
 
. 
In a cubic crystal system (formed from cubic unit cells placed at the lattice points) there are four C
3
 axes
placed in a tetrahedral arrangement. In Fig 16.1(a) the line joining the points 3 and 5, for example is a C
3
axis. What this means is that if the unit cell/crystal is rotated by 120
o
, 240
o
 and 360
o
 (three angles, multiples
of 120
o
), we get an arrangement which is indistinguishable from the original arrangement. Having only a C
1
axis is as good as having no symmetry at all because every object has a C
1
 axis of symmetry, i.e., if you
rotate it with respect to any axis by 360
o
, you will recover the original arrangement. A triclinic crystal has no
symmetry or has only a C
1
 symmetry axis
 
The symmetry elements of the seven crystal systems are given in Table 16.1
 
 Table 16.1 Essential symmetries of the seven crystal systems.
 
Sr no System Symmetries
   
1 Cubic four C
3
 axes tetrahedrally arranged
   
2 Hexagonal one C
6
 axes
   
3 Tetragonal one C
4
 axes
   
4 Rhombohedral one C
3
 axes
   
5 Orthorhombic three perpendicular C
2
 axes
   
6 Monoclinic one C
2
 axes
   
7 Triclinic none ! or only C
1
 axes
  
 
The seven cubic systems can also be classified in terms of the relations between their unit cell parameters a,
b, c and , and . These relations are shown in Table 16.2
  
 The parameters characterizing the seven crystal systems are given in Table 16.2
  
 Table 16.2 Seven crystal systems or Bravais unit cells
 
Sr no Type/category cubic Edge lengths Internal angles
    
1 Cubic a = b= c
 =  =  = 90
o
    
2 Hexagonal a = b c
 =  = 90
o
,  = 120
o
    
3 Tetragonal a = b c
 =  =  = 90
o
    
4 Rhombohedral a = b = c
 = =     90
o
    
5 Orthorhombic a  b c
 =  =  = 90
o
    
6 Monoclinic a  b c
 = = 90
o
,   
    
7 Triclinic a  b c
            90
o
  
 
The cubic unit cell can be further categorized as simple cubic, face centered cubic and body centered cubic.
The fourteen Bravais lattices are shown in fig 16.2
  
 
  
 
                                                    Figure 16.2 The 14 Bravais lattices
  
 
In the following table (16.3), we list some common substances which are found in the 14 Bravais lattices. How
a given substance chooses to be found in a given lattice type is determined by the sizes of the constituent
particles and the detailed interactions between them.
  
 Table 16.3 Common substances in seven crystal systems.
1     Cubic Metal like Ni, Ag, Au, Cu, Al; Na
+
 ions in NaCl lattice [FCC] one form of
Fe, V, Cr, Mo, W; CsCl, taking Cs
+
 and Cl
 -
 together [BCC], -
 
Polonium, Cl
-
 lattice in a CsCl lattice [simple cubic]
2 Hexagonal
Metallic Be, SiO
2,
 Mg
2
SiO
4
, corundum, quartz, ruby
3 Orthorhombic
Epsom salts, ancylite, sulphur, pyrite, Hg(II)chloride
4 Tetragonal
Zircon, Tellurium oxide, PbTiO
3
5 Triclinic
Pentahydrate form of Cu (II) sulphate, serandite, feldspar, n- alkanes,
lysozyme crystals grown at pH 4.5
6 Mono clinic
Gypsum, sulphur, jadite, nephrite, K
3
Fe(CN)
6
 Hg(II) chloride,
potassium chlorate
7 Rhombohedral
La cerite, dolomite, calcite, ruby, MgCO
3
   
  
16.2  
Ionic lattices and Lattice Energies.
  
 
We have studied in Module 2 the details regarding intermolecular interactions. In Fig 16.3, the total interaction
is shown as a sum of the attractive (Coulombic) and repulsive interactions.
  
               
 
Figure 16.3 Interaction energy between a positive ion and a negative ion as a function of interionic
distance r.
  
 
The repulsive term is dominated by exchange interactions(see lecture 10). The above form is for a pair of
ions. When ion pairs are stable at a distance r
0
, why are lattices fromedat all? To see why, consider a one
dimensional chain of sodium and chloride ions as shown in fig 16.4.
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