The document discusses the structure of crystal lattices. It begins by defining a crystal as a solid whose constituents are arranged in a highly ordered microscopic structure called a crystal lattice. There are seven basic crystal systems based on symmetry elements. The document then discusses various types of unit cells including primitive and centered unit cells. It describes the 14 Bravais lattices and provides examples of body centered cubic, face centered cubic, and end centered cubic lattices. Finally, it summarizes the structures of sodium chloride and calcium fluoride crystals including their chemical formulas and coordination numbers.

1. STRUCTURES OF CRYSTAL
LATTICE
K ADITHI PRABHU
199316
MSc. ANALYTICAL CHEMISTRY

2. INTRODUCTION
•Crystal structure is one of the most important aspects of material
science and many properties of materials depend on their crystal
structures.
•The basic principles of many materials characterization techniques such
as X-RAY diffraction (XRD), Transmission electron microscopy (TEM) are
based on crystallography.
•Therefore, understanding the basic of crystal structures is of paramount
importance.

3. CRYSTAL
• A crystal or crystalline solid is solid material whose constituents such as
atoms , molecules or ions are arranged in a highly ordered microscopic
structure , forming a crystal lattice that extends in all directions.
• In addition , macroscopic single crystals are usually identifiable by their
geometrical shape , consisting of flat faces with specific , characteristic
orientations.
•The scientific study of crystals and crystal formation is known as
crystallography.
•The process of crystal formation via mechanisms of crystal growth is
called crystallization. ex: diamonds, table salt etc.

4. CRYSTAL STRUCTURE

5. SYMMETRYIN CRYSTALLATTICE
The most important property of crystals is their symmetry. “All
the crystals of the substance have the same elements of
symmetry”. There are three important types of symmetries in a
crystal. They are ,
❖ Plane of symmetry
❖ Axis of symmetry (line of symmetry)
❖ Centre of symmetry

6. PLANEOF SYMMETRY
➢ When an imaginary plane passing through the centre can divide a crystal into two
parts such that one is the exact mirror image of the other, the crystal is said to have a
plane of symmetry.
➢ Two portions obtainedby imagining a plane passing rectangularlythrough the
centre, then it is called rectangularplane of symmetry.
➢Two portionsobtained by imagining a plane passing diagonally through the centre,
then it is a diagonal plane of symmetry.

7. AXISOF SYMMETRY
❑ It is an imaginary axis about which the crystal may be rotatedsuch that it presents
the same appearance more than once during a completerotation.
❑ If the appearance is the same only twice in a rotation, the axis of rotation is said to
have two-fold symmetry.
❑ If the repetitionis three times, the axis has three-foldsymmetry.
❑ If the repetitionis four times, the axis has four-fold symmetry.

8. CENTREOF SYMMETRY
• Centre of symmetry of a crystal is such a point that any line drawn through it
intersects the surface of crystal at equal distance in both the directions.
• Below figure representsthe Centre of symmetry for a cube.
• A crystal may have only one Centre of symmetry.

9. LATTICEPOINTSANDCRYSTALLATTICE
• The constituentparticles (atoms , ions or molecules)of a crystalline solid are
arranged in a regular order.
• The positionof these particles in the crystalline solid , relative to one another,
is usually represented by points(.) which are called “latticepoints”or lattice
sites.
• The orderly or regular arrangement of infinite set of these lattice points which
shows how the constituentparticles present in the crystal lattice are arranged
in three dimensional space is called “crystallattice”or “spacelattice”.

10. UNITCELL
➢ In the figure it is found that crystal lattice is
composed of an infinite number of small units
which are adjacent to one another, in three
dimensions.
➢ Each of these small units is called a unit as
shown in the figure.
➢ Thus a unit cell can be defined as the smallest unit of the crystal lattice which, when
repeated again and again gives the entire crystal of the given substance.
➢ A unit cell is characterised by :-
1. Its dimensions along three edges a, b and c. These edges may or may not be mutually
perpendicular.
2. Angles between the edges , α (between b & c) β (between a & c) γ (between a & b).
3. Thus a unit cell is characterised by six parameters a, b, c, α, β and γ.

11. SEVENCRYSTALSYSTEMS
1. It canbe showngeometricallythattheoretically therecanbe 32 differentcombinationsofelementsof
symmetryofa crystal.
2. Thesearecalled32 pointsgroupsor 32 systems.
3. Someof thesesystemshavebeengroupedintosevendifferent categorieswhichareknownas sevenbasic
crystalsystems.
4. Thesesevencrystalsystemsarecubic, orthorhombic, tetragonal,monoclinic, triclinic,hexagonaland
rhombohedral. Thefigureof thesecrystalsis givenbelow.

12. CLASSIFICATIONOF UNIT CELL
Unit cell can be divided into two categories, they are primitive unit cell and centredunit
cell.
a) PRIMITIVE UNIT CELL :-
When constituentparticles are present only on the corner positions of a unit cell, it
is called as primitive unit cell.
b) CENTRED UNIT CELL :-
When a unit cell containsone or more constituentparticles present at positions
other than corners in addition to those at corners, it is called a centredunit cell.
Centred unit cell

13. BRAVAISLATTICE
❖The French crystallographer Augusta Bravais in 1848 showed from geometrical
considerationsthat there can be 14 differentways in which similar pointscan be
arranged in a three-dimensional space.
❖Thus, the total number of space latticesbelonging to all the seven basic crystal
systems put together is only 14. These latticesare called Bravais lattices.
❖ Cubic unit cell has three differenttypes of arrangement of latticepoints which give
rise to three differentBravais lattices.
❖ They are:-
1) Body centredcubic lattice
2) Face centred cubic lattice
3) End centred cubic lattice

14. BODY CENTRED CUBIC LATTICE
• In this structure, there are eight spheres at 8 corners of a cube and one sphere is in
the centre or body of the cube.
• In this structure, each sphere is in contactwith 8 other spheres and hence the
coordinationnumber of each sphere is 8 in this structure.
• The total number of atoms present at the one body centredcubic unit cell is,
= 1/8 * number of atoms placed at 8 corners of the cube + no. of atoms situated
at the centreof the cube
= (1/8 * 8)+ 1
= 1 + 1
= 2
• Body centred latticeis labelled as ‘I’ and is present only in cubic, orthorhombicand
tetragonal crystal lattice.

15. FACE CENTRED CUBIC LATTICE
o In this structure, the atoms are present at all the eight corners of the cube and at
the centre of each face.
o In this structure, each atom is in contactwith 12 other atom and hence the
coordinationnumber of the atom is 12.
o The face centredatom in the front face is in contactwith four corner atoms and four
other face centred atoms behind it and is also touchingfour face centred atoms of
the unit cell in front of it.
o The total number of atoms present in one face centred cubic unit cell is,
= 1/8 * no. of atoms placed at eight corners of the cube + 1/2 * no. of atoms
placed at the six centresof the six faces of the cube.
= (1/8 * 8) + (1/2 * 6)
= 4
o This type of lattice is labelled as ‘F’ and is present only in cubic and orthorhombic
crystal systems.

16. END CENTRED CUBIC LATTICE
▪ In this structure, the atoms are present at all the eight corners of the cube and at the
centresof the two end faces of the unit cell.
▪ There are total 10 atoms. Eight atoms each at the corner of the cube and 2 atoms at
centresof the two end faces of the cubes.
▪ In this structure, each atom is surrounded by 12 other atoms and hence the
coordinationnumber of the atom is 12.
▪ This type of lattice is labelled as ‘C’ and is present only in orthorhombic and
monoclinic crystal systems.
▪ Total number of atoms present in one end centred cubic unit cell is
= 1/8 * no. of atoms placed at 8 corners of the cube + 1/2 * no. of atoms placed at
the centre of the 2 faces of the cube.
= (1/8 * 8) + (1/2 * 2)
= 2

17. BRAVAISLATTICECRYSTALS:-
➢ The Bravais space latticesassociated with
various crystal systems are given below.
➢ The parameters of unit cell, i.e., the cell
dimensions a , b , c and the interfacial
angles α ,β ,γ are also shown
in each case.

18. PROPERTIES OFTHECRYSTAL SYSTEM
NAMEOF THE
CRYSTAL SYSTEM
PARAMETERS OR DIMENSIONSOF THE
UNITCELL
TYPESOF BRVIAS
LATTICESPRESENT
IN THESYSTEM
NUMBEROF
BRAVIAS LATTICES
MINIMUM
SYMMETRY
ELEMENTS
EXAMPLES
INTERCEPTS CRYSTAL ANGELS
1. Cubic a = b = c α = β = γ = 90⁰ 1.Simpleor
primitive.
2.Bodycentred
cubic(bcc)
3.Facecentred
cubic(fcc)
3 Four 3-fold NaCl, KCl, ZnS,
diamond, Ag , Au ,
Hg , Pb
2.Orthorhombic a ≠ b ≠ c α = β = γ = 90⁰ 1.Simpleor
primitive.
2.Bodycentred
orthorhombic.
3.Facecentred
orthorhombic.
4.End centred
orthorhombic.
4 Threemutually
perpendicular 2-fold
KNO₃, K₂SO₄ BaSO₄
MgSO₄ Mg₂SiO₄
3. Tetragonal a = b ≠ c α = β = γ = 90⁰ 1.Simpleor
primitive.
2.Bodycentred
orthorhombic.
2 One4-fold Sn , NiSO₄, SnO₂,
TiO₂

19. 4.Monoclinic a ≠ b ≠ c α = γ = 90⁰
Β ≠ 90⁰
1.Simple
monoclinic.
2.End centred
monoclinic.
2 One 2 - fold Na₂SO₄,
CaSO₄, 2H₂O.
FeSO₄
5. Triclinic a ≠ b ≠ c α ≠ β ≠ α ≠ 90⁰ Simple
triclinic
1 One 1 - fold CuSO₄,
5H₂O.H₃BO₃
6. Hexagonal a = b ≠ c α = γ = 90⁰
β = 120⁰
Simple
hexagonal
1 One 6 - fold AgI , graphite ,
ZnO, CdS, ice
7. Rhombohedral a = b = c α = β = γ = 90⁰ Simple
trigonal
1 One 3 – fold NaNO₃ , ICI ,
calcite, Sb , Bi
, magnesite.

20. STRUCTURES OF SOME OF THE
CRYSTALS
1. STUCTURE OF SODIUM CHLORIDE CRYSTAL ( ROCK SALT) :-
The cubic unit cell of NaCl crystal is shown in which the circle with negative unit
cell representsCl⁻ and the circles having positive sign representsNa⁺ ions.

21. ▪ ArrangementofNa⁺andCl⁻ions:-
➢In this unit cell of NaCl crystal, Cl⁻ ions have cubic close-packed(ccp)
arrangement i.e., in the NaCl crystal Cl⁻ ions are present at all the 8
corners of the cube and at the centre of each of the 6 faces. Thus there
are 14 Cl⁻ ions.
➢ One Na⁺ ion is present at all the centre of the cubic unit cell and 12 Na⁺
ions are present at centre of 12 edges. Thus there are 13 Na⁺ ions.
➢Thus the NaCl crystal contains alternate Na⁺ and Cl⁻ ions each of which
has face-centred cubic arrangement.
➢When this type of arrangement takes place we say that Cl⁻ ions have
CCP arrangement and Na⁺ ions occupy all the available octahedral sites
created by the CCP of Cl⁻ ions.

22. ▪ FORMULA OF NaCl CRYSTAL :-
➢Since in the crystal of sodium chloride, Cl⁻ ions have ccp arrangement,
each Cl⁻ ion is associated with only one octahedral hole and this hole is
occupied by Na⁺ ions.
➢Thus there is only one Na⁺ ion for one Cl ⁻ ion in the rock salt.
➢ Therefore the formula of rock salt crystal is NaCl.

23. ▪ COORDINATION NUMBER OFNa⁺AND Cl⁻:-
➢The coordination number of each ion in NaCl crystal is equal
to 6.
➢Thus each Na⁺ ion is surrounded octahedrally by six Cl⁻ ions.
➢Similarly each Cl⁻ ion is also surrounded octahedrally by six
Na⁺ ions. Thus NaCl is a 6:6 ionic crystal.
➢The octahedral arrangement of 6 Na⁺ ions round a Cl⁻ ion
and octahedral arrangement of 6 Cl⁻ ions round a Na⁺ ion
have been shown in the diagram.

24. 2.STRUCTURE OFCALCIUM FLUORIDE(CaF₂) CRYSTAL:-
The unit cell of CaF₂ crystal is shown in which Ca²⁺ and F⁻ ions have been represented by
circles having orange and yellow colours respectively.

25. Silentfeaturesof CaF₂isgivenbelow:-
❑In CaF₂, Ca²⁺ions havecubicclosepackingarrangement.
❑Ca²⁺ionsareat all the8 cornersandat allthe6 centresof thesixfacesof thecube.F⁻ ionsoccupyallthe
availabletetrahedralsites.
❑Eachof theCa²⁺ionshave2 tetrahedralsitesandboththesetetrahedralsitesareoccupiedby twoF⁻ Ions.
❑Thusthereare2 F⁻ ionsfor1 Ca²⁺ions. Theformula ofcalciumfluorideis CaF₂.
❑Theradiusratiovalueof CaF₂crystalis 0.73.
❑HencethecoordinationnumberofCa²⁺ionin CaF₂crystalis 8 . i.e.,eachCa²⁺ionis surroundedby 8 F⁻ ions
whichis presentat eight cornerof thecube.
❑EachF⁻ ionis surroundedby Ca²⁺ionswhicharesituatedat the 4 cornersof thetetrahedron.Thereforethe
coordinationnumberofF⁻ ioninCaF₂is 4.
❑ThusCaF₂ isan8 : 4 crystal.

26. REFERENCES :-
1) ADVANCED INORGANIC CHEMISTRY
VOLUME – 1
BY :- S CHAND
2) PRINCIPLES OF PHYSICAL CHEMISTRY
BY :- PURI SHARMA