solid state P353

1. Teaching Staff
Dr. Mohamed Ahmed Sabet Hammam
Please don’t hesitate to contact me
Email: mas_hammam@aun.edu.eg
Whatsapp: 01006773115
Office hours: 2 hrs/week – fifth floor – physics dept.
Condensed Matter Physics
P 353
1
Assiut University
Faculty of Science
Physics Department

2. Topics:
1.1 Introduction
1.2 The crystalline state
1.3 Basic definitions
1.4 The fourteen Bravais lattices and the seven crystal
systems
1.5 Elements of symmetry
1.6 Crystal directions & planes and Miller indices
1.7 Examples of some simple structures
1.9 Interatomic forces
1.10 Types of bonding
CHAPTER 1
Crystal Structure and Interatomic Forces
2

3. Solid State Physics is largely concerned with the remarkable properties
exhibited by atoms and molecules because of their-regular arrangement in
crystals.
Charles Kittle - 1976
Preface,5th Ed. Introduction to solid state physics
1.1 Introduction
Solid State Physics
3
Solid state physics is concerned with the properties, often astonishing and
often of great utility, that results from the distribution of electrons in
metals semiconductors and insulators
Preface,7th Ed. Introduction to solid state physics
Charles Kittle - 1996

4. Solids
To the naked eye, a solid is a continuous rigid body.
Solids are, however, composed of discrete basic units (atoms).
Crystalline solid
The atoms are arranged in a highly ordered manner relative to
each other.
Non crystalline solids (Amorphous)
The atoms are randomly arranged.
This course is mainly dealing with
crystals and electrons in crystals.
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5. 1.2 The crystalline state
Crystal: A solid is said to be a crystal, if the atoms are arranged
in such away that their positions are exactly periodic.
Distance between two nearest neighbors (NN):
along x is a
along y is b
A crystalline solid (two dimensional)
Note that: All atoms are arranged periodically.
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6. •A perfect crystal maintains its periodicity (or repetitively) in all
directions from -∞ to +∞
• Here, All atoms (A,B,C etc.) are equivalent.
• The crystal appears exactly the same to “an observer” located
at any atomic site.
Perfect crystal
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7. Transitional Symmetry
• If the crystal is transitioned by any vector 𝑅 joining two
atoms, the crystal appears exactly the same.
• The crystal is said to possess a transitional symmetry or the
crystal remains invariant under any such transition.
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8. Perfect Versus Real
One can not prepare a perfect crystal. Why?
1. Surface effects
Atoms near the surface “see” less neighbors than an atom deep
inside the crystal.
2. Imperfections (Defects)
Actual (real) crystals always contain impurities or vacancies.
Defects interrupt the periodicity and they modify the physical properties.
Impurity
Vacancy
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9. 3. Thermal Distortion
At T > 0K, atoms vibrate around their equilibrium position, the
crystal is distorted due to this vibration.
Real crystals
They are the crystals that contain few amount of defects that can
be treated as small perturbations in the crystalline structure.
T > 0K
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10. 1.3 Basic Definitions
A regular array of points in space or a pattern of points having
the same geometrical properties of the crystal
Points not atoms
(lattice sites )
Crystalline lattice (lattice)
A crystal is formed when atoms are attached identically to every
point.
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The crystal lattice

11. Classes of lattices:
1- Bravais lattice 2- Non Bravais lattice
1- Bravais lattice
All lattice points are equivalent and all atoms in the crystal are
of the same kind.
2- Non Bravais lattice
Lattice points are not equivalent
Non Bravais Lattice
One dimensional Bravais lattice
A atoms
A atoms
B atoms
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12. In two dimensions
B atom
Bravais lattice
Non Bravais lattice
Non Bravais lattice
It may be considered as a combination of two or more than two
interpenetrating Bravais lattices with fixed orientation relative to
each other.
A atom
A atom
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13. Representing lattice vectors by basis vectors
Consider the lattice shown below:
A: is the origin of the coordinate system
The position vector of any point can be written as:
b
n
a
n
R 2
1 

• (n1, n2 ) pair of integers whose values depend on the lattice
point.
• (a ,b ) set of basis vectors of the lattice.
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14. • Point A has (n1, n2) = (0,0)
• Point B has (n1, n2) =(1,0)
R1 from A to B R1 = 1 a +0 b
• Point C has (n1, n2) =(1,1)
R2 from A to C R2 = 1 a +1 b
• Point D has (n1, n2 )=(2,3)
R3 from A to D R3 = 2 a +3b
a from b are called basis vectors of the lattice.
R1, R2, R3 are called lattice vectors.
D
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15. Quick Quiz I
Is the choice of basis vectors unique ?
No, the choice of basis vectors is not unique.
You can take a and b' as basis vectors.
To go from A to X :
b = a + b a a
b
b
b
A
X
b
a
R 

 2
0
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16. The unit cell
Unit cell:
Smallest area which produces coverage of the whole lattice when
translated by 𝑅
a
b
b
a
A unit cell
•If the unit cell is translated by the lattice vector ,
the area of the whole lattice is covered.
b
n
a
n
R 2
1 

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•The area of the parallelogram whose sides are the basis vectors.

17. Quick Quiz II
Is the choice of unit cell unique ? Why ?
No, it is not unique because the choice of basis vectors (sides of unit cell) is not
unique.
S is a unit cell area
b
a
S 

S is another unit cell
b
a
b
a
b b
b
a
S 



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18. Quick Quiz III
Is the area of the unit cell unique ? Does all unit cells possess the
same area ?
Answer: Left to you
Hint: You can use Vectors Algebra to compare S and S in the
previous example.
a
b b b
a
b
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19. Quick Quiz IV
Calculate the number of lattice point per unit cell in the lattice
shown below?
b
a
Unit cell
We have four points, each point is shared by four different unit
cells
No. of lattice points 1
4
1
4 


Each unit cell has only one lattice point.
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20. Primitive Versus Non Primitive Cell
Primitive Unit Cell
A unit cell that contains only one lattice point.
Non Primitive Unit Cell
A larger cell and contains more than one lattice point.
• S1 is a primitive cell. Why ?
• S2 is a Non primitive cell. Why ?
• Show using simple geometry or
Vector Algebra that S2=2S1 ?
S2
a1
a
b
a2
S1
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21. Three dimensions
A three dimensional unit cell
 All previous discussion can be extended to three dimensions.
 Basis vectors three non coplanar vectors
 Lattice vector : has three components.
 Unit cell : parallelepiped whose sides are .
 Volume: not an Area .
 All unit cells have the same volume .
c
b
a



and
,
c
n
b
n
a
n
R



3
2
1 


c
b
a and
,
c

a
 b

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22. The fourteen Bravais lattices & the seven crystal
systems
Two dimensional Bravais lattices
• In two dimensions, there are only 5 possible Bravais lattices.
These are: Oblique, Square, Hexagonal, Rectangular and
Centered Rectangular lattices .
Three dimensional Bravias lattices
• There are only fourteen(14) three dimensional Bravais lattices
The reduction of the number of Bravais lattices to 5 (in 2 dim.) &
14 (in 3 dim.) is a consequence of the translational symmetry
condition obeyed by lattice.
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23. The fourteen Bravais lattices & the seven crystal
systems
23

24. The fourteen Bravais lattices & the seven crystal
systems
24

25. The 5 lattice types in two dimensions
a
2) Square lattice
b

a
b
1) Oplique lattice
b
a
b
a
b
a
5) Base Centered
Rectangular
o
b
a 90
|,
|
|
| 




Base centered
o
b
a 90
|,
|
|
| 




3) Hexagonal lattice 0
120
|,
|
|
| 




b
a
0
90
|,
|
|
| 




b
a
4) Rectangular lattice
Unit cell
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26. Can we construct a two – dimensional lattice whose unit cell is a
regular pentagon ? Why?
1
5
4
3
2
Answer: No !
One can draw an isolated pentagon, but one cannot place many
such pentagons side by side so that they fit tightly and cover the
whole area.
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27. The 14 lattice types in three dimensions
The 14 lattices are grouped into seven crystal systems according
to the shape and symmetry of the unit cell .
These systems are:
1) Triclinic Has 1 Bravais lattice
2) Monoclinic Has 2 Bravais lattice
3) Ortheorhombic Has 4 Bravais lattice
4) Tetragonal Has 2 Bravais lattice
5) Cubic Has 3 Bravais lattice
6) Trigonal
(Rhombohedral)
Has 1 Bravais lattice
7) Hexagonal Has 1 Bravais lattice
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28. 1) Triclinic Has 1 Bravais lattice
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29. 2) Monoclinic Has 2 Bravais lattice
29

30. 3) Ortheorhombic Has 4 Bravais lattice
30

31. 4) Tetragonal Has 2 Bravais lattice
31

32. 5) Cubic Has 3 Bravais lattice
32

33. 6) Trigonal
(Rhombohedral)
Has 1 Bravais lattice
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34. 7) Hexagonal Has 1 Bravais lattice
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35. Note that
A simple lattice has points only at the corners, a body-centered
lattice has one additional point at the center of the cell and a face-
centered lattice has six additional points on each face
In this course we will be mainly dealing with:
Face-centered cubic fcc.
base-centered cubic bcc.
Hexagonal system
The unit cell for a simple lattice is primitive.
The unit cell of all non simple lattices are non primitive.
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36. Table 1.1
The seven crystal systems divided into fourteen Bravais lattice
System Bravais lattice Unit cell
characteristics
Characteristic symmetry
elements
Triclinic Simple a ≠b ≠ c
α≠ β≠ γ≠900
None
Monoclinic Simple
Base-centered
a ≠b ≠ c
α= β= 900 ≠ γ
One 2-fold rotation axis
Orthorhombic Simple
Base-centered
Body- centered
Face- centered
a ≠b ≠ c
α= β = γ =900
Three mutually
orthogonal 2-flod
rotation axes
Tetragonal Simple
Body -centered
a =b ≠ c
α= β = γ =900
One 4- fold rotation axis
Cubic Simple
Body -centered
Face- centered
a =b = c
α= β = γ =900
Four 3- fold rotation axes
(along cube diagonal)
Trigonal
(rhombohedral)
Simple a =b = c
α= β = γ ≠900
One 3- fold rotation axis
Hexagonal Simple a =b ≠ c
α= β =900
γ =120º
One 3-fold rotation axis
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