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Crystal Structure of Solids Books for Solid state physics Course: -

Introduction to Solid State Physics by Charles Kittel Solid State Physics By A J Dekker

Solid State Physics By Ashcroft and Mermin

Dr. Sukanta De

How do atoms assemble into solid structures? Crystalline Solids :

Atoms are arranged in regular manner , form 3-D pattern. (by 3-D repetition of a certain pattern unit.)

PERIODIC ARRANGEMENT OF ATOMS/IONS OVER LARGE ATOMIC DISTANCES

Leads to structure displaying LONG-RANGE ORDER that is Measurable and Quantifiable

All metals, many ceramics, and some polymers

When the periodicity of the pattern extends throughout a certain piece of material

Single Crystal

When the periodicity of the pattern interrupted at grain boundary and grain size is at least several Angstroms

Polycrystalline materials If the grain size is comparable to the size of pattern unit Amorphous materials Materials Lacking Long range order Example: Ceramic GLASS and many “plastics”

POLYCRYSTALLINE MATERIALS • “Nuclei” form during solidification, each of which grows into crystals

Ideal Crystal • An ideal crystal is a periodic array of structural units, such as atoms or molecules. • It can be constructed by the infinite repetition of these identical structural units in space. • Structure can be described in terms of a lattice, with a group of atoms attached to each lattice point. The group of atoms is the basis.

Bravais Lattice • An infinite array of discrete points with an arrangement and orientation that appears exactly the same, from any of the points the array is viewed from. • A three dimensional Bravais lattice consists of all points with position vectors R that can be written as a linear combination of primitive vectors. The expansion coefficients must be integers.

Lattice - infinite, perfectly periodic array of points in a space

Not a lattice:

We abstracted points from the shape:

Now we abstract further:

Now we abstract further:

This is a UNIT CELL

Now we abstract further:

This is a UNIT CELL Represented by two lengths and an angle

…….or, alternatively, by two vectors

Basis vectors and unit cells

b a T

T = ma + nb

a and b are the basis vectors for the lattice

In 3-D: c b a

a, b, and c are the basis vectors for the lattice

In 3-D: c b a

T

T = m1a + m2b + m3c a, b, and c are the basis vectors for the lattice

Crystal Systems – Some Definitional information Unit cell: smallest repetitive volume which contains the complete lattice pattern of a crystal. 7 crystal systems of varying symmetry are known

These systems are built by changing the lattice parameters: a, b, and c are the edge lengths , , and are interaxial angles Fig. 3.4, Callister 7e.

Crystal Systems Crystal structures are divided into groups according to unit cell geometry (symmetry).

CRYSTAL SYMMETRY

Symmetry defines the order resulting from how atoms are arranged and oriented in a crystal The definite ordered arrangement of the faces and edges of a crystal known as Crystal Symmetry

The Symmetry Operation is one that leaves the crystal and its environment invariant. i.e., actions which result in no change to the order of atoms in the crystal structure

Imagine that this object will be rotated (maybe)

Was it?

The object is obviously symmetric…it has symmetry

The object is obviously symmetric…it has symmetry Can be rotated 90° w/o detection

…………so symmetry is really

doing nothing

Symmetry is doing nothing - or at least doing something so that it looks like nothing was done!

What kind of symmetry does this object have?

What kind of symmetry does this object have?

4

What kind of symmetry does this object have?

4

m

What kind of symmetry does this object have?

4

m

What kind of symmetry does this object have?

4

m

What kind of symmetry does this object have?

4

4mm m

Another example:

Another example:

6

6mm m

Why Is Symmetry Important? • Identification of Materials • Prediction of Atomic Structure • Relation to Physical Properties – Optical – Mechanical – Electrical and Magnetic

Symmetry operations performed about a point or a line are called point group symmetry operations

Point group symmetry elements exhibited by crystals are 1. The inversion

(centre of symmetry)

2. Reflection symmetry (The mirror reflection) 3. Rotation Symmetry

Inversion A crystal will possess an inversion centre if for every lattice point given by a position vector r there will be a corresponding lattice point at the position -r

Crystallographic symmetry element : Centre of Symmetry

1

1

Reflection Symmetry In this operation, the reflection of a structure at a mirror plane m

passing through a lattice point leaves the crystal unchanged. The mirror plane may or may not be composed of the atoms lying on the concerned imaginary plane.

Crystallographic symmetry element : Plane of Symmetry

Rotational Symmetry If a crystal left invariant after a rotation about an axis , is said to possess rotational symmetry. The axis is called Axis of Symmetry The axis is called `n-fold, axis’ if the angle of rotation is 3600/n

.

If equivalent configuration occurs after rotation of 180º,

120º and 90º, the axes of rotation are known as two-fold, threefold and four-fold axes of symmetry respectively.

If n=1, the crystal has to be rotated through an angle = 360º, about an axis to achieve self coincidence. Such an axis is called an `identity axis’. Each crystal possesses an infinite number of such axes.

Symmetry – the rules behind the shapes

Symmetry elements For a simple Cubic Lattice

One CENTRE OF SYMMETRY

Plane of Symmetry:

9 planes of Symmetry

There are three planes of symmetry parallel to the faces of the cube and six diagonal planes of symmetry

If n=4, for every 90º rotation, coincidence is achieved and the axis is termed `tetrad axis’.It is discussed already that a cube has `three’ tetrad axes.

If n=3, the crystal has to be rotated through an angle = 120º about an axis to achieve self coincidence. Such an axis is called is `triad axis’. In a cube, the axis passing through a solid diagonal acts as a triad axis. Since there are 4 solid diagonals in a cube, the number of triad axis is four.

If n=2, the crystal has to be rotated through an angle = 180º about an axis to achieve self coincidence. Such an axis is called a `diad axis’.Since there are 12 such edges in a cube, the number of diad axes is six.

Total 13 axes of rotational symmetry for a Cube

SYMMETRICAL ELEMENTS OF CUBE

(a) Centre of symmetry (b) Planes of symmetry (Straight planes -3,Diagonal planes -6) (c) Diad axes (d) Triad axes (e) Tetrad axes

1 9

6 4 3 ---Total number of symmetry elements = 23 ---Thus the total number of symmetry elements of a cubic structure is 23.

N-fold axes with n=5 or n>6 does not occur in crystals

Adjacent spaces must be completely filled (no gaps, no overlaps).

ABSENCE OF 5 FOLD SYMMETRY

We have seen earlier that the crystalline solids show only 1,2,3,4 and 6-fold axes of symmetry and not 5-fold axis of symmetry or symmetry axis higher than 6. The reason is that, a crystal is a one in which the atoms or molecules are internally arranged in a very regular and periodic fashion in a three dimensional pattern, and identical repetition of an unit cell can take place only when we consider 1,2,3,4 and 6-fold axes.

MATHEMATICAL VERIFICATION Let us consider a lattice P Q R S as shown in figure

θ

θ

P

Q a

R

S

Let this lattice has n-fold axis of symmetry and the lattice parameter be equal to ‘a’.

Let us rotate the vectors Q P and R S through an

angle = 3600/n , in the clockwise and anti clockwise

directions respectively. After rotation the ends of the vectors be at x and y. Since the lattice PQRS has n-fold axis of symmetry,

the points x and y should be the lattice points. y

x

θ

θ

P

Q a

R

S

Further the line xy should be parallel to the line PQRS. Therefore the distance xy must equal to some integral multiple of the lattice parameter ‘a’ say, m a. i.e., xy = a + 2a cos = ma (1) Here, m = 0, 1, 2, 3, .................. From equation (1), 2a cos = m a – a i.e., 2a cos = a (m - 1) (or) cos = m 1 N 2

Here,

2

N = 0, 1, 2, 3, .....

(2)

since (m-1) is also an integer, say N. We can determine the values of which are allowed in a lattice by solving the equation (2) for all values of N. For example, if N = 0, cos = 0 i.e., = 90o n = 4. In a similar way, we can get four more rotation axes in a lattice, i.e., n = 1, n = 2, n = 3, and n = 6. Since the allowed values of cos have the limits –1 to +1, the solutions of the equation (2) are not possible for N > 2. Therefore only 1, 2, 3, 4 and 6 fold symmetry axes can exist in a lattice.

Lattice Sites in Cubic Unit Cell

Crystal Structure

53

Crystal Directions • We choose one lattice point on the line as an origin, say the point O. Choice of origin is completely arbitrary, since every lattice point is identical. • Then we choose the lattice vector joining O to any point on the line, say point T. This vector can be written as; R = n1 a + n2 b + n3c • To distinguish a lattice direction from a lattice point, the triple is enclosed in square brackets [ ...] is used.[n1n2n3] • [n1n2n3] is the smallest integer of the same relative ratios.

Crystal Structure

Fig. Shows [111] direction

54

Examples

210

X=½ ,Y=½,Z=1 [½ ½ 1] [1 1 2]

X=1,Y=½,Z=0 [1 ½ 0] [2 1 0] Crystal Structure

55

Negative directions • When we write the direction [n1n2n3] depend on the origin, negative directions can be written as

Z direction

[n1n2 n3 ]

(origin) O

- X direction

- Y direction

Y direction [n1n2 n3 ]

X direction - Z direction

Crystal Structure

56

Examples of crystal directions

X=1,Y=0,Z=0

[1 0 0]

X = -1 , Y = -1 , Z = 0

Crystal Structure

[110]

57

Crystal Planes • Within a crystal lattice it is possible to identify sets of equally spaced parallel planes. These are called lattice planes.

The set of planes in 2D lattice.

b

b

a

a

Crystal Structure

58

Miller Indices Miller Indices are a symbolic vector representation for the orientation of an atomic plane in a crystal lattice and are defined as the reciprocals of the fractional intercepts which the plane makes with the crystallographic axes. To determine Miller indices of a plane, take the following steps; 1) Determine the intercepts of the plane along each of the three crystallographic directions

2) Take the reciprocals of the intercepts 3) If fractions result, multiply each by the denominator of the smallest fraction

Crystal Structure

59

Example-1

Axis

X

Y

Z

Intercept points

1

∞

∞

Reciprocals Smallest Ratio

(1,0,0)

1/1 1/ ∞ 1/ ∞ 1

Miller İndices

Crystal Structure

0

0

(100)

60

Example-2 Axis

X

Y

Z

Intercept points

1

1

∞

Reciprocals

(0,1,0)

Smallest Ratio

1/1 1/ 1 1/ ∞ 1

Miller İndices

1

0 (110)

(1,0,0)

Crystal Structure

61

Example-3

(0,0,1)

Axis

X

Y

Z

Intercept points

1

1

1

Reciprocals (0,1,0)

(1,0,0)

Smallest Ratio

1/1 1/ 1 1/ 1 1

Miller İndices

Crystal Structure

1

1 (111)

62

Example-4 Axis

X

Y

Z

Intercept points

1/2

1

∞

Reciprocals (0,1,0) (1/2, 0, 0)

Smallest Ratio

1/(½) 1/ 1 1/ ∞ 2

Miller İndices

Crystal Structure

1

0

(210)

63

Example-5 Axis

a

b

c

Intercept points

1

∞

½

Reciprocals

1/1

1/ ∞

1/(½)

Smallest Ratio

1

0

2

Miller İndices

Crystal Structure

(102)

64

Example-6 Axis

a

b

c

Intercept points

-1

∞

½

Reciprocals

1/-1

1/ ∞

1/(½)

Smallest Ratio

-1

0

2

Miller İndices

Crystal Structure

(102)

65

Miller Indices [2,3,3] 2

c

3a , 2b , 2c 1 1 1 Reciprocal numbers are: , , 3 2 2

Plane intercepts axes at

Indices of the plane (Miller): (2,3,3)

b 2

a

Indices of the direction: [2,3,3]

3

(200) (110) (100)

Crystal Structure

(111)

(100)

66

Example-7

Crystal Structure

67

Indices of a Family or Form • Sometimes when the unit cell has rotational symmetry, several nonparallel planes may be equivalent by virtue of this symmetry, in which case it is convenient to lump all these planes in the same Miller Indices, but with curly brackets.

{100} (100), (010), (001), (0 1 0), (00 1 ), ( 1 00) {111} (111), (11 1 ), (1 1 1), ( 1 11), ( 1 1 1 ), ( 1 1 1), ( 1 1 1 ), (1 1 1 )

Thus indices {h,k,l} represent all the planes equivalent to the plane (hkl) through rotational symmetry. Crystal Structure

68

Coordinatıon Number • Coordinatıon Number (CN) : The Bravais lattice points closest to a given point are the nearest neighbours. • Because the Bravais lattice is periodic, all points have the same number of nearest neighbours or coordination number. It is a property of the lattice. • A simple cubic has coordination number 6; a bodycentered cubic lattice, 8; and a face-centered cubic lattice,12.

Crystal Structure

69

Atomic Packing Factor

• Atomic Packing Factor (APF) is defined as the volume of atoms within the unit cell divided by the volume of the unit cell.

1-CUBIC CRYSTAL SYSTEM a- Simple Cubic (SC)

Simple Cubic has one lattice point so its primitive cell. In the unit cell on the left, the atoms at the corners are cut because only a portion (in this case 1/8) belongs to that cell. The rest of the atom belongs to neighboring cells. Coordinatination number of simple cubic is 6.

b

c a Crystal Structure

71

Atomic Packing Factor (APF) APF =

Volume of atoms in unit cell* Volume of unit cell

*assume hard spheres • APF for a simple cubic structure = 0.52 atoms unit cell

a R=0.5a close-packed directions contains (8 x 1/8) = 1 atom/unit cell Adapted from Fig. 3.23, Callister 7e.

APF =

volume atom 4 p (0.5a) 3 1 3

a3

volume unit cell

Here: a = Rat*2 Where Rat is the ‘handbook’ atomic radius

b-Body Centered Cubic (BCC)

BCC has two lattice points so BCC is a non-primitive cell.

BCC has eight nearest neighbors. Each atom is in contact with its neighbors only along the bodydiagonal directions.

Many metals (Fe,Li,Na..etc), including the alkalis and several transition elements choose the BCC structure. Crystal Structure

b

c a 73

Atomic Packing Factor: BCC 3a

a 2a

R

Close-packed directions: length = 4R = 3 a

a

atoms

unit cell APF = Adapted from Fig. 3.2(a), Callister 7e.

2

4 3

p ( 3 a/4 ) 3 a3

volume atom volume

unit cell • APF for a body-centered cubic structure = 0.68

c- Face Centered Cubic (FCC) • There are atoms at the corners of the unit cell and at the center of each face. • Face centered cubic has 4 atoms so its non primitive cell. • Many of common metals (Cu,Ni,Pb..etc) crystallize in FCC structure.

Crystal Structure

75

3 - Face Centered Cubıc

Crystal Structure

Atoms are all same.

76

Atomic Packing Factor: FCC • APF for a face-centered cubic structure = 0.74 The maximum achievable APF! Close-packed directions: length = 4R = 2 a

2a

(a = 22*R)

Unit cell contains: 6 x 1/2 + 8 x 1/8 = 4 atoms/unit cell

a Adapted from Fig. 3.1(a), Callister 7e.

atoms unit cell APF =

4

4

3

p ( 2 a/4 ) 3 a3

volume atom volume unit cell

Unit cell contents Counting the number of atoms within the unit cell Atoms corner face centre body centre edge centre lattice type P I F C

Shared Between: 8 cells 2 cells 1 cell 2 cells

Each atom counts: 1/8 1/2 1 1/2

cell contents 1 [=8 x 1/8] 2 [=(8 x 1/8) + (1 x 1)] 4 [=(8 x 1/8) + (6 x 1/2)] 2 [=(8 x 1/8) + (2 x 1/2)] Crystal Structure

78

Theoretical Density, r Density = r =

r =

where

Mass of Atoms in Unit Cell Total Volume of Unit Cell

nA VC NA

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

Theoretical Density, r

R atoms unit cell

r= volume unit cell

• Ex: Cr (BCC) A = 52.00 g/mol R = 0.125 nm n=2 a = 4R/3 = 0.2887 nm a

rtheoretical = 7.18 g/cm3

2 52.00 a 3 6.023 x 1023

g mol ractual atoms mol

= 7.19 g/cm3

The Concept of the reciprocal lattice devised to tabulate two important properties of crystal planes: Their slopes and their inter planer distance. The reciprocal space lattice is a set of imaginary points constructed in such a way that the direction of vector from one point to another coincides with the direction of a normal to the real space planes and the separation of those points (absolute value of the vector) is equal to the reciprocal of the real interplaner distance.

Reciprocal Lattice Vectors The electronic number density is a periodic function in space with a period equal to the lattice translation vector T, i.e. n(r T) n(r ) This means that one can use a Fourier series expansion to represent in 1D n(x) as where:

n( x) n0 C p cos(2ppx / a) S p sin( 2ppx / a) n p ei 2ppx / a p 0

p

a 1 n p a dxn( x)e i 2ppx / a 0

In 3D, we have n(r ) nG e

iGr

G

a

nG V1 dVn(r )e iGr c 0

The set of reciprocal lattice vectors that lead to electron density invariant under lattice translations is found from the condition: n(r T) nG eiG(r T) nG eiGr eiGT n(r ) when eiGT 1 G

G

The reciprocal lattice vectors that satisfy the above requirement are of the form G v1b1 v2b 2 v3b3 where v1, v2 and v3 are integers and b i 2p

a j ak

ai a j a k

, i x, y, z bi a j 2pij

HW:

Reciprocal of a reciprocal lattice is the direct lattice

Wigner-Seitz Method A simply way to find the primitive cell which is called Wigner-Seitz cell can be done as follows; 1. Choose a lattice point. 2. Draw lines to connect these lattice point to its neighbours. 3. At the mid-point and normal to these lines draw new lines. The volume enclosed is called as a Wigner-Seitz cell. Crystal Structure

97

Wigner-Seitz Cell - 3D

Crystal Structure

98

X-ray Diffraction

E hc / Typical interatomic distances in solid are of the order of an angstrom. Thus the typical wavelength of an electromagnetic probe of such distances Must be of the order of an angstrom.

Upon substituting this value for the wavelength into the energy equation, We find that E is of the order of 12 thousand eV, which is a typical X-ray Energy. Thus X-ray diffraction of crystals is a standard probe.

X-Rays to Determine Crystal Structure • Incoming X-rays diffract from crystal planes.

extra distance traveled by wave “2”

d

Measurement of critical angle, c, allows computation of planar spacing, d. For Cubic Crystals:

d hkl

reflections must be in phase for a detectable signal! Adapted from Fig. 3.19, Callister 7e.

spacing between planes

X-ray intensity (from detector)

n d 2 sin c

a

h2 k 2 l 2

h, k, l are Miller Indices

c

Figure 3.34 (a) An x-ray diffractometer. (Courtesy of Scintag, Inc.) (b) A schematic of the experiment.

X-Ray Diffraction Pattern z

z

Intensity (relative)

c a x

z

c b

y (110)

a x

c b

y

a x (211)

b

(200)

Diffraction angle 2

Diffraction pattern for polycrystalline -iron (BCC) Adapted from Fig. 3.20, Callister 5e.

y

INTERATOMIC FORCES What kind of forces hold the atoms together in a solid? Energies of Interactions Between Atoms

Ionic bonding NaCl Covalent bonding Comparison of ionic and covalent bonding Metallic bonding Van der waals bonding Hydrogen bonding

Energies of Interactions Between Atoms • The energy of the crystal is lower than that of the free atoms by an amount equal to the energy required to pull the crystal apart into a set of free atoms. This is called the binding (cohesive) energy of the crystal. – NaCl is more stable than a collection of free Na and Cl. – Ge crystal is more stable than a collection of free Ge.

Cl

Na

NaCl

Types of Bonding Mechanisms It is conventional to classify the bonds between atoms into different types as • • • • •

Ionic Covalent Metallic Van der Waals Hydrogen All bonding is a consequence of the electrostatic interaction between the nuclei and electrons.

IONIC BONDING

Ionic bonding is the electrostatic force of attraction between positively and negatively charged ions (between non-metals and metals).

All ionic compounds are crystalline solids at room temperature.

NaCl is a typical example of ionic bonding.

Metallic elements have only up to the valence electrons in their outer shell. When losing their electrons they become positive ions.

Electronegative elements tend to acquire additional electrons to become negative ions or anions.

Na

Cl

• When the Na+ and Cl- ions approach each other closely enough so that the orbits of the electron in the ions begin to overlap with each other, then the electron begins to repel each other by virtue of the repulsive electrostatic coulomb force. Of course the closer together the ions are, the greater the repulsive force.

• Pauli exclusion principle has an important role in repulsive force. To prevent a violation of the exclusion principle, the potential energy of the system increases very rapidly.

COVALENT BONDING • Covalent bonding takes place between atoms with small differences in electronegativity which are close to each other in the periodic table (between non-metals and non-metals). • The covalent bonding is formed when the atoms share the outer shell electrons (i.e., s and p electrons) rather than by electron transfer. • Noble gas electron configuration can be attained.

• Each electron in a shared pair is attracted to both nuclei involved in the bond. The approach, electron overlap, and attraction can be visualized as shown in the following figure representing the nuclei and electrons in a hydrogen molecule.

e e

Comparison of Ionic and Covalent Bonding

METALLIC BONDING • Metallic bonding is found in metal elements. This is the electrostatic force of attraction between positively charged ions and delocalized outer electrons. • The metallic bond is weaker than the ionic and the covalent bonds.

+

+

+

•

A metal may be described as a lowdensity cloud of free electrons.

+

+

+

•

Therefore, metals have high electrical and thermal conductivity.

+

+

+

VAN DER WAALS BONDING • These are weak bonds with a typical strength of 0.2 eV/atom. • Van Der Waals bonds occur between neutral atoms and molecules. • Weak forces of attraction result from the natural fluctuations in the electron density of all molecules that cause small temporary dipoles to appear within the molecules.

• It is these temporary dipoles that attract one molecule to another. They are called van der Waals' forces.

• The shape of a molecule influences its ability to form temporary dipoles. Long thin molecules can pack closer to each other than molecules that are more spherical. The bigger the 'surface area' of a molecule, the greater the van der Waal's forces will be and the higher the melting and boiling points of the compound will be. • Van der Waal's forces are of the order of 1% of the strength of a covalent bond.

Homonuclear molecules, such as iodine, develop temporary dipoles due to natural fluctuations of electron density within the molecule

Heteronuclear molecules, such as H-Cl have permanent dipoles that attract the opposite pole in other molecules.

• These forces are due to the electrostatic attraction between the nucleus of one atom and the electrons of the other.

Van der waals interaction occurs generally between atoms which have noble gas configuration.

van der waals bonding

HYDROGEN BONDING • A hydrogen atom, having one electron, can be covalently bonded to only one atom. However, the hydrogen atom can involve itself in an additional electrostatic bond with a second atom of highly electronegative character such as fluorine or oxygen. This second bond permits a hydrogen bond between two atoms or strucures. • The strength of hydrogen bonding varies from 0.1 to 0.5 ev/atom.

Hydrogen bonds connect water molecules in ordinary ice. Hydrogen bonding is also very important in proteins and nucleic acids and therefore in life processes.