Solid state physics d r joshi

Introduction to

SOLID STATE PHYSICSA Random Walk

Dr. Dattu JoshiApplied Physics Department

Faculty of Tech. & Engg.The M S University of Baroda

Vadodara-390 00103/11/2011

Intro to Solid State by Dr Dattu Joshi, MSU, Vadodara

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03/11/2011Intro to Solid State by Dr Dattu Joshi,

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INTRODUCTIONINTRODUCTION

AIM OF SOLID STATE PHYSICS WHAT IS SOLID STATE PHYSICS? CONTENTS APPLICATIONS AND RESEARCH


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Aim of Solid State Physics

Solid state physics (SSP) explains the properties of solid materials as found on earth.

The properties are expected to follow from Schrödinger’s eqn. for a collection of atomic nuclei and electrons interacting with electrostatic forces.

The fundamental laws governing the behaviour of solids are known and well tested.


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Crystalline Solids

We will deal with crystalline solids, that is solids with an atomic structure based on a regular repeated pattern.

Many important solids are crystalline.

More progress has been made in understanding the behaviour of crystalline solids than that of non-crystalline materials since the calculation are easier in crystalline materials.


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EP364 SOLID STATE PHYSICS INTRODUCTION

What is solid state physics?

Solid state physics, also known as condensed matter physics, is the study of the behaviour of atoms when they are placed in close proximity to one another.

In fact, condensed matter physics is a much better name, since many of the concepts relevant to solids are also applied to liquids, for example.


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What is the point?

Understanding the electrical properties of solids is right at the heart of modern society and technology.

The entire computer and electronics industry relies on tuning of a special class of material, the semiconductor, which lies right at the metal-insulator boundary.

Solid state physics provide a background to understand what goes on in semiconductors.


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Electrical resistivity of three states of solid matter

How can this be? After all, they each contain a system of atoms and especially electrons of similar density. And the plot thickens: graphite is a metal, diamond is an insulator and buckminster-fullerene is a superconductor.

They are all just carbon!


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Among our aims - understand why one is a metal and one an insulator, and then the physical origin of the marked features.

Also think about thermal properties etc. etc.


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Solid State PhysicsCrystal Structure


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Solid State PhysicsCrystal Diffraction and the Reciprocal Lattice


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Solid State PhysicsImperfections in Crystals


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Solid State PhysicsCrystal Bonding


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Solid State PhysicsMagnetic Materials


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Elastic constants and Elastic Waves Lattice Vibrations and Phonons Thermal Properties of Solids Free Electron Theory of Metals Transport Properties Band Theory of Solids Semiconductors Superconductivity Dielectrics Optical Phenomena in insulators etc.


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CRYSTAL STRUCTURE

Elementary Crystallography Solid materials (crystalline, polycrystalline,

amorphous) Crystallography Crystal Lattice Crystal Structure Types of Lattices Unit Cell Directions-Planes-Miller Indices in Cubic Unit

Cell Typical Crystal Structures

(3D– 14 Bravais Lattices and the Seven Crystal System)

Elements of Symmetry


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X-RAY CRYSTALLOGRAPHY

X-ray Diffraction

Bragg equation X-ray diffraction methods

Laue Method Rotating Crystal Method Powder Method

Neutron & electron diffraction


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CRYSTAL STRUCTURE

Elementary CrystallographyTypical Crystal Structures

Elements Of Symmetry


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Crystal Structure 20

Objectives

By the end of this section you should:

be able to identify a unit cell in a symmetrical pattern

know that there are 7 possible unit cell shapes

be able to define cubic, tetragonal, orthorhombic and hexagonal unit cell shapes


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mattermatter


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Crystal Structure

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Gases Gases have atoms or molecules that do not

bond to one another in a range of pressure, temperature and volume.

These molecules haven’t any particular order and move freely within a container.


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Liquids and Liquid Crystals

Similar to gases, liquids haven’t any atomic/molecular order and they assume the shape of the containers.

Applying low levels of thermal energy can easily break the existing weak bonds.

Liquid crystals have mobile molecules, but a type of long range order can exist; the molecules have a permanent dipole. Applying an electric field rotates the dipole and establishes order within the collection of molecules.

+

-

+

-

+

-

+

-

+

-

+

-

+

-

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Crytals Solids consist of atoms or molecules

executing thermal motion about an equilibrium position fixed at a point in space.

Solids can take the form of crystalline, polycrstalline, or amorphous materials.

Solids (at a given temperature, pressure, and volume) have stronger bonds between molecules and atoms than liquids.

Solids require more energy to break the bonds.


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SOLID MATERIALS

CRYSTALLINE POLYCRYSTALLINE AMORPHOUS(Non-crystalline)

Single Crystal

ELEMENTARY CRYSTALLOGRAPHYELEMENTARY CRYSTALLOGRAPHY


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Types of Solids

Single crsytal, polycrystalline, and amorphous, are the three general types of solids.

Each type is characterized by the size of ordered region within the material.

An ordered region is a spatial volume in which atoms or molecules have a regular geometric arrangement or periodicity.


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Crystalline Solid Crystalline Solid is the solid form of a substance

in which the atoms or molecules are arranged in a definite, repeating pattern in three dimension.

Single crystals, ideally have a high degree of order, or regular geometric periodicity, throughout the entire volume of the material.


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Crystalline Solid

Single Crystal

Single Pyrite Crystal

AmorphousSolid

Single crystal has an atomic structure that repeats periodically across its whole volume. Even at infinite length scales, each atom is related to every other equivalent atom in the structure by translational symmetry


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Polycrystalline Solid

PolycrystallinePyrite form

(Grain)

Polycrystal is a material made up of an aggregate of many small single crystals (also called crystallites or grains).

Polycrystalline material have a high degree of order over many atomic or molecular dimensions.

These ordered regions, or single crytal regions, vary in size and orientation wrt one another.

These regions are called as grains ( domain) and are separated from one another by grain boundaries. The atomic order can vary from one domain to the next.

The grains are usually 100 nm - 100 microns in diameter. Polycrystals with grains that are <100 nm in diameter are called nanocrystalline


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Amorphous Solid Amorphous (Non-crystalline) Solid is composed of

randomly orientated atoms , ions, or molecules that do not form defined patterns or lattice structures.

Amorphous materials have order only within a few atomic or molecular dimensions.

Amorphous materials do not have any long-range order, but they have varying degrees of short-range order.

Examples to amorphous materials include amorphous silicon, plastics, and glasses.

Amorphous silicon can be used in solar cells and thin film transistors.


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Departure From Perfect Crystal

Strictly speaking, one cannot prepare a perfect crystal. For example, even the surface of a crystal is a kind of imperfection because the periodicity is interrupted there.

Another example concerns the thermal vibrations of the atoms around their equilibrium positions for any temperature T>0°K.

As a third example, actual crystal always contains some foreign atoms, i.e., impurities. These impurities spoils the perfect crystal structure.


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Distinction Between Crystalline and Amorphous Solids

Crystalline Solids• Have a regular arrangement

of particles• Have different physical

properties (thermal conductivity, electrical conductivity, refractive index etc.) in different directions i.e. Anisotropic

• Melting point is very sharp

Amorphous Solids• Have completely random

particle arrangement• Have physical properties same

in all directions, i.e. isotropic• Do not have sharp melting

point e.g. as the temperature of glass is gradually raised, it softens and starts flowing without any sharp change from solid state to liquid state


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•The cooling curve for crystalline substance has breaks, see curve 1 in the fig., the middle of which corresponds to the process of crystallization.In the process of crystallization some energy is liberated which compensates the loss of heat and hence temperature remains constant.

Crystalline Solid

Amorphous Solid

• The cooling curve for amorphous substance is smooth, see curve 2 in the fig.


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CRYSTALLOGRAPHY

What is crystallography?

The branch of science that deals with the geometric description of crystals and their internal arrangement.


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Science of Crystallography

The study of the geometric form and other physical properties of crystalline solids by using X-rays, electron beams and neutron beams etc., constitute the science of crystallography.


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The


Crystallography is essential for solid state physics

Symmetry of a crystal can have a profound influence on its properties.

Any crystal structure should be specified completely, concisely and unambiguously.

Structures should be classified into different types according to the symmetries they possess.

Crystallography


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A basic knowledge of crystallography is essential for solid state physicists; to specify any crystal structure and to classify the solids into different types

according to the symmetries they possess.

Symmetry of a crystal can have a profound influence on its properties.

We will concern in this course with solids with simple structures.

ELEMENTARY CRYSTALLOGRAPHY


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CRYSTAL LATTICE

What is crystal (space) lattice?

In crystallography, only the geometrical properties of the crystal are of interest, therefore one replaces each atom by a geometrical point located at the equilibrium position of that atom.

Platinum Platinum surface Crystal lattice and structure of Platinum(scanning tunneling microscope)


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An infinite array of points in space,

Each point has identical surroundings to all others.

Arrays are arranged exactly in a periodic manner.

Crystal Lattice

α

a

bCB ED

O A

y

x


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Crystal Structure

Crystal structure can be obtained by attaching atoms, groups of atoms or molecules which are called basis (motif) to the lattice sides of the lattice point.Crystal Structure = Crystal Lattice + Basis


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A two-dimensional Bravais lattice with different choices for the basis


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E

H

O A

CB

Fb G

D

x

y

a

α

a

bCB ED

O A

y

x

b) Crystal lattice obtained by identifying all the atoms in (a)

a) Situation of atoms at the corners of regular hexagons

Basis A group of atoms which describe crystal structureA group of atoms which describe crystal structure


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Crystal structure

Don't mix up atoms with lattice points

Lattice points are infinitesimal points in space

Lattice points do not necessarily lie at the centre of atoms

Crystal Structure = Crystal Lattice + Basis


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Crystal Lattice

Bravais Lattice (BL) Non-Bravais Lattice (non-BL)

All atoms are of the same kind All lattice points are equivalent

Atoms can be of different kind Some lattice points are not equivalentA combination of two or more BL


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Types Of Crystal Lattices

1) Bravais lattice is an infinite array of discrete points with an arrangement and orientation that appears exactly the same, from whichever of the points the array is viewed. Lattice is invariant under a translation.

Nb film


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Types Of Crystal Lattices

The red side has a neighbour to its immediate left, the blue one instead has a neighbour to its right.

Red (and blue) sides are equivalent and have the same appearance

Red and blue sides are not equivalent. Same appearance can be obtained rotating blue side 180º.

2) 2) Non-Bravais LatticeNon-Bravais LatticeNot only the arrangement but also the orientation must appear exactly the same from every point in a bravais lattice.

Honeycomb


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Translational Lattice Vectors – 2D

A space lattice is a set of points such that a translation from any point in the lattice by a vector;

Rn = n1 a + n2 b

locates an exactly equivalent point, i.e. a point with the same environment as P . This is translational symmetry. The vectors a, b are known as lattice vectors and (n1, n2) is a pair of integers whose values depend on the lattice point.

P

Point D(n1, n2) = (0,2)

Point F (n1, n2) = (0,-1)


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The two vectors a and b form a set of lattice vectors for the lattice.

The choice of lattice vectors is not unique. Thus one could equally well take the vectors a and b’ as a lattice vectors.

Lattice Vectors – 2D


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Lattice Vectors – 3D

An ideal three dimensional crystal is described by 3 fundamental translation vectors a, b and c. If there is a lattice point represented by the position vector r, there is then also a lattice point represented by the position vector where u, v and w are arbitrary integers.

r’ = r + u a + v b + w c (1)


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Five Bravais Lattices in 2D


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Unit Cell in 2D The smallest component of the crystal (group of

atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal.

S

a

b

S

S

S

S

S

S

S

S

S

S

S

S

S

S


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Unit Cell in 2D

The smallest component of the crystal (group of atoms, ions or molecules), which when stacked together with pure translational repetition reproduces the whole crystal.

S

S

The choice of unit cell

is not unique.

a

Sb

S


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2D Unit Cell example -(NaCl)

We define lattice points ; these are points with identical environments


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Choice of origin is arbitrary - lattice points need not be atoms - but unit cell size should always be the same.


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This is also a unit cell - it doesn’t matter if you start from Na or Cl


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- or if you don’t start from an atom


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This is NOT a unit cell even though they are all the same - empty space is not allowed!


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In 2D, this IS a unit cellIn 3D, it is NOT


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Why can't the blue triangle

be a unit cell?


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Unit Cell in 3D


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Unit Cell in 3D


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Three common Unit Cell in 3D


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UNIT CELL

Primitive Conventional & Non-primitive

Single lattice point per cell Smallest area in 2D, orSmallest volume in 3D

More than one lattice point per cell Integral multibles of the area of primitive cell

Body centered cubic(bcc)Body centered cubic(bcc)

Conventional Conventional ≠ Primitive cell≠ Primitive cellSimple cubic(sc)Simple cubic(sc)

ConventionalConventional = Primitive cell = Primitive cell


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The Conventional Unit Cell

A unit cell just fills space when translated through a subset of Bravais lattice vectors.

The conventional unit cell is chosen to be larger than the primitive cell, but with the full symmetry of the Bravais lattice.

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Primitive and conventional cells of FCC


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Primitive and conventional cells of BCC

Primitive Translation Vectors:


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a

b c

Simple cubic (sc): primitive cell=conventional cell

Fractional coordinates of lattice points:000, 100, 010, 001, 110,101, 011, 111

Primitive and conventional cells

Body centered cubic (bcc): conventional primitive cell

a

b cFractional coordinates of lattice points in conventional cell: 000,100, 010, 001, 110,101, 011, 111, ½ ½ ½


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Body centered cubic (bcc): primitive (rombohedron) conventional cell

a

bc

Fractional coordinates: 000, 100, 101, 110, 110,101, 011, 211, 200

Face centered cubic (fcc): conventional primitive cell

a

bc

Fractional coordinates: 000,100, 010, 001, 110,101, 011,111, ½ ½ 0, ½ 0 ½, 0 ½ ½ ,½1 ½ , 1 ½ ½ , ½ ½ 1

Primitive and conventional cells


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Hexagonal close packed cell (hcp): conventional primitive cell Fractional coordinates: 100, 010, 110, 101,011, 111,000, 001

points of primitive cell

a

b

c

120

o

Primitive and conventional cells-hcp


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The unit cell and, consequently, the entire lattice, is uniquely determined by the six lattice constants: a, b, c, α, β and γ.

Only 1/8 of each lattice point in a unit cell can actually be assigned to that cell.

Each unit cell in the figure can be associated with 8 x 1/8 = 1 lattice point.

Unit CellUnit Cell


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A primitive unit cell is made of primitive translation vectors a1 ,a2, and a3 such that there is no cell of smaller volume that can be used as a building block for crystal structures.

A primitive unit cell will fill space by repetition of suitable crystal translation vectors. This defined by the parallelpiped a1, a2 and a3. The volume of a primitive unit cell can be found by

V = a1.(a2 x a3) (vector products)

Cubic cell volume = a3

Primitive Unit Cell and vectors


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The primitive unit cell must have only one lattice point.

There can be different choices for lattice vectors , but the volumes of these primitive cells are all the same.

P = Primitive Unit CellNP = Non-Primitive Unit Cell

Primitive Unit Cell

1a


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Wigner-Seitz MethodA simply way to find the

primitivecell which is called Wigner-

Seitzcell 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.

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Wigner-Seitz Cell - 3D


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Lattice Sites in Cubic Unit Cell


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Crystal Directions

Fig. Shows [111] direction

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.


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210

X = 1 , Y = ½ , Z = 0[1 ½ 0] [2 1 0]

X = ½ , Y = ½ , Z = 1[½ ½ 1] [1 1 2]

Examples


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Negative directions

When we write the

direction [n1n2n3] depend on the origin, negative directions can be written as

R = n1 a + n2 b + n3c

Direction must be

smallest integers.

Y direction

(origin) O

- Y direction

X direction

- X direction

Z direction

- Z direction

][ 321 nnn

][ 321 nnn


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X = -1 , Y = -1 , Z = 0 [110]

Examples of crystal directions

X = 1 , Y = 0 , Z = 0 [1 0 0]


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Examples

X =-1 , Y = 1 , Z = -1/6[-1 1 -1/6] [6 6 1]

We can move vector to the origin.


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Crystal Planes Within a crystal lattice it is possible to identify sets of

equally spaced parallel planes. These are called lattice planes.

In the figure density of lattice points on each plane of a set is the same and all lattice points are contained on each set of planes.

b

a

b

a

The set of planes in 2D lattice.


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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


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Axis X Y Z

Intercept points 1 ∞ ∞

Reciprocals 1/1 1/ ∞ 1/ ∞Smallest

Ratio 1 0 0

Miller İndices (100)

Example-1

(1,0,0)


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Axis X Y Z

Intercept points 1 1 ∞

Reciprocals 1/1 1/ 1 1/ ∞Smallest

Ratio 1 1 0


Example-2

(1,0,0)

(0,1,0)


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Axis X Y Z

Intercept points 1 1 1

Reciprocals 1/1 1/ 1 1/ 1Smallest

Ratio 1 1 1

Miller İndices (111)(1,0,0)

(0,1,0)

(0,0,1)

Example-3


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Axis X Y Z

Intercept points 1/2 1 ∞

Reciprocals 1/(½) 1/ 1 1/ ∞Smallest

Ratio 2 1 0

Miller İndices (210)(1/2, 0, 0)

(0,1,0)

Example-4


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Axis a b c

Intercept points 1 ∞ ½

Reciprocals 1/1 1/ ∞ 1/(½)

Smallest Ratio 1 0 2


Example-5


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Example-6


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Miller Indices

Reciprocal numbers are: 2

1 ,

2

1 ,

3

1Plane intercepts axes at cba 2 ,2 ,3

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

(100)

(200)

(110)(111)

(100)

Indices of the direction: [2,3,3]a3

2

2

bc

[2,3,3]


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Example-7


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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.

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

)111(),111(),111(),111(),111(),111(),111(),111(}111{

)001(),100(),010(),001(),010(),100(}100{


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Find out which one is wrong?


There are only seven different shapes of unit cell which can be stacked together to completely fill all space (in 3 dimensions) without overlapping. This gives the seven crystal systems, in which all crystal structures can be classified.

Cubic Crystal System (SC, BCC,FCC) Hexagonal Crystal System (S) Triclinic Crystal System (S) Monoclinic Crystal System (S, Base-C) Orthorhombic Crystal System (S, Base-C, BC, FC) Tetragonal Crystal System (S, BC) Trigonal (Rhombohedral) Crystal System (S)

3D – 14 BRAVAIS LATTICES AND THE SEVEN CRYSTAL SYSTEM

TYPICAL CRYSTAL STRUCTURES


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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 body-centered cubic lattice, 8; and a face-centered cubic lattice,12.


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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.


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1-CUBIC CRYSTAL SYSTEM

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.

a- Simple Cubic (SC)

a

b c


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a- Simple Cubic (SC)


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Atomic Packing Factor of SC


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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 body-diagonal directions.

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

a

b c


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2 (0.433a)

Atomic Packing Factor of BCC


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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.


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4 (0.353a)

FCC 0.74

Atomic Packing Factor of FCC


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Atoms Shared Between: Each atom counts:corner 8 cells 1/8face centre 2 cells 1/2body centre 1 cell 1edge centre 2 cells 1/2

lattice type cell contentsP 1 [=8 x 1/8]I 2 [=(8 x 1/8) + (1 x 1)]F 4 [=(8 x 1/8) + (6 x 1/2)]C 2 [=(8 x 1/8) + (2 x 1/2)]

Unit cell contentsCounting the number of atoms within the unit cell


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Example; Atomic Packing Factor


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2 - HEXAGONAL SYSTEM

A crystal system in which three equal coplanar axes intersect at an angle of 120 , and a perpendicular to the others, is of a different length.


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2 - HEXAGONAL SYSTEM

Atoms are all same.03/11/2011


111


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3 - TRICLINIC 3 - TRICLINIC 4 - MONOCLINIC CRYSTAL 4 - MONOCLINIC CRYSTAL SYSTEMSYSTEM

Triclinic minerals are the least symmetrical. Their three axes are all different lengths and none of them are perpendicular to each other. These minerals are the most difficult to recognize.

Triclinic (Simple) ß 90

oa b c

Monoclinic (Simple) = = 90o, ß 90o

a b c

Monoclinic (Base Centered) = = 90o, ß 90o

a b c,


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5 - ORTHORHOMBIC SYSTEM

Orthorhombic (Simple) = ß = = 90o

a b c

Orthorhombic (Base-centred)

= ß = = 90o

a b c

Orthorhombic (BC) = ß = = 90o

a b c

Orthorhombic (FC) = ß = = 90o

a b c


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6 – TETRAGONAL SYSTEM

Tetragonal (P) = ß = = 90o

a = b c

Tetragonal (BC) = ß = = 90o

a = b c


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7 - Rhombohedral (R) or Trigonal

Rhombohedral (R) or Trigonal (S) a = b = c, = ß = 90o


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THE MOST IMPORTANT CRYSTAL STRUCTURES

Sodium Chloride Structure Na+Cl-

Cesium Chloride Structure Cs+Cl-

Hexagonal Closed-Packed Structure Diamond Structure Zinc Blende


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1 – Sodium Chloride Structure

Sodium chloride also crystallizes in a cubic lattice, but with a different unit cell.

Sodium chloride structure consists of equal numbers of sodium and chlorine ions placed at alternate points of a simple cubic lattice.

Each ion has six of the other kind of ions as its nearest neighbours.


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Sodium Chloride Structure

If we take the NaCl unit cell and remove all the red Cl ions, we are left with only the blue Na. If we compare this with the fcc / ccp unit cell, it is clear that they are identical. Thus, the Na is in a fcc sublattice.


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Sodium Chloride Structure This structure can be

considered as a face-centered-cubic Bravais lattice with a basis consisting of a sodium ion at 0 and a chlorine ion at the center of the conventional cell,

LiF,NaBr,KCl,LiI,etc The lattice constants are in

the order of 4-7 angstroms.

)(2/

zyxa


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2-Cesium Chloride Structure Cs+Cl-

Cesium chloride crystallizes in a cubic lattice. The unit cell may be depicted as shown. (Cs+ is teal, Cl- is gold).

Cesium chloride consists of equal numbers of cesium and chlorine ions, placed at the points of a body-centered cubic lattice so that each ion has eight of the other kind as its nearest neighbors.


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Cesium Chloride Structure Cs+Cl-

The translational symmetry of this structure is that of the simple cubic Bravais lattice, and is described as a simple cubic lattice with a basis consisting of a cesium ion at the origin 0 and a chlorine ion at the cube center

CsBr,CsI crystallize in this structure.The lattice constants are in the order of 4 angstroms.

)(2/

zyxa


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8 cell

Cesium Chloride Cs+Cl-


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3–Hexagonal Close-Packed Str.

This is another structure that is common, particularly in metals. In addition to the two layers of atoms which form the base and the upper face of the hexagon, there is also an intervening layer of atoms arranged such that each of these atoms rest over a depression between three atoms in the base.


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Bravais Lattice : Hexagonal LatticeHe, Be, Mg, Hf, Re (Group II elements)ABABAB Type of Stacking

Hexagonal Close-packed Structure

a=b a=120, c=1.633a, basis : (0,0,0) (2/3a ,1/3a,1/2c)


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A A

AA

AA

A

AAA

AA

AAA

AAA

B B

B

B

B B

B

B

B

BB

C C C

CC

C

C

C C C

Sequence ABABAB..-hexagonal close packSequence ABCABCAB..

-face centered cubic close pack

Close pack

B

AA

AA

A

A

A

A A

B

B B

Sequence AAAA…- simple cubic

Sequence ABAB…- body centered cubic

Packing


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4 - Diamond Structure

The diamond lattice is consist of two interpenetrating face centered bravais lattices.

There are eight atom in the structure of diamond. Each atom bonds covalently to 4 others equally spread

about atom in 3d.


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4 - Diamond Structure

The coordination number of diamond structure is 4.

The diamond lattice is not a Bravais lattice.

Si, Ge and C crystallizes in diamond structure.


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5- Zinc Blende

Zincblende has equal numbers of zinc and sulfur ions distributed on a diamond lattice so that each has four of the opposite kind as nearest neighbors. This structure is an example of a lattice with a basis, which must so described both because of the geometrical position of the ions and because two types of ions occur.

AgI,GaAs,GaSb,InAs,03/11/2011


131

5- Zinc Blende


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5- Zinc Blende

Zinc Blende is the name given to the mineral ZnS. It has a cubic close packed (face centred) array of S and the Zn(II) sit in tetrahedral (1/2 occupied) sites in the lattice.


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Each of the unit cells of the 14 Bravais lattices has one or more types of symmetry properties, such as inversion, reflection or rotation,etc.

SYMMETRY

INVERSION REFLECTION ROTATION

ELEMENTS OF SYMMETRY


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Lattice goes into itself through Symmetry without translation

Operation Element

Inversion Point

Reflection Plane

Rotation Axis

Rotoinversion Axes


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Inversion Center A center of symmetry: A point at the center of the

molecule.(x,y,z) --> (-x,-y,-z)

Center of inversion can only be in a molecule. It is not necessary to have an atom in the center (benzene, ethane). Tetrahedral, triangles, pentagons don't have a center of inversion symmetry. All Bravais lattices are inversion symmetric. Mo(CO)6


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Reflection Plane

A plane in a cell such that, when a mirror reflection in this plane is performed, the cell remains invariant.


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Examples

Triclinic has no reflection plane. Monoclinic has one plane midway between and

parallel to the bases, and so forth.


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We can not find a lattice that goes into itself under other rotations

• A single molecule can have any degree of rotational symmetry, but an infinite periodic lattice – can not.

Rotation Symmetry


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Rotation Axis

This is an axis such that, if the cell is rotated around it through some angles, the cell remains invariant.

The axis is called n-fold if the angle of rotation is 2π/n.

90°

120° 180°


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Axis of Rotation


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Axis of Rotation


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Can not be combined with translational periodicity!

5-fold symmetry


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Symmetry Elements for Cubic System

Axis of Symmetry present in cubic system

3-Tetrads4-triads6-diadsTotal-13 axes of symmetry

Total =13+9+1=23 elements of symmetry

Centre of symmetry


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The characteristic symmetry elements in each of the seven groups are listed below

Cubic Three triads

Hexagonal One hexad (// z)

Tetragonal One tetrad (// z)

Trigonal One triad (// [111])

Orthorhombic Three perpendicular diads (// x, y and z)

Monoclinic One diad (// y)

Triclinic -

The characteristic symmetry elements in each of the seven groups are listed below


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Concept Map


Group discussion

Kepler wondered why snowflakes have 6 corners, never 5 or 7.By considering the packing of polygons in 2 dimensions, demonstrate why pentagons and heptagons shouldn’t occur.

Empty space not allowed


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90°

Examples

Triclinic has no axis of rotation. Monoclinic has 2-fold axis (θ= 2π/2 =π) normal to

the base.


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IMPERFECTIONS IN CRYSTALS

Lattice Defect or Imperfection

• An important feature of crystals is their regular atomic arrangement but no crystal is perfectly regular.

• Any deviation from this perfect atomic periodicity is called an imperfection or lattice defect.

• A lattice defect is a state in which the atomic arrangement in the small region (of a size of only a few lattice constants) of a crystal has departed from regularity.

Electrical properties gets affected

• The electrical resistance of the crystal is greatly affected.

• These defects scatter the conduction electrons in a metal and thus increase its electrical resistance.

• Especially in case of alloys this increase in electrical resistance is several tens of percentage.

Elastic properties get affected

• The strength of crystals: • Certain kinds of defects exist very rarely but

they decrease the strength of the crystal by a factor of several hundreds or thousands

• Such properties that are greatly affected by the defects are called defect or structure sensitive properties.

• The strength of crystals: • Certain kinds of defects exist very rarely but

they decrease the strength of the crystal by a factor of several hundreds or thousands

• Such properties that are greatly affected by the defects are called defect or structure sensitive properties.

CLASSIFICATION OF IMPERFECTIONS

There are three types of imperfections exist in general.

(A) Crystal Imperfections or atomic imperfections)

(B) Electronic Imperfections (C) Transient Imperfections

(A) Crystal Imperfections (or atomic imperfections) :

(A) Crystal Imperfections (or atomic imperfections) :

Concerned with this types of imperfections. To list them they are: (1) Thermal vibrations,

(2) Point defects,

(i) Vacancies,(ii) Interstitials,(iii) Isolated impurities.

(3) Line defects; the dislocation: Edge and Screw dislocations,

(4) Surface defects,(i) External surfaces of solids(ii) Internal surfaces; grain boundaries and other internal boundaries.

Concerned with this types of imperfections. To list them they are: (1) Thermal vibrations,

(2) Point defects,

(i) Vacancies,(ii) Interstitials,(iii) Isolated impurities.

(3) Line defects; the dislocation: Edge and Screw dislocations,

(4) Surface defects,(i) External surfaces of solids(ii) Internal surfaces; grain boundaries and other internal boundaries.

(B) Electronic Imperfections:

They are the defects in electronic structure e.g.,• (i) conduction electron• (ii) hole, which are excited thermally from filled bands or

impurity levels. These defects are responsible for important

electrical and magnetic properties,

(C) Transient Imperfections : (C) Transient Imperfections :

These defects are introduced into the crystal from external sources and are, for example

(i) Photons are bombarded on crystals(ii) Beam of charged particles like electrons,

protons, and mesons etc.(iii) Beam of neutral particles e.g., neutrons

and neutral atoms.Are bombarded on crystals.

These defects are introduced into the crystal from external sources and are, for example

(i) Photons are bombarded on crystals(ii) Beam of charged particles like electrons,

protons, and mesons etc.(iii) Beam of neutral particles e.g., neutrons

and neutral atoms.Are bombarded on crystals.

Different types of point defects in crystals

Vacancy

Point defects in ionic crystals

CRYSTALLOGRAPHIC IMPERFECTIONS:

CRYSTALLOGRAPHIC IMPERFECTIONS:

• To discuss the defects that arise due to the

departure from perfect periodicity of an atomic array in a crystal

• —the so called lattice defects. • They can then be classified according as the

periodic regularity is interrupted in zero, one, two and three dimensions.

(1) Point Defects:

• A lattice defect which spreads out very little in (zero dimension) is called a point defect.

• They are of following types:(i) Interstitial atoms(ii) Vacancy –also known as Schottky defects(iii)Impurity atom(iv) Interstitial + Vacancy = Frenkel defects

• A lattice defect which spreads out very little in (zero dimension) is called a point defect.

• They are of following types:(i) Interstitial atoms(ii) Vacancy –also known as Schottky defects(iii)Impurity atom(iv) Interstitial + Vacancy = Frenkel defects

(i) Interstitial atom:

• This is an extra atom inserted into the voids (called interstice of the lattice) between the regularly occupied sites.

• Thus such an atom does not occupy regular lattice sites.

• This extra atom may be an impurity atom or an atom of the same types as on the regular lattice sites.

• This is an extra atom inserted into the voids (called interstice of the lattice) between the regularly occupied sites.

• Thus such an atom does not occupy regular lattice sites.

• This extra atom may be an impurity atom or an atom of the same types as on the regular lattice sites.

(ii) Vacancies : (ii) Vacancies :

• These are the lattice sites from which the atoms are missing.

• Such a vacancy is also called Schottky defect. • But if a vacancy is created by transferring an

atom from a regular lattice site to an interstitial position then it is called Frenkel defect.

• In this case, therefore, - two imperfections are created—vacancy as well as an interstitial atom.

• These are the lattice sites from which the atoms are missing.

• Such a vacancy is also called Schottky defect. • But if a vacancy is created by transferring an

atom from a regular lattice site to an interstitial position then it is called Frenkel defect.

• In this case, therefore, - two imperfections are created—vacancy as well as an interstitial atom.

Point defects in elemental solids

Frenkel defects in ionic crystals

Cation and Anion vacancy

(iii) Impurity atom :

• This is a defect in which a foreign atom occupies a regular lattice site.

• Point defects• The simplest point defects are as follows: • Vacancy – missing atom at a certain crystal lattice

position;• Interstitial impurity atom – extra impurity atom in

an interstitial position;• Self-interstitial atom – extra atom in an interstitial

position;• Substitution impurity atom – impurity atom,

substituting an atom in crystal lattice;• Frenkel defect – extra self-interstitial atom,

responsible for the vacancy nearby.

• Point defects• The simplest point defects are as follows: • Vacancy – missing atom at a certain crystal lattice

position;• Interstitial impurity atom – extra impurity atom in

an interstitial position;• Self-interstitial atom – extra atom in an interstitial

position;• Substitution impurity atom – impurity atom,

substituting an atom in crystal lattice;• Frenkel defect – extra self-interstitial atom,

responsible for the vacancy nearby.

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Line defectsLine defects

• Linear crystal defects are edge and screw dislocations.

• Edge dislocation is an extra half plane of atoms “inserted” into the crystal lattice.

• Due to the edge dislocations metals possess high plasticity characteristics: ductility and malleability.

• Linear crystal defects are edge and screw dislocations.

• Edge dislocation is an extra half plane of atoms “inserted” into the crystal lattice.

• Due to the edge dislocations metals possess high plasticity characteristics: ductility and malleability.

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Screw Dislocation

• Screw dislocation forms when one part of crystal lattice is shifted (through shear) relative to the other crystal part. It is called screw as atomic planes form a spiral surface around the dislocation line.

• For quantitative characterization of a difference between a crystal distorted by a dislocation and the perfect crystal the Burgers vector is used.

• Screw dislocation forms when one part of crystal lattice is shifted (through shear) relative to the other crystal part. It is called screw as atomic planes form a spiral surface around the dislocation line.

• For quantitative characterization of a difference between a crystal distorted by a dislocation and the perfect crystal the Burgers vector is used.

• The dislocation density is a total length of dislocations in a unit crystal volume.

• The dislocation density of annealed metals is about 1010 - 1012 m−².

• After work hardening the dislocation density increases up to 1015-1016 m-².

• Further increase of dislocation density causes cracks formation and fracture.

• The dislocation density is a total length of dislocations in a unit crystal volume.

• The dislocation density of annealed metals is about 1010 - 1012 m−².

• After work hardening the dislocation density increases up to 1015-1016 m-².

• Further increase of dislocation density causes cracks formation and fracture.

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(2) Line Defects:

• When a lattice defect is confined to a small region in one dimension, it is called a line defect. In this type of defect, called dislocation, part of the lattice undergoes a shearing strain equal to one lattice vector (called a Burgers vector).

• They are of two types :• (1) Edge dislocation : This type of

dislocation is created by a missing half plane of atoms.

• (ii) Screw dislocation : It can be thought of as created by cutting the crystal part way and shearing down one part relative to other by one atomic spacing.

• They are of two types :• (1) Edge dislocation : This type of

dislocation is created by a missing half plane of atoms.

• (ii) Screw dislocation : It can be thought of as created by cutting the crystal part way and shearing down one part relative to other by one atomic spacing.

Planar defects

• Planar defect is an imperfection in the form of a plane between uniform parts of the material. The most important planar defect is a grain boundary.

• Planar defect is an imperfection in the form of a plane between uniform parts of the material. The most important planar defect is a grain boundary.

http://www.substech.com/dokuwiki/doku.php?id=grain_structure

http://www.substech.com/dokuwiki/doku.php?id=grain_structure

• Formation of a boundary between two grains may be imagined as a result of rotation of crystal lattice of one of them about a specific axis. Depending on the rotation axis direction, two ideal types of a grain boundary are possible:

• Tilt boundary – rotation axis is parallel to the boundary plane;

• Twist boundary - rotation axis is perpendicular to the boundary plane:

• An actual boundary is a “mixture” of these two ideal types.

• Formation of a boundary between two grains may be imagined as a result of rotation of crystal lattice of one of them about a specific axis. Depending on the rotation axis direction, two ideal types of a grain boundary are possible:




• Grain boundaries are called large-angle boundaries if misorientation of two neighboring grains exceeds 10°-15°.

• Grain boundaries are called small-angle boundaries if misorientation of two neighboring grains is 5° or less.




• Grain boundaries are called large-angle boundaries if misorientation of two neighboring grains exceeds 10°-15°.

• Grain boundaries are called small-angle boundaries if misorientation of two neighboring grains is 5° or less.




• Grains, divided by small-angle boundaries are also called subgrains.

• Grain boundaries accumulate crystal lattice defects (vacancies, dislocations) and other imperfections, therefore they effect on the metallurgical processes, occurring in alloys and their properties.

• Since the mechanism of metal deformation is a motion of crystal dislocations through the lattice, grain boundaries, enriched with dislocations, play an important role in the deformation process.

• Grains, divided by small-angle boundaries are also called subgrains.

• Grain boundaries accumulate crystal lattice defects (vacancies, dislocations) and other imperfections, therefore they effect on the metallurgical processes, occurring in alloys and their properties.

• Since the mechanism of metal deformation is a motion of crystal dislocations through the lattice, grain boundaries, enriched with dislocations, play an important role in the deformation process.

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• Diffusion along grain boundaries is much faster, than throughout the grains.

• Segregation of impurities in form of precipitating phases in the boundary regions causes a form of corrosion, associated with chemical attack of grain boundaries. This corrosion is called Intergranular corrosion.

• Diffusion along grain boundaries is much faster, than throughout the grains.

• Segregation of impurities in form of precipitating phases in the boundary regions causes a form of corrosion, associated with chemical attack of grain boundaries. This corrosion is called Intergranular corrosion.

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(3) Plane Defects: (3) Plane Defects:

• When a lattice defect is confined to a small region only in two dimensions; it is called a plane defect.

• When line defects cluster together in a plane, they can form a plane which is described as follows :

• When a lattice defect is confined to a small region only in two dimensions; it is called a plane defect.

• When line defects cluster together in a plane, they can form a plane which is described as follows :

• (i) Lineage Boundary: • It is boundary between two adjacent perfect regions in

the same crystal that are slightly tilted with respect to each other.

• (ii) Grain boundary: • A crystal is made up of a large number of small grains

or crystallites which are single crystals, (i.e., all molecules in a crystallite are oriented in the same direction). Generally, these crystallites in. the crystal of a solid remain oriented indiscriminately in random directions unless special precautions are taken during the crystal growth. Such crystals are called polycrystalline. Grain boundary is the boundary between two crystals in a polycrystalline solid.

(iii) Stacking fault:• It is possible for ‘mistakes’ to occur in the

stacking sequence of hexagonal close packed layers. The plane separating the two incorrectly juxtaposed layers is called stacking fault.

We take the blue atoms as the base plane for what we are going to built on it, we will call it the "A - plane".

The next layer will have the center of the atoms right over the depressions of the A - plane; it could be either the B - or C - configuration. Here the pink layer is in the "B" position

If you pick the B - configuration (and whatever you pick at this stage, we can always call it the B - configuration), the third layer can either be directly over the A - plane and then is also an A - plane (shown for one atom), or in the C - configuration.

If you chose "C", you get the face centered cubic lattice (fcc)

If you chose "A"; you obtain the hexagonal close packed lattice (hcp),

The stacking sequences of the two close-packed lattices therefore are

fcc: ABCABCABCA...

hcp: ABABABA...

Trends of Research In Crystal Growth

Why to Grow and study crystals?

• Various device fabrication requires crystal and various properties are exploited for that

• Following Table gives some of the Uses of crystals

• Some devices in the table marked with an asterisk use crystals with controlled additions of impurities.

• In the complex structures, the necessary impurities can either be incorporated in a series of growth processes or can be added after growth by diffusion or by ion implantation.

• Various device fabrication requires crystal and various properties are exploited for that

• Following Table gives some of the Uses of crystals

• Some devices in the table marked with an asterisk use crystals with controlled additions of impurities.

• In the complex structures, the necessary impurities can either be incorporated in a series of growth processes or can be added after growth by diffusion or by ion implantation.

Properties exploited Device Crystal

1 Uniformity alone X-ray prismsNeutron collimators

Lithium fluoride

2 Uniformity givingreproducible mechanicalproperties and abrasionresistance

Turbine bladesGramophone styliBearingsTape-recorder heads Wire drawing dies

MetalsSapphireRubyFerrites Diamond

3 Uniformity eliminating scattering ofelectromagnetic waves

Lenses, prisms and optical windowsLasers*Microwave filters

Alkali and alkaline earth halidesYttrium aluminum garnetYttrium iron garnet

4 Uniformity reducing charged carrier scattering

Transistors*, Diodes* Thyristors *Photocells

Silicon, germanium and gallium arsenideCadmium suiphide

5 Uniformity reducing scattering of sound waves

Resonant filters Delay lines

QuartzLithium niobateZinc oxide

6 Uniformity allowingexploitation of tensor properties

Nicol prismUltrasonic transducersGramophone pick-ups

FluoriteRochelle saltLithium sulphate

CRYSTAL GROWTH METHODS

• MELT GROWTH METHODS • SOLUTION GROWTH METHODS • VAPOR PHASE GROWTH METHOD • MODIFICATION OF CRYSTAL

GROWTH METHODS

MELT GROWTH METHODS – Horizontal Boat Growth Methods

• Horizontal Gradient Freezing (HGF) method• Horizontal Bridgman (HB) method• Horizontal Zone Melting (HZM) method

– Vertical Boat Growth Methods • Vertical Bridgman (VB) method• Vertical Gradient Freezing (VGF) method• Vertical Zone Melting (VZM) method

– Pulling Methods • Czochralski (CZ) method• Liquid Encapsulated Czochralski (LEC) method• Kyropolous and Liquid Encapsulated Kyropolous (LEK) methods

– Floating Zone (FZ) Method– Other Methods

• Shaped Crystal Growth Method• Heat Exchange Method (HEM)

– Horizontal Boat Growth Methods • Horizontal Gradient Freezing (HGF) method• Horizontal Bridgman (HB) method• Horizontal Zone Melting (HZM) method

– Vertical Boat Growth Methods • Vertical Bridgman (VB) method• Vertical Gradient Freezing (VGF) method• Vertical Zone Melting (VZM) method

– Pulling Methods • Czochralski (CZ) method• Liquid Encapsulated Czochralski (LEC) method• Kyropolous and Liquid Encapsulated Kyropolous (LEK) methods

– Floating Zone (FZ) Method– Other Methods

• Shaped Crystal Growth Method• Heat Exchange Method (HEM)

SOLUTION GROWTH METHODS SOLUTION GROWTH METHODS

– Simple Solution Growth Method– Traveling Heater Method (THM)– Solute Solution Diffusion (SSD) Method– Solvent Evaporation (SE) Method– Temperature Difference Method under

Controlled Vapor Pressure (TDM-CVP)– Hydrothermal Synthesis Method

– Simple Solution Growth Method– Traveling Heater Method (THM)– Solute Solution Diffusion (SSD) Method– Solvent Evaporation (SE) Method– Temperature Difference Method under

Controlled Vapor Pressure (TDM-CVP)– Hydrothermal Synthesis Method

VAPOR PHASE GROWTH METHOD

– Direct Synthesis (DS) Method– Physical Vapor Transport (PVT) Method

• Open tube method• Closed tube method

– Chemical Vapor Transport (CVT) Method– Solid Phase Reaction (Solid State

Recrystallization)

– Direct Synthesis (DS) Method– Physical Vapor Transport (PVT) Method

• Open tube method• Closed tube method

– Chemical Vapor Transport (CVT) Method– Solid Phase Reaction (Solid State

Recrystallization)

MODIFICATION OF CRYSTAL GROWTH METHODS

–In-Situ Synthesis–Vapor Pressure Control–Magnetic Field Application–Accelerated Crucible Rotation

Technique (ACRT)

Survey of the methods of crystal growth

Growth from Approximate % growth

Melt 80

Vapour 7

Low Temperature solution 5

High Temperature Solution 5

Solid 3

Hydrothermal 2In some cases huge quantities of crystals are grown annually e.g. silicon, quartz, germanium, Rubby, and di-hydrogen phosphates of potassium and ammonium

In some cases huge quantities of crystals are grown annually e.g. silicon, quartz, germanium, Rubby, and di-hydrogen phosphates of potassium and ammonium

Classification of Growth TechniquesClassification of Growth Techniques

Growth from the pure melt

Growth from Solution

Single Crystals for Research PurposesCrystal Doping Agent Uses

-Al2O3, TiO2, CaF2 Transition elements Paramagnetic studies

CaWO4, etc. Rare Earths and Actinides Masers; Lasers

ZnS, CdS, Organic Crystals Cr, Mn, Cu, Ag, Tl, etc. FluorescencePhotoconductivityPhotoelectricity

Ge, Si, InSb, GaAs, SiC, PbTe, Bi2Te3

Donor or acceptor impurities

Semiconductivity, Thermoelectric, Galvanomagnetic effects

Fe3O4, MFe2O4, BaFe12O19, Y3Fe5O12

Paramagnetic substituents Magnetic studies

BeO, MgO, -Al2O3, UO2 Pure Reactor material

Al2SiO5, aluminosilicates, ZrSiO4, C, BN, WC, ThO2

ZrO2, Si3N4, etc.

Pure Refractories, abrasives, Structural materials

Alkali halides, -SiO2, CaF2, SrTiO3

Pure Optical materials

Methods of Crystal AssessmentsMethod Destructive or

Non-destructive

Information given

1 Chemical analysisSpectrographic analysis

D Composition

2 X-ray analysis N Structure (Some information on composition)

3 X-ray fluorescence spectroscopy N Composition

4 Electron diffraction N Structure, surface detail

5 Electron microscopy N Surface detail

6 Electron beam X-ray spectroscopy N Composition

7 Optical spectroscopy, IRUV N Structure and composition

8 Electron spin resonance N Purity (structure)

9 Optical examination N Imperfections, Surface detail

10 Etching, decorating N/D Perfection

11 Measurement of specialized physical properties (electrical or magnetic)

N Perfection, purity

Requirements for growth control

From theory and practice

• As a particle settles on growing crystal surface a finite time is necessary for the particle to move to an available and proper site.

• Growth rates must therefore be slow enough to allow this surface diffusion to be effective.

• The most rapid growth is thus allowed at the melting point of a material, and the growth of crystals of the same material at lower temperatures (by solution techniques) must be correspondingly slower.

• As a particle settles on growing crystal surface a finite time is necessary for the particle to move to an available and proper site.

• Growth rates must therefore be slow enough to allow this surface diffusion to be effective.

• The most rapid growth is thus allowed at the melting point of a material, and the growth of crystals of the same material at lower temperatures (by solution techniques) must be correspondingly slower.

Typical example of growing corundum (-Al2O3) crystals by different

techniquesGrowth rates for corundum

Method Temperature Linear growth rate

Hydrothermal 650°C 0.1 mm/day

Fluxed-melt 1200°C 1 mm/day

Flame fusion 2100°C 450 mm/day

Any fluctuation, irregularity or temporary halt in the growth process

is reflected in the crystal obtainedThe result may appear as

• Included material (‘ghosting’)• Variations in dislocation densities• Lattice irregularities • Inhomogeneity of composition

The result may appear as• Included material (‘ghosting’)• Variations in dislocation densities• Lattice irregularities • Inhomogeneity of composition

growth from the melt

• fast (~mm/hr) growth rate is limited by heat transfer, not by mass transfer

• allows for a large variety of techniques

• Verneuil• Bridgman-Stockbarger• Czochralski-Kyropoulos• zone melting and floating zone

• fast (~mm/hr) growth rate is limited by heat transfer, not by mass transfer

• allows for a large variety of techniques

• Verneuil• Bridgman-Stockbarger• Czochralski-Kyropoulos• zone melting and floating zone

characteristicscharacteristics

Verneuil1902, Auguste Verneuil

characteristics: no crucible contamination highly pure starting material (>99.9995%) strict control of flame temperature precise positioning of melted region

vibration

growth

The Verneuil method. :• A fine dry powder of the material

to be grown is shaken through the wire mesh and falls through the oxy-hydrogen flame in which it melts.

• A film of liquid is formed on top of the seed crystal.

• This freezes progressively as the crystal is slowly lowered (a few mm/hr).

• To maintain symmetry the seed is rotated (usually at about 10 r.p.m.)

• The art of the method is to balance the rate of powder feed and the rate of lowering to maintain a constant growth rate and diameter.

• The method is used extensively for the production of ruby crystals

• A fine dry powder of the material to be grown is shaken through the wire mesh and falls through the oxy-hydrogen flame in which it melts.

• A film of liquid is formed on top of the seed crystal.

• This freezes progressively as the crystal is slowly lowered (a few mm/hr).

• To maintain symmetry the seed is rotated (usually at about 10 r.p.m.)

• The art of the method is to balance the rate of powder feed and the rate of lowering to maintain a constant growth rate and diameter.

• The method is used extensively for the production of ruby crystals

temperature

Tmelt

Bridgman-Stockbarger

characteristics:characteristics: charge and seed are placed into the crucible no material is added or removed (conservative process) axial temperature gradient along the crucible

charge and seed are placed into the crucible no material is added or removed (conservative process) axial temperature gradient along the crucible

• As the crucible is lowered, solid forms first at the pointed tip of the crucible.

• If this tip is correctly shaped, usually only one crystal will be formed initially, and single crystal growth will generally continue if the conditions have been correctly chosen.

• The latent heat of solidification, which is evolved as the crystal grows, is removed by conduction through the crystal and the crucible.

• The principal characteristic of this method is that at least some part of the solid—liquid interface is in contact with the crucible.

• As the crucible is lowered, solid forms first at the pointed tip of the crucible.

• If this tip is correctly shaped, usually only one crystal will be formed initially, and single crystal growth will generally continue if the conditions have been correctly chosen.

• The latent heat of solidification, which is evolved as the crystal grows, is removed by conduction through the crystal and the crucible.

• The principal characteristic of this method is that at least some part of the solid—liquid interface is in contact with the crucible.

Bridgman-Stockbarger

•The shape of the crystal is defined by the container

•No radial temperature gradients are needed to control the crystal shape.

•Low thermal stresses result in low level of stress-induced dislocations.

•Crystals may be grown in sealed ampules (easy control of stoichiometry)

•Relatively low level of natural convection

•Easy control and maintenance

•The shape of the crystal is defined by the container

•No radial temperature gradients are needed to control the crystal shape.

•Low thermal stresses result in low level of stress-induced dislocations.

•Crystals may be grown in sealed ampules (easy control of stoichiometry)

•Relatively low level of natural convection

•Easy control and maintenance

AdvantagesAdvantages

•Confined growth (crucible may induce stresses during cooling)

•Difficult to observe seeding and growing processes

•Changes in natural convection as the melt is depleted

•Delicate crucible and seed preparation, sealing, etc.

•Confined growth (crucible may induce stresses during cooling)

•Difficult to observe seeding and growing processes

•Changes in natural convection as the melt is depleted

•Delicate crucible and seed preparation, sealing, etc.

Drawbacks

Bridgman-StockbargerBridgman-Stockbarger

melts with volatile constituents:III-V compounds (GaAs, lnP, GaSb)II-VI compounds (CdTe)

ternary compounds:Ga1-xlnxAs, Ga1-xlnxSb, Hg1-xCdxTe

melts with volatile constituents:III-V compounds (GaAs, lnP, GaSb)II-VI compounds (CdTe)

ternary compounds:Ga1-xlnxAs, Ga1-xlnxSb, Hg1-xCdxTe

applications

reduced nucleationreduced thermal stressesreduced evaporationprevents contact between crucible and melt

reduced nucleationreduced thermal stressesreduced evaporationprevents contact between crucible and melt

improvement example (liquid encapsulation)improvement example (liquid encapsulation)

crucible

encapsulant

melt

crystallow vapor pressuremelting temperature lower than the crystaldensity lower than the density of the meltno reaction with the melt or crucible

low vapor pressuremelting temperature lower than the crystaldensity lower than the density of the meltno reaction with the melt or crucible

encapsulant characteristicsencapsulant characteristics

B2O3

LiCl, KCl, CaCl2, NaCl

Czochralski-Kyropoulos A seed crystal mounted on a rod is dipped into the molten material. The seed crystal's rod is pulled upwards and rotated at the same time. By precisely controlling the temperature gradients, rate of pulling and speed of rotation, a single-crystal cylindrical ingot is extracted from the melt. The process may be peformed in controlled atmosphere and in inert chamber.

A seed crystal mounted on a rod is dipped into the molten material. The seed crystal's rod is pulled upwards and rotated at the same time. By precisely controlling the temperature gradients, rate of pulling and speed of rotation, a single-crystal cylindrical ingot is extracted from the melt. The process may be peformed in controlled atmosphere and in inert chamber.

Jan Czochralski (1885 - 1953)

characteristics:

charge and seed are separated at start no material is added or removed (conservative process) charge is held at temperature slightly above melting point crystal grows as atoms from the melt adhere to the seed

charge and seed are separated at start no material is added or removed (conservative process) charge is held at temperature slightly above melting point crystal grows as atoms from the melt adhere to the seed

seed

grown crystal

molten raw

material

heating elements

seed

grown crystal

molten raw material

Kyropoulos

Czochralski1918

1926

Pulling direction of seed on rod

Heater

CZOCHRALSKI

Crucible

Inert atmosphere under pressure prevents material loss and unwanted reactions

Layer of molten oxide like B2O3 prevents preferential volatilization of one component - precise stoichiometry control

Melt just above mp

Growing crystal

Crystal seed

Counterclockwise rotation of melt and crystal being pulled from melt, helps unifomity of temperature and homogeneity of crystal growth

• Molten material is held in a crucible at a temperature just above its melting point.

• Heat is abstracted through a water-cooled seed and crystallization occurs on the seed which grows down into the melt.

• Temperature control of the furnace largely determines the diameter of the growing crystal, and some adjustment of the seed position relative to the crucible may be necessary if a large volume change occurs on solidification.

• In practice the crystals are removed from the furnace for annealing, although this may be done in situ.

• Molten material is held in a crucible at a temperature just above its melting point.

• Heat is abstracted through a water-cooled seed and crystallization occurs on the seed which grows down into the melt.

• Temperature control of the furnace largely determines the diameter of the growing crystal, and some adjustment of the seed position relative to the crucible may be necessary if a large volume change occurs on solidification.

• In practice the crystals are removed from the furnace for annealing, although this may be done in situ.Kyropoulous apparatus

• The technique is mainly used for the production of large alkali halide crystals for optical use.

• Growth rates of about 1 cm/hr are obtainable with gradients of the order of 50°C/cm.

• Although optically of acceptable quality, the crystals contain numerous low-angle boundaries.

• For high- purity materials, crucible contamination is a serious problem.

• The technique is mainly used for the production of large alkali halide crystals for optical use.

• Growth rates of about 1 cm/hr are obtainable with gradients of the order of 50°C/cm.

• Although optically of acceptable quality, the crystals contain numerous low-angle boundaries.

• For high- purity materials, crucible contamination is a serious problem.

Czochralski-Kyropoulos

•Growth from free surface (stress free)•Crystal can be observed during the growth process•Forced convection easy to impose•Large crystals can be obtained•High crystalline perfection can be achieved•Good radial homogeneity

AdvantagesAdvantages

•Delicate start (seeding, necking) and sophisticated further control•Delicate mechanics (the crystal has to be rotated; Rotation of the crucible is desirable)•Cannot grow materials with high vapor pressurebatch process (axial segregation, limited productivity)

•Delicate start (seeding, necking) and sophisticated further control•Delicate mechanics (the crystal has to be rotated; Rotation of the crucible is desirable)•Cannot grow materials with high vapor pressurebatch process (axial segregation, limited productivity)

DrawbacksDrawbacks

zone meltingzone melting

ultra-pure silicon

characteristics:characteristics: only a small part of the charge is molten material is added to molten region (nonconservative process) molten zone is advanced by moving the charge or the gradient axial temperature gradient is imposed along the crucible

only a small part of the charge is molten material is added to molten region (nonconservative process) molten zone is advanced by moving the charge or the gradient axial temperature gradient is imposed along the crucible

zone melting

•Charge is purified by repeated passage of the zone (zone refining).•Crystals may be grown in sealed ampules or without containers (floating zone).•Steady-state growth possible.•Zone leveling is possible; can lead to superior axial homogeneity.•Process requires little attention (maintenance).•Simple: no need to control the shape of the crystal.•Radial temperature gradients are high.

•Charge is purified by repeated passage of the zone (zone refining).•Crystals may be grown in sealed ampules or without containers (floating zone).•Steady-state growth possible.•Zone leveling is possible; can lead to superior axial homogeneity.•Process requires little attention (maintenance).•Simple: no need to control the shape of the crystal.•Radial temperature gradients are high.

advantages

•Confined growth (except in floating zone).•Hard to observe the seeding process and the growing crystal.•Forced convection is hard to impose (except in floating zone).•In floating zone, materials with high vapor pressure can not be grown.

•Confined growth (except in floating zone).•Hard to observe the seeding process and the growing crystal.•Forced convection is hard to impose (except in floating zone).•In floating zone, materials with high vapor pressure can not be grown.

drawbacksdrawbacks

other methods (1)

melt non congruently decompose before melting have very high melting point undergo solid state phase transformation between melting point and room temperature

growth from solutions

key requirementhigh purity solvent

insoluble in the crystal

oxides with very high melting points

PbO, PbF2, B2O3, KF

very slow, borderline purity, platinum crucibles, stoichiometry hard to control

carried on at much lower temperature than melting point

typical solvents:

main advantage:

limitations:

molten salt (flux) growtha liquid reaction

medium that dissolves the reactants and

products, but do not participate in the

reaction

flux:

other methods (2)

high quality layers of III-V compounds (Ga1-xlnxAs, GaAsxP1-x) GaAs and GaSb from Ga solution

liquid phase epitaxy advantagelower temperatures

than melt growth

aqueous solution at high temperature and pressure

typical example: quartz industry

SiO2 is grown by hydrothermal growth at 2000 bars and 400°C because of α-β quartz transition at 583°C

hydrothermal growth

limitationvery slow, small crystals

or thin layers

crystal purity (1)

Solubility of possible impurity is different in crystal than melt, the ratio between respective concentrations is defined as segregation coefficient (k0)

L

S

C

Ck 0

impurity equilibrium concentration in crystal

impurity equilibrium concentration in melt

1

00

0

01

k

LS M

MCkC

As the crystal is pulled impurity concentration will change in the melt (becomes larger if segregation coefficient is <1). Impurity concentration in crystal after solidifying a weight fraction M/M0 is:

Simple laboratory techniques you can also try

• Crystal growing is an art, and there are as many variations to the basic crystal growing recipes as there are crystallographers.

• The recipes given below are ones which I have either tried or I have read about and sound reasonable.

• The techniques chosen will largely depend on the chemical properties of the compound of interest: – Is the compound air sensitive, – moisture sensitive? – Is it hygroscopic? etc. etc.

• Crystal growing is an art, and there are as many variations to the basic crystal growing recipes as there are crystallographers.

• The recipes given below are ones which I have either tried or I have read about and sound reasonable.

• The techniques chosen will largely depend on the chemical properties of the compound of interest: – Is the compound air sensitive, – moisture sensitive? – Is it hygroscopic? etc. etc.

Slow Evaporation. Slow Evaporation. The simplest way to grow crystals and works best for

compounds which are not sensitive to ambient conditions in the laboratory.

• Prepare a solution of the compound in a suitable solvent. • The solution should be saturated or nearly saturated. • Transfer the solution to a CLEAN crystal growing dish and

cover. • The covering for the container should not be air tight. • Aluminium foil with some holes poked in it works well, or

a flat piece of glass with microscope slides used as a spacer also will do the trick.

• Place the container in a quiet out of the way place and let it evaporate.

• This method works best where there is enough material to saturate at least a few milliliters of solvent.

The simplest way to grow crystals and works best for compounds which are not sensitive to ambient conditions in the laboratory.

• Prepare a solution of the compound in a suitable solvent. • The solution should be saturated or nearly saturated. • Transfer the solution to a CLEAN crystal growing dish and

cover. • The covering for the container should not be air tight. • Aluminium foil with some holes poked in it works well, or

a flat piece of glass with microscope slides used as a spacer also will do the trick.

• Place the container in a quiet out of the way place and let it evaporate.

• This method works best where there is enough material to saturate at least a few milliliters of solvent.

Slow Cooling. This is good for solute-solvent systems which are less than

moderately soluble and the solvent's boiling point is less than 100°C.

• Prepare a saturated solution of the compound where the solvent is heated to just it's boiling point or a just below it.

• Transfer the solution to a CLEAN large test tube and stopper.

• Transfer the test tube to a Dewar flask in which hot water (heated to a temperature of a couple of degrees below the solvent boiling point).

• The water level should exceed the solvent level in the test tube, but should not exceed the height of the test tube.

• Stopper the Dewar flask with a cork stopper and let the vessel sit for a week.

• A more elaborate version of this involves a thermostated oven rather than a Dewar flask.

This is good for solute-solvent systems which are less than moderately soluble and the solvent's boiling point is less than 100°C.

• Prepare a saturated solution of the compound where the solvent is heated to just it's boiling point or a just below it.

• Transfer the solution to a CLEAN large test tube and stopper.

• Transfer the test tube to a Dewar flask in which hot water (heated to a temperature of a couple of degrees below the solvent boiling point).

• The water level should exceed the solvent level in the test tube, but should not exceed the height of the test tube.

• Stopper the Dewar flask with a cork stopper and let the vessel sit for a week.

• A more elaborate version of this involves a thermostated oven rather than a Dewar flask.

Variations on Slow Evaporation and Slow Cooling

Variations on Slow Evaporation and Slow Cooling

If the above two techniques do yield suitable crystals from single solvent systems, one may expand these techniques to binary or tertiary solvent systems.

• The basic rationale for this is by varying the solvent composition one may inhibit growth of certain crystal faces and promote the growth of other faces, yielding crystals of suitable morphology and size.

• If you choose this route for growing crystals, it is absolutely necessary to record the solvent composition you use!

• If crystal growing is an art, growing crystals from binary or tertiary solvent mixtures is that much more imprecise.

• Remember reproducibility is paramount in science.

crystal purity (2)

As a consequence, floating zone method will give crystals with lower concentration of impurities having k<1 than Czochralski growth

L

xk

eS

e

ekCC 110

The effective segregation coefficient (ke):

D

ve

ekk

kk

)1( 00

0

multiple pass may be run in order to achieve the required impurity concentration

there is no contamination from crucible

crystals for DBDDBD application constraints

ββ emitters of experimental interest

IsotopeIsotopic

abundance (%)

half life (y)

48Ca 0.0035

~ 4.0

1019

76Ge 7.8

~ 1.4

1021

82Se 9.2

~ 0.9

1020

96Zr 2.8

~ 2.1

1019

100Mo 9.6

~ 8.0

1018

116Cd 7.5

~ 3.3

1019

128Te 31.7

~ 2.5

1024

130Te 34.5

~ 0.9

1021

136Xe 8.9 ?150Nd 5.6

~ 7.0

1018

i

i

i

i

i

ii

T

T

m

m

T

T

m

m

p

p

impurity allowed (g/g):

T = 1018 – 1024 yrusual Ti < 1012 yr

1210m

mi

close to detection limit of the most sensitive techniques used for quantitative elemental analysis (NAA, ICP-MS)

T

T

m

m ii

TeO2 crystal (1)

Paratellurite Tellurite

tetragonal orthorhombic-dipyramidal

greyish-white, opaque

white to yellow, subtranslucent to opaque

1960, Mexico 1842, Romania

Characteristic valueChemical Formula TeO2

Molecular Weight 159.61

Crystal Class Tetragonal

Density (g/cm3 at 20 °C) 6

Melting Point (°C) 733°CHardness (Mohs) 4Solubility in water NoneColor ClearTransmittance Range (μm) 0.33-5.0

no=2.3194

ne=2.4829Thermal Expansion (1/K at 0°C)

normal to <001> 19.5 x 10-6

parallel to <001> 6.10 x 10-6

Refractive index (λ=500nm)

relatively low melting pointdistorted rutile (TiO2) structure

anisotropy of expansion coefficient

TeO2 (paratellurite)

a = 4.8088 Åc = 7.6038 Å

short:: 1.88 Å

long:: 2.12 Å

TeO2 crystal (2)

raw material preparation

TeO2+HCl→TeCl4+H2O

TeO2

2Te+9HNO3 → Te2O3(OH)NO3+8NO2+4H2OTe2O3(OH)NO3→2 TeO2+HNO3

TeCl4+4NH4OH→Te(OH)4+4NH4ClTe(OH)4→TeO2+H2O

HNO3

TeO2

HClTeCl4

TeCl4

NH4OHTeO2

TeO2

Te

TeO2 99.999%

washing

filtering

washing

drying

TeO2 crystal (3)

seed

grown Xtal

molten TeO2

heating

Czochralski

molten TeO2

Bridgman

seed

grown Xtal

Bridgman grown crystals are more stressed than Czochralski ones annealing at about 550°C helps in removing the residual stresses

TeO2 crystal is particularly repellent to impurities most of radioactive isotopes have ionic characteristics incompatible with substitutional incorporation in TeO2

crystal growth

TeO2 crystal (4)

At At. Nb. Oxi. No. Coo. No. IonicR (Å)Stable

isot. (%)Dev. IonR

(%)

Ag 47 3 4 0.67 100.000 1.52

Au 79 3 4 0.68 100.000 3.03

Co 27 2; 2 6; 5 0.65; 0.67 100.000 -1.52; 1.52

Fe 26 2; 3 4; 6 0.64; 0.65 100.000 -3.03; -2.27

Ir 77 3 6 0.68 100.000 3.03

Mg 12 2 5 0.66 100.000 0

Mn 25 2; 2; 3 6; 4; 6 0.66; 0.67; 0.65 100.000 0; 1.52; -2.27

Mo 42 4; 5 6; 4 0.65; 0.65 75.530 -1.52; -1.52

Nb 41 4; 5 6; 6 0.68; 0.64 100.000 3.03

Pb 82 2; 4 4; 4 0.64; 0.65 98.600 -3.03; -1.52

Pd 46 2 4 0.64 100.000 -3.03

Rh 45 3 6 0.665 100.000 0.76

Ru 44 3 6 0.68 100.000 3.03

Ta 73 4; 5 6 0.68; 0.64 99.988 3.03; -3.03

Te 52 4 4 0.66 33.606 0

Ti 22 3 6 0.67 100.000 1.51

V 23 3 6 0.64 99.750 -3.03

W 74 4 6 0.66 55.440 0

Zr 40 4 5 0.66 97.200 0

Te possible substitutional ions in TeO2

NAA XRF ICP-MS(ng) (µg/g) (pg/g)

Cs 0.1 5 0.1Co 0.1 1 3Pb -- 1 10Mo 1 0.5 2Pd 1 5 0.1K 1 10 200

Ra 0.1 5 1Ta 0.1 2 0.01Th 0.1 1 0.01W 0.01 1 1U 0.01 1 0.3V 0.1 1 1Zr 10 0.5 1

Approximate Detection LimitSymbol

238U (T=4.5·109 yr) 1210T

Ti

184W (T=3·1017 yr) 410T

Ti

TeO2 crystal (5)radiopurity

main radioactive series

crucible materialactivation products

ox. No.

coo. No

ion rad. (Å)

Te 4 4 0.66Co 3 6 0.61Pa 4 6 0.90Th 4 6 0.94U 4 6 0.89

3 6 0.644 5 0.534 6 0.583 6 0.704 6 0.70

V

Pt

natural radioactivity

decay mode

energy (MeV)

T1/2

40K beta 1.311 1.277E11 y

190Pt alpha 3.249 6.5E11 y

e- capture 2.208 1.4E17 y

beta 1.037 1.4E17 y

50V

conclusion

shares of 20 000 tons, world crystals production in 1999

60%12%

10%

10%5% 3%

semiconductors scintillation crystals

optical crystals acousto-optics crystals

laser and nonlinear crystals jewlery and watch industry

tonssemiconductors 12000scintillation crystals 2400optical crystals 2000acousto-optics crystals 2000laser and nonlinear crystals 1000jewlery and watch industry 600

60%12%

10%

10%5% 3%

semiconductors scintillation crystals

optical crystals acousto-optics crystals

laser and nonlinear crystals jewlery and watch industry

ECAL-CMS: (80 tons PWO)/2000-2006

CUORE: (1 ton TeO2)/?

Solid state physics d r joshi

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