Section-A

Crystal structure

Some Basic Definitions

LATTICE = An infinite array of points in space, in which each point has identical surroundings to all others.

CRYSTAL STRUCTURE = The periodic arrangement of atoms in the crystal.

It can be described by associating with each lattice point a group of atoms called the MOTIF (BASIS)

UNIT CELL = The smallest component of the crystal, which when stacked together with pure translational repetition reproduces the whole crystal

Primitive (P)unit cells contain only a single lattice point

2D LATTICES

e.g. the fused hexagonal pattern of a single layer of GRAPHITE

Counting Lattice Points/Atoms in 2D Lattices

Unit cell is Primitive (1 lattice point) but contains TWO atoms in the Motif Atoms at the corner of the 2D unit cell contribute only 1/4 to unit cell count Atoms at the edge of the 2D unit cell contribute only 1/2 to unit cell count Atoms within the 2D unit cell contribute 1 (i.e. uniquely) to that unit cell

Analysing a 3D solid

e.g. Graphite = a staggered arrangement of stacked hexagonal layers

Perspective: Clinographic views of solids

Projection onto a Plane: Plan views of solids

GRAPHITE

Unit Cell Dimensions

• a, b and c are the unit cell edge lengths

• , and are the angles ( between b and c, etc....)

Counting Atoms in 3D Cells

Atoms in different positions in a cell are shared by differing numbers of unit cells

Vertex atom shared by 8 cells 1/8 atom per cell

Edge atom shared by 4 cells 1/4 atom per cell

Face atom shared by 2 cells 1/2 atom per cell

Body unique to 1 cell 1 atom per cell

A substance is said to be crystalline when the arrangement of the units (atoms, molecules or ions) of matter inside it is regular and periodic. Crystal structure is a unique arrangement of atoms or molecules in a crystalline liquid or solid. A crystal structure is composed of a pattern, a set of atoms arranged in a particular way, and a lattice exhibiting long-range order and symmetry. Patterns are located upon the points of a lattice, which is an array of points repeating periodically in three dimensions. The points can be thought of as forming identical tiny boxes, called unit cells, that fill the space of the lattice. The lengths of the edges of a unit cell and the angles between them are called the lattice parameters. The symmetry properties of the crystal are embodied in its space group.

A crystal's structure and symmetry play a role in determining many of its physical properties, such as cleavage, electronic band structure, and optical transparency.

Unit cell The smallest portion of the crystal which can generate the complete crystal by repeating its own dimensions in various directions is called unit cell. The position vector R for any lattice point in a space lattice can be written as R= n1a+n2b+n3c Where a,b and c are the basis vector set. The angles between the vectors b and c, c and a, a and b are denoted as, and and are called interfacial angles. The three basis vectors and the three

interfacial angles, form a set of six parameters that define the unit cell, and are called lattice parameters.

U nit cell: The crystal structure of a material or the arrangement of atoms within a given type of crystal structure can be described in terms of its unit cell. The unit cell is a tiny box containing one or more atoms, a spatial arrangement of atoms. The unit cells stacked in three-dimensional space describe the bulk arrangement of atoms of the crystal. The crystal structure has a three dimensional shape. The unit cell is given by its lattice parameters, the length of the cell edges and the angles between them, while the positions of the atoms inside the unit cell are described by the set of atomic positions (xi , yi , zi) measured from a lattice point.

Simple cubic (P) Body-centered cubic (I) Face-centered cubic (F)

Unit cell

The smallest three-dimensional portion of a complete space lattice, which when repeated over and again in different

directions produces the complete space lattice.

The size and shape of a unit cell is determined by the lengths of the edges of the unit cell (a, b and c) and by the

angles a, b and g between the edges b and c, c and a, and a and b respectively.

If we take into consideration, the symmetry of the axial distances (a, b, c) and also the axial angles between the

edges (a, b and g), the various crystals can be divided into seven systems. These are also called crystal habits.

The constituent particles of a crystalline solid are arranged in a definite fashion in the three dimensional space.

One such arrangement by representing the particles with points is shown below:

Such a regular arrangement of the constituent particles of a crystal in a three dimensional space is called crystal

lattice or space lattice.

From the complete space lattice, it is possible to select a smallest three dimensional portion which repeats itself in

different directions to generate the complete space lattice. This is called a Unit Cell

Primitive cell A primitive cell is a minimum volume unit cell. Consider a bravais lattice (in two dimensions) as shown below: We can imagine two ways of identifying the unit cell in this structure. One is, with a1 and b1 as the basis vectors in which case, the unit cell will be a parallelogram. Here four lattice points are located at the vertices. This is a primitive cell. Other one is with the basis vectors a2 and b2

which would make a rectangle for the unit cell. Here in addition to the 4 points at the corners, one lattice point is at the centre. This is a nonprimitive cell.)

Primitive cell

Aprimitive cell is a minimum cell corresponding to a single lattice point of a structure with translational symmetry in 2 dimensions, 3 dimensions, or other dimensions. A lattice can be characterized by the geometry of its primitive cell.

The primitive cell is a fundamental domain with respect to translational symmetry only. In the case of additional symmetries a fundamental domain is smaller.

A crystal can be categorized by its lattice and the atoms that lie in a primitive cell (the basis). A cell will fill all the lattice space without leaving gaps by repetition of crystal translation operations.

Primitive translation vectors are used to define a crystal translation vector, , and also gives a lattice cell of smallest volume for a particular lattice. The lattice and translation vectors , ,

and are primitive if the atoms look the same from any lattice points using integers u1, u2, and u3.

The primitive cell is defined by the primitive axes (vectors) , , and . The volume, Vc, of the primitive cell is given by the parallelepiped from the above axes as,

Translational SymmetryThe crystalline state of substances is different from other states (gaseous, liquid, and amorphous)In that the atoms are in an ordered and symmetrical arrangement called thecrystal lattice. The lattice is characterized by space periodicity or translational symmetry.In an unbounded crystal we can define three noncoplanar vectors a1, a2, a3,such that displacement of the crystal by the length of any of these vectors brings itback on itself. The unit vectors aα, α = 1, 2, 3 are the shortest vectors by which acrystal can be displaced and be brought back into itself.The crystal lattice is thus a simple three-dimensional network of straight lineswhose points of intersection are called the crystal lattice1. If the origin of the coordinatesystem coincides with a site the position vector of any other site is writtenasR = Rn = R(n) =3Σα=1nαaα, n = (n1, n2, n3), (0.1.1)where nα are integers. The vector R is said to be a translational vector or a translationalperiod of the lattice. According to the definition of translational symmetry, thelattice is brought back onto itself when it is translated along the vector R.We can assign a translation operator to the translation vector R(n). A set of allpossible translations with the given vectors aα forms a discrete group of translations.Since sequential translations can be carried out arbitrarily, a group of transformations

is commutative (Abelian). A group of symmetry transformations can be used to explaina number of qualitative physical properties of crystals independently of their

specific structure.Miller indices

Planes with different Miller indices in cubic crystals

Vectors and atomic planes in a crystal lattice can be described by a three-value Miller index notation (ℓmn). The ℓ, m and n directional indices are separated by 90°, and are thus orthogonal. In fact, the ℓ component is mutually perpendicular to the m and n indices.

By definition, (ℓmn) denotes a plane that intercepts the three points a1/ℓ, a2/m, and a3/n, or some multiple thereof. That is, the Miller indices are proportional to the inverses of the intercepts of the plane with the unit cell (in the basis of the lattice vectors). If one or more of the indices is zero, it simply means that the planes do not intersect that axis (i.e. the intercept is "at infinity").

Considering only (ℓmn) planes intersecting one or more lattice points (the lattice planes), the perpendicular distance d between adjacent lattice planes is related to the (shortest) reciprocal lattice vector orthogonal to the planes by the formula:

Planes and directions

The crystallographic directions are fictitious lines linking nodes (atoms, ions or molecules) of a crystal. Similarly, the crystallographic planes are fictitious planes linking nodes. Some directions and planes have a higher density of nodes. These high density planes have an influence on the behavior of the crystal as follows:

Optical properties : Refractive index is directly related to density (or periodic density fluctuations).

Adsorption and reactivity: Physical adsorption and chemical reactions occur at or near surface atoms or molecules. These phenomena are thus sensitive to the density of nodes.

Surface tension : The condensation of a material means that the atoms, ions or molecules are more stable if they are surrounded by other similar species. The surface tension of an interface thus varies according to the density on the surface.

Cubic structures

For the special case of simple cubic crystals, the lattice vectors are orthogonal and of equal length (usually denoted a); similarly for the reciprocal lattice. So, in this common case, the Miller indices (ℓmn) and [ℓmn] both simply denote normals/directions in Cartesian coordinates. For cubic crystals with lattice constant a, the spacing d between adjacent (ℓmn) lattice planes is (from above):

Because of the symmetry of cubic crystals, it is possible to change the place and sign of the integers and have equivalent directions and planes:

Coordinates in angle brackets such as <100> denote a family of directions which are equivalent due to symmetry operations, such as [100], [010], [001] or the negative of any of those directions.

Coordinates in curly brackets or braces such as 100 denote a family of plane normals which are equivalent due to symmetry operations, much the way angle brackets denote a family of directions.

For face-centered cubic (fcc) and body-centered cubic (bcc) lattices, the primitive lattice vectors are not orthogonal. However, in these cases the Miller indices are conventionally defined relative to the lattice vectors of the cubic supercell and hence are again simply the Cartesian directions.

Lattice systems

These lattice systems are a grouping of crystal structures according to the axial system used to describe their lattice. Each lattice system consists of a set of three axes in a particular geometrical arrangement. There are seven lattice systems. They are similar to but not quite the same as the seven crystal systems and the six crystal families.

The 7 lattice systems (From least to most symmetric)

The 14 Bravais Lattices Examples

1. triclinic(none)

2. monoclinic(1 diad)

simple base-centered

3. orthorhombic(3 perpendicular diads)

simple base-centered body-centered face-centered

4. rhombohedral(1 triad)

5. tetragonal(1 tetrad)

simple body-centered

6. hexagonal(1 hexad)

7. cubic(4 triads)

simple (SC) body-centered (bcc)

face-centered (fcc)

The simplest and most symmetric, the cubic (or isometric) system, has the symmetry of a cube, that is, it exhibits four threefold rotational axes oriented at 109.5° (the tetrahedral angle) with respect to each other. These threefold axes lie along the body diagonals of the cube. The other six lattice systems, are hexagonal, tetragonal, rhombohedral (often confused with the trigonal crystal system), orthorhombic, monoclinic and triclinic. On combining 7 Crystal Classes with 4 possible unit cell types Symmetry indicates that only 14 3-D lattice types occur

The 14 possible BRAVAIS LATTICES

note that spheres in this picture represent lattice points, not atoms!

Examine the 14 Bravais Lattices in Detail

Cubic-P, Cubic-I, Cubic-F, Tetragonal-P, Tetragonal-I, Orthorhombic-P, Orthorhombic-I, Orthorhombic-F, Orthorhombic-C, Hexagonal-P, Trigonal-P,

Monoclinic-P, Monoclinic-C, Triclinic-P

Combining these 14 Bravais lattices with all possible symmetry elements

230 different Space Groups

A single layer of spheres is closest-packed with a HEXAGONAL coordination of each sphere

A second layer of spheres is placed in the indentations left by the first layer

space is trapped between the layers that is not filled by the spheres

TWO different types of HOLES (so-called INTERSTITIAL sites) are left

o OCTAHEDRAL (O) holes with 6 nearest sphere neighbours o TETRAHEDRAL (T±) holes with 4 nearest sphere neighbour

P = sphere, O = octahedral hole, T+ / T- = tetrahedral holes)

When a third layer of spheres is placed in the indentations of the second layer there are TWO choices

The third layer lies in indentations directly in line (eclipsed) with the 1st layer

o Layer ordering may be described as ABA The third layer lies in the alternative indentations leaving it staggered with

respect to both previous layers

Layer ordering may be described as ABC

Close-Packed Structures

Features of Close-Packing

Coordination Number = 12 74% of space is occupied Largest interstitial sites are:-

o octahedral (O) ( r = 0.414) ~ 1 per sphere o tetrahedral (T±) (r = 0.225) ~ 2 per sphere

Simplest Close-Packing Structures

ABABAB.... repeat gives Hexagonal Close-Packing (HCP) o Unit cell showing the full symmetry of the arrangement is Hexagonal

Hexagonal: a = b, c = 1.63a, = = 90°, = 120° 2 atoms in the unit cell: (0, 0, 0) (2/3, 1/3, 1/2)

ABCABC.... repeat gives Cubic Close-Packing (CCP) o Unit cell showing the full symmetry of the arrangement is Face-Centred Cubic

Cubic: a = b =c, = = = 90° 4 atoms in the unit cell: (0, 0, 0) (0, 1/2, 1/2) (1/2, 0, 1/2) (1/2, 1/2, 0)

2 atoms in the unit cell (0, 0, 0) (2/3, 1 /3, 1 /2)

4 atoms in the unit cell (0, 0, 0) (0, 1 /2, 1 /2) (1 /2, 0, 1 /2) (1 /2, 1 /2, 0)

The most common close-packed structures are METALS

Atomic coordination

By considering the arrangement of atoms relative to each other, their coordination numbers (or number of nearest neighbors), interatomic distances, types of bonding, etc., it is possible to form a general view of the structures and alternative ways of visualizing them.

HCP lattice (left) and the fcc lattice (right)

Close packing

The principles involved can be understood by considering the most efficient way of packing together equal-sized spheres and stacking close-packed atomic planes in three dimensions. For example, if plane A lies beneath plane B, there are two possible ways of placing an additional atom on top of layer B. If an additional layer was placed directly over plane A, this would give rise to the following series :

...ABABABAB....

This type of crystal structure is known as hexagonal close packing (hcp).

If however, all three planes are staggered relative to each other and it is not until the fourth layer is positioned directly over plane A that the sequence is repeated, then the following sequence arises:

...ABCABCABC...

This type of crystal structure is known as cubic close packing (ccp)

The unit cell of the ccp arrangement is the face-centered cubic (fcc) unit cell. This is not immediately obvious as the closely packed layers are parallel to the 111 planes of the fcc unit cell. There are four different orientations of the close-packed layers.

The packing efficiency could be worked out by calculating the total volume of the spheres and dividing that by the volume of the cell as follows:

The 74% packing efficiency is the maximum density possible in unit cells constructed of spheres of only one size. Most crystalline forms of metallic elements are hcp, ccp or bcc (body-centered cubic). The coordination number of hcp and fcc is 12 and its atomic packing factor (APF) is the number mentioned above, 0.74.

Space lattice

An array of points which describe the three dimensional arrangement of particles (atoms, molecules or ions) in a crystal structure is called space lattice. Here environment about each point should be identical. Basis A crystal structure is formed by associating with every lattice point a unit assembly of units or molecules identical in composition. This unit assembly is called basis. A crystal structure is formed by the addition of a basis to every lattice point. I.e., lattice + Basis = crystal structure. Thus the crystal structure is real and the crystal lattice is imaginary. Bravais latticesFor a crystal lattice, if each lattice point substitutes for an identical set of one or more atoms, then the lattice points become equivalent and the lattice is called Bravais lattice. On the other hand, if some of the lattice points are non-equivalent, then it is said to be a non-Bravais lattice.

When the crystal systems are combined with the various possible lattice centerings, we arrive at the Bravais lattices. They describe the geometric arrangement of the lattice points, and thereby the translational symmetry of the crystal. In three dimensions, there are 14 unique Bravais lattices which are distinct from one another in the translational symmetry they contain. All crystalline materials recognized until now (not including quasicrystals) fit in one of these arrangements. The fourteen three-dimensional lattices, classified by crystal system, are shown above. The Bravais lattices are sometimes referred to as space lattices.

The crystal structure consists of the same group of atoms, the basis, positioned around each and every lattice point. This group of atoms therefore repeats indefinitely in three dimensions according to the arrangement of one of the 14 Bravais lattices. The characteristic rotation and mirror symmetries of the group of atoms, or unit cell, is described by its crystallographic point group.

Crystal structures

Close packed structures: hexagonal close packing and cubic close packing

Many crystal structures can be described using the concept of close packing. This concept requires that the atoms (ions) are arranged so as to have the maximum density. In order to understand close packing in three dimensions, the most efficient way for equal sized spheres to be packed in two dimensions must be considered.

The most efficient way for equal sized spheres to be packed in two dimensions is shown in Figure 4, in which it can be seen that each sphere (the dark gray shaded sphere) is surrounded by, and is in contact with, six other spheres (the light gray spheres in Figure 4). It should be noted that contact with six other spheres the maximum possible is the spheres are the same size,

although lower density packing is possible. Close packed layers are formed by repetition to an infinite sheet. Within these close packed layers, three close packed rows are present, shown by the dashed lines in Figure 4.

Figure 4: Schematic representation of a close packed layer of equal sized spheres. The close

packed rows (directions) are shown by the dashed lines.

The most efficient way for equal sized spheres to be packed in three dimensions is to stack close packed layers on top of each other to give a close packed structure. There are two simple ways in which this can be done, resulting in either a hexagonal or cubic close packed structures.

Hexagonal close packed

If two close packed layers A and B are placed in contact with each other so as to maximize the density, then the spheres of layer B will rest in the hollow (vacancy) between three of the spheres in layer A. This is demonstrated in Figure 5. Atoms in the second layer, B (shaded light gray), may occupy one of two possible positions (Figure 5a or b) but not both together or a mixture of each. If a third layer is placed on top of layer B such that it exactly covers layer A, subsequent placement of layers will result in the following sequence ...ABABAB.... This is known as hexagonal close packing or hcp.

Figure 5: Schematic representation of two close packed layers arranged in A (dark grey) and B (light grey) positions. The alternative stacking

of the B layer is shown in (a) and (b).

The hexagonal close packed cell is a derivative of the hexagonal Bravais lattice system (Figure 1) with the addition of an atom inside the unit cell at the coordinates (1/3,2/3,1/2). The basal plane of the unit cell coincides with the close packed layers (Figure 6). In other words the close packed layer makes-up the 001 family of crystal planes.

Figure 6: A schematic projection of the basal plane of the hcp unit cell on the close packed

layers.

The “packing fraction” in a hexagonal close packed cell is 74.05%; that is 74.05% of the total volume is occupied. The packing fraction or density is derived by assuming that each atom is a hard sphere in contact with its nearest neighbors. Determination of the packing fraction is accomplished by calculating the number of whole spheres per unit cell (2 in hcp), the volume occupied by these spheres, and a comparison with the total volume of a unit cell. The number gives an idea of how “open” or filled a structure is. By comparison, the packing fraction for body-centered cubic (Figure 1) is 68% and for diamond cubic (an important semiconductor structure to be described later) is it 34%.

Cubic close packed: face-centered cubic

In a similar manner to the generation of the hexagonal close packed structure, two close packed layers are stacked (Figure 4) however, the third layer (C) is placed such that it does not exactly cover layer A, while sitting in a set of troughs in layer B (Figure 7), then upon repetition the packing sequence will be ...ABCABCABC.... This is known as cubic close packing or ccp.

Figure 7: Schematic representation of the three close packed layers in a cubic close packed

arrangement: A (dark grey), B (medium grey), and C (light grey).

The unit cell of cubic close packed structure is actually that of a face-centered cubic (fcc) Bravais lattice. In the fcc lattice the close packed layers constitute the 111 planes. As with the hcp lattice packing fraction in a cubic close packed (fcc) cell is 74.05%. Since face centered

cubic or fcc is more commonly used in preference to cubic close packed (ccp) in describing the structures, the former will be used throughout this text.

Coordination number

The coordination number of an atom or ion within an extended structure is defined as the number of nearest neighbor atoms (ions of opposite charge) that are in contact with it. A slightly different definition is often used for atoms within individual molecules: the number of donor atoms associated with the central atom or ion. However, this distinction is rather artificial, and both can be employed.

The coordination numbers for metal atoms in a molecule or complex are commonly 4, 5, and 6, but all values from 2 to 9 are known and a few examples of higher coordination numbers have been reported. In contrast, common coordination numbers in the solid state are 3, 4, 6, 8, and 12. For example, the atom in the center of body-centered cubic lattice has a coordination number of 8, because it touches the eight atoms at the corners of the unit cell, while an atom in a simple cubic structure would have a coordination number of 6. In both fcc and hcp lattices each of the atoms have a coordination number of 12.

Diamond Cubic

The diamond cubic structure consists of two interpenetrating face-centered cubic lattices, with one offset 1/4 of a cube along the cube diagonal. It may also be described as face centered cubic lattice in which half of the tetrahedral sites are filled while all the octahedral sites remain vacant. The diamond cubic unit cell is shown in Figure 8. Each of the atoms (e.g., C) is four coordinate, and the shortest interatomic distance (C-C) may be determined from the unit cell parameter (a).

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Figure 8: Unit cell structure of a diamond cubic lattice showing the two interpenetrating face-

centered cubic lattices.

Zinc blende

This is a binary phase (ME) and is named after its archetype, a common mineral form of zinc sulfide (ZnS). As with the diamond lattice, zinc blende consists of the two interpenetrating fcc lattices. However, in zinc blende one lattice consists of one of the types of atoms (Zn in ZnS), and the other lattice is of the second type of atom (S in ZnS). It may also be described as face centered cubic lattice of S atoms in which half of the tetrahedral sites are filled with Zn atoms.

All the atoms in a zinc blende structure are 4-coordinate. The zinc blende unit cell is shown in Figure 9. A number of inter-atomic distances may be calculated for any material with a zinc blende unit cell using the lattice parameter (a).

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Figure 9: Unit cell structure of a zinc blende (ZnS) lattice. Zinc atoms are shown in green

(small), sulfur atoms shown in red (large), and the dashed lines show the unit cell.

Chalcopyrite

The mineral chalcopyrite CuFeS2 is the archetype of this structure. The structure is tetragonal (a = b ≠ c, α = β = γ = 90°, and is essentially a superlattice on that of zinc blende. Thus, is easiest to imagine that the chalcopyrite lattice is made-up of a lattice of sulfur atoms in which the tetrahedral sites are filled in layers, ...FeCuCuFe..., etc. (Figure 10). In such an idealized structure c = 2a, however, this is not true of all materials with chalcopyrite structures.

Figure 10: Unit cell structure of a chalcopyrite lattice. Copper atoms are shown in blue, iron

atoms are shown in green and sulfur atoms are shown in yellow. The dashed lines show the

unit cell.

Rock salt

As its name implies the archetypal rock salt structure is NaCl (table salt). In common with the zinc blende structure, rock salt consists of two interpenetrating face-centered cubic lattices. However, the second lattice is offset 1/2a along the unit cell axis. It may also be described as face centered cubic lattice in which all of the octahedral sites are filled, while all the tetrahedral sites remain vacant, and thus each of the atoms in the rock salt structure are 6-coordinate. The rock salt unit cell is shown in Figure 11. A number of inter-atomic distances may be calculated for any material with a rock salt structure using the lattice parameter (a).

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Figure 11: Unit cell structure of a rock salt lattice. Sodium ions are shown in purple (small

spheres) and chloride ions are shown in red (large spheres).

Cinnabar

Cinnabar, named after the archetype mercury sulfide, HgS, is a distorted rock salt structure in which the resulting cell is rhombohedral (trigonal) with each atom having a coordination number of six.

Wurtzite

This is a hexagonal form of the zinc sulfide. It is identical in the number of and types of atoms, but it is built from two interpenetrating hcp lattices as opposed to the fcc lattices in zinc blende. As with zinc blende all the atoms in a wurtzite structure are 4-coordinate. The wurtzite unit cell is shown in Figure 12. A number of inter atomic distances may be calculated for any material with a wurtzite cell using the lattice parameter (a).

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However, it should be noted that these formulae do not necessarily apply when the ratio a/c is different from the ideal value of 1.632.

Figure 12: Unit cell structure of a wurtzite lattice. Zinc atoms are shown in green (small

spheres), sulfur atoms shown in red (large spheres), and the dashed lines show the unit

cell.

Cesium Chloride

The cesium chloride structure is found in materials with large cations and relatively small anions. It has a simple (primitive) cubic cell (Figure 1) with a chloride ion at the corners of the cube and the cesium ion at the body center. The coordination numbers of both Cs+ and Cl-, with the inner atomic distances determined from the cell lattice constant (a).

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β-Tin.

The room temperature allotrope of tin is β-tin or white tin. It has a tetragonal structure, in which each tin atom has four nearest neighbors (Sn-Sn = 3.016 Å) arranged in a very flattened tetrahedron, and two next nearest neighbors (Sn-Sn = 3.175 Å). The overall structure of β-tin consists of fused hexagons, each being linked to its neighbor via a four-membered Sn4 ring.

Bonding in solidsSolids can be classified according to the nature of the bonding between their atomic or molecular components. The traditional classification distinguishes four kinds of bonding[1]:

Covalent bonding , which forms network covalent solids (sometimes called simply "covalent solids")

Ionic bonding , which forms ionic solids

Metallic bonding , which forms metallic solids

Weak intermolecular bonding , which forms molecular solids (sometimes anomalously called "covalent solids")

Typical members of these classes have distinctive electron distributions [2], thermodynamic, electronic, and mechanical properties. In particular, the binding energies of these interactions vary widely. Bonding in solids can be of mixed or intermediate kinds, however, hence not all solids have the typical properties of a particular class, and some can be described as intermediate forms.

Basic classes of solids

Network covalent solids

A network covalent solid consists of atoms held together by a network of covalent bonds (pairs of electrons shared between atoms of similar electronegativity), and hence can be regarded as a single, large molecule[1]. The classic example is diamond; other examples include silicon [3] , quartz, and graphite.

Covalent network solids typically have high strength, high elastic modulus, and high melting temperatures. Their strength, stiffness, and high melting points are consequences of the strength and stiffness of the covalent bonds that hold them together. They are also characteristically brittle because the directional nature of covalent bonds strongly resists the shearing motions associated with plastic flow, and are, in effect, broken when shear occurs. This property results in brittleness for reasons studied in the field of fracture mechanics. Network covalent solids vary from insulating to semiconducting in their behavior, depending on the band gap of the material.

Ionic solids

A standard ionic solid consists of atoms held together by ionic bonds, that is, by the electrostatic attraction of opposite charges (the result of transferring electrons from lower- to higher-electronegativity atoms). Among the ionic solids are compounds formed by alkali and alkaline earth metals in combination with halogens; a classic example is table salt, sodium chloride.

Ionic solids are typically of intermediate strength and extremely brittle. Melting points are typically moderately high, but some combinations of molecular cations and anions yield an ionic liquid with a freezing point below room temperature. Vapor pressures in all instances are extraordinarily low; this is a consequence of the large energy required to move a bare charge (or charge pair) from an ionic medium into free space. Ionic solids have large band gaps, and hence are insulators.

Metallic solids

Metallic solids are held together by a high density of shared, delocalized electrons, resulting in metallic bonding. Classic examples are metals such as copper and aluminum, but some materials are metals in an electronic sense but have negligible metallic bonding in a mechanical or thermodynamic sense (see intermediate forms). Metallic solids have, by definition, no band gap at the Fermi level and hence are conducting.

Solids with purely metallic bonding are characteristically ductile and, in their pure forms, have low strength; melting points can be very low (e.g., Mercury melts at 234 K (–39°C). These

properties are consequences of the non-directional and non-polar nature of metallic bonding, which allows atoms (and planes of atoms in a crystal lattice) to move past one another without disrupting their bonding interactions. Metals can be strengthened by introducing crystal defects (for example, by alloying) that interfere with the motion of dislocations that mediate plastic deformation. Further, some transition metals exhibit directional bonding in addition to metallic bonding; this increases shear strength and reduces ductility, imparting some of the characteristics of a covalent solid (an intermediate case below).

Molecular solids

A classic molecular solid consists of small, non-polar covalent molecules, and is held together by London dispersion forces (van der Waals forces); a classic example is paraffin wax. These forces are weak, resulting in pairwise interatomic binding energies on the order of 1/100 those of covalent, ionic, and metallic bonds. Binding energies tend to increase with increasing molecular size and polarity (see intermediate forms).

Solids that are composed of small, weakly bound molecules are mechanically weak and have low melting points; an extreme case is solid molecular hydrogen, which melts at 14 K (–259°C). The non-directional nature of dispersion forces typically allows easy plastic deformation, as planes of molecules can slide over one another without seriously disrupting their attractive interactions. Molecular solids are typically insulators with large band gaps.

Solids of intermediate kindsThe four classes of solids permit six pairwise intermediate forms:

Ionic to network covalent

Covalent and ionic bonding form a continuum, with ionic character increasing with increasing difference in the electronegativity of the participating atoms. Covalent bonding corresponds to sharing of a pair of electrons between two atoms of essentially equal electronegativity (for example, C–C and C–H bonds in aliphatic hydrocarbons). As bonds become more polar, they become increasingly ionic in character. Metal oxides vary along the iono-covalent spectrum [2]. The Si–O bonds in quartz, for example, are polar yet largely covalent, and are considered to be of mixed character[3]; the bonds between Mg and O in magnesium oxide, by contrast, are chiefly ionic in character.

Metallic to network covalent

What is in most respects a purely covalent structure can support metallic delocalization of electrons; metallic carbon nanotubes are one example. Transition metals and intermetallic compounds based on transition metals can exhibit mixed metallic and covalent bonding[4], resulting in high shear strength, low ductility, and elevated melting points; a classic example is tungsten.

Molecular to network covalent

Materials can be intermediate between molecular and network covalent solids either because of the intermediate organization of their covalent bonds, or because the bonds themselves are of an intermediate kind.

Intermediate organization of covalent bonds:

Regarding the organization of covalent bonds, recall that classic molecular solids, as stated above, consist of small, non-polar covalent molecules. The example given, paraffin wax, is a member of a family of hydrocarbon molecules of differing chain lengths, with high-density polyethylene at the long-chain end of the series. High-density polyethylene can be a strong material: when the hydrocarbon chains are well aligned, the resulting fibers rival the strength of steel. The covalent bonds in this material form extended structures, but do not form a continuous network. With cross-linking, however, polymer networks can become continuous, and a series of materials spans the range from Cross-linked polyethylene, to rigid thermosetting resins, to hydrogen-rich amorphous solids, to vitreous carbon, diamond-like carbons, and ultimately to diamond itself. As this example shows, there can be no sharp boundary between molecular and network covalent solids.

Intermediate kinds of bonding:

A solid with extensive hydrogen bonding will be considered a molecular solid, yet strong hydrogen bonds can have a significant degree of covalent character. As noted above, covalent and ionic bonds form a continuum between shared and transferred electrons; covalent and weak bonds form a continuum between shared and unshared electrons. In addition, molecules can be polar, or have polar groups, and the resulting regions of positive and negative charge can interact to produce electrostatic bonding resembling that in ionic solids.

Molecular to ionic

A large molecule with an ionized group is technically an ion, but its behavior may be largely the result of non-ionic interactions. For example, sodium stearate (the main constituent of traditional soaps) consists entirely of ions, yet it is a soft material quite unlike a typical ionic solid. There is a continuum between ionic solids and molecular solids with little ionic character in their bonding.

Metallic to molecular

Metallic solids are bound by a high density of shared, delocalized electrons. Although weakly bound molecular components are incompatible with strong metallic bonding, low densities of shared, delocalized electrons can impart varying degrees of metallic bonding and conductivity overlaid on discrete, covalently-bonded molecular units, especially in reduced-dimensional systems. Examples include charge transfer complexes.

Metallic to ionic

The charged components that make up ionic solids cannot exist in the high-density sea of delocalized electrons characteristic of strong metallic bonding. Some molecular salts, however, feature both ionic bonding among molecules and substantial one-dimensional conductivity, indicating a degree of metallic bonding among structural components along the axis of conductivity. Examples include tetrathiafulvalene salts.

Q 1.1 What you understand by bonds in solids? What are the main causes and conditions for bondformation?Answer: The forces which keep or hold together the atoms or molecules of a substance in the form ofgroups are called bonds. The atoms or molecules in the gaseous and liquid states are loosely-packed anda very little binding force exists among them. Therefore, gases and liquids do not possess any definiteshape. If a gas (or liquid) is heated, it expands out indefinitely, showing that little binding force existsamong its various atoms. However, atoms and molecules in a solid are closely-packed and are heldtogether by strong mutual forces of attraction. Therefore, solids have definite shape and occupy welldefined space. If a solid is heated, it does not change its shape easily, showing that a very big force existsthat binds the various atoms and molecules. In other words the bonds in solids are very strong comparedwith that in gases and liquids. The law of nature is to make every system to attain a stable state byacquiring minimum potential energy. When two atoms come closer and unite to form molecules, theirelectrons rearrange themselves in such a way so as to form a stable state.InferenceThe formation of bonds between atoms is mainly due to their tendency to attain minimum potentialenergy. When two atoms tend to form a bond, their valence electrons rearrange themselves so as toreach a stable state by acquiring minimum potential energy. In the process, the two atoms lose someenergy. The strength of the bond between two atoms would obviously depend upon the energy lost inthe process.Q 1.2 Describe ionic or electrovalent bonds in solids with suitable examples.Answer: The bond formed between two atoms by the total transfer of valence electrons from one atomto the other is called an ionic or electrovalent bond. Here one or more electrons from an atom maytransfer to the other atom and the resulting positive and negative ions attract each other. A typicalexample of an ionic bond is sodium chloride (NaCl) where the bond exists between Na+ and Cl– ions.When sodium is burnt in an atmosphere of chlorine, the sodium gives up its valence electron to thechlorine, each of the resulting ions then has a stable filled shell of outer electrons, and a strong electrostaticattraction is set up that bonds the Na+ cation and the Cl– anion into a very stable molecule NaCl at theequilibrium spacing. The relevant equation is:Q 1.3 Discuss the variation of interatomic force between atoms with spacing between them with asuitable graph. Compute the cohesive energy of this system by drawing a similar curve between potentialenergy and spacing.Answer: We assume here that in a solid material the following two types of forces act between theatoms:(i) attractive forces which keep the atoms together forcing them to form a solid.(ii) repulsive forces which come into play when a solid is compressed.Such forces, however, act in the case of liquids also and even in single molecule. But mere existenceof these forces between atoms does not guarantee the formation of a stable chemical bond. This may beestablished by considering two atoms say A and B exerting attractive and repulsive forces on each othersuch that the bonding force F, between the atoms may be represented as:The first term represents the attractive force and the second term the repulsive force. Near theequilibrium position the second term must increase more rapidly for diminishing value of r than doesthe first, and N is necessarily greater than M.

Q 1.4 Write a note on the properties of ionic crystals.Answer: (i) Crystal structure: Most of the ionic solids have fine crystalline structure. It has beenfound by x-ray diffraction that the constituents of these crystals are ions and not atoms. For instance, inthe case of NaCl, each Na+ ion is surrounded by six Cl– ions at equal distances. Similarly, each Cl– issurrounded by six Na+ ions. The result is, we get crystal lattice of NaCl.(ii) Melting and boiling points: Ionic solids have high melting and boiling points. It is becauseconsiderable external energy is required to overcome the electrostatic forces existing between the ionsin such a solid.(iii) Electrical conductivity: Pure and dry ionic solids are good insulators because all the electrons aretightly bound with the ions involved in the bond formation. However, such solids show electricalconductivity when;(a) the temperature is raised. At high temperature, the electrostatic forces between the ions aregreatly reduced so that some of the ions themselves transport the charge in the material.(b) dissolve easily in solvents like water. When an ionic solid is dissolved easily in water, theelectrostatic forces are considerably weakened (by 80 times) due to high permittivity of water.The result is that the ions become free and wander about in the solution. If now a field is

applied, these ions will themselves carry the charge in the solution (electrolysis). Thepermittivity of water is about 80.(iv) Solubility: Ionic compounds easily dissolve in solvents like water (H2O). It is because moleculesof water strongly interact with the crystal ions to destroy the forces of attraction between the ions. Ioniccompounds are insoluble in non-polar solvents like benzene (C6H6), carbon tetrachloride (CCl4), becausetheir dielectric constants are very low.(v) Other properties: Reaction between ionic compounds in solution state is always fast. This is becausein a solution, ionic substances exist as ions and chemical reactions take place between the ions. Ioniccrystals are transparent for all frequencies up to the value called the fundamental absorption frequency.At frequencies higher than this, they are opaque. High hardness and low conductivity are typical propertiesof these solids. When subjected to stresses, ionic crystals tend to cleave (break) along certain planes ofatoms rather than to deform in a ductile fashion as metals do.Below are given, some important relations used in the study of other properties of ionic crystals.(a) The expressions for bulk modulus and compressibility are respectively listedQ 1.6 Briefly explain the properties of covalent compounds.Answer:(i) Covalent compounds may be solids, liquids or gases. Generally those substances which havehigh molecular weights exist as solids. Covalent solids are hard as well as brittle.(ii) Covalent solids have crystalline structure i.e., atoms or molecules are arranged in someregular repeatable pattern in the three dimensions.(iii) Pure covalent solids are good insulators. The reason is that all the valence electrons are tightlyheld in the covalent bonds. However, when certain impurities are added to such solids, theybecome reasonably good conductors and are termed as semiconductors.(iv) Since covalent bonds are comparatively weak, therefore, covalent solids have low meltingand boiling points(v) Covalent solids are not readily soluble in water. However, they are easily soluble in organicsolvents like benzene.(vi) A very interesting property of covalent compounds is the apparent lack of sensitivity of theirphysical properties to their bonding type. For example, carbon in the diamond structure is thehardest substance and has a very high melting point of 3280 K. The hardness and meltingpoint then decrease as we proceed to other elements in column IV of the periodic table fromsilicon to lead. Tin, for example, is very soft and has a low melting point. The variation in theelectrical properties is also pronounced. Diamond is a very good insulator. Silicon andgermanium are well known semiconductors while tin is a good conductor. Depending on thenumber of electrons shared, the bond length and bond energy vary. When the number ofelectrons shared is more, the bond length between the atoms is decreased and bond energy isincreased.Diamond, silicon, germanium, silicon carbide, tin and rutile are some examples of covalentcrystals.Q 1.7 Explain the nature of the bonds that exist in metals. Discuss the important physical properties ofmetals.Answer: Metallic elements have low ionisation energies and hence, in this bonding, atoms of the sameelement or different elements give their valance electrons to form an electron cloud or say ‘electron gas’throughout the space occupied by the atoms. Having given up their valence electrons, the atoms are inreality positive ions. These ions are held together by forces that are similar to those of ionic bond in thatthey are primarily electrostatic, but are between the ions and the electrons. Most of the atoms in metalshave one or two valence electrons. These electrons are loosely held by their atoms and therefore can beeasily released to the common pool to form an electron cloud. The electrostatic interaction between thepositive ions and the electron gas holds the metal together. The high electrical and thermal conductivitiesof metals follow from the ability of the free electrons to migrate through their crystal lattices while all ofthe electrons in ionic and covalent crystals are bound to particular atoms.Unlike other crystals, metals may be deformed without fracture, because the electron gas permitsatoms to slide fast one another by acting as a lubricant. As we have seen carbon can exist in the covalentform and so it is an extremely poor conductor. However, it may also exist in an alternate form asgraphite. In this case, bonds are formed in which covalency is not fully achieved and these bonds canbreak and reform fairly easily as in metallic bond. For this reason graphite is a conductor. If a potentialdifference is applied between any two points in a metal piece, the electron gas flows from negatively

charged part to the positively charged part, constituting electric current.Properties of Metallic Crystals(i) Bonding energies and melting temperatures for some metals are given in Table 1.A. Bondingenergies may be weaker or stronger, energies range from 64 × 103 kJ/kmol (0.7 eV/atom) for(Negative) Electron cloud(Positive) Metal ion14 Rudiments of Materials Sciencemercury to 850 × 103 kJ/kmol (8.8 eV/atom) for tungsten. Their respective melting points are–39°C and 3410°C.(ii) Due to the symmetrical arrangements of the positive ions in a space lattice, metals are highlycrystalline.(iii) Metallic bonds being weak, metals have a melting point moderate to high. i.e., the meltingpoints of metallic crystals are lower than those of the electrovalent crystals.(iv) Since a large number of free electrons are available, metallic crystals have high electricalconductivity.(v) Metallic crystals have higher thermal conductivity because of the availability of large numberof free electrons which act as carriers of heat.(vi) They are mechanically strong.Copper, sodium, aluminum and silver are some examples.Q 1.8 Discuss briefly molecular bonds. Also write a short note on hydrogen bonding.Answer: The bonds between atoms of those substances whose electrons have little transferability areknown as molecular bonds.Molecular bonds are formed for those elements or compounds whose electronic configuration issuch that there is little transfer of electrons between their atoms. (e.g. noble gases like argon, neon, etc.).Unlike the three bonds considered above, in which electrons are either exchanged or shared, molecularbonds involve no transfer or exchange of charge. Rather the bond arises from the van der Waals forcesof attraction which exist between various atoms as explained below:All noble gases (neon, argon, etc.) have their last orbits complete. Obviously, they cannot formbonds by exchange or sharing of electrons. Hence, atoms of noble gases have little attraction for eachother and consequently they remain in atomic state under ordinary conditions of temperature and pressure.However, at very low temperature, condensation of these gases takes place. This condensation wouldnot have been possible if there are no interatomic forces, however weak. These interatomic forces ofattraction are called van der Waals forces.van der Waals ForcesAn atom is neutral from a distance only. However, close to it, there is always a net charge at any time asseen by a neighbouring atom. This is because all the electrons are not concentrated at one end in anatom. Thus there exists forces of electrostatic attraction between the nucleus of one atom and the electronsof the other. These forces are called van der Waals forces. i.e., the forces of attraction between twoneighbouring atoms due to the resultant charge are called van der Waals forces. These forces are veryweak and were first discovered by van der Waals.Characteristics of Molecular Solids(i) They exist in crystalline form as well as non-crystalline solids(ii) They have no melting point as the binding arises from van der Waals forces which are quiteweak(iii) They are good insulators as free electrons are not available(iv) They are insoluable in water.Hydrogen BondingCovalently bonded atoms some times produce molecules that behave as permanent dipoles. For examplein water molecule, the oxygen atom shares two half filled p-orbitals with two hydrogen atoms. A simpleway of describing the situation is, the electrons shared between these atoms spend more “time” inbetween the two atoms so that the oxygen atom tends to act as +ve end of the dipole. So in the formationof ice the bonding tends to become more ionic by the +ve and the –ve ions being arranged alternatelyforming long chains as shown below:H– O – H– O – H

Table 1.F Comparison between ionic bonds and metallic bondsProperties Ionic Bonds Metallic BondsBonding force The bonds exist due to electrostatic The bonds exist due to electro static

force of attraction between positive force of attraction between the electronand negative ions of different elements cloud of valence electrons and positiveions of the same or different metallicelementsBond formation Ionic bonds are most easily formed This type of bond is characteristic ofwhen one of the atoms has smaller the elements having smaller number ofnumber of valence electrons, such as valence electrons, which are looselythe alkali metals and alkali earths held, so that they can be released to thecommon poolConductivity Low conductivity is the property of the Good thermal and electrical conductivitysolids formed by ionic bonding is the property of most of the solidsformed by metallic bondingMechanical Solids formed have high hardness. Solids formed mostly have good ductilityproperties Ionic crystals tend to cleave (break)along certain planes of atoms ratherthan to deform in a ductile fashionwhen subjected to stressesBond strength These bonds are generally stronger These bonds are generally less strongerthan the metallic bonds than ionic bondsOBJECTIVE QUESTIONS1. Ionization energy of sodium atom is(a) 5 joule (b) 8 × 10–19 joule(c) 200 × 1010 joule (d) 1420 joule2. The net energy required for creating a positive sodium ion and a negative chlorine ion is(a) 5.1 eV (b) 3.6 eV(c) 1.5 eV (d) 2.2 eV3. Which of the following classes is most likely to produce a semiconductor?(a) ionic (b) covalent(c) metallic (d) van der Waals4. At high temperature, some of the ions in an ionic crystal transport charge because of interaction(a) true (b) false18 Rudiments of Materials Science5. When an ionic solid is dissolved in water the ions are free and wander in the solution. This isbecause(a) high dielectric constant of water (b) low dielectric constant of water(c) low density of water(d) none of these6. Ionic compounds are insoluble in carbon tetrachloride because(a) it is a polar solvent (b) it is a non-polar solvent(c) it is an organic solvent (d) none of these7. Which of the following solids are always opaque to visible radiation?(a) covalent (b) metallic(c) ionic (d) none of these8. Which of the following has the hydrogen bonding?(a) CH4 (b) CsCl(c) NaCl (d) HF10. The total number of Cl– ions in a unit cell of NaCl crystal is(a) 4 (b) 2(c) 8 (d) 1011. If the distance between a Na ion and a chlorine ion in NaCl crystal is 0.28 nm, then the latticeparameter of the crystal is(a) 0.14 nm (b) 0.8 nm(c) 56 Ao (d) 0.56 nm12. At the equilibrium spacing of a diatomic molecule, the resultant force is(a) zero (b) minimum(c) maximum (d) unity13. The potential energy in the above said spacing is(a) zero (b) minimum

(c) maximum (d) unityBonding in Solids 1914. The maximum number of covalent bonds that can be formed by a carbon atom is(a) 2 (b) 8(c) 4 (d) 115. Which of the interatomic bonds are directional?(a) covalent (b) metallic(c) ionic (d) van der Waals16. The energy required to break the bond (H –– Cl) is 4 eV. This is equal to(a) 420 × 106 J/kmol (b) 420 × 104 kJ/kmol(c) 120 J/kmol (d) 420 kJ/kmol17. The number of unshared electrons by each oxygen atom with the carbon atom to form CO2 is(a) 2 (b) 3(c) 4 (d) 518. Covalent crystals are hard and brittle(a) true (b) false19. The properties of covalent crystals are not sensitive to the nature of the bonding that exist(a) true (b) false20. The absence of electrostatic interaction between the electron gas and positive ions holds the metaltogether(a) true (b) false21. The endless symmetrical arrangements of positive ions in metals in three dimension is the maincause for the formation of a metal to be single crystal(a) true (b) false22. The electrostatic attraction between the nucleus of one atom and the electrons of the other is called(a) coulomb forces (b) gravitational(c) van der Waals forces (d) none of these23. Cohesive energy in the case of van der Waals bonding lies in the range(a) 8 – 10 eV (b) 6 – 8 eV(c) 0.1 – 0.5 eV (d) 0.002 – 0.1 eV24. The co-ordination number of Na+ and Cl– in the rock salt structure is respectively(a) 8 and 6 (b) 6 and 8(c) 6 and 6 (d) 4 and 4

X-ray analysis of crystals

The incoming beam (coming from upper left) causes each scatterer to re-radiate a small portion of its intensity as a spherical wave. If scatterers are arranged symmetrically with a separation d, these spherical waves will be in sync (add constructively) only in directions where their path-length difference 2d sin θ equals an integer multiple of the wavelength λ. In that case, part of the incoming beam is deflected by an angle 2θ, producing a reflection spot in the diffraction pattern.

Crystals are regular arrays of atoms, and X-rays can be considered waves of electromagnetic radiation. Atoms scatter X-ray waves, primarily through the atoms' electrons. Just as an ocean wave striking a lighthouse produces secondary circular waves emanating from the lighthouse, so an X-ray striking an electron produces secondary spherical waves emanating from the electron. This phenomenon is known as elastic scattering, and the electron (or lighthouse) is known as the scatterer. A regular array of scatterers produces a regular array of spherical waves. Although these waves cancel one another out in most directions through destructive interference, they add constructively in a few specific directions, determined by Bragg's law:

Here d is the spacing between diffracting planes, θ is the incident angle, n is any integer, and λ is the wavelength of the beam. These specific directions appear as spots on the diffraction pattern called reflections. Thus, X-ray diffraction results from an electromagnetic wave (the X-ray) impinging on a regular array of scatterers (the repeating arrangement of atoms within the crystal).

X-rays are used to produce the diffraction pattern because their wavelength λ is typically the same order of magnitude (1-100 Ångströms) as the spacing d between planes in the crystal. In principle, any wave impinging on a regular array of scatterers produces diffraction, as predicted first by Francesco Maria Grimaldi in 1665. To produce significant diffraction, the spacing between the scatterers and the wavelength of the impinging wave should be similar in size. For illustration, the diffraction of sunlight through a bird's feather was first reported by James Gregory in the later 17th century. The first artificial diffraction gratings for visible light were constructed by David Rittenhouse in 1787, and Joseph von Fraunhofer in 1821. However, visible light has too long a wavelength (typically, 5500 Ångströms) to observe diffraction from crystals. Prior to the first X-ray diffraction experiments, the spacings between lattice planes in a crystal were not known with certainty.

As described in the mathematical derivation below, the X-ray scattering is determined by the density of electrons within the crystal. Since the energy of an X-ray is much greater than that of a valence electron, the scattering may be modeled as Thomson scattering, the interaction of an electromagnetic ray with a free electron. This model is generally adopted to describe the polarization of the scattered radiation. The intensity of Thomson scattering declines as 1/m² with the mass m of the charged particle that is scattering the radiation; hence, the atomic nuclei, which are thousands of times heavier than an electron, contribute negligibly to the scattered X-rays.

The potential of X-ray crystallography for determining the structure of molecules and minerals — then only known vaguely from chemical and hydrodynamic experiments — was realized immediately. The earliest structures were simple inorganic crystals and minerals, but even these revealed fundamental laws of physics and chemistry. The first atomic-resolution structure to be "solved" (i.e. determined) in 1914 was that of table salt.[22][23][24] The distribution of electrons in the table-salt structure showed that crystals are not necessarily composed of covalently bonded molecules, and proved the existence of ionic compounds.[25] The structure of diamond was solved in the same year,[26][27] proving the tetrahedral arrangement of its chemical bonds and showing that the length of C–C single bond was 1.52 Ångströms. Other early structures included copper,[28] calcium fluoride (CaF2, also known as fluorite), calcite (CaCO3) and pyrite (FeS2)[29] in 1914; spinel (MgAl2O4) in 1915;[30][31] the rutile and anatase forms of titanium dioxide (TiO2) in 1916;[32] pyrochroite Mn(OH)2 and, by extension, brucite Mg(OH)2 in 1919;.[33][34] Also in 1919

sodium nitrate (NaNO3) and cesium dichloroiodide (CsICl2) were determined by Ralph Walter Graystone Wyckoff, and the wurtzite (hexagonal ZnS) structure became known in 1920.[35]

Powder diffraction is a scientific technique using X-ray, neutron, or electron diffraction on powder or microcrystalline samples for structural characterization of materials.[2]

Explanation

Ideally, every possible crystalline orientation is represented very equally in a powdered sample. The resulting orientational averaging causes the three dimensional reciprocal space that is studied in single crystal diffraction to be projected onto a single dimension. The three dimensional space can be described with (reciprocal) axes x*, y* and z* or alternatively in spherical coordinates q, φ*, χ*. In powder diffraction intensity is homogeneous over φ* and χ* and only q remains as an important measurable quantity. In practice, it is sometimes necessary to rotate the sample orientation to eliminate the effects of texturing and achieve true randomness.

Two-dimensional powder diffraction setup with flat plate detector(Ref.)

When the scattered radiation is collected on a flat plate detector the rotational averaging leads to smooth diffraction rings around the beam axis rather than the discrete Laue spots as observed for single crystal diffraction. The angle between the beam axis and the ring is called the scattering angle and in X-ray crystallography always denoted as 2θ. (In scattering of visible light the convention is usually to call it θ). In accordance with Bragg's law, each ring corresponds to a particular reciprocal lattice vector G in the sample crystal. This leads to the definition of the scattering vector as:

Powder diffraction data are usually presented as a diffractogram in which the diffracted intensity I is shown as function either of the scattering angle 2θ or as a function of the scattering vector q. The latter variable has the advantage that the diffractogram no longer depends on the value of the wavelength λ. The advent of synchrotron sources has widened the choice of wavelength considerably. To facilitate comparability of data obtained with different wavelengths the use of q is therefore recommended and gaining acceptability.

An instrument dedicated to perform powder measurements is called a powder diffractometer.

Uses

Relative to other methods of analysis, powder diffraction allows for rapid, non-destructive analysis of multi-component mixtures without the need for extensive sample preparation.[3] This gives laboratories around the world the ability to quickly analyse unknown materials and perform materials characterization in such fields as metallurgy, mineralogy, forensic science, archeology, condensed matter physics, and the biological and pharmaceutical sciences. Identification is performed by comparison of the diffraction pattern to a known standard or to a database such as the International Centre for Diffraction Data's Powder Diffraction File (PDF) or the Cambridge Structural Database (CSD). Advances in hardware and software, particularly improved optics and fast detectors, have dramatically improved the analytical capability of the technique, especially relative to the speed of the analysis. The fundamental physics upon which the technique is based provides high precision and accuracy in the measurement of interplanar spacings, sometimes to fractions of an Ångström, resulting in authoritative identification frequently used in patents, criminal cases and other areas of law enforcement. The ability to analyze multiphase materials also allows analysis of how materials interact in a particular matrix such as a pharmaceutical tablet, a circuit board, a mechanical weld, a geologic core sampling, cement and concrete, or a pigment found in an historic painting. The method has been historically used for the identification and classification of minerals, but it can be used for any materials, even amorphous ones, so long as a suitable reference pattern is known or can be constructed.

Phase identification

The most widespread use of powder diffraction is in the identification and characterization of crystalline solids, each of which produces a distinctive diffraction pattern. Both the positions (corresponding to lattice spacings) and the relative intensity of the lines are indicative of a particular phase and material, providing a "fingerprint" for comparison. A multi-phase mixture, e.g. a soil sample, will show more than one pattern superposed, allowing for determination of relative concentration.

J.D. Hanawalt, an analytical chemist who worked for Dow Chemical in the 1930s, was the first to realize the analytical potential of creating a database. Today it is represented by the Powder Diffraction File (PDF) of the International Centre for Diffraction Data (formerly Joint Committee for Powder Diffraction Studies). This has been made searchable by computer through the work of global software developers and equipment manufacturers. There are now over 550,000 reference materials in the 2006 Powder Diffraction File Databases, and these databases are interfaced to a wide variety of diffraction analysis software and distributed globally. The Powder Diffraction File contains many subfiles, such as minerals, metals and alloys, pharmaceuticals, forensics, excipients, superconductors, semiconductors etc., with large collections of organic, organometallic and inorganic reference materials.

Crystallinity

In contrast to a crystalline pattern consisting of a series of sharp peaks, amorphous materials (liquids, glasses etc.) produce a broad background signal. Many polymers show semicrystalline behavior, i.e. part of the material forms an ordered crystallite by folding of the molecule. One and the same molecule may well be folded into two different crystallites and thus form a tie between the two. The tie part is prevented from crystallizing. The result is that the crystallinity will never reach 100%. Powder XRD can be used to determine the crystallinity by comparing the integrated intensity of the background pattern to that of the sharp peaks. Values obtained from

powder XRD are typically comparable but not quite identical to those obtained from other methods such as DSC.

Lattice parameters

The position of a diffraction peak is independent of the atomic positions within the cell and entirely determined by the size and shape of the unit cell of the crystalline phase. Each peak represents a certain lattice plane and can therefore be characterized by a Miller index. If the symmetry is high, e.g. cubic or hexagonal it is usually not too hard to identify the index of each peak, even for an unknown phase. This is particularly important in solid-state chemistry, where one is interested in finding and identifying new materials. Once a pattern has been indexed, this characterizes the reaction product and identifies it as a new solid phase. Indexing programs exist to deal with the harder cases, but if the unit cell is very large and the symmetry low (triclinic) success is not always guaranteed.

Expansion tensors, bulk modulus

Thermal expansion of a sulfur powder

Cell parameters are somewhat temperature and pressure dependent. Powder diffraction can be combined with in situ temperature and pressure control. As these thermodynamic variables are changed, the observed diffraction peaks will migrate continuously to indicate higher or lower lattice spacings as the unit cell distorts. This allows for measurement of such quantities as the thermal expansion tensor and the isothermal bulk modulus, as well determination of the full equation of state of the material.

Phase transitions

At some critical set of conditions, for example 0 °C for water at 1 atm, a new arrangement of atoms or molecules may become stable, leading to a phase transition. At this point new diffraction peaks will appear or old ones disappear according to the symmetry of the new phase. If the material melts to an isotropic liquid, all sharp lines will disappear and be replaced by a broad amorphous pattern. If the transition produces another crystalline phase, one set of lines will suddenly be replaced by another set. In some cases however lines will split or coalesce, e.g. if the material undergoes a continuous, second order phase transition. In such cases the symmetry may change because the existing structure is distorted rather than replaced by a completely different one. E.g. the diffraction peaks for the lattice planes (100) and (001) can be found at two different values of q for a tetragonal phase, but if the symmetry becomes cubic the two peaks will come to coincide.

Crystal structure refinement and determination

Crystal structure determination from powder diffraction data is extremely challenging due to the overlap of reflections in a powder experiment. The crystal structures of known materials can be refined, i.e. as a function of temperature or pressure, using the Rietveld method. The Rietveld method is a so-called full pattern analysis technique. A crystal structure, together with instrumental and microstructural information is used to generate a theoretical diffraction pattern that can be compared to the observed data. A least squares procedure is then used to minimise the difference between the calculated pattern and each point of the observed pattern by adjusting model parameters. Techniques to determine unknown structures from powder data do exist, but are somewhat specialised.[4] A number of program that can be used in structure determination are TOPAS, GSAS, Fox, EXPO2004, and a few others.

Size and strain broadening

There are many factors that determine the width B of a diffraction peak. These include:

1. instrumental factors2. the presence of defects to the perfect lattice3. differences in strain in different grains4. the size of the crystallites

It is often possible to separate the effects of size and strain. Where size broadening is independent of q (K=1/d), strain broadening increases with increasing q-values. In most cases there will be both size and strain broadening. It is possible to separate these by combining the two equations in what is known as the Hall-Williamson method:

Thus, when we plot vs. we get a straight line with slope and intercept .

The expression is a combination of the Scherrer Equation for size broadening and the Stokes and Wilson expression for strain broadening. The value of η is the strain in the crystallites, the value of D represents the size of the crystallites. The constant k is typically close to unity and ranges from 0.8-1.39.

Comparison of X-ray and Neutron Scattering

X-ray photons scatter by interaction with the electron cloud of the material, neutrons are scattered by the nuclei. This means that, in the presence of heavy atoms with many electrons, it may be difficult to detect light atoms by X-ray diffraction. In contrast, the neutron scattering length of most atoms are approximately equal in magnitude. Neutron diffraction techniques may therefore be used to detect light elements such as oxygen or hydrogen in combination with heavy atoms. The neutron diffraction technique therefore has obvious applications to problems such as determining oxygen displacements in materials like high temperature superconductors and ferroelectrics, or to hydrogen bonding in biological systems.

A further complication in the case of neutron scattering from hydrogenous materials is the strong incoherent scattering of hydrogen (80.27(6) barn). This leads to a very high background in neutron diffraction experiments, and may make structural investigations impossible. A common solution is deuteration, i.e. replacing the 1-H atoms in the sample with deuterium (2-H). The incoherent scattering length of deuterium is much smaller (2.05(3) barn) making structural investigations significantly easier. However, in some systems, replacing hydrogen with deuterium may alter the structural and dynamic properties of interest.

As neutrons also have a magnetic moment, they are additionally scattered by any magnetic moments in a sample. In the case of long range magnetic order, this leads to the appearance of new Bragg reflections. In most simple cases, powder diffraction may be used to determine the size of the moments and their spatial orientation.

Aperiodically-arranged clusters

Predicting the scattered intensity in powder diffraction patterns from gases, liquids, and randomly-distributed nano-clusters in the solid state[5] is (to first order) done rather elegantly with the Debye scattering equation[6]:

where the magnitude of the scattering vector q is in reciprocal lattice distance units, N is the number of atoms, fi(q) is the atomic scattering factor for atom i and scattering vector q, while rij is the distance between atom i and atom j. One can also use this to predict the effect of nano-crystallite shape on detected diffraction peaks, even if in some directions the cluster is only one atom thick.

Devices

Cameras

The simplest cameras for X-ray powder diffraction consist of a small capillary and either a flat plate detector (originally a piece of X-ray film, now more and more a flat-plate detector or a CCD-camera) or a cylindrical one (originally a piece of film in a cookie-jar, now more and more a bent position sensitive detector). The two types of cameras are known as the Laue and the Debye-Scherrer camera.

In order to ensure complete powder averaging, the capillary is usually spun around its axis.

For neutron diffraction vanadium cylinders are used as sample holders. Vanadium has a negligible absorption and coherent scattering cross section for neutrons and is hence nearly invisible in a powder diffraction experiment. Vanadium does however have a considerable incoherent scattering cross section which may cause problems for more sensitive techniques such as neutron inelastic scattering.

A later development in X-ray cameras is the Guinier camera. It is built around a focusing bent crystal monochromator. The sample is usually placed in the focusing beam., e.g. as a dusting on

a piece of sticky tape. A cylindrical piece of film (or electronic multichannel detector) is put on the focusing circle, but the incident beam prevented from reaching the detector to prevent damage from its high intensity.

Diffractometers

Diffractometers can be operated both in transmission and in reflection configurations. The reflection one is more common. The powder sample is filled in a small disc like container and its surface carefully flattened. The disc is put on one axis of the diffractometer and tilted by an angle θ while a detector (scintillation counter) rotates around it on an arm at twice this angle. This configuration is known under the name Bragg-Brentano.

Another configuration is the theta-theta configuration in which the sample is stationary while the X-ray tube and the detector are rotated around it. The angle formed between the tube and the detector is 2theta. This configuration is most convenient for loose powders.

The availability of position sensitive detectors and CCD-cameras is making this type of equipment more and more obsolete.

Neutron diffraction

Sources that produce a neutron beam of suitable intensity and speed for diffraction are only available at a small number of research reactors and spallation sources in the world. Angle dispersive (fixed wavelength) instruments typically have a battery of individual detectors arranged in a cylindrical fashion around the sample holder, and can therefore collect scattered intensity simultaneously on a large 2θ range. Time of flight instruments normally have a small range of banks at different scattering angles which collect data at varying resolutions.

X-ray tubes

Laboratory X-ray diffraction equipment relies on the use of an X-ray tube, which is used to produce the X-rays.

For more on how X-ray tubes work, see for example here or X-ray.

The most commonly used laboratory X-ray tube uses a Copper anode, but Cobalt, Molybdenum are also popular. The wavelength in nm varies for each source.

Advantages and disadvantagesAlthough it possible to solve crystal structures from powder X-ray data alone, its single crystal analogue is a far more powerful technique for structure determination. This is directly related to the fact that much information is lost by the collapse of the 3D space onto a 1D axis. Nevertheless powder X-ray diffraction is a powerful and useful technique in its own right. It is mostly used to characterize and identify phases rather than solving structures.

The great advantages of the technique are:

simplicity of sample preparation

rapidity of measurement the ability to analyse mixed phases, e.g. soil samples

By contrast growth and mounting of large single crystals is notoriously difficult. In fact there are many materials for which despite many attempts it has not proven possible to obtain single crystals. Many materials are readily available with sufficient microcrystallinity for powder diffraction, or samples may be easily ground from larger crystals. In the field of solid-state chemistry that often aims at synthesizing new materials, single crystals thereof are typically not immediately available. Powder diffraction is therefore one of the most powerful methods to identify and characterize new materials in this field.

Particularly for neutron diffraction, which requires larger samples than X-Ray Diffraction due to a relatively weak scattering cross section, the ability to use large samples can be critical, although new more brilliant neutron sources are being built that may change this picture.

Since all possible crystal orientations are measured simultaneously, collection times can be quite short even for small and weakly scattering samples. This is not merely convenient, but can be essential for samples which are unstable either inherently or under X-ray or neutron bombardment, or for time-resolved studies. For the latter it is desirable to have a strong radiation source. The advent of synchrotron radiation and modern neutron sources has therefore done much to revitalize the powder diffraction field because it is now possible to study temperature dependent changes, reaction kinetics and so forth by means of time dependent powder diffraction.

The powder method is used to determine the value of the lattice parameters accurately. Lattice parameters are the magnitudes of the unit vectors a, b and c which define the unit cell for the crystal.

If a monochromatic x-ray beam is directed at a single crystal, then only one or two diffracted beams may result.

If the sample consists of some tens of randomly orientated single crystals, the diffracted beams are seen to lie on the surface of several cones. The cones may emerge in all directions, forwards and backwards.

A sample of some hundreds of crystals (i.e. a powdered sample) show that the diffracted beams form continuous cones.

A circle of film is used to record the diffraction pattern as shown. Each cone intersects the film giving diffraction lines. The lines are seen as arcs on the film.

For every set of crystal planes, by chance, one or more crystals will be in the correct orientation to give the correct Bragg angle to satisfy Bragg's equation. Every crystal plane is thus capable of diffraction. Each diffraction line is made up of a large number of small spots, each from a separate crystal. Each spot is so small as to give the appearance of a continuous line. If the crystal is not ground finely enough, the diffraction lines appear speckled.

We shall now consider the powder patterns from a sample crystal. The sample is known to have a cubic structure, but we don't know which one.

We remove the film strip from the Debye camera after exposure, then develop and fix it. From the strip of film we make measurements of the position of each diffraction line. From the results it is possible to associate the sample with a particular type of cubic structure and also to determine a value for its lattice parameter.

• When the film is laid flat, S1 can be measured. This is the distance along the film, from a diffraction line, to the centre of the hole for the transmitted direct beam.

• For back reflections, i.e. where 2 > 90° you can measure S2 as the distance from the beam entry point.

• The distance S1 corresponds to a diffraction angle of 2. The angle between the diffracted and the transmitted beams is always 2. We know that the distance between the holes in the film, W, corresponds to a diffraction angle of = . So we can find from:

or

• We know Bragg's Law: n = 2dsin

and the equation for interplanar spacing, d, for cubic crystals is given by:

where a is the lattice parameter

this gives:

• From the measurements of each arc we can now generate a table of S1, and sin2.

• If all the diffraction lines are considered, then the experimental values of sin2 should form a pattern related to the values of h, k and l for the structure.

• We now multiply the values of sin2 by some constant value to give nearly integer values for all the h2+ k2+ l2 values. Integer values are then assigned.

• The integer values of h2+ k2+ l2 are then equated with their hkl values to index each arc, using the table shown below:

• For some structures e.g. bcc, fcc, not all planes reflect, so some of the arcs may be missing.

• It is then possible to identify certain structures, in this case fcc (- the planes have hkl values: all even, or all odd in the table above).

• For each line we can also calculate a value for a, the lattice parameter. For greater accuracy the value is averaged over all the lines.

The Laue method is mainly used to determine the orientation of large single crystals. White radiation is reflected from, or transmitted through, a fixed crystal.

The diffracted beams form arrays of spots, that lie on curves on the film. The Bragg angle is fixed for every set of planes in the crystal. Each set of planes picks out and diffracts the particular wavelength from the white radiation that satisfies the Bragg law for the values of d and involved. Each curve therefore corresponds to a different wavelength. The spots lying on any one curve are reflections from planes belonging to one zone. Laue reflections from planes of the same zone all lie on the surface of an imaginary cone whose axis is the zone axis.

ExperimentalThere are two practical variants of the Laue method, the back-reflection and the transmission Laue method. You can study these below:

Back-reflection Laue

In the back-reflection method, the film is placed between the x-ray source and the crystal. The beams which are diffracted in a backward direction are recorded.

One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an hyperbola.

Transmission Laue

In the transmission Laue method, the film is placed behind the crystal to record beams which are transmitted through the crystal.

One side of the cone of Laue reflections is defined by the transmitted beam. The film intersects the cone, with the diffraction spots generally lying on an ellipse.

Crystal orientation is determined from the position of the spots. Each spot can be indexed, i.e. attributed to a particular plane, using special charts. The Greninger chart is used for back-reflection patterns and the Leonhardt chart for transmission patterns.

The Laue technique can also be used to assess crystal perfection from the size and shape of the spots. If the crystal has been bent or twisted in anyway, the spots become distorted and smeared out.

Free electron theory Free electron model

In three dimensions, the density of states of a gas of fermions is proportional to the square root of the kinetic energy of the particles.

In solid-state physics, the free electron model is a simple model for the behaviour of valence electrons in a crystal structure of a metallic solid. It was developed principally by Arnold Sommerfeld who combined the classical Drude model with quantum mechanical Fermi-Dirac statistics and hence it is also known as the Drude–Sommerfeld model. It forms the basis of the band structure model known as nearly-free electron model. Given its simplicity, it is surprisingly successful in explaining many experimental phenomena, especially

the Wiedemann-Franz law which relates electrical conductivity and thermal conductivity; the temperature dependence of the heat capacity; the shape of the electronic density of states; the range of binding energy values; electrical conductivities; thermal electron emission and field electron emission from bulk metals.

Ideas and assumptions

As in the Drude model, valence electrons are assumed to be completely detached from their ions (forming an electron gas). As in an ideal gas, electron-electron interactions are completely neglected (they are weak because of the shielding effect).

The crystal lattice is not explicitly taken into account. A quantum-mechanical justification is given by Bloch's Theorem: an unbound electron moves in a periodic potential as a free electron in vacuum, except for the electron mass m becoming an effective mass m* which may deviate considerably from m (one can even use negative effective mass to describe conduction by electron holes). Effective masses can be derived from band structure computations. While the static lattice does not hinder the motion of the electrons, they can well be scattered by impurities and by phonons; these two interactions determine electrical and thermal conductivity (superconductivity requires a more refined theory than the free electron model).

According to the Pauli exclusion principle, each phase space element (Δk)3(Δx)3 can be occupied only by two electrons (one per spin quantum number). This restriction of available electron states is taken into account by Fermi-Dirac statistics (see also Fermi gas). Main predictions of the free-electron model are derived by the Sommerfeld expansion of the Fermi-Dirac occupancy for energies around the Fermi level.

Energy and wave function of a free electron

Plane wave traveling in the x-direction

For a free particle the potential is . The Schrödinger equation for such a particle, like the free electron, is[1][2][3]

The wave function can be split into a solution of a time dependent and a solution of a time independent equation. The solution of the the time dependent equation is

with energy

The solution of the time independent equation is

with a wave vector . Ωr is the volume of space occupied by the electron. The electron has a kinetic energy

The plane wave solution of this Schrödinger equation is

For solid state and condensed matter physicists the time independent solution is of major interest. It is the basis of electronic band structure models that are widely used in solid-state physics for model Hamiltonians like the nearly free electron model and the Tight binding model and different models that use a Muffin-tin approximation. The eigenfunctions of these Hamiltonians are Bloch waves which are modulated plane waves.

Empty Lattice ApproximationPeriodic potential

The periodic potential of the crystal lattice in a free electron band structure model is not more precisely defined than "periodic". Implicitly it is assumed that the potential is weak, otherwise the electron wouldn't be free, but it is just strong enough to scatter the electrons. How strong must a potential be to be able to scatter an electron? The answer is that it depends on the topology of the system how large topologically defined parameters, like scattering cross sections in three dimensions], depend on the magnitude of the potential and the size of the potential well. One thing is clear for currently known 1, 2 and 3-dimensional spaces: potential wells do always scatter waves no matter how small their potentials are or what their sign is and how limited their sizes are. For a particle in a one-dimensional lattice, like the Kronig-Penney model, it is easy to substitute the values for the potential and the size of the potential well. If the values of the potential and size of the potential wells are reduced to infinitesimal values the band structure of the Empty Lattice Approximation [4] is obtained.

Nearly free electron model

In the NFE model the Fourier transform, , of the lattice potential, , in the NFE Hamiltonian, can be reduced to an infinitesimal value. When the values of the off-diagonal elements in the Hamiltonian almost go to zero, the magnitude of the band gap collapses. The division of k-space in Brillouin zones still remains however because the electrons will still be weakly scattered. In theory the lattice is infinitely large, so a weak periodic scattering potential will eventually be strong enough to reflect the wave. The dispersion relation is

and consists of a increasing number of free electron bands when the energy rises. is the reciprocal lattice vector to which the band belongs.

Second, third and higher Brillouin zones

"Free electrons" that move through the lattice of a solid with wave vectors far outside the first Brillouin zone are still reflected back into the first Brillouin zone. See the external links section for sites with examples and figures.

Dielectric function of the electron gas

On a scale much larger than the inter atomic distance a solid can be viewed as an aggregate of a negatively charged plasma of the free electron gas and a positively charged background of atomic cores. The background is the rather stiff and massive background of atomic nuclei and core electrons which we will consider to be infinitely massive and fixed in space. The negatively charged plasma is formed by the valence electrons of the free electron model that are uniformly distributed over the interior of the solid. If an oscillating electric field is applied to the solid, the negatively charged plasma tends to move a distance x apart from the positively charged background. As a result the sample is polarized and there will be an excess charge at the opposite surfaces of the sample. The surface charge density is

ρs = − nex

which produces a restoring electric field in the sample

The dielectric function of the sample is expressed as

where D(ω) is the electric displacement and P(ω) is the polarization density.

The electric field and polarization densities are

and the polarization per atom with n electrons is

P = − nex

The force F of the oscillating electric field causes the electrons with charge e and mass m to accelerate with an acceleration a

which, after substitution of E, P and x, yields an harmonic oscillator equation.

After a little algebra the relation between polarization density and electric field can be expressed as

The frequency dependent dielectric function of the solid is

At a resonance frequency ωp, called the plasma frequency, the the dielectric function changes sign from negative to positive and real part of the dielectric function drops to zero.

This is a plasma oscillation resonance or plasmon. The plasma frequency is a direct measure of the square root of the density of valence electrons in a solid. Observed values are in reasonable agreement with this theoretical prediction for a large number of materials.[4] Below the plasma frequency, the dielectric function is negative and the field cannot penetrate the sample. Light with angular frequency below the plasma frequency will be totally reflected. Above the plasma frequency the light waves can penetrate the sample.

Solution of the Schrödinger equation

The Schrödinger equation

For a free particle the potential is , so the Schrödinger equation for the free electron is[1][2][3]

This is a type of wave equation that has numerous kinds of solutions. One way of solving the equation is splitting it in a time-dependent oscillator equation and a space-dependent wave equation like

and

and substituting a product of solutions like

The Schrödinger equation can be split in a time dependent part and a time independent part.

Solution of the time dependent equation

The peculiar time dependent part of the Schrödinger equation is, unlike the Klein-Gordon equation for pions and most of the other well known wave equations, a first order in time differential equation with a 90o out of phase driving mechanism, while most oscillator equations are second order in time differential equations with 180o out of phase driving mechanisms.

The equation that has to be solved is

.

The complex (imaginary) exponent is proportional to the energy

The imaginary exponent can be transformed to an angular frequency

The wave function now has a stationary and an oscillating part

The stationary part is of major importance to the physical properties of the electronic structure of matter.

[edit] Solution of the time independent equation

The wave function of free electrons is in general described as the solution of the time independent Schrödinger equation for free electrons

The Laplace operator in Cartesian coordinates is

The wave function can be factorized for the three Cartesian directions

Now the time independent Schrödinger equation can be split in three independent parts for the three different Cartesian directions

As a solution an exponential function is substituted in the time independent Schrödinger equation

φx(x) = Nxeκx

The solution of

gives the exponent

which yields the wave equation

and the energy

With the normalization

and the wave vector length

we arrive at the plane wave solution with a wave function

for free electrons with a wave vector and a kinetic energy

in which Ωr is the volume of space occupied by the electron.

Traveling plane waves restlessly heading for their Final Destination

The traveling plane wave solution

The product of the time independent stationary wave solution and time dependent oscillator solution

gives the traveling plane wave solution

which is the final solution for the free electron wave function.

FREE ELECTRON THEORY.Aim: To introduce the free electron model for the physical properties of metals. It is the simplest

theory for these materials, but still gives a very good description of many properties of metals

which depend on the dynamics of the electrons.

We assume a free electron gas in the metals. The ”ions”, i.e atomic nuclei with inner electrons (in

closed shells) are immersed in a gas of freely moving conduction electrons. This assumption works

best for the simple metals, i.e. the alkali metals (Li, Na, K...), but also for the noble metals and Al.

How can the electrons be considered as free? 1. We will see later that matter waves can propagate

freely in a periodic structure (lattice). 2. Electron-electron interactions are effectively screened due

to the high density of electrons. 3. The Pauli principle also acts to diminish electron-electron

scattering.

The most important scattering mechanism for the electrons is instead from the phonons (lattice

vibrations).

1. Basics of the free electron model (H p. 86-90).

We assume that the electrons move in a box potential, i.e. the potential in the metal is taken to be

zero and ∞ outside of the metal (an infinite barrier at the surface). We also employ the

independent electron approximation. This means that each electron is considered to move

independently of the others. We now proceed as follows:

a) Solve the Schrödinger equation for one electron. From that we find the orbitals, which are

defined as solutions for a system of one electron.

b) Boundary conditions: As usual in solid state physics we may choose periodic boundary

conditions. From the periodic boundary conditions we obtain the set of k-values that are allowed.

The procedure is the same as the one we earlier applied for the lattice vibrations.

The general solution of the free electron Schrödinger equation is a plane wave, ψk(r) ~= exp(ik.r).

c) The components of k, together with the spin quantum number specify the different energy

levels. The Pauli principle says that we can only have two electrons, one with each spin, in each

energy level. We now fill the obtained energy levels with conduction electrons, starting from the

lowest one. At T=0 the energy of the highest filled level is called the Fermi energy, EF. In three

dimensions, the points with E=EF are on the surface of a sphere, the Fermi sphere, in reciprocal

space. At higher temperature, the occupancy of the levels is smeared out around the Fermi level, in

accordance with the Fermi-Dirac distribution (H p. 89).

d) Next we compute the energy as a function of k (the dispersion relation of this problem),

E(k) = (h2 / 2m)k 2 ,

as well as other basic quantities like the Fermi velocity and Fermi temperature. The Fermi energy

corresponds to a Fermi wave-vector of magnitude kF = (3π2n)1/3, where n is the electron

concentration. The Fermi velocity, i.e. the velocity of an electron moving on the Fermi surface, is

given by vF

2 = 2EF/m. The Fermi energy corresponds to a Fermi temperature, typically of the order

of 104 – 105 K. The density of electron states as a function of energy g(E), can be obtained by a

procedure rather similar to the one used for the Debye model before (H p. 89). We obtain

3 / 2 1/ 2

2 2 (2 )

2

g(E) V m E

π h= ,

Which can be rewritten by using the relation between the Fermi energy and the electron

concentration n=N/V. As a special case we obtain the density of states at the Fermi energy as:

g(EF) = 3N / 2E F ,

where N is the number of electrons. For numerical estimates one may also conveniently express

the density of states per atom by replacing N with the number of conduction electrons per atom N0

and the volume V by the volume per atom, Ω. Hence the electron density can be expressed as

n=N/V = N0/Ω.

Below we illustrate how the distribution in energy of the conduction electrons is obtained from the

product of the density of states, g(E), and the Fermi-Dirac distribution (adapted from Myers).

How large is the electron density of states?

Example: Al has three conduction electrons per atom and a Fermi energy of 11.6 eV. The relation

above directly gives g(EF) = 0.39 eV-1 atom –1. On the other hand one mole of Al (a macroscopic

solid) has, at the Fermi energy, 2.34 1023 levels per eV with an average spacing of 4.3 10-24 eV.

Hence the energy levels of a macroscopic metal are so close (”quasi-continuous” and certainly

broadened in reality) that they must be viewed as a continuous energy band.

2. Electronic heat capacity (H p. 90-91).

In metals, the free electrons give a contribution to the heat capacity, in addition to that of the lattice

vibrations. The total energy of the electron gas is obtained by integrating over the electron energy

distribution (see figure above), and the derivative with respect to temperature gives the electronic

heat capacity. The contributing electrons are those that can be excited to an empty state at

temperature T. They are within an interval ±kBT of the Fermi energy. In this interval the density of

states is approximately equal to its value at the Fermi energy and the Fermi-Dirac function is 1/2.

Each electron is excited by the thermal energy 3kBT/2. The resulting heat capacity becomes

C = 3 kB

2 g(EF) T,

which is very close to the result of a more rigorous calculation (H eq.(6.18)).

The figure shows the heat capacity divided by T as a function of the square of T for a typical

insulator, KCl, and for a metal, Cu. The insulator has only a contribution from the lattice

vibrations; at these low temperatures the Debye T3-law is obeyed. The metal has an electronic

contribution in addition to the one from the lattice vibrations. It is seen from the intersection of the

curve with the y-axis. The electronic contribution is linearly dependent on T, as expected.

The deviation of the experimental values from the free electron ones are taken into account by

introducing a thermal effective mass of the conduction electrons. Note that γ ~ g(EF) ~ 1/EF ~ m.

The effective mass may be lower or higher than the electron mass depending on if the

experimentally observed γ is lower or higher than the free electron value.

3. Electrical properties (H p. 71-76).

The high electrical conductivity of metals is a direct consequence of the relatively free motion of

the electrons.

a) The equation of motion of the free electrons under the force due to applied electric (E) can

easily be written down, eq (5.2). It seems that the electron velocity increases indefinitely, eq.(5.3).

b) This is not possible in reality. It is evident that scattering processes limit the electron velocity

and hence the electrical conductivity of metals. The most important ones are collisions of the

electrons with phonons, impurities and defects. The scattering is described by a relaxation time, τ.

The mean free path is equal to the Fermi velocity times the relaxation time. A viscous term must

be introduced in the equation of motion, which becomes:

m dv/dt + mv/τ = -eE.

The stationary state in a constant field is obtained by putting the first term equal to zero. Then we

find that v = (-eτ/m) E. The electrical conductivity, σ = j / E, is obtained as,

σ = -nev/E,

where n is the electron concentration. Below we discuss the resistivity

ρ = 1/σ = m / ne2τ .

c) Matthiessen’s rule states that the contribution of different scattering processes to the resistivity

are additive. The figure below (adapted from Myers) illustrates the general behaviour of the

resistivity of metals. At high temperature scattering by phonons dominates. The phonon

concentration is approximately proportional to temperature (Planck’s law), and hence so is the

collision rate, 1/τ. This leads to a resistivity proportional to T. At low temperatures the resistivity

approaches a constant value. This is due to scattering from impurities, defects etc, whose

concentration is independent of temperature. This is illustrated in the right figure for silver with

two impurity concentrations.

4. Hall effect (H p. 75-76).

We now have to solve the equation of motion taking into account both applied electric and

magnetic fields. As discussed above a collision term also has to be included. We only consider a

special case in which the magnetic field is directed along the z-axis. The equation of motion can be

separated into one equation for each of the three directions, x, y and z.

An application of this theory is the Hall effect. In a rod-shaped sample a current (current density J)

is flowing in the x-direction and a magnetic field is applied in the z-direction. The charge carriers

are forced by the magnetic field to deflect in the (negative) y-direction.

The sides of the rod in the y-direction will be oppositely charged and this sets up a transverse Hall

field in the y-direction. This field is directed in the negative y-direction for negative charge

carriers (electrons) and in the positive y-direction in the case of positive charge carriers. The Hall

coefficient is given by

RH = EH/JxB = 1/nq.

This is hence negative for free electrons, since they have negative charge q=-e. Hall measurements

allow one to determine the sign of the charge carriers and their density, n. It is seen that the free

electron model is good for many simple metals. However, in some cases the Hall coefficient is

positive, suggesting positive charge carriers. This is a signal of a breakdown of the free electron

model and we need to go into the energy band theory of metals to understand this behaviour.

5. Thermal conductivity of metals (H p. 79-80, 92).

In pure metals the electrons make the dominant contribution to the thermal conductivity. The

expression is analogous to that for the phonon contribution and follows from kinetic gas theory:

κe = Ce vF l.

The difference is that we have to use the electronic contribution to the specific heat as well as the

Fermi velocity. It is found that the ratio of thermal and electric conductivities is proportional to

temperature in the free electron model. This is the Wiedemann-Franz law. It is in good agreement

with experimental data in a region above the Debye temperature, where the thermal conductivity is

almost independent of temperature (since the electrical conductivity ~1/T). This is often so at room

temperature and above; here the Wiedemann-Franz law is quite good. The Lorentz number,

K/σT, of many metals is close to the prediction of free electron theory at ambient temperatures and

above.

The figure below shows a qualitative picture of the temperature dependence of the thermal

conductivity of a metal. Just as for the electrical conductivity, phonon scattering of the electrons

dominates except at low temperatures. At temperatures above θD, the number of phonons is

proportional to T, the mean free path l~T-1 and Ce ~T, hence the thermal conductivity of metals

becomes constant. At lower temperatures considerably below θD, the fraction of excited phonon

modes scales as (T/θD)3 (see the Debye sphere), and hence l~T-3, and K~T-2. At very low

temperatures both the thermal and electrical conductivity are limited by scattering by defects,

impurities etc. Here the electrical conductivity is constant, the thermal conductivity is ~T and the

Wiedemann-Franz law is again applicable.

The thermal conductivity of Cu is about 400 W/m K at room temperature and at the maximum at

low temperatures of the order 2500 W/m K. Metals have a higher thermal conductivity than most

insulators (not diamond!) at room temperature, but at the low temperature peak in insulators values

of the order of a few thousand W/m K can be found.

FREE ELECTRON THEORY Classical free electron theory of metals

This theory was developed by Drude and Lorentz and hence is also known as Drude-Lorentz theory. According to

this theory, a metal consists of electrons which are free to move about in the crystal like molecules of a gas in a

container. Mutual repulsion between electrons is ignored and hence potential energy is taken as zero. Therefore the

total energy of the electron is equal to its kinetic energy.

Drift velocity

If no electric field is applied on a conductor, the free electrons move in random directions. They collide with each

other and also with the positive ions. Since the motion is completely random, average velocity in any direction is

zero. If a constant electric field is established inside a conductor, the electrons experience a force F = -eE due to

which they move in the direction opposite to direction of the field. These electrons undergo frequent collisions with

positive ions. In each such collision, direction of motion of electrons undergoes random changes. As a result, in

addition to the random motion, the electrons are subjected to a very slow directional motion. This motion is called

drift and the average velocity of this motion is called drift velocity vd.

Consider a conductor subjected to an electric field E in the x-direction. The force on the electron due to the electric

field = -eE.

By Newton’s law, -eE = mdvd/dt

dvd = -eEdt/m

Integrating,

Vd = -eEt/m + Constant

When t = 0, vd = 0 Therefore Constant = 0

Vd = -eEt/m --------------- (1)

Electrical conductivity

Consider a wire of length ‘dl’ and area of cross section ‘A’ subjected

to an electric field E. If ‘n’ is the concentration of the electrons, the number of electrons flowing through the wire in

dt seconds = nAvddt.

The quantity of charge flowing in time dt = nAvddt.e

Therefore I = dq/dt = neAvd

Current density J = I/A = nevd

Subsittuting the value of vd from (1),

J = nee Et/m = ne2Et/m --------------- (2)

By Ohm’s law, J = s E

Therefore s = J/E = ne2t/m -------------- (3)

Mobility of a charge carrier is the ratio of the drift mobility to the electric field.

µ = vd/E m2/Volt-Sec

Substituting vd from (1),

µ = et/m -------------- (4)

Substituting this in equation (3),

s = neµ ------------- (5)

Relaxation time and mean free path

When the field E is switched off, due to the collision of the electrons with lattice ions and lattice defects, their

velocity will start to decrease. This process is called relaxation. The relaxation time(t) is the time required for the

drift velocity to reduce to 1/e of its initial value. The average distance traveled by an electron between two

consecutive collisions is called mean free path (l) of the electron.

l = vdt -------------- (6)

Temperature dependence

The free electron theory is based on Maxwell-Boltzmann statistics.

Therefore Kinetic energy of electron = ½ mvd2 = 3/2 KBT

Vd = Ö 3KBT/m

Substituting this in equation (6),

t = lÖ m/3KBT -------------- (7)

Since s = ne2t/m, s is proportional to Ö1/T

Or r is proportional to ÖT.

Wiedmann-Franz law

The ratio of thermal conductivity to electrical conductivity of a metal is directly

proportional to absolute temperature.

K/s is proportional to T

Or, K/sT = L, a constant called Lorentz number.

L = 3KB2/2e2

Drawbacks of Classical free electron theory

1) According to this theory, r is proportional to ÖT. But experimentally it was found that r is proportional to T.

2) According to this theory, K/sT = L, a constant (Wiedmann-Franz law) for all temperatures. But this is not true at

low temperatures.

3) The theoretically predicted value of specific heat of a metal does not agree with the experimentally obtained

value.

4) This theory fails to explain ferromagnetism, superconductivity, photoelectric effect, Compton effect and

blackbody radiation.

Quantum free electron theory

Classical free electron theory could not explain many physical properties. In 1928, Sommerfeld developed a new

theory applying quantum mechanical concepts and Fermi-Dirac statistics to the free electrons in the metal. This

theory is called quantum free electron theory.

Classical free electron theory permits all electrons to gain energy. But quantum free electron theory permits only a

fraction of electrons to gain energy. In order to determine the actual number of electrons in a given energy

range(dE), it is necessary to know the number of states(dNs) which have energy in that range. The number of states

per unit energy range is called the density of states g(E).

Therefore, g(E) = dNs/dE

According to Fermi-Dirac statistics, the probability that a particular energy statewith energy E is occupied by an

electron is given by,

f(E) = 1 / [1+e(E-EF/KT) ]

where EF is called Fermi level. Fermi level is the highest filled energy level at 0 K. Energy corresponding to Fermi

level is known as Fermi energy. Now the actual number of electrons present in the energy range dE,

dN = f(E) g(E)dE

Effect of temperature on Fermi-Dirac distribution function

Fermi-Dirac distribution function is given by,

f(E) = 1 / [1+e(E-EF/KT) ]

At T=0K, for EEF, f(E)=0

At T=0K, for E=EF, f(E)=indeterminate

At T>0K, for E=EF, f(E)=1/2

All these results are depicted in the figure.

Classification of semiconductors on the basis of fermi level

and fermi energy

In intrinsic semiconductors, the fermilevel lie exactly at the centre of the forbidden energy gap. In n-type

semiconductors fermilevel lie near the conduction band. In p-type semiconductors fermilevel lie near the valence

band.

Impurity levels

In extrinsic semiconductors, addition of impurities introduces new allowed quantum energy states in the forbidden

energy band. The quantum state which appears as a single energy level is known as impurity level.

Impurity level is called donor level (Ed) in n-type semiconductors and lie just below the conduction band. Impurity

level is called acceptor level (Ea) in p- type semiconductor and lie just above the valence band.

Band theory of solids

The atoms in the solid are very closely packed. The nucleus of an atom is so heavy that it considered to be at rest

and hence the characteristic of an atom are decided by the electrons. The electrons in an isolated atom have different

and discrete amounts of energy according to their occupations in different shells and sub shells. These energy values

are represented by sharp lines in an energy level diagram.

During the formation of a solid, energy levels of outer shell electrons got split up. As a result, closely packed energy

levels are produced. The collection of such a large number of energy levels is called energy band. The electrons in

the outermost shell are called valence electrons. The band formed by a series of energy levels containing the valence

electrons is known as valence band. The next higher permitted band in a solid is the conduction band. The electrons

occupying this band are known as conduction electrons.

Conduction band valence band are separated by a gap known as forbidden energy gap. No electrons can occupy

energy levels in this band. When an electrons in the valence band absorbs enough energy, it jumps across the

forbidden energy gap and enters the conduction band, creating a positively charged hole in the valence band.the

hole is basically the deficiency of an electron.

Classification of solids on the basis of energy bands

Insulator

Insulators are very poor conductors of electricity with resitivity ranging 103 – 1017 Ωm. In this case Eg ≈ 6eV. For

E.g. carbon.

Semiconductor

A semiconductor material is one whose electrical properties lie between that of insulators and good conductors.

Their forbidden band is small and resistivity ranges between 10-4to 103Ωm. Ge and Si are examples with forbidden

energy gap 0.7eV and 1.1eV respectively. An appreciable number of electrons can be excited across the gap at room

temperature. By adding impurities or by thermal excitation, we can increase the electrical conductivity in

semiconductors

Conductor

Here valence band and conduction band overlap and there is no forbidden energy gap. Resistivity ranges between

10=9 to 10-4 Ωm. Here plenty of electrons are available for electrical conduction. The electrons from valence band

can freely enter the conduction band.

Intrinsic semiconductor

A pure semiconductor free from any impurity is called intrinsic semiconductor. Here charge carriers (electrons and

holes) are created by thermal excitation. Si and Ge are examples. Both Si and Ge are tetravalent. I.e. each has four

valence electrons in the outermost shell. Consider the case of Ge. It has a total of 32 electrons. Out of these 32

electrons, 28 are tightly bound to the nucleus, while the remaining 4 electrons (valence electrons) revolve in the

outermost orbit. In a solid, each atom shares its 4 valence electrons with its nearest neighbors to form covalent

bonds.The energy needed to liberate an electron from Ge atom is very small, of the order of 0.7 eV. Thus even at

room temperature, a few electrons can detach from its bonds by thermal excitation. When the electron escapes from

the covalent bond, an empty space or a hole is created. The number of free electrons is always equal to the number

of holes.

Extrinsic semiconductor

Extrinsic semiconductors are formed by adding suitable impurities to the intrinsic semiconductor. This process of

adding impurities is called doping. Doping increases the electrical conductivity in semiconductors. The added

impurity is very small, of the order of one atom per million atoms of the pure semiconductor. The added impurity

may be pentavalent or trivalent. Depending on the type of impurity added, the extrinsic semiconductors can be

divided into two classes: n-type and p-type.

n-type semiconductor

When pentavalent impurity is added to pure semiconductor, it results in n-type semiconducutor. Consider the case

when pentavalent Arsenic is added to pure Ge crystal. As shown in the figure, four electrons of Arsenic atom form

covalent bonds with the four valence electrons of neighbouring Ge atoms. The fifth electron of Arsenic atom is not

covalently bonded, but it is loosely bound to the Arsenic atom. Now by increasing the thermal energy or by

applying electric field, this electron can be easily excited from the valence band to the conduction band. Thus every

Arsenic atom contributes one conduction electron without creating a positive hole. Hence Arsenic is called donor

element since it donates free electrons. Since current carriers are negatively charged particles, this type of

semiconductor is called n-type semiconductor.

p-type semiconductor

When trivalent impurity is added to pure semiconductor, it results in p-type semiconducutor. Consider the case

when trivalent Boron is added to pure Ge crystal. As shown in the figure, three valence electrons of Boron atom

form covalent bonds with the three neighbouring Ge atoms. There is a deficiency of one electron (hole) in the

bonding with the fourth Ge atom. The Ge atom will steal an electron from the neighbouring Ge atom to form a

covalent bond. Due to this stealing action, a hole is created in the adjascent atom. This process continues. Impurity

atoms that contribute hole in this manner are called acceptors. Since current carriers are positively charged particles,

this type of semiconductor is called p-type semiconductor.

Section-B

Section-C

Properties of solids

SUPERCONDUCTIVITY The electrical resistivity of many metals and alloys drops suddenly

to zero when their specimens are cooled to a sufficiently low temperature, offer a temperature in the liquid

helium range. This phenomenon is called superconductivity. This was first observed by Kamerlingh Onnes in

1911 while measuring the resistivity of mercury at low temperatures.Transition temperature The transition temperature (Tc) is the critical temperature at which the

resistivity of the material suddenly changes to zero.

Properties of superconductors

Critical field (Hc) When the superconducting materials are subjected to a strong magnetic field, it will result in

the destruction of the superconducting property. i.e, they return to the normal state. The minimum field required to

destroy the superconducting property is called the critical field (Hc). The variation of Hc with temperature is as

shown.

Critical current density (Jc)

The application of a large value of electric current to a superconducting material destroys its superconducting

property. The current required for this is called critical current (Ic) and corresponding current density is called

critical current density (Jc).

Specific heat Specific heat of superconductors undergoes a discontinuous change below transition temperature.

Isotope effect It has been observed that the superconducting transition temperature for various isotopes of a

superconductor is different. This effect is known as isotope effect. This effect was discovered in isotopes of mercury

by Maxwell and Reynolds in the year 1950.

Meissner effect Meissner and Ochsenfeld in 1933 found that if a superconductor is cooled in a magnetic field

below the transition temperature, the magnetic flux lines are pushed out of the body of the superconductor as

shown:

This phenomenon is called Meissner effect which establishes that a

superconductor is a perfect diamagnet.Therefore inside the specimen B=0

But B=H+4PM

Therefore H+4PM=0

H=-4PM

Susceptibility=M/H=-1/4P

TypeI and typeII Superconductors In type I superconductors, magnetization curve is as shownThey are

completely diamagnetic or exhibits complete Meissener effect up to critical field Hc. They are also called soft

superconductors. Eg: Al, In, Sn, Pb etc.

In type II superconductors, magnetization is as shown

For applied fields below Hc1 the specimen is diamagnetic,

exhibiting complete Meissner effect. Hc1, the flux begins to penetrate the specimen and the penetration increases

until Hc2 is related. Here Meissner effect is incomplete and the specimen is said to be in a vertex (mixed) state. At

Hc2, the specimen becomes a normal conductor. Hc2 is called upper critical field. They are also called hard

superconductors. Eg: Nb3Sn, Nb3Ge, Nb3H etc.

BCS Theory of Superconductivity This theory was developed by Bardeen, Cooper and Schrieffer in 1957 based

on electron- lattice- electron interaction. According to this theory, an electron attracts lattice ions towards itself, so

that it is surrounded by a region of positive charges. Another gets attracted to this region at high positive ion

concentration. Thus an electron- lattice- electron interaction results in an electron pair formation. These pairs are

called cooper pairs. They can be scattered only if the energy involved is sufficient to break it up into two single

electrons.

Copper pair electrons are of opposite momenta and spin (K and –K¯). In addition, a cooper pair does not obey

Pauli’s exclusion principle and hence any number of cooper pairs can be accommodated into a single quantum state.

Since an electron pair has a lower energy than the two normal electrons, there is an energy gap between the paired

and the two single electrons.

As long as Cooper pair electrons remain in Cooper pair states, they do not suffer scattering and hence resistivity will

be zero. When the temperature is raised, to overcome the energy gap, Cooper pair electrons gets separated to normal

single electrons which may undergo scattering due to the presence of imperfections in the crystal or lattice

vibrations which leads to a finite resistivity.

Josephson effect

This effect was first predicted by Josephson in 1962. The experimental arrangement was a Josephson junction

which consists of a thin insulator sandwiched between two superconductors as shown:

If the insulator layer is very thin, of the order of 10-50 Å

in thickness, a tunneling phenomenon called Josephson tunneling (Josephson effect) takesplace through the

insulator. Thus the insulator turns into a superconductor.

I-V characteristics of a Josephson junction is as shown:

With no applied voltage, a dc current (ic) will flow across

the junction. This is called dc Josephson effect. When a small voltage is applied across the junction, current

oscillates with a frequency w = 4peV/h. This is called ac Josephson effect. If the applied voltage is increased

beyond the critical voltage (vc), the current attains an ohmic behaviour.

Applications of superconductors

For the production of high magnetic fields.

In high energy physics experiments.

In NMR imaging.

In magnetohydrodynamic power generation.

In magnetic separation for refining ores and chemicals.

As memory storage element in computers.

In superconducting generators and motors.

In superconducting fuses, switches and cables.

In Superconducting Quantum Interference Device (SQUID).

In levitating trains

Hall effect

Consider a rectangular slab that carries a current I

in the X-direction. A uniform magnetic field of flux density B is applied along the Z-direction. The current carriers

experience a force (Lorents force) in the downward direction. This leads to an accumulation of electrons in the

lower face of the slab. This makes the lower face negative. Similarly the deficiency of electrons make the upper face

positive. As a result, an electric field is developed along Y-axis. This effect is called Hall effect and the emf thus

developed is called Hall voltage VH. The electric field developed is called Hall field EH.

If ‘e’ is the charge of carrier and ‘v’ the velocity, then Lorentz force acting on it due to magnetic field is given by,

Fm = -e(vxB)

Since v and B are perpendicular to each other,

Fm = -evB ---------- (1)

The electric field exerts a force on charge carriers. It is given by,

Fe = EHe --------- (2)

At equilibrium,

Fm = Fe

Ie, -evB = EHe

Or EH = -Bv -------- (3)

But current density J=nev where n is the electron density.

Or v = J/ne

Substituting this in equation (3),

EH = -BJ/ne

Or EH/BJ = -1/ne = RH --------- (4)

RH is called Hall coefficient.

If the carriers are holes,

RH = 1/pe --------- (5)

Where p is the hole density.

If ‘d’, ‘w’ and ‘A’ represent the thickness, width and area of cross section of the slab respectively, then,

EH = VH/d --------- (6)

J = I/A = I/wd --------- (7)

Substituting (6) and (7) in equation (4),

RH = EH/BJ = VHw/BI

Or VH = BIRH/w = BI/wne ------------ (8) Measurement of Hall voltage and Hall coefficient

A rectangular slab of the given material having a thickness ‘d’ and width ‘w’ is placed between the pole pieces of an

electromagnet with magnetic flux density B coinciding z-axis. A current I that coincides with X-axis is allowed to

pass through the sample by connecting it to a battery.

The Hall voltage is measured by placing two probes at the centres of the bottom and top surfaces of the sample.

Hall coefficient RH = VHw/BI

Importance of Hall effect

Hall effect proved that band theory of solids is more accurate than free electron theory. Hall effect proved that

electrons are the majority carriers in all the metals and n-type semiconductors. In p-type semiconductors, holes are

the majority carriers.

Applications of Hall effect

To determine the type ( n-type or p-type) of semiconductors.

To determine the concentration of the carriers.

In nondestructive testing.

In Hall generators.

Section-D

FIBRE OPTICS Conventional methods of long distance communication use radio waves (~ 106 Hz) and

micro waves (~1010 Hz) as carrier waves. A light beam acting as carrier waves is capable of carrying far more

information since optical frequencies are extremely large (~1015 Hz).

Soon after the discovery of laser, some preliminary experiments in propogation of information carrying light waves

through the open atmosphere wave carried out, but it was realized that the unwanted elements such as rain, fog etc.

leads to adverse effects. Thus in order to have an efficient and dependable communication system one would require

a guiding medium in which the information carrying light waves could be transmitted. This resulted in the

development of optical fibre which is an efficient guiding medium for laser light.

Basic principle-total internal reflection The basic principle of optical fibre is multiple total internal

reflection. When a ray of light travel from denser to a rarer medium, at an angle of incidence greater than critical

angle qc, the ray is not reflected but it is reflected into the same denser medium. This property is called total internal

reflection. Light signals are transmitted through optic fibres by multiple total internal reflection.

Fibre construction and fibre

dimension

An optical fibre consists of a vary thin transparent cylindrical core having refractive index n1 surrounded by a

cylindrical shell called cladding of slightly lower refractive index n2. The core cladding system is surrounded by

plastic jackets.

Optical fibres are hair thin threads of glass or plastic. Plastic fibres have the advantage of more flexibility than glass

fibres but attenuation is greater in plastic fibres, comparing with glass fibres.

Core diameter ranges from 5 to 600µm. Cladding diameter ranges from 125 to 750µm. The thickness of the jacket is

around 100µm. totally optical fibre has an outer diameter ranging from 0.1mm to 1.5 mm.

Light propogation in optical fibres

Consider an optical fibre with n1 and n2 as the refractive indices of core and cladding material respectively.

Consider a ray of light entering through one end making an angle q with the axis.

By Snell’s law, n1=sini/ sinq

Or sinq=sini/ n1 ------(1)

At core –cladding interface, refractive index µ=n1/n2

Total internal reflection takes place at core-cladding interface if the angle of incidence at core-cladding interface is

equal or greater than critical angleÆc.

We have µ=n1/n2=1/sinÆc

sinÆc=n2/n1 -----(2)

For total internal reflection to take place,

Æ > Æc

or, sinÆ > sinÆc

i.e, sinÆ > n2/n2 (from equ(2))

But from figure, sinÆ =cosq

cosq > n2/n1

i.e, (1-sin2q)1/2 > n2/n1

1-sin2q > (n2/n1)2

sin2q < sinq =" sini/n1">Acceptance angle

The maximum angle of incidence at which light may enter the fibre in order to be propagated is, im= sin-1(n2-

n22)1/2

This angle is called acceptance angle for the fibre.

Numerical aperture

Numerical aperture of an optical fibre is a measure of its gathering capacity and it is denoted as the sine of

acceptance angle.

i.e, NA=sinim=(n2-n22)1/2

Step index and graded index fibre

If the optical fibre has a core of uniform constant refractive index n1 and a cladding of slightly lower refractive

index n2, it is called a step index fibre. The cross sectional refractive index profile is as shown:

If the core of optic fibre has a non-uniform refractive index that decreases gradually from the

centre towards the core-cladding boundary, it is called a graded index fibre. The cladding surrounding the core has a

uniform refractive index, slightly lower than the refractive index of the core. The cross sectional refractive index

profile is as shown:

Single mode and multimode fibre

In single mode fibres, only one mode of light rays are guided through the fibre. For this, a step index fibre with a

small core diameter is used.

In multimode fibres, a number of modes of light rays guided through the fibres. For this, a step index fibre with a

large core diameter or a graded index fibre is used.

Signal distortion and transmission losses

1) Attenuation

Absorption and scattering of light traveling through a fibre leads to decrease the strength of the signal which is

referred as attenuation of the signal.

2) Intermodal dispersion

The intermodal dispersion occurs in multimode fibres where rays associate with various modes travel different

distances through the fibre. As a result, the signal broadens and the output signal is no longer identical with the

input signal. Signal broadening is less in graded index and step index single mode fibres.

3) Material dispersion or chromatic dispersion

If we use white light, all the colours of the input radiation are not reacting the other end at the same time since they

travel with different velocities. This type of signal distortion is called material dispersion or chromatic dispersion.

4) Waveguide dispersion

In single mode fibres, 80% of signal travels through the core. The remaining 20% travels through the cladding,

move faster than the signal through the core. The signal distortion occurring due to this is called waveguide

dispersion. Waveguide dispersion is small in multimode fibres and can be ignored.

Light wave communication using optic fibre

A simple block diagram of fibre optic communication system is shown below:

Optical transmitter

A light emitting diode (LED) or a semiconductor laser can be used as optical source. Modulation modulates the

input signal and optical signal and then transmitted through optical fibre cables to the receiver.

Optical receiver

A photodiode can be used as optical detector. The detected wave is demodulated to extract the signal.

Advantage of fibre optic communication

1) Wide band width.

2) Low attenuation and other transmission losses.

3) Small size and weight.

4) Safe from electrical interference caused by lightning, electric motors, fluorescent tube and other electrical noise

sources.

5) Lack of cross talk between parallel fibres.

6) Easy installation and easy maintenance.

7) Flexible.

8) Temperature resistance.

9) Highly economical.

10) High degree of signal security.

11) Longer life span.

Fibre amplifier

Fibre loss in optical fibres is the main disadvantage of fibre optic communication system. To overcome this, optical

amplifiers are used. There are 4 types of optical amplifiers:

Erbium doped fibre amplifier (EDFA)

Fibre Raman amplifier (FRA)

Semiconductor laser amplifier (SLA) and

Integrated optical amplifier (IOA)

Erbium doped fibre amplifier (EDFA)

Here Erbium doped silica fibres are used. When incident photon energy is incident on a doped fibre, Erbium ions in

the medium are made to move to higher energy levels. The Erbium ions in the excited state return to the ground

state either spontaneously or by stimulation. Erbium doped fibres have long metastable states leading to coherent

amplification. A practical configuration of EDFA is as shown:

Advantages of EDFA

High gain

High output power

Low noise

Less gain variation

Wide bandwidth

Compatible to transmission fibre with minimum loss

Cross talk immunity and

Low power consumption.

LASER

LASER IS THE SHORT FORM OF LIGHT AMPLIFICATION BY STIMULATED EMISSION OF RADIATION.CHARACTERISTICS OF A LASER BEAM1.DIRECTIONALITY:LASER BEAM IS HIGHLY DIRECTIONAL.IT CAN BE FOCUSSED TO A FINE POINT.HENCE LASER IS USED FOR SURGICAL APPLICATIONS.2.MONOCHROMATICITY:LASER BEAM IS HIGHLY MONOCHROMATIC.LINE WIDTH IS NARROW COMPARING WITH CONVENTIONAL LIGHT SOURCES.FOR RUBY LASER LINE WIDTH = 5 ANGSTROM.3.COHERENCE:LASER BEAM IS HIGHLY COHERENT.TWO INDEPENDENT LASER SOURCES CAN PRODUCE INTERFERENCE EFFECTS.4.BRIGHTNESS:LASER IS HIGHLY INTENSE BEAM OF LIGHT.IT IS USED IN INDUSTRY FOR CUTTING,WELDING AND DRILLING OPERATIONS.Basic concepts of laser

Interaction of radiation with matter

Consider a system having two energy levels E1 and E2 with E2-E1=hÖ. When it is exposed to radiation having a

stream of photons, each with energy hÖ, three district processes can take place. They are 1) Absorption

2) Spontaneous emission and

3) Stimulated emission.Absorption

An atom in the ground state E1 can absorb a portion of energy hÖ and go to the higher energy state E2. This

process is known as absorption and is illustrated in figure.Rate of absorption R12 is proportional to population

(number of available atoms per unit volume) of the lower energy level N1 and u(Ö), the energy density of radiation

u(Ö).i.e, R12 a N1u(Ö)R12 = B12N1u(Ö) ------------(1)

Where B12 is called Einstein coefficient.

Spontaneous Emission In spontaneous emission, the atoms in the higher energy state E2 eventually

return to the ground state by emitting their excess energy spontaneously. This process is independent of the external

radiation. The rate of spontaneous emission R21 is directly proportional to the population of the energy level E2

(N2).i.e, R21 a N2R21 = A21N2 --------------(2)

Where A21 is called Einstein coefficient.

Stimulated emission In stimulated emission, a photon having energy hÖ(E2-E1) stimulate an atom in the

higher state E2 to make a transition to the lower state E1 with the creation of a second photon. The rate of

stimulated emission R121 is proportional to population at the energy level E2(N2) and energy density of radiation

u(Ö).i.e, R121 a N2u(Ö)

Semiconductor laser (Diode laser/GaAs laser)

GaAs is a direct bandgap semiconductor. Laser transition is possible only in direct bandgap semiconductors. Si and

Ge do not give laser transition since they are indirect bandgap semiconductors.

Fermilevel (EF) is the highest filled energy level at absolute zero. A semiconductor in which Fermi level lies the

conduction band (in n type) or valence band (in p type) is called a degenerate semiconductor. A p-n junction is used

for the fabrication of semiconductor laser. Both p and n regions are made degenerate by heavy doping.

With a forward bias, depletion region (active region) contains a high concentration of electrons in the conduction

band and holes in the valence band. Population inversion has occurred in the sense that more states are occupied in

the conduction band than in the valence band. At low bias currents, electron-hole recombination takes place

spontaneously resulting in a spontaneous emission of photons. This is the principle of a light emitting diode (LED).

As the diode current increases, a point is reached, where significant population inversion exist near the junction

resulting in a stimulated emission.

One pair of faces perpendicular to the

junction is polished so that they act as resonant cavity. The remaining faces are roughened to eliminate laser action

in those directions.

In a semiconductor laser, the transitions are

associated with the electron states in the conduction band and valence band. The upper and lower energy states are

continuous and hence the output is not sharp. Thus coherence and monochromaticity of a GaAs laser are poor. But

they have a few advantages. They are

*Portable since compact and small.

*High efficiency

*Highly economical

*Can produce both continuous wave and pulsed laser.

*Tuning of output is easily possible.

Applications of laser

1. Industrial application: Welding, drilling and cutting.

2. Medical applications: In dermatology, dentistry, ophthalmology, in surgery of tumours, kidney stone and for

cancer treatment.

3. For making sensors.

4. In holography.

5. In laser printers.

6. In research.

7. In microelectronics.

8. In accelerating certain chemical reactions.

9. In fibre optic communication.

10. In underwater communication.

11. In military applications.

Uses

Lasers range in size from microscopic diode lasers (top) with numerous applications, to football field sized neodymium glass lasers (bottom) used for inertial confinement fusion, nuclear weapons research and other high energy density physics experiments.

Laser applications

When lasers were invented in 1960, they were called "a solution looking for a problem".[23] Since then, they have become ubiquitous, finding utility in thousands of highly varied applications in every section of modern society, including consumer electronics, information technology, science, medicine, industry, law enforcement, entertainment, and the military.

The first use of lasers in the daily lives of the general population was the supermarket barcode scanner, introduced in 1974. The laserdisc player, introduced in 1978, was the first successful consumer product to include a laser but the compact disc player was the first laser-equipped device to become common, beginning in 1982 followed shortly by laser printers.

Some other uses are:

Medicine : Bloodless surgery, laser healing, surgical treatment, kidney stone treatment, eye treatment, dentistry

Industry : Cutting, welding, material heat treatment, marking parts, non-contact measurement of parts

Military : Marking targets, guiding munitions, missile defence, electro-optical countermeasures (EOCM), alternative to radar, blinding troops.

Law enforcement : used for latent fingerprint detection in the forensic identification field [24][25]

Research : Spectroscopy, laser ablation, laser annealing, laser scattering, laser interferometry, LIDAR, laser capture microdissection, fluorescence microscopy

Product development/commercial: laser printers, optical discs (e.g. CDs and the like), barcode scanners, thermometers, laser pointers, holograms, bubblegrams.

Laser lighting displays : Laser light shows Cosmetic skin treatments: acne treatment, cellulite and striae reduction, and hair removal.

Spontaneous Emission

• Excited atoms normally emit light spontaneously• Photons are uncorrelated and independent

• Incoherent light Spontaneous emission is the process by which a light source such as an atom, molecule, nanocrystal or nucleus in an excited state undergoes a transition to a state with a lower energy, e.g., the ground state and emits a photon. Spontaneous emission of light or luminescence is a fundamental process that plays an essential role in many phenomena in nature and forms the basis of many applications, such as fluorescent tubes, older television screens (cathode ray tubes), plasma display panels, lasers (for startup - normal continuous operation works by stimulated emission instead) and light emitting diodes.

If a light source ('the atom') is in the excited state with energy E2, it may spontaneously decay to a lower lying level (e.g., the ground state) with energy E1, releasing the difference in energy between the two states as a photon. The photon will have angular frequency ω and energy (= hν, where h is the Planck constant and ν is the frequency):

where is the reduced Planck constant. The phase of the photon in spontaneous emission is random as is the direction the photon propagates in. This is not true for stimulated emission. An energy level diagram illustrating the process of spontaneous emission is shown below:

If the number of light sources in the excited state is given by N, the rate at which N decays is:

where A21 is the rate of spontaneous emission. In the rate-equation A21 is a proportionality constant for this particular transition in this particular light source. The constant is referred to as the Einstein A coefficient, and has units s − 1[1]. The above equation can be solved to give:

where N(0) is the initial number of light sources in the excited state, t is the time and Γrad is the radiative decay rate of the transition. The number of excited states N thus decays exponentially with time, similar to radioactive decay. After one lifetime, the number of excited states decays to

36.8% of its original value ( -time). The radiative decay rate Γrad is inversely proportional to the lifetime τ12:

Stimulated Emission• Excited atomscan be stimulatedinto duplicating

passing light • Photons arecorrelated andidentical• Coherent light

quantum mechanical effects force electrons to take on discrete positions in orbitals. Thus, electrons are found in specific energy levels of an atom, two of which are shown below:

When an electron absorbs energy either from light (photons) or from heat (phonons), it will receive that incident quanta of energy. But transitions are only allowed in between discrete energy levels such as the two shown above. This leads to emission lines and absorption lines.

When an electron is excited from a lower to a higher energy level, it will not stay that way forever. An electron in an excited state may decay to a lower energy state which is not occupied, according to a particular time constant characterizing that transition. When such an electron decays without external influence, emitting a photon, that is called "spontaneous emission". The phase associated with the photon that is emitted is random. A material with many atoms in such an excited state may thus result in radiation which is very spectrally limited (centered around one wavelength of light), but the individual photons would have no common phase relationship and would emanate in random directions. This is the mechanism of fluorescence and thermal emission.

An external electromagnetic field at a frequency associated with a transition can affect the quantum mechanical state of the atom. The atom will act like a small electric dipole oscillating in response to the external field. A consequence of this oscillation is that the rate of transitions between two states is enhanced beyond that due to spontaneous emission. Such a transition to the higher state is called absorption, destroying an incident photon. A transition from the higher to a lower energy state, however, produces an additional photon; this is the process of stimulated emission.

Mathematical Model

Stimulated emission can be modelled mathematically by considering an atom that may be in one of two electronic energy states, a lower level state (possibly the ground state) (1) and an excited state (2), with energies E1 and E2 respectively.

If the atom is in the excited state, it may decay into the lower state by the process of spontaneous emission, releasing the difference in energies between the two states as a photon. The photon will have frequency ν and energy hν, given approximately by:

where h is Planck's constant.

Alternatively, if the excited-state atom is perturbed by an electric field of frequency ν0, it may emit an additional photon of the same frequency and in phase, thus augmenting the external field, leaving the atom in the lower energy state. This process is known as stimulated emission.

In a group of such atoms, if the number of atoms in the excited state is given by N2, the rate at which stimulated emission occurs is given by:

where the proportionality constant B21 is known as the Einstein B coefficient for that particular transition, and ρ(ν) is the radiation density of the incident field at frequency ν. The rate of emission is thus proportional to the number of atoms in the excited state N2, and to the density of incident photons.

At the same time, there will be a process of atomic absorption which removes energy from the field while raising electrons from the lower state to the upper state. Its rate is given by an essentially identical equation:

.

The rate of absorption is thus proportional to the number of atoms in the lower state, N1. Einstein showed that the coefficient for this transition must be identical to that for stimulated emission:

B12 = B21 .

Thus absorption and stimulated emission are reverse processes proceeding at somewhat different rates. Another way of viewing this is to look at the net stimulated emission or absorption viewing it as a single process. The net rate of transitions from E2 to E1 due to this combined process can be found by adding their respective rates, given above:

.

Thus a net power is released into the electric field equal to the photon energy hν times this net transition rate. In order for this to be a positive number, indicating net stimulated emission, there must be more atoms in the excited state than in the lower level: ΔN > 0. Otherwise there is net absorption and the power of the wave is reduced during passage through the medium. The special condition N2 > N1 is known as a population inversion, a rather unusual condition that must be effected in the gain medium of a laser.

The notable characteristic of stimulated emission compared to everyday light sources (which depend on spontaneous emission) is that the emitted photons have the same frequency, phase, polarization, and direction of propagation as the incident photons. The photons involved are thus mutually coherent. When a population inversion (ΔN > 0) is present, therefore, optical amplification of incident radiation will take place.

Although energy generated by stimulated emission is always at the exact frequency of the field which has stimulated it, the above rate equation refers only to excitation at the particular optical frequency ν0 corresponding to the energy of the transition. At frequencies offset from ν0 the strength of stimulated (or spontaneous) emission will be decreased according to the so-called "line shape". Considering only homogeneous broadening affecting an atomic or molecular resonance, the spectral line shape function is described as a Lorentzian distribution:

where is the full width at half maximum or FWHM bandwidth.

The peak value of the Lorentzian line shape occurs at the line center, ν = ν0. A line shape function can be normalized so that its value at ν0 is unity; in the case of a Lorentzian we obtain:

.

Thus stimulated emission at frequencies away from ν0 is reduced by this factor. In practice there may also be broadening of the line shape due to inhomogeneous broadening, most notably due to the Doppler effect resulting from the distribution of velocities in a gas at a certain temperature. This has a Gaussian shape and reduces the peak strength of the line shape function. In a practical problem the full line shape function can be computed through a convolution of the individual line shape functions involved. Therefore optical amplification will add power to an incident optical field at frequency ν at a rate given by:

LASER ACTION

Lasers are possible because of the way light interacts with electrons. Electrons exist at

specific energy levels or states characteristic of that particular atom or molecule. The

energy levels can be imagined as rings or orbits around a nucleus. Electrons in outer rings

are at higher energy levels than those in inner rings. Electrons can be bumped up to

higher energy levels by the injection of energy-for example, by a flash of light. When an

electron drops from an outer to an inner level, "excess" energy is given off as light. The

wavelength or color of the emitted light is precisely related to the amount of energy

released. Depending on the particular lasing material being used, specific wavelengths of

light are absorbed (to energize or excite the electrons) and specific wavelengths are

emitted (when the electrons fall back to their initial level).

In a cylinder a fully reflecting mirror is placed on one end and a partially reflecting

mirror on the other. A high-intensity lamp is spiraled around the ruby cylinder to provide

a flash of white light that triggers the laser action. The green and blue wavelengths in the

flash excite electrons in the atoms to a higher energy level. Upon returning to their

normal state, the electrons emit their characteristic ruby-red light. The mirrors reflect

some of this light back and forth inside the ruby crystal, stimulating other excited

chromium atoms to produce more red light, until the light pulse builds up to high power

and drains the energy stored in the crystal. High-voltage electricity causes the quartz flash

tube to emit an intense burst of light, exciting some of the atoms in the ruby crystal to

higher energy levels. At a specific energy level, some atoms emit particles of light called

photons. At first the photons are emitted in all directions. Photons from one atom

stimulate emission of photons from other atoms and the light intensity is rapidly

amplified. Mirrors at each end reflect the photons back and forth, continuing this process

of stimulated emission and amplification. The photons leave through the partially silvered

mirror at one end. This is laser light.