In this article we will discuss about:- 1. Concept of Atom 2. Radiations Emitted from the Spectrum of Hydrogen Atom 3. Maximum Number of Electrons in Sub-Shell 4. Crystal Structure 5. Lattice Parameters of a Unit Cell 6. Crystal Systems of Atoms 7. Bravais Lattices 8. Crystal Structures for Metallic Elements 9. Crystal Symmetry 10. Co-Ordination Number 11. Number of Atoms Per Unit Cell 12. Atomic Packing Factor (A.P.F.).
Concept of Atom:
All substances are made up of atoms. Each atom consists of the following:
1. Nucleus
2. Electrons.
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Central nucleus is surrounded by orbital electrons which move in concentric spherical shells.
1. Nucleus:
The nucleus is at the centre of the atom and comprises protons and neutrons. Its diameter is 1/10000th of the atom as a whole. Almost the entire mass of a given atom is concentrated in its nucleus. Due to the presence of protons nucleus is positively charged. The number of protons in the nucleus is equal (numerically) to the atomic number of the element. A neutron is an uncharged particle and has same mass as the hydrogen nucleus.
The weight of the atom and its radioactive properties are associated with the nucleus. The chemical properties and spectrum on the other hand depend on the planetary electrons.
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2. Electrons:
An electron is a negatively charged particle present in an atom. The number of electrons which surround the nucleus of a neutral atom is equal to the number of protons within the nucleus, i.e., the atomic number. The electrons move about the nucleus.
The orbit or shell nearest to the nucleus known as the K-orbit, contains a maximum of two, the next (L- orbit) eight and next (M-orbit) eighteen electrons and so on. In fact, the number of electrons in any orbit is equal to 2n2 where n is the serial number of the orbit taking first orbit nearest to the nucleus, with the exception that the outermost orbit cannot have more than eight. In a given atom all orbits may not be complete.
Fig. 6 (a, b, c) illustrates atomic structures of some elements viz. lithium, sodium and argon.
The main electron shells are designated by the letters K, L, M, N, O, P and Q and by the corresponding principal quantum numbers, n = 1, 2 3, 4, 5, 6 and 7. The K shell, for example- has the principal quantum number n = 1.
The sub-shells in each main shell are designated by small letters, s, p, d and with f with quantum numbers l = 0, 1, 2 and 3 respectively. The number of sub-shells in a given main shell is the same as its principal quantum number, n (although in none of the known elements do any of the last three shells O, P, Q, contain more than three sub-shells).
Thus the number of electrons that can occupy the subshell, s = 2; p = 6; d = 10; f = 14. The capacities thus obtained, apply only to inner, completed main shells. The outermost main shell never has more than 8 electrons.
As soon as this number is reached the shell becomes stable even though the sum of its sub-shell capacities may be more than 8. The next electrons to be added start to fill the next main shell, taking position in its ’s’ sub-shell. Only after both spaces in this’s’ sub- shell have been filled does the inner complete main shell go on filing up.
Radiations Emitted from the Spectrum of Hydrogen Atom:
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As and when hydrogen atom is excited (the atom becomes excited when the electron absorbs energy in some way or the other), it comes back to its unexcited (or ground) state by emitting the energy (absorbed earlier). The raised electron gives out this energy in the form of radiations of different wavelengths as it jumps from a higher to a lower orbit.
These different wavelengths constitute spectral series which are characteristics of the atom emitting them. Through a spectroscope these relations are imaged as sharp and straight vertical lines of a single colour.
These frequency of the emitted radiations can be found from the following relations:
In the hydrogen spectrum following series are observed:
1. Lyman series
2. Balmer series
3. Paschen series
4. Brackett series
5. Pfund series.
1. Lyman Series:
Lyman series consists of all those wavelengths which are emitted when electron jumps from different higher orbits to the final orbit with n = 1 (or all those electronic jumps which end at K-orbit give rise to Lyman series).
2. Balmer Series:
Balmer series consists of all those wavelength which are emitted when different electrons jumps end at orbit or level with n=2 (or all those electronic jumps which end at L-orbit give rise to Balmer series).
3. Paschen Series:
Note:
To find the value of λ, put n2 = 4 for first member, n2 = 5 for second member and so on.
4. Brackett Series:
5. Pfund Series:
Paschen, Brackett, Pfund series lie in the Infrared region of the spectrum.
The combined formula from which all series can be found is as under:
Maximum Number of Electrons in Sub-Shell:
In a sub-shell, the maximum number of electrons is = 2(2l + 1)
Where, l = Number of subshell.
Thus for:
First subshell – l = 0:
Maximum number of electrons 2 (2l + 1) = 2 (2 x 0 + 1) = 2
This subshell is known as s-sub-shell. The symbols 2s, 3s, 4s, … denote the s-sub-shell of the first, second, third and fourth energy levels respectively.
Second sub-shell – I = 1:
Maximum number of electrons = 2 (2×1 + 1) = 6
This sub-shell is known as p-sub-shell. The symbols 2p, 3p, 4p,… denote the p-sub-shell of the second, third and fourth energy levels respectively.
Third sub-shell – I = 2:
Maximum number of electrons = 2 (2 x 2 + 1) = 10
This sub-shell is known as d-sub-shell. The symbols 2d, 3d, 4d,… denote the d-sub-shell of the third, fourth and fifth energy levels respectively.
Fourth sub-shell – I = 3:
Maximum number of electrons = 2 (2 x 3 + 1) = 14
This sub-shell is known as f-sub-shell. The symbols 4f, 5f, …denote the f-sub-shell of the fourth and fifth energy levels respectively.
The following points are worth noting:
1. The maximum number of electrons in the first principal energy level is 2. Therefore it will have only one sub-shell (1s) containing two electrons. The first energy level cannot have any other sub-shell.
2. The maximum number of electrons in second principal energy level are 8. Therefore it will have two sub-shells namely 2s (containing two electrons), and 2p (containing 6 electrons).
3. The maximum number of electrons in the third principal energy level are 18. Therefore, it will have three sub-shells namely 3s (containing two electrons), 3p (containing 6 electrons) and 4d (containing 10 electrons).
4. The maximum number of electrons in the fourth principal energy level are 32. Therefore, it will have four sub-shells namely 4s (containing 2 electrons), 4p (containing 6 electrons), 4d (containing 10 electrons) and 4f (containing 14 electrons).
Crystal Structure:
Materials can be broadly classified as crystalline and non-crystalline solids. In a crystal, the arrangement of atoms is in a periodically repeating pattern whereas no such regularity of arrangement is found in a non-crystalline material. A crystalline solid can either be a single crystal, where the entire solid consists of only one crystal, or an aggregate of many crystals separated by well-defined boundaries. In the latter form, the solid is said to be poly- crystalline.
Unit Cell and Space Lattice:
1. The metallic crystals can be considered as consisting of tiny blocks which are repeated in three dimensional pattern. The tiny block formed by the arrangement of a small group of atoms is called the unit cell. It is that volume of solid from which the entire crystal can be constructed by translational repetition in three dimensions.
2. If each atom in a lattice is replaced by a point; then each point is called a lattice point and the arrangement of the points is referred to as the (three dimensional) lattice array. Thus a space lattice, is defined as an array of points in three dimensions in which every point has surroundings identical to that every other point in the array.
The distance between the atoms-points is called inter-atomic or lattice spacing. A space lattice is a conventional geometrical basis by which crystal structures can be described. Every point of space lattice has identical surroundings.
3. The length of the side of a unit cell is the distance between the atoms of the same kind. In the case of pure metals whose crystals have simple cubic structure [Fig. 9(a)], It is equal to the basis distance ‘a’ only. However, in case of crystals of chemical compound like the cubic crystals of sodium chloride (NaCl), it is twice the distance ‘d’ between two adjacent atoms as shown in Fig. 9(b).
Lattice Parameter and Crystallographic Planes:
1. Lattice parameter means the dimensions of the unit cell in any of the crystallographic arrangements. In case of a cubic symmetry, the size of the lattice is fixed by the length of the cubic unit. In other case (with symmetry other than cubic) more than one lattice parameter has to be specified.
For a metal, the lattice parameter can be measured by X-ray diffraction studies and results are reported in ‘angstrom’ (Å). The metallic structure is built up of series of unit cells, each face of the cell being shared by the adjacent units. A crystal, thus consists of a regular assembly of atoms which may or may not be of the same species.
2. Each atom has a site defined by its geometrical centre of the relationship with the crystal lattice and this is the mean position of the atom. The position of the atom is less well defined, because it undergoes thermal vibrations within the space available to it between its neighbours.
3. The crystal structure can be regarded as an array of atoms built of layers one over the other, each layer consisting of simple arrangement of uniform rows of atoms, which mate or key with the atoms in adjacent layers. Some of layers, depending on the lattice arrangement and angle of view, shall have high-density of atoms per unit area and others may not be so dense. Such layers of atoms are called crystallographic planes and the planes and the interaction forces in a solid vary greatly with the geometry of these planes.
Lattice Parameters of a Unit Cell:
Fig. 10 shows a parallelopiped having the unit cell of a three dimensions space lattice formed by means of primitives, a, b and c as its adjacent sides (A primitive cell is one which has got points of atoms only at the corners of the unit cell). The angles between the three axes X, Y and Z are called interfacial angles. The primitives and interfacial angles constitute the lattice parameters of the unit cell (also called geometrical constants).
By knowing the lattice parameters of the linear dimensions a, b and c as well as interfacial angles α, β, γ between them, the form and actual size of the unit cell can be known. But if we do not know actual values of primitives but only their ratio and the values of interfacial angles, then we can only determine the form of the unit cell and not its actual size.
Crystal Systems:
The space lattice of a crystal is described by means of a 3-dimensional coordinate system in which the coordinate axes coincide with any three edges of the crystal intersecting at one point. To describe basic crystal structure seven different coordinate systems of reference axes are required.
Bravais Lattices:
The idea of space lattice was introduced by Bravais in 1880. According to him there are 14 possible types of space lattices in the seven basic crystal system- one triclinic, two monoclinic, four orthorhombic, two tetragonal, one hexagonal, one rhombohedral and three cubic.
Crystal Structures for Metallic Elements:
The crystals of most metals have highly symmetrical structure with closed packed atoms.
The most common types of space lattice (or unit cells) with which metallic elements crystallise are given below:
1. Body centered cubic structure (B.C.C.).
2. Face centered cubic structure (F.C.C.).
3. Hexagonal close-packed structure (H.C.P.).
1. Body Centered Cubic Structure (B.C.C.):
Here the atoms are located at the corners of the cube and one atom at its centre. This type of unit cell is found in metals like lithium, sodium, potassium, barium, vanadium, molybdenum etc.
2. Face Centered Cubic Structure (F.C.C.):
Here the atoms are located at the corners of the cube and one atom at the centre of each face. This type of unit cell is found in metals like copper, silver, gold, calcium, aluminium, lead etc.
3. Hexagon Close Packed Structure (H.C.P.):
Here the unit cell has an atom at each of the twelve corners of the hexagonal prism, one atom at the centre of each of the two hexagonal faces and three atoms in the body of the cell. This type of unit cell is found in metals like zinc, lithium, magnesium, beryllium etc.
Crystal Symmetry:
The symmetry of crystal is that characteristic of a crystal which indicates that if the parts of an ideal crystal are interchanged the result produced is just like the original crystal. The different ways in which the symmetry of a crystal can be specified and combined together is known as “symmetry class”.
Symmetry results from the ordered arrangement of atoms, ions or molecules in the internal structure of a crystal. Symmetry of a crystal form is determined by the position of similar faces, edges etc.
This is discussed below:
Simple Form:
A crystal is said to have a simple form if it’s all faces are similar.
Example-
Cube, octahedron. The faces of a cube, are identical square while the faces of an octahedron are identical triangles.
Combination Form:
A crystal is said to have a combination form when it has sets of faces corresponding to two or more simple forms.
The fundamental symmetry elements are as follows:
(i) Plane of Symmetry:
Plane of symmetry is plane drawn in a crystal, which divides a crystal into two similar and similarly-faced halves where one-half of the crystal is the reflection of the other half. The plane of symmetry of a crystal also contains its centre. There are nine planes of symmetry in cube.
(ii) Axis of Symmetry:
It is an imaginary line through the centre of the crystal about which the crystal may be rotated so that it presents exactly the same appearance more than once in the course of a complete revolution. In general, if a rotation of 360°/n around an axis brings the figure into coincidence, then the axis is an n-fold axis of symmetry.
Thus if a crystal is brought into coincidence with its initial position on rotation through 180°, 120°, 90°, 60° about an axis, the axis of rotation or symmetry are 2, 3, 4 and 6 fold axis. It is found that crystalline solids show only 1, 2, 3, 4 and 6 fold symmetry. They do not show 5 fold or any symmetry axis higher than 6.
(iii) Centre of Symmetry:
The centre of symmetry is a point such that any line drawn through it will intersect at the surface of the crystal at equal distances on either side. A crystal may have one or more planes, and one or more axes of symmetry but never more than one centre of symmetry. In fact, many crystals are not centre-symmetrical since they develop differently at opposite ends.
Co-Ordination Number:
It is defined as the number of nearest atoms which are directly surrounding a given atom.
Let us consider the following cases:
1. Simple Cubic (S.C.) Structure:
Here there are atoms one each at the corners of a unit cell. Any corner atom has four nearest neighbours in the same plane plus two nearest neighbours are exactly above and other exactly below that of corner atom, giving a total of six nearest neighbouring atoms. Thus the co-ordination number of S.C. structure is six.
2. Body Centered Cubic (B.C.C.) Structure:
Here there are 8 atoms, one each at the corners of a unit cell, one atom is at the centre of the cube. For any corner atom of the unit cell, the nearest atoms are the body centered atoms and the corresponding to corner atom of the unit cell, there are 8 unit cells in neighbour, which are having 8-body centered atoms. Thus the co-ordination number of B.C.C. structure is eight.
3. Face Centered Cubic (F.C.C.) Structure:
The co-ordination number of such structure is twelve. Hence the nearest neighbours of any corner atom are the face-centered atoms of surrounding unit cells. Any corner atom has four such atoms in its own plane, four in a plane above it and four in a plane below it, giving a total of twelve neighbouring atoms.
Atomic Radius:
Atomic radius is defined as half the distance between nearest neighbours in a crystal of a pure element. It is possible to calculate the atomic radius by assuming that atoms are spheres in contact in a crystal if the structure and the lattice parameters are known.
(i) Simle Cubic (S.C.) Structure:
Refer to Fig. 16 (a, b). In this structure atoms touch each other along the lattice.
Therefore, a = 2r
Or, r = a/2
(ii) Body Centre Cubic (B.C.C.) Structure:
Refer to Fig. 17 (a, b). In this structure the atoms touch each other along the diagonal of the cube. Therefore, diagonal of the cube (AC) in this case is 4r.
(iii) Face Centered Cubic (F.C.C.) Structure:
Refer to Fig. 18 (a, b). In this case atoms touch each other along the diagonal of any face of the cube. Obviously the length of diagonal of the face = 4r.
Number of Atoms Per Unit Cell:
The number of atoms possessed can be calculated if the arrangement of atoms inside the unit cell is known.
For three types of cube crystal, the number of atoms are calculated as follows:
1. Simple Cube:
Refer to Fig. 19. In this case there are 8 atoms one at each corner of the cube and all of them are shared by adjoining or surrounding cube. Hence the share of cube = 1/8th of each corner atom.
2. Body Centered Cube:
Refer to Fig. 20. In this case there are 8 atoms one at each corner and are shared by 8 surrounding cubes, plus one centre atom at the centre of the cube.
3. Face Centered Cube:
Refer Fig. 21. In this case there are 8 atoms one at each corner of the cube plus 6 face centered atoms at 6 planes of the cube. Each corner atom is shared by 8 surroundings cubes, and each face centered atom is shared by 2 surroundings cubes.
Atomic Packing Factor (A.P.F.):
Atomic packing factor (also known as density of facing) is defined as the ratio of the volume of atoms per unit cell to the total volume occupied by the unit cell.
1. Simple Cube:
Atoms per unit cell = 1
2. Body Centered Cube:
Atomic per unit cell = 2
3. Face Centered Cube:
Atoms per unit cell = 4
4. Hexagonal Closed Packed (H.C.P.) Structure:
Fig. 22 shows a specific hexagonal structure of magnesium.
The unit cell contains:
A. One atom at each corner,
B. One atom each at the centre of the hexagonal faces, and
C. Three more atoms within the body of the cell.
Each atom touches three atoms in the layer below its plane, six atoms in-its own plane, and three atoms in the layer above. Hence coordination number of this structure is 12.
Further, the atoms touch each other along the edge of the hexagon. Thus, a = 2r.
The top layer contains seven atoms. Each corner atom is shared by six surrounding hexagon cells and the centre atom is shared by two surrounding cells. The three within the body of the cell are fully contributing to the cell.
Magnesium, zinc and cadmium crystallize in this structure.
The packing factor of F.C.C. and H.C.P. is 74% and hence they are known as “closest packing structures”.
5. Diamond Cubic (D.C.) Structure:
In the case of diamond each atom forms four bonds due to hybridization. It is an F.C.C. crystal with four more carbon atoms symmetrically existing inside the unit cell. The four bonds produce a three dimensional network of primary bonds. The bond angle is about 109°. The atoms being tetravalent, all the eight valence electrons are to be shared for two carbon atoms. Silicon, germanium have D.C. structure.
To saturate the bond take one corner atom at three face atoms. This will give that coordination number is four and consequently the Atomic Packing Factor (A.P.F.) is very low (34%).
No. of atoms per unit cell for D.C. type will be 8 (4 for F.C.C. and 4 inside the unit cell).
Relation between the density of crystal material and lattice constant in a cubic lattice:
