SOLID STATE By: Dr.DEPINDER KAUR

SPACE LATTICE The regular three-dimensional arrangement of atoms or ions in a crystal is usually described in terms of a space lattice and a unit cell. Each of these three structures as a large number of repetitions in two directions of the parallel-sided figure shown immediately below each pattern.

This parallel-sided figure is the unit cell This parallel-sided figure is the unit cell. It represents the simplest, smallest shape from which the overall structure can be constructed. The pattern of points made by the comers of the unit cells when they are packed together is called the space lattice. The lines joining the points of the space lattice are shown in color. Without some experience, it is quite easy to pick the wrong unit cell for a given structure. Some incorrect choices are shown immediately below the correct choice in the figure. Note in particular that the unit cell for structure b, in which each circle is surrounded by six others at the comers of a hexagon, is not a hexagon, but a parallelogram of equal sides (a rhombus) with angles of 60 and 120°.

LAW OF CRYSTALLOGRAPHY

Through studies of external forms and angular relationships between the crystal faces, some fundamental laws have been estab­lished, which govern the whole crystallography. They are as follows:   Law of constancy of interfacial angle. Law of rational indices. Law of constancy of symmetry.

Law of constancy of 'interfacial-angle': Interfacial angle may more generally be defined as the angle between any two adjacent faces of a crystal. In crystallography, however, the inter­facial angle to a crystal is the angle subtended between the normal drawn on the two faces concerned.

Law of constancy of interfacial-angle states that 'measured at the same temperature, similar angles on crystals of the same substance remain constant, regardless the size and the shape of the crystals.

 This law states that angle between the adjacent corresponding faces is the inter facial angles of the crystal of the particular substance is always constant inspite of having different shapes and sizes and mode of growth of crystal. The size and shape of the crystal depend upon the conditions of the crystallisation. This law is also called as Steno's Law.  

2. Law of rational-indices: Two crystals of the same substance may differ considerably in appearance that in number, size and shape of the individual faces. In order to describe the external form of crystals, a mathematical method of relating plane; to certain imagi­nary lines in space is used.

The Law of constancy of symmetry :  In accordance to this law, all the crystals of a substance have the same elements of the symmetry is the plane of symmetry, the axis of symmetry and the centre of symmetry.

Symmetry Crystals, and therefore minerals, have an ordered internal arrangement of atoms.  This ordered arrangement shows symmetry, i.e. the atoms are arranged in a symmetrical fashion on a three dimensional network referred to as a lattice.  When a crystal forms in an environment where there are no impediments to its growth, crystal faces form as smooth planar boundaries that make up the surface of the crystal.  These crystal faces reflect the ordered internal arrangement of atoms and thus reflect the symmetry of the crystal lattice.  To see this, let's first imagine a small 2 dimensional crystal composed of atoms in an ordered internal arrangement as shown below.  Although all of the atoms in this lattice are the same, I have colored one of them gray so we can keep track of its position.

If we rotate the simple crystals by 90o notice that the lattice and crystal look exactly the same as what we started with.  Rotate it another 90o and again its the same.  Another 90o rotation again results in an identical crystal, and another 90o rotation returns the crystal to its original orientation.  Thus, in 1 360orotation, the crystal has repeated itself, or looks identical 4 times.  We thus say that this object has 4-fold rotational symmetry.

Symmetry Operations and Elements A Symmetry operation is an operation that can be performed either physically or imaginatively that results in no change in the appearance of an object.  Again it is emphasized that in crystals, the symmetry is internal, that is it is an ordered geometrical arrangement of atoms and molecules on the crystal lattice.  But, since the internal symmetry is reflected in the external form of perfect crystals, we are going to concentrate on external symmetry, because this is what we can observe. 

There are 3 types of symmetry operations: rotation, reflection, inversion.

Bragg's Law Deriving Bragg's Law using the reflection geometry and applying trigonometry. The lower beam must travel the extra distance (AB + BC) to continue traveling parallel and adjacent to the top beam.

Bragg's Law can easily be derived by considering the conditions necessary to make the phases of the beams coincide when the incident angle equals and reflecting angle. The rays of the incident beam are always in phase and parallel up to the point at which the top beam strikes the top layer at atom z (Fig. 1). The second beam continues to the next layer where it is scattered by atom B. The second beam must travel the extra distance AB + BC if the two beams are to continue traveling adjacent and parallel. This extra distance must be an integral (n) multiple of the wavelength (λ) for the phases of the two beams to be the same: nλ = AB +BC  

Recognizing d as the hypotenuse of the right triangle Abz, we can use trigonometry to relate d and q to the distance (AB + BC). The distance AB is opposite Θ so, AB = d sinΘ. Because AB = BC, therefore, nλ = 2AB Further nλ = 2 d sinΘ, and Bragg's Law has been derived. The location of the surface does not change the derivation of Bragg's Law.

Crystal structure of NaCl

Crystal structure of KCl

Crystal structure of CsCl