Introduction of crystal structure and space lattice

In this particular article Introduction of crystal structure and space lattice, we are going to discuss about introduction of crystal structure and space lattice. We will also discuss about primitive cell.

Crystal structure

The identification of solid state material are hardness, rigidity and in compressibility etc. Hence it clear that in solids elementary particles are closest to the position of distance and formed condensed structure and strong bond is obtained. Hence they are divided broadly in two classes namely amorphous solid and crystalline solids.

Amorphous Solids

Some solids like glass, plastic, rubber, resein etc in which do not have any definite regular geometry. There solids are called amorphous solid. And in these solid, which do not have any regular and periodic arrangement of structure as shown in fig.

Introduction of crystal structure and space lattice

Crystalline solids

Some solids like NaCl, KCl, diamond , sulphur, sugar etc in which have a definite regular geometry. And these solids are called crystalline solids. Consequently the solids which have regular and periodic arrangement of elementary particles in a definite and long range order as shown in fig.

A crystal is made up of a three dimensional array of points such that each of surrounded by the neighbouring points in an identical way. And such an array of points is known as single periodic lattice. Thus lattice is a regular periodic arrangement of points extended repeatedly in space. Now the crystal structure is formed by associating with every lattice point. Hence this group is called the basis. Each basis is identical in composition, arrangement and orientation with any other basis. The actual crystal structure is formed when a basis of atoms or ions or molecule or radical is attached identically to each lattice point. Thus, defination of notation.
Lattice + Basis = Crystal

Space Lattice

We need a co-ordinate system to express lattice point in the space, so that we can express all lattice point after translation of these non- coplanor unit ventors. These unit vectors are called basis vector or fundamental vectors.

Introduction of crystal structure and space lattice
(a) linear lattice(b) two dimensional lattice(c) space lattice

A Crystal has definite and regular shape, so is made up of regular and periodic arrangements of atoms or molecules or ions. The crystal structure may be describe in terms of an idealized geometrical concept of group is called lattice. The lattice is defined as an array of points in space. Such that the environment about each point is same with the environment about any other point. Such an array of infinite points in one dimension shown in fig is called linear lattice. If these points are repeated in another direction by a regular interval of distance b as shown in fig. Then they form two dimensional lattice called plane lattice.

Theory

Now if the points of two dimensional lattice are repeated in another non-coplaner direction by a regular interval of distance C as show in fig, then they form three dimensional lattice called space lattice. If a , b and c are vector distances between the point along crystallographic axis x, y and z respectively, then the position of any lattice point can be expressed as
T = n₁ a + n₂ b + n₃ c ………(1)

Where the vectors a , b , c are called fundamental translation vectors and n₁, n₂ , n₃ arbitrary integers.
The translation operation means that when the operation T is applied to any point r in the material. The resulting point r’ is given by

r’ = r + T = r + n₁ a + n₂ b + n₃ c ………(2)
If r’ is not identical to r for any arbitrary choice of n₁ , n₂ , n₃ , the vectors a , b , c are not translation vectors. In order for an assembly of atom to be classified as a crystal structure, it must be possible to find three translation vector which satisfy that r’ is in distinguishable from r.

 Primitive Cell

If in lattice, with respect to position of points r and r’ the atoms configuration in crystal look like same and n₁ , n₂ , n₃ are the integers of any set which satisfy the equation then the translation a , b , c is called primitive vectors.
r’ = r + n₁ a + n₂ b + n₃ c
A primitive cell is the parallelopiped formed by the primitive axes a , b , c in the crystal lattice. Volume of primitive is equal to the vectors triple product a .( b x c). It is a type of unit cell which will fit all space on in definite repetitions in all the three direction.

In fact a primitive cell is a minimum – volume unit cell containing one lattice point each at the corners only. Thus, there is a density of one lattice point per primitive cell. There are lattice points at the eight corners of the primitive cell but each corner point is shared among the eight cells which touch there as shown in fig.

Introduction of crystal structure and space lattice
Two dimensional primitive cell

A primitive cell choosen by a particular procedure
1. Firstly lines are drawn to connect a given lattice point to all near by lattice point.
2. Normal are drawn at the midpoints of the lines drawn above. Mark the smallest volume enclosed in this way. This is the wigner-seitz primitive cell. And only one lattice point is exist in this cell.

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