Steps to Draw a Gigen Crystalographic Plane

In this article we will discuss about:- 1. Concept of Miller Indices 2. Important Features of Miller Indices three. Spacing of Planes 4. Relation between Interplanar Spacing 'd' and Cube Border 'a'.

Concept of Miller Indices :

Miller indices is a system of notation of planes within a crystal of space lattice. They are based on the intercepts of plane with the three crystal axes, i.e., edges of the unit cell. The intercepts are measured in terms of the edge lengths or dimensions of the unit cell which are unit of measurement distances from the origin along three axes.

Procedure for finding miller indices:

The Miller indices of a crystal plane are determined every bit follows: (Refer to Fig. 25)

Step i:

Find the intercepts of the plane along the axes 10, y, z (The intercepts are measured every bit multiples of the fundamental vector). …4, ii, 3.

Step 2:

Take reciprocals of the intercepts. 1/iv, i/ii, 1/3

Step three:

Convert into smallest integers in the same ration. …three 6 four

Stride four:

Enclose in parentheses. … (3 six 4)

The factor that results in converting the reciprocals of integers may be indicated exterior the brackets, only information technology is usually omitted.

Important Note:

The directions in infinite are represented by square brackets [ ]. The commas inside the square brackets are used separately and not combined. Thus [ane 1 0] is read as "One-one-zero" and not "one hundred x". Negative indices are represented by putting a bar over digit, e.g., [1 ane 0].

The general way of representing the indices of a direction of a line is [10 y z]. The indices of a plane are represented by a small bracket, (h, g I). Sometimes the notations < > and ( ) or { } are besides used for representing planes and directions _x respectively.

The following process is adopted for sketching any direction:

i. First of all sketch the plane with the given Miller indices.

ii. At present through the origin, draw a line normal to the sketched aeroplane, which will give the required management.

Important Features of Miller Indices :

Some of the of import features of Miller indices (peculiarly for the cubic system) are detailed below:

1. A aeroplane which is parallel to any ane of the co-ordinate axes has an intercept of infinity (∞) and therefore, the Miller index for that axis is zero.

ii. All equally spaced parallel planes with a detail orientation have aforementioned index number (h yard I).

iii. Miller indices do not only ascertain item plane merely a ready of parallel planes.

iv. It is the ratio of indices which is only of importance. The planes (211) and (422) are the same.

5. A plane passing through the origin is divers in terms of a parallel airplane having non­cypher intercepts.

vi. All the parallel equidistant planes have the same Miller indices. Thus the Miller indices define a fix of parallel planes.

7. A airplane parallel to i of the coordinate axes has an intercept of infinity.

8. If the Miller indices of two planes take the same ratio (i.e., 844 and 422 or 211), then the planes are parallel to each other.

ix. If (h k I) are the Miller indices of a airplane, so the plane cuts the axes into a/h, b/one thousand and c/l equal segments respectively.

ten. When the integers used in the Miller indices incorporate more than one digit, the indices must be separated by commas for clarity, east.g., (3, 11, 12).

11. The crystal directions of a family are non necessarily parallel to one some other. Similarly, not all members of a family unit of planes are parallel to one another.

12. Past changing the signs of all the indices of a crystal direction, we obtain the antiparallel or reverse direction. Past changing the signs of all the indices of a plane, nosotros obtain a plane located at the same distance on the other side of the origin.

13. The normal to the plane with indices (hkl) is the management [hkl].

14. The distance d betwixt next planes of a set of parallel planes of the indices (h grand I) is given by-

Where a is the border of the cube.

Normally the planes with low index numbers have wide interplanar spacing compared with those having high index numbers. Moreover, low index planes accept a higher density of atoms per unit surface area than the high index plane. In fact, information technology is the low alphabetize planes which play an of import role in determining the physical and chemical properties of solids.

15. The bending between the normals to the two planes (h_one 1000₁ l_i) and (h_ii chiliad_two l₂) is-

16. A negative Miller index shows that the aeroplane (hkl) cuts the 10-axis on the negative side of the origin.

17. Miller indices are proportional to the direction consines of the normal to all respective plane.

xviii. The purpose of taking reciprocals in the present scheme is to bring all the planes within a single unit cell and so that we tin can discuss all crystal planes in terms of the planes passing through a single unit jail cell.

nineteen. About planes which are important in determining the physical and chemical properties of solids are those with low index numbers.

20. The plane (hkl) is parallel to the line [uvw] if hu + kv + Iw = 0.

21. Ii planes (h_one 1000₁ l_i) and (h₂ chiliad_ii Z₂) both contain line [uvw] if u = m₁ 50₂ – grand₂ l₁, v = fifty₁ h₂ – l₂ h₁ and w = h₁ grand_two – h₂ chiliad₁

So both the planes are parallel to the line [uvw] and therefore, their intersection is parallel to [uvw] which defines the zone axis.

22. The plane (hkl) belongs to two zones [u₁ 5₁ westward₁] and [u₂ v₂ w₂] if h = five_ane w_ii – five₂ w₁, k = v₁ due west₂ – v₂ w₁ and I = v₁ w_two – v_two w₁.

23. The airplane (h_iii k₃ l_three) will exist among those belonging to the same zone every bit (h₁ yard_ane l₁) and (h₂ k₂ 50_two) if h₃ = h₁ ± h₂, k₃ = thou₁ ± k_ii and l₃ = fifty₁ ± l_ii.

24. The angle between the two directions [u₁ five_ane due west₁] and [u₂ v_two w₂] for orthorhombic system is-

Given Miller Indices How to Draw the Plane:

For the given Miller indices, the plane tin can be drawn equally follows:

Step ane:

Detect the reciprocal of the given Miller indices. These reciprocals requite the intercepts made by the plane on X, Y and Z axes respectively.

Stride 2:

Draw the cube and select a proper origin and testify X, Y and Z axes respectively.

Footstep three:

With respect to origin mark these intercepts and join through straight lines. The aeroplane obtained is the required airplane.

Following points are worth noting:

(i) Have lattice constant as ane unit.

(ii) If the intercept for an axis is infinity then proceed parallel to that centrality till you achieve the next lattice signal.

(iii) Try to get ii points and join them starting time.

Fig. 26 (a) and (b) shows important planes of cube. Thick lines with arrows point the directions.

Spacing of Planes:

In social club to place different types of crystals information technology is essential to take noesis of spacing of planes. It is then considering for each crystal at that place exists a definite ratio betwixt the spacing of planes which are rich in atoms. Refer to Fig. 27. (a).

Bragg past carrying out experiments on different crystals with X-rays not only verified the to a higher place ratio merely besides employed them to determine whether the crystal was simple cubic or B.C.C. type.

Relation between Interplanar Spacing 'd' and Cube Edge 'a':

Permit united states of america assume that the plane shown in Fig. 28 belongs to a family of planes whose Miller indices are <h 1000 l>. The perpendicular ON from origin to the plane represents the interplanar spacing d of this family of planes.

Let the management cosines of ON be cos α', cos β' and cos γ'.

The intercepts of the plane on the 3 axes are:

rollinsandelibubled.blogspot.com

Source: https://www.engineeringenotes.com/engineering/miller-indices/miller-indices-of-a-plane-feature-and-spacing-engineering/42324

Share this post