How to Draw Plane From Miller Indices

In this article we will discuss near:- 1. Concept of Miller Indices 2. Of import Features of Miller Indices 3. Spacing of Planes 4. Relation between Interplanar Spacing 'd' and Cube Edge 'a'.

Concept of Miller Indices :

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

Procedure for finding miller indices:

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

Step 1:

Find the intercepts of the plane along the axes x, y, z (The intercepts are measured as multiples of the fundamental vector). …4, 2, 3.

Step two:

Accept reciprocals of the intercepts. 1/iv, 1/2, 1/3

Step iii:

Convert into smallest integers in the same ration. …3 6 iv

Step 4:

Enclose in parentheses. … (3 6 4)

The cistron that results in converting the reciprocals of integers may be indicated outside the brackets, merely information technology is commonly omitted.

Important Annotation:

The directions in space are represented past square brackets [ ]. The commas inside the square brackets are used separately and not combined. Thus [i 1 0] is read every bit "One-one-null" and not "1 hundred x". Negative indices are represented by putting a bar over digit, eastward.g., [i ane 0].

The general way of representing the indices of a direction of a line is [ten y z]. The indices of a plane are represented past a small bracket, (h, k I). Sometimes the notations < > and ( ) or { } are too used for representing planes and directions _ten respectively.

The post-obit process is adopted for sketching any management:

1. Starting time of all sketch the plane with the given Miller indices.

2. Now through the origin, draw a line normal to the sketched airplane, which will give the required direction.

Of import Features of Miller Indices :

Some of the important features of Miller indices (particularly for the cubic system) are detailed below:

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

2. All equally spaced parallel planes with a detail orientation accept aforementioned index number (h one thousand I).

3. Miller indices exercise not only define particular plane just a set of parallel planes.

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

5. A aeroplane passing through the origin is defined in terms of a parallel plane having non­zero intercepts.

6. All the parallel equidistant planes have the aforementioned Miller indices. Thus the Miller indices define a set of parallel planes.

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

eight. If the Miller indices of ii planes have the same ratio (i.e., 844 and 422 or 211), and so the planes are parallel to each other.

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

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

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

12. Past changing the signs of all the indices of a crystal management, we obtain the antiparallel or reverse management. By 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 direction [hkl].

14. The altitude d between adjacent planes of a set of parallel planes of the indices (h k I) is given past-

Where a is the border of the cube.

Ordinarily the planes with depression index numbers have wide interplanar spacing compared with those having loftier index numbers. Moreover, depression index planes take a higher density of atoms per unit expanse than the high index aeroplane. In fact, it is the low index planes which play an important role in determining the concrete and chemic backdrop of solids.

15. The angle between the normals to the ii planes (h_one k_one l_i) and (h_ii k₂ 50₂) is-

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

17. Miller indices are proportional to the management consines of the normal to all corresponding plane.

xviii. The purpose of taking reciprocals in the present scheme is to bring all the planes inside a single unit cell then that we can hash out all crystal planes in terms of the planes passing through a unmarried unit of measurement cell.

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

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

21. Two planes (h_one thou₁ fifty_i) and (h₂ k₂ Z_ii) both contain line [uvw] if u = one thousand₁ l₂ – k₂ l₁, v = 50₁ h₂ – l₂ h_one and w = h₁ k₂ – h₂ yard₁

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

22. The aeroplane (hkl) belongs to two zones [u₁ 5_one due west₁] and [u₂ v_ii due west_two] if h = v₁ due west₂ – v₂ w₁, k = v₁ w_ii – v₂ westward₁ and I = v_i west₂ – five₂ w₁.

23. The airplane (h₃ chiliad₃ l₃) will be among those belonging to the same zone as (h₁ k₁ fifty₁) and (h_ii k_two l₂) if h₃ = h₁ ± h₂, k_iii = k₁ ± chiliad₂ and 50₃ = l_i ± fifty₂.

24. The angle between the two directions [u₁ 5₁ w_one] and [u₂ v_two w_ii] for orthorhombic system is-

Given Miller Indices How to Draw the Plane:

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

Step 1:

Notice the reciprocal of the given Miller indices. These reciprocals give the intercepts made past the plane on X, Y and Z axes respectively.

Pace 2:

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

Footstep 3:

With respect to origin marking these intercepts and join through direct lines. The airplane obtained is the required plane.

Following points are worth noting:

(i) Accept lattice constant as i unit.

(ii) If the intercept for an axis is infinity then keep parallel to that axis till y'all reach the side by side lattice point.

(iii) Try to get 2 points and join them beginning.

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

Spacing of Planes:

In order to place dissimilar types of crystals it is essential to have knowledge of spacing of planes. It is then because for each crystal there exists a definite ratio between the spacing of planes which are rich in atoms. Refer to Fig. 27. (a).

Bragg past carrying out experiments on dissimilar crystals with X-rays non but verified the to a higher place ratio but as well employed them to determine whether the crystal was unproblematic cubic or B.C.C. type.

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

Let us assume that the airplane shown in Fig. 28 belongs to a family of planes whose Miller indices are <h yard fifty>. The perpendicular ON from origin to the aeroplane represents the interplanar spacing d of this family of planes.

Allow the direction cosines of ON be cos α', cos β' and cos γ'.

The intercepts of the plane on the iii axes are:

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

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