How to Draw Plane Given Miller Indices

In this article we will discuss about:- ane. Concept of Miller Indices 2. Important 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 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.due east., edges of the unit cell. The intercepts are measured in terms of the border lengths or dimensions of the unit cell which are unit of measurement distances from the origin along three axes.

Process for finding miller indices:

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

Step 1:

Discover the intercepts of the plane along the axes x, y, z (The intercepts are measured every bit multiples of the fundamental vector). …iv, ii, 3.

Footstep 2:

Take reciprocals of the intercepts. one/4, i/2, i/3

Stride 3:

Convert into smallest integers in the same ration. …three six 4

Step four:

Enclose in parentheses. … (3 6 4)

The factor that results in converting the reciprocals of integers may exist indicated outside the brackets, but information technology is usually omitted.

Important Note:

The directions in space are represented by square brackets [ ]. The commas inside the square brackets are used separately and not combined. Thus [1 one 0] is read as "Ane-1-zero" and non "1 hundred 10". Negative indices are represented by putting a bar over digit, e.m., [1 one 0].

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

The following procedure is adopted for sketching whatever management:

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

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

Important Features of Miller Indices :

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

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

ii. All equally spaced parallel planes with a particular orientation have aforementioned alphabetize number (h g I).

3. Miller indices do non only define detail aeroplane but a ready of parallel planes.

4. Information technology is the ratio of indices which is only of importance. The planes (211) and (422) are the same.

5. A aeroplane passing through the origin is divers in terms of a parallel plane having not­goose egg intercepts.

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

7. A airplane parallel to 1 of the coordinate axes has an intercept of infinity.

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

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

10. When the integers used in the Miller indices incorporate more than ane digit, the indices must exist separated by commas for clarity, e.g., (3, eleven, 12).

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

12. By changing the signs of all the indices of a crystal management, nosotros obtain the antiparallel or opposite direction. Past changing the signs of all the indices of a aeroplane, we obtain a airplane located at the same distance on the other side of the origin.

xiii. The normal to the aeroplane with indices (hkl) is the direction [hkl].

xiv. The distance d between adjacent planes of a set of parallel planes of the indices (h k I) is given by-

Where a is the border of the cube.

Normally the planes with low index numbers have broad interplanar spacing compared with those having loftier index numbers. Moreover, low index planes accept a higher density of atoms per unit expanse than the high index aeroplane. In fact, it is the low alphabetize planes which play an of import function in determining the physical and chemical backdrop of solids.

15. The angle between the normals to the ii planes (h1 k1 l1) and (hii g2 l2) is-

16. A negative Miller alphabetize shows that the airplane (hkl) cuts the x-axis on the negative side of the origin.

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

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

19. Most planes which are of import 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. 2 planes (h1 g1 50one) and (h2 k2 Z2) both contain line [uvw] if u = kone lii – one thousandtwo li, five = l1 hii – fifty2 h1 and w = hone chiliad2 – hii yardi

And 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 v1 w1] and [u2 v2 wtwo] if h = v1 due west2 – 5ii w1, k = vane wtwo – v2 westward1 and I = v1 wtwo – vii westward1.

23. The plane (hiii k3 fifty3) will be among those belonging to the same zone every bit (h1 k1 fifty1) and (h2 one thousandii l2) if h3 = hi ± h2, grand3 = grand1 ± 10002 and l3 = fifty1 ± l2.

24. The angle between the ii directions [u1 v1 w1] and [utwo v2 w2] for orthorhombic system is-

Given Miller Indices How to Draw the Plane:

For the given Miller indices, the aeroplane tin can be fatigued equally follows:

Step 1:

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

Step 2:

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

Step iii:

With respect to origin mark these intercepts and bring together through straight lines. The plane obtained is the required plane.

Following points are worth noting:

(i) Take lattice constant equally one unit.

(ii) If the intercept for an axis is infinity and then continue parallel to that centrality till you reach the side by side lattice point.

(3) Attempt to get two points and join them first.

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

Spacing of Planes:

In society to identify different types of crystals it is essential to take knowledge of spacing of planes. Information technology is and so 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 conveying out experiments on dissimilar crystals with X-rays not but verified the in a higher place ratio but besides employed them to make up one's mind whether the crystal was simple cubic or B.C.C. type.

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

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

Let the direction cosines of ON exist cos α', cos β' and cos γ'.

The intercepts of the plane on the three 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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