The index of the planes. Miller index

The first way. If the plane cuts on the coordinate axis segments x, y, z in the units of the lattice constants, then there is a number S such that

h = S / x; k = S / y; l = S / z.

In result we have h, k, l how minimum whole numbers.

The numbers h, k, l are the index of the plane or Miller index of the plane – (h k l).

The second way. In the common case we must to decide 3-th order determinant

m – m ₁ n – n ₁ p – p ₁

m ₂ – m ₁ n ₂ – n ₁ p ₂ – p ₁ = 0 (2.2)

m ₃ – m ₁ n ₃ – n ₁ p ₃ – p ₁

where [[ m ₁ n ₁ p ₁]], [[ m ₂ n ₂ p ₂]], [[ m ₃ n ₃ p ₃]] are coordinates of three point, which belong this plane. When we had decided this determinant in result we have

hm + kn + lp = 0 (2.3)

Thus we have (h k l).

For the cubic system:

a) normal to the plane with index (h k l) lies in the direction [ h k l ];

b) distance d between neighbor planes with index (h k l) is

, (2.4)

where a is the length of the cubic edge. The distance between planes with large index is little compare with distance between planes with little index.

The planes with little index have more high atoms density. The density packing of atoms is the quantity of atoms per unit an area.

At first we must to sketching all possible planes with different indexes. For instance

After that we determine the quantity of the atoms belonged to these planes:

a) N = 1 b) N = 2 c) N = 1/2.

Figure 2.4

Then we calculate the area of these planes S and density n

(2.5)

Compare different n _i it is possible to find n _max. For description crystal structure we must determine the coordination number. It is the number of the nearest neighbors.

And also the interatomic distance is the shortest distance between two atoms in a crystal.

Control questions

1. What is crystal structure?

2. What is unit sell?

3. What is coordination number?

4. What is the density packing of atoms?

5. What is interatomic distance?

Literature

1. Ч. Киттель, Элементарная физика твёрдого тела. – М.: Наука, 1965.

2. Епифанов Г.М. Физика твёрдого тела. – М.: Высшая школа, 1977.

3. Gevorkjan R.G., Shepel V.V. A Course of General Physics. – Moscow: Higher School, 1967. – 550 p.

Translator: Loskutov S.V., professor, doctor of physical and mathematical sciences.

Reviewer: Lushchin S.P., the reader, candidate of physical and mathematical sciences.

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