Question

4. What is Miller indices? Write the procedure for obtaining Miller indices.

5. Define Bragg's law by clearly explaining each term in the equation.

Answer 1

Answer-4 :-

Miller indices, group of three numbers that indicates the orientation of a plane or set of parallel planes of atoms in a crystal.

If each atom in the crystal is represented by a point and these points are connected by lines, the resulting lattice may be divided into a number of identical blocks, or unit cells; the intersecting edges of one of the unit cells defines a set of crystallographic axes, and the Miller indices are determined by the intersection of the plane with these axes.

The reciprocals of these intercepts are computed, and fractions are cleared to give the three Miller indices (hkl). For example, a plane parallel to two axes but cutting the third axis at a length equal to one edge of a unit cell has Miller indices of (100), (010), or (001), depending upon the axis cut; and a plane cutting all three axes at lengths equal to the edges of a unit cell has Miller indices of (111).

Procedure for finding Miler Indices :-

• Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions.
• Take the reciprocals
• Clear fractions
• Reduce to lowest terms

For example, if the x-, y-, and z- intercepts are 2, 1, and 3, the Miller indices are calculated as:

• Take reciprocals: 1/2, 1/1, 1/3
• Clear fractions (multiply by 6): 3, 6, 2
• Reduce to lowest terms (already there)

Thus, the Miller indices are 3,6,2. If a plane is parallel to an axis, its intercept is at infinity and its Miller index is zero. A generic Miller index is denoted by (hkl).

If a plane has negative intercept, the negative number is denoted by a bar above the number. Never alter negative numbers. For example, do not divide -1, -1, -1 by -1 to get 1,1,1. This implies symmetry that the crystal may not have!

For example :-   The (112) surface

Answer-5 :-

The structures of crystals and molecules are often being identified using x-ray diffraction studies, which are explained by Bragg’s Law. The law explains the relationship between an x-ray light shooting into and its reflection off from crystal surface.

The Bragg's law states that when the x-ray is incident onto a crystal surface, its angle of incidence, , will reflect back with a same angle of scattering, . And, when the path difference, d is equal to a whole number, n, of wavelength, , a constructive interference will occur.

where;

• is the wavelength of the x-ray,
• is the spacing of the crystal layers (path difference),
• is the incident angle (the angle between incident ray and the scatter plane), and
• is an integer

Consider a single crystal with aligned planes of lattice points separated by a distance d. Monochromatic X-rays A, B, and C are incident upon the crystal at an angle θ. They reflect off atoms X, Y, or Z.

  

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Answer 2

Answer-4 :-

Miller indices, group of three numbers that indicates the orientation of a plane or set of parallel planes of atoms in a crystal.

If each atom in the crystal is represented by a point and these points are connected by lines, the resulting lattice may be divided into a number of identical blocks, or unit cells; the intersecting edges of one of the unit cells defines a set of crystallographic axes, and the Miller indices are determined by the intersection of the plane with these axes.

The reciprocals of these intercepts are computed, and fractions are cleared to give the three Miller indices (hkl). For example, a plane parallel to two axes but cutting the third axis at a length equal to one edge of a unit cell has Miller indices of (100), (010), or (001), depending upon the axis cut; and a plane cutting all three axes at lengths equal to the edges of a unit cell has Miller indices of (111).

Procedure for finding Miler Indices :-

• Determine the intercepts of the face along the crystallographic axes, in terms of unit cell dimensions.
• Take the reciprocals
• Clear fractions
• Reduce to lowest terms

For example, if the x-, y-, and z- intercepts are 2, 1, and 3, the Miller indices are calculated as:

• Take reciprocals: 1/2, 1/1, 1/3
• Clear fractions (multiply by 6): 3, 6, 2
• Reduce to lowest terms (already there)

Thus, the Miller indices are 3,6,2. If a plane is parallel to an axis, its intercept is at infinity and its Miller index is zero. A generic Miller index is denoted by (hkl).

If a plane has negative intercept, the negative number is denoted by a bar above the number. Never alter negative numbers. For example, do not divide -1, -1, -1 by -1 to get 1,1,1. This implies symmetry that the crystal may not have!

For example :-   The (112) surface

Answer-5 :-

The structures of crystals and molecules are often being identified using x-ray diffraction studies, which are explained by Bragg’s Law. The law explains the relationship between an x-ray light shooting into and its reflection off from crystal surface.

The Bragg's law states that when the x-ray is incident onto a crystal surface, its angle of incidence, , will reflect back with a same angle of scattering, . And, when the path difference, d is equal to a whole number, n, of wavelength, , a constructive interference will occur.

where;

• is the wavelength of the x-ray,
• is the spacing of the crystal layers (path difference),
• is the incident angle (the angle between incident ray and the scatter plane), and
• is an integer

Consider a single crystal with aligned planes of lattice points separated by a distance d. Monochromatic X-rays A, B, and C are incident upon the crystal at an angle θ. They reflect off atoms X, Y, or Z.

  

(Note :- If you liked the answer, then please give a thumbs up. If you have any doubts, then please comment.)