1.4 Miller indices

Miller indices are a crucial tool in crystallography, helping us describe and analyze crystal structures. They provide a simple way to identify planes and directions within crystals, using just three numbers to represent complex geometric relationships.

Understanding Miller indices is key to interpreting diffraction patterns and determining crystal structures. They're used in various applications, from X-ray diffraction to electron microscopy, and play a vital role in surface and interface studies.

Definition of Miller indices

• Miller indices are a notation system used to describe the orientation of crystal planes and directions in a crystal lattice
• Provide a concise way to identify and label specific planes and directions within a crystalline structure
• Essential tool in crystallography for characterizing and analyzing the geometric properties of crystals

Notation for Miller indices

(hkl) format

• Miller indices are typically expressed as a set of three integers enclosed in parentheses: (hkl)
• Each integer represents a reciprocal of the fractional intercept along one of the three principal axes (x, y, z) of the crystal lattice
• Convention is to write the indices in the order of (hkl) without commas or spaces between the integers

Meaning of h, k, and l

• h, k, and l are the Miller indices corresponding to the x, y, and z axes, respectively
• They indicate the reciprocal of the fractional intercepts of the plane with each axis
• For example, (100) represents a plane that intercepts the x-axis at 1 and is parallel to both the y and z axes

Conventions for negative indices

• Negative Miller indices are denoted by placing a bar over the corresponding integer (e.g., $(\bar{1}00)$)
• Negative indices indicate that the plane intercepts the negative side of the respective axis
• Planes with negative indices are crystallographically equivalent to their positive counterparts but have opposite orientations

Relationship to crystal planes

Plane orientation

• Miller indices define the orientation of a crystal plane relative to the crystallographic axes
• Planes with the same Miller indices are parallel to each other and have the same orientation within the crystal lattice
• Different sets of Miller indices correspond to different plane orientations

Interplanar spacing

• Miller indices are related to the interplanar spacing ($d_{hkl}$) between parallel planes in a crystal
• The interplanar spacing depends on the lattice parameters and the values of h, k, and l
• For cubic crystals, the interplanar spacing is given by: $d_{hkl} = \frac{a}{\sqrt{h^2 + k^2 + l^2}}$, where $a$ is the lattice constant

Parallel planes

• Planes with the same Miller indices or integer multiples of the same indices are parallel to each other
• For example, (100), (200), and (300) planes are all parallel to each other
• Parallel planes have the same orientation but different interplanar spacings

Calculation of Miller indices

Intercepts on crystallographic axes

• To determine the Miller indices of a plane, first find the intercepts of the plane with the x, y, and z axes
• Intercepts are expressed as fractions of the lattice constants (a, b, c) along each axis
• For example, if a plane intercepts the x, y, and z axes at (2a, 3b, ∞c), the fractional intercepts are (1/2, 1/3, 0)

Conversion to smallest integer values

• Take the reciprocal of the fractional intercepts (excluding any zeros)
• Clear fractions by multiplying all values by the least common denominator
• Reduce the indices to the smallest integer values while maintaining their ratio

Examples for cubic crystals

• For a plane intercepting the axes at (a, a, a), the fractional intercepts are (1, 1, 1), and the Miller indices are (111)
• A plane intercepting at (a, a, ∞) has fractional intercepts (1, 1, 0) and Miller indices (110)
• Intercepts at (a, 2a, ∞) give fractional intercepts (1, 1/2, 0) and Miller indices (210)

Significance in crystallography

Identification of crystal planes

• Miller indices provide a standardized way to identify and label specific planes within a crystal structure
• Enable crystallographers to communicate and compare plane orientations across different crystals and studies
• Facilitate the analysis and interpretation of crystallographic data

Determination of crystal structure

• Miller indices play a crucial role in determining the crystal structure through techniques like X-ray diffraction
• The positions and intensities of diffraction peaks are related to the Miller indices of the diffracting planes
• By analyzing the diffraction pattern and indexing the peaks, the crystal structure can be elucidated

Relation to diffraction patterns

• In diffraction experiments (e.g., X-ray, electron, or neutron diffraction), the observed diffraction patterns are directly related to the Miller indices of the diffracting planes
• Each diffraction spot or peak corresponds to a specific set of Miller indices
• The arrangement and intensities of the diffraction spots provide information about the crystal structure and symmetry

Miller indices in different crystal systems

Cubic vs non-cubic lattices

• In cubic crystal systems (e.g., simple cubic, face-centered cubic, body-centered cubic), the Miller indices directly correspond to the reciprocal of the fractional intercepts along the axes
• For non-cubic lattices (e.g., tetragonal, orthorhombic, hexagonal), the relationship between Miller indices and plane orientation is more complex due to different lattice parameters and symmetry

Special cases for hexagonal crystals

• Hexagonal crystals require four indices (hkil) to uniquely describe plane orientations
• The first three indices (hki) are related to the reciprocal of the fractional intercepts along the a1, a2, and a3 axes in the hexagonal plane
• The fourth index (l) corresponds to the reciprocal of the fractional intercept along the c-axis, which is perpendicular to the hexagonal plane
• The indices satisfy the condition: $h + k + i = 0$

Applications of Miller indices

X-ray diffraction analysis

• Miller indices are essential in X-ray diffraction (XRD) analysis of crystalline materials
• XRD patterns consist of diffraction peaks at specific angles, which are related to the interplanar spacings and Miller indices of the diffracting planes
• By indexing the diffraction peaks and determining the corresponding Miller indices, the crystal structure, lattice parameters, and phase composition can be determined

Electron microscopy

• In electron microscopy techniques such as transmission electron microscopy (TEM) and scanning electron microscopy (SEM), Miller indices are used to describe the orientation of crystal planes and zone axes
• Electron diffraction patterns obtained in TEM are indexed using Miller indices to identify the crystal structure and orientation
• High-resolution TEM images can be interpreted based on the Miller indices of the observed lattice fringes

Surface and interface studies

• Miller indices are used to specify the orientation of crystal surfaces and interfaces
• Surface properties, such as atomic arrangement, reactivity, and electronic structure, depend on the specific Miller indices of the exposed planes
• Studies of surface reconstruction, adsorption, and catalytic behavior often refer to surfaces using their Miller indices (e.g., (100), (110), (111) surfaces)
• Interface phenomena, such as grain boundaries and heterostructures, are also described using the Miller indices of the adjoining planes