Miller Indices and Lattice Planes
Have you ever wondered how scientists talk about the microscopic building blocks of materials? Enter the world of Miller indices and lattice levels, a language that reveals the secrets of atomic arrangements within crystals. Imagine visualizing levels of atoms stacked like bricks, each level described by a unique set of numbers that reveal its orientation and spacing.
What are Miller Indices?
Miller indices are a set of three integers (h, k, and l) used to represent planes and directions within a crystal lattice. Think of a crystal as a neatly stacked arrangement of atoms, like bricks in a wall. Miller indices provide a way to identify and describe specific sets of these planes. They are commonly used in crystallography to represent the orientation and spacing of atomic planes within a crystal structure, which are crucial for understanding a crystal's properties and behavior.
How to determine Miller Indices?
- Imagine the crystal lattice with three axes: These axes represent the basic building blocks of the crystal's structure. In a cubic crystal, for example, the axes are all perpendicular to each other and of equal length.
- Identify a plane: This plane can cut through the lattice in any direction, intersecting the axes at different points.
- Find the intercepts: For each axis, measure the distance from the origin to the point where the plane intersects it. If the plane doesn't intersect an axis, then the intercept is considered to be "at infinity" and is represented by a zero.
- Take the reciprocals: For each non-zero intercept, take the reciprocal (flip it over).
- Simplify and normalize: If any of the reciprocals have a common factor, divide them all by that factor to get the smallest possible integers.
- Put the numbers together: The resulting three integers, enclosed in parentheses (h k l), are the Miller indices for that plane.
The plane in Figure 1 has the intercepts 1,∞, ∞ for the x, y, and z axes, respectively. We take reciprocals of these intercepts to obtain the Miller indices, which are therefore 1, 0, 0. Since, these numbers do not contain fractions, and the Miller indices for this plane are (100), which is read as one-zero-zero plane.
The plane shown in Figure 2 has the intercepts 1, 1, ∞. Since the reciprocals of these numbers are 1, 1, 0, which do not involve fractions, the Miller indices of this plane are (110).
The plane in Figure 3 has the intercepts 1, 1, 1. Since the reciprocals of these numbers are 1, 1, 1, which do not involve fractions, the Miller indices of this plane are (111).
The plane in Figure 4 has the intercepts 1/3, 2/3, 1. The reciprocals of these intercepts are 3, 3/2, 1. Since fractional intercepts are not allowed, these fractional intercepts must be multiplied by 2 to clear the 3/2 fraction. Thus, the reciprocal intercepts become 6, 3, 2, and the Miller indices are (632).
Note that:
- If the crystal plane being considered passes through the origin so that one or more intercepts are zero, the plane must be moved to an equivalent position in the same unit cell, and the plane must remain parallel to the original plane. This is possible because all equispaced parallel planes are indicated by the same Miller indices.
- If sets of equivalent lattice planes are related by the symmetry of the crystal system, they are called planes of a family or form, and the indices of one plane of the family are enclosed in braces as {h k l} to represent the indices of a family of symmetrical planes. For example, the Miller indices of the cubic surface planes (100), (010), and (001) are designated collectively as a family or form by the notation {100}.
a. First determine the reciprocals of the Miller indices of the (101) plane. These are 1, ∞, 1. The (101) plane must pass through a unit cube at intercepts x = 1 and z = 1 and be parallel to the y axis.
b. First determine the reciprocals of the Miller indices of the (1 ¯ 1 0 ) plane. These are 1, −1, ∞. The (1 ¯ 1 0 ) plane must pass through a unit cube at intercepts x = 1 and y = −1 and be parallel to the z axis. Note that the origin of axes must be moved to the lower-right back side of the cube on the y axis which intercepts negative number.
Why are Miller indices important?
- They provide a universal language for describing crystal planes, regardless of the specific crystal system or lattice parameters.
- They are used in X-ray crystallography, a technique for determining the structure of crystals by analyzing the way X-rays diffract off different planes.
- They are crucial for understanding many of the physical properties of crystals, such as their cleavage planes, electrical conductivity, and optical properties.
In short, Miller indices are a powerful tool for understanding the inner workings of crystals, playing a vital role in various fields like materials science, engineering, and chemistry.
Distance between two parallel lattice planes
In cubic crystal structures, the interplanar spacing between two closest parallel planes with the same Miller indices is designated dhkl, where h, k, and l are the Miller indices of the planes. This spacing represents the distance from a selected origin containing one plane and another parallel plane with the same indices that is closest to it. For example, the distance between (110) planes 1 and 2, d110, in Figure 5 is AB.
Also, the distance between (110) planes 2 and 3 is d110 and is length BC. From simple geometry, the distance between two parallel lattice planes formula is:
where:
- dhkl = interplanar spacing between parallel closest planes with Miller indices h, k, and l.
- a = lattice constant (edge of unit cube).
- h, k, l = Miller indices of cubic planes being considered.
- An important relationship for the cubic system, and only the cubic system, is that the direction indices of a direction perpendicular to a crystal plane are the same as the Miller indices of that plane. For example, the [100] direction is perpendicular to the (100) crystal plane.
Copper has an FCC crystal structure and a unit cell with a lattice constant of 0.361 nm. What is its interplanar spacing d220?
Solution
