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Mobile App Chapter 1: Problem 21
Determine the Miller indices of a plane that makes an intercept of \(3 \mathrm{~A}, 4 \mathrm{~A}\), and \(5 \mathrm{~A}\) on the coordinate axes of an orthorhombic crystal with \(a: b: c=1: 2: 5 .\) Ans. (236)
Short Answer
Expert verified
(236)
Step by step solution
01
- Understand the intercepts
Determine the intercepts the plane makes with the coordinate axes: The plane intercepts the x-axis at 3 Å, the y-axis at 4 Å, and the z-axis at 5 Å.
02
- Normalize intercepts by lattice parameters
For the orthorhombic crystal, the lattice parameters are given by the ratios \(a: b: c = 1: 2: 5\). Therefore, normalize the intercepts: \[ \frac{3}{a} = \frac{3}{1} = 3, \quad \frac{4}{b} = \frac{4}{2} = 2, \quad \frac{5}{c} = \frac{5}{5} = 1 \]
03
- Take the reciprocal
Take the reciprocal of the normalized intercepts: \[ \frac{1}{3}, \quad \frac{1}{2}, \quad \frac{1}{1} = 1 \]
04
- Clear the fractions
Multiply each reciprocal by a common factor to get a set of integers. In this case, multiplying by 6 (the least common multiple of 3, 2, and 1) gives:\[ \frac{1}{3} \times 6 = 2, \quad \frac{1}{2} \times 6 = 3, \quad 1 \times 6 = 6 \]
05
Final Step - Write the Miller Indices
Combine the integers into the Miller indices in the form (hkl): The Miller indices are (236).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
crystallography
Crystallography is the study of crystals and their structures. It helps us understand how atoms are arranged inside a crystal. These arrangements affect the material's properties, such as hardness or conductivity.
A crystal is composed of repeating units called unit cells. Each unit cell is defined by its lattice parameters, which describe the cell's size and shape. Understanding these basic concepts allows us to delve deeper into specific structures and their unique characteristics.
A crystal is composed of repeating units called unit cells. Each unit cell is defined by its lattice parameters, which describe the cell's size and shape. Understanding these basic concepts allows us to delve deeper into specific structures and their unique characteristics.
orthorhombic crystal
An orthorhombic crystal is a type of crystal system that features three mutually perpendicular axes of different lengths. In other words, if you imagine a box, each side has a different length.
This makes orthorhombic crystals unique compared to cubic crystals, where all sides are equal. Common materials with orthorhombic structures include certain minerals and engineered materials. These differences in axis lengths are captured in the lattice parameters, denoted as 'a', 'b', and 'c'. These parameters play a crucial role in determining the Miller indices of planes in the crystal.
This makes orthorhombic crystals unique compared to cubic crystals, where all sides are equal. Common materials with orthorhombic structures include certain minerals and engineered materials. These differences in axis lengths are captured in the lattice parameters, denoted as 'a', 'b', and 'c'. These parameters play a crucial role in determining the Miller indices of planes in the crystal.
lattice parameters
Lattice parameters are the dimensions that define the unit cell of a crystal. For an orthorhombic crystal system, we have three parameters: 'a', 'b', and 'c', representing the lengths of the unit cell edges.
These parameters can be ratios, like in our example: a : b : c = 1 : 2 : 5. These ratios help us understand the relative dimensions along each axis. They are essential when calculating the Miller indices, as they modify the intercepts of a plane with the axes. Incorrect interpretation of lattice parameters can lead to wrong calculations of Miller indices.
These parameters can be ratios, like in our example: a : b : c = 1 : 2 : 5. These ratios help us understand the relative dimensions along each axis. They are essential when calculating the Miller indices, as they modify the intercepts of a plane with the axes. Incorrect interpretation of lattice parameters can lead to wrong calculations of Miller indices.
reciprocal intercepts
Reciprocal intercepts are crucial when converting the intercepts of a plane into Miller indices. First, you need to normalize the intercepts by dividing them by the lattice parameters.
For example, if a plane intersects the axes at 3 Å, 4 Å, and 5 Å, and the lattice parameters are 1, 2, and 5, you would divide the intercepts as follows: 3/1, 4/2, and 5/5. This gives you 3, 2, and 1. Taking the reciprocal of these values results in 1/3, 1/2, and 1. The reciprocal process simplifies the indices and brings a level of standardization useful in crystallography.
For example, if a plane intersects the axes at 3 Å, 4 Å, and 5 Å, and the lattice parameters are 1, 2, and 5, you would divide the intercepts as follows: 3/1, 4/2, and 5/5. This gives you 3, 2, and 1. Taking the reciprocal of these values results in 1/3, 1/2, and 1. The reciprocal process simplifies the indices and brings a level of standardization useful in crystallography.
Miller indices
Miller indices are a set of three numbers (hkl) used to denote the orientation of a plane within a crystal lattice. They are derived through a systematic process:
This standardized notation helps scientists and engineers to communicate the structural aspects of crystals efficiently.
- Determine the intercepts on the axes.
- Normalize these intercepts by the corresponding lattice parameters.
- Take the reciprocals of the normalized intercepts.
- Clear the fractions by multiplying by a common factor.
This standardized notation helps scientists and engineers to communicate the structural aspects of crystals efficiently.
