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Chapter 29: Problem 20

Determine the Miller indices of the plane that intersects the crystal axes at (a) \((a, 2 b, 3 c)\) (b) \((a, b,-c),\) and (c) \((2 a, b, c)\).

Short Answer

Expert verified
(a) (6, 3, 2) (b) (1, 1, -1) (c) (2, 1, 1)

Step by step solution

01

Introduction to Miller Indices

Miller indices are a notation system in crystallography for planes in crystal (Bravais) lattices. A set of lattice planes is denoted by three integers (h, k, l), which are derived from the intercepts of the plane with the crystallographic axes.
02

Identify Intercepts for Part (a)

The plane intersects the axes at \( x = a \), \( y = 2b \), and \( z = 3c \). Here the intercepts are \( a, 2b, \) and \( 3c \). Let's write these in terms of the corresponding lattice parameters: \( x/a = 1, y/(2b) = 1, z/(3c) = 1 \).
03

Reciprocal of Intercepts for Part (a)

Take the reciprocals of the intercepts: \( 1/1 = 1 \), \( 1/2 = 0.5 \), and \( 1/3 = 0.333 \).
04

Clear Fractions for Part (a)

Multiply the reciprocals by the smallest common multiple to clear fractions. In this case, multiply by 6: \( 6 \times 1 = 6, 6 \times 0.5 = 3, 6 \times 0.333 \approx 2 \). Hence, the Miller indices are \( (6, 3, 2) \).
05

Identify Intercepts for Part (b)

The axes intersections are at \( x = a \), \( y = b \), and \( z = -c \). The intercepts are \( a, b, -c \). Writing these in terms of lattice parameters: \( x/a = 1, y/b = 1, z/(-c) = -1 \).
06

Reciprocal of Intercepts for Part (b)

Take the reciprocals of the intercepts: \( 1/1 = 1 \), \( 1/1 = 1 \), and \( 1/(-1) = -1 \).
07

Clear Fractions for Part (b)

The integers are already clear of fractions, so the Miller indices are \( (1, 1, -1) \).
08

Identify Intercepts for Part (c)

The axes intersections are at \( x = 2a \), \( y = b \), and \( z = c \). The intercepts are \( 2a, b, c \). Writing these in terms of lattice parameters: \( x/(2a) = 0.5, y/b = 1, z/c = 1 \).
09

Reciprocal of Intercepts for Part (c)

Take the reciprocals of the intercepts: \( 1/0.5 = 2 \), \( 1/1 = 1 \), and \( 1/1 = 1 \).
10

Clear Fractions for Part (c)

The integers are already clear of fractions, so the Miller indices are \( (2, 1, 1) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Crystallography
Crystallography is the scientific study of crystals and their structures. It involves understanding how atoms are arranged in solid materials. These arrangements can affect the physical properties of the material, like strength, flexibility, and melting point.
In crystallography, each crystal is made up of a repeating pattern called a lattice, which extends in all three dimensions. The way these atoms are packed can be represented by a geometric network called a "unit cell."
Crystallography involves:
  • Analyzing crystal lattices and their symmetry.
  • Determining how atoms interact within these structures.
  • Understanding how these structures influence the properties of the crystal.
By exploring crystallography further, scientists can create new materials with desirable features for various applications, from medicine to technology.
Lattice Planes
Lattice planes are imaginary flat surfaces that pass through points of a crystal lattice. These planes are essential in the study of crystals because they reflect how crystals interact with external factors, like X-rays.
Lattice planes can be defined by intercepts they make with the crystal axes. When considering how these planes intersect with the crystal, we use a system called Miller indices. A set of numbers denotes the orientation of a lattice plane, which helps in identifying and categorizing these planes.
Lattice planes are integral in:
  • Understanding crystal cleavage, or how crystals break along certain planes.
  • Predicting how crystals will behave when subjected to stress and external forces.
  • Determining the patterns of atomic layers within the crystal.
This concept is crucial for material scientists and physicists who wish to manipulate crystal properties for various uses.
Crystal Axes
Crystal axes are the imaginary lines used to reference points and planes within a crystal's structure. Think of them like a coordinate system oriented along the symmetry directions of the crystal lattice.
When working with crystal structures, these axes provide a framework to describe how the crystal lattice extends in space. The common axes are labeled as x, y, and z, representing length, width, and height respectively.
The understanding of crystal axes includes:
  • Using axes to define orientation and dimensions of crystal structures.
  • Applying axes as reference points for identifying planes using Miller indices.
  • Enhancing the understanding of crystal symmetry and geometry.
Crystal axes are instrumental in both theoretical studies and practical applications, helping to categorize and utilize different crystal forms.
Reciprocal Lattice
The reciprocal lattice is a powerful tool in crystallography, helping to simplify the mathematics involved in studying crystal structures. Essentially, it's an inverted representation of the real space lattice, used especially in the analysis of diffraction patterns.
Understanding the reciprocal lattice can assist in visualizing how waves, like X-rays, interact with the crystal lattice. This is important for techniques like X-ray diffraction, which is used to determine crystal structures.
  • It allows for easier computational work when dealing with periodicities in crystallography.
  • Provides insight into the possible directions of wave propagation.
  • Simplifies the calculation of properties like density of states in electronic structures.
By utilizing the reciprocal lattice, scientists can more efficiently analyze and predict crystal behavior, driving advancements in materials science and physics.

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Most popular questions from this chapter

The density of tantallum at \(20^{\circ} \mathrm{C}\) is \(16.69 \mathrm{g} \cdot \mathrm{cm}^{-3},\) and its unit cell is cubic. Given that the first five observed Bragg diffraction angles are \(\theta=19.31^{\circ}, 27.88^{\circ}, 34.95^{\circ}, 41.41^{\circ}\) and \(47.69^{\circ},\) find the type of unit cell and its length. Take the wavelength of the X-radiation to be \(\lambda=154.433 \mathrm{pm}\).

The density of silver at \(20^{\circ} \mathrm{C}\) is \(10.50 \mathrm{g} \cdot \mathrm{cm}^{-3},\) and its unit cell is cubic. Given that the first five observed Bragg diffraction angles are \(\theta=19.10^{\circ}, 22.17^{\circ}, 32.33^{\circ}, 38.82^{\circ},\) and \(40.88^{\circ},\) find the type of unit cell and its length. Take the wavelength of the X-radiation to be \(\lambda=154.433 \mathrm{pm}\).

Consider a surface-catalyzed bimolecular reaction between molecules \(A\) and \(B\) that has a rate law of the form \\[ v=k_{3} \theta_{\mathrm{A}} \theta_{\mathrm{B}} \\] where \(\theta_{\mathrm{A}}\) is the fraction of surface sites occupied by reactant \(\mathrm{A}\) and \(\theta_{\mathrm{B}}\) is the fraction of surface sites occupied by reactant \(\mathrm{B}\). A mechanism consistent with this reaction is as follows: Take \(K_{\mathrm{A}}\) and \(K_{\mathrm{B}}\) to be the equilibrium constants for Equations 1 and \(2,\) respectively. Derive expressions for \(\theta_{\mathrm{A}}\) and \(\theta_{\mathrm{B}}\) in terms of \([\mathrm{A}],[\mathrm{B}], K_{\mathrm{A}},\) and \(K_{\mathrm{B}} .\) Use your results to show that the rate law can be written as \\[ v=\frac{k_{3} K_{\mathrm{A}} K_{\mathrm{B}}[\mathrm{A}][\mathrm{B}]}{\left(1+K_{\mathrm{A}}[\mathrm{A}]+K_{\mathrm{B}}[\mathrm{B}]\right)^{2}} \\]

1\. In this problem, we will derive the structure factor for a sodium chloride-type unit cell. First, show that the coordinates of the cations at the eight corners are (0,0,0),(1,0,0),(0,1,0) \((0,0,1),(1,1,0),(1,0,1),(0,1,1,),\) and (1,1,1) and those at the six faces are \(\left(\frac{1}{2}, \frac{1}{2}, 0\right),\left(\frac{1}{2}, 0, \frac{1}{2}\right)\) \(\left(0, \frac{1}{2}, \frac{1}{2}\right),\left(\frac{1}{2}, \frac{1}{2}, 1\right),\left(\frac{1}{2}, 1, \frac{1}{2}\right),\) and \(\left(1, \frac{1}{2}, \frac{1}{2}\right) .\) Similarly, show that the coordinates of the anions along the 12 edges are \(\left(\frac{1}{2}, 0,0\right),\left(0, \frac{1}{2}, 0\right),\left(0,0, \frac{1}{2}\right),\left(\frac{1}{2}, 1,0\right),\left(1, \frac{1}{2}, 0\right),\left(0, \frac{1}{2}, 1\right),\left(\frac{1}{2}, 0,1\right),\left(1,0, \frac{1}{2}\right)\) \(\left(0,1, \frac{1}{2}\right),\left(\frac{1}{2}, 1,1\right),\left(1, \frac{1}{2}, 1\right),\) and \(\left(1,1, \frac{1}{2}\right)\) and those of the anion at the center of the unit cell are \(\left(\frac{1}{2}, \frac{1}{2}, \frac{1}{2}\right) .\) Now show that \\[ \begin{aligned} F(h k l)=& \frac{f_{+}}{8}\left[1+e^{2 \pi i h}+e^{2 \pi i k}+e^{2 \pi i l}+e^{2 \pi i(h+k)}+e^{2 \pi i(h+l)}+e^{2 \pi i(k+l)}+e^{2 \pi i(h+k+l)}\right] \\ &+\frac{f_{+}}{2}\left[e^{\pi i(h+k)}+e^{\pi i(h+l)}+e^{\pi i(k+l)}+e^{\pi i(h+k+2 l)}+e^{\pi i(h+2 k+l)}+e^{\pi i(2 h+k+l)}\right] \\ &+\frac{f}{4}\left[e^{\pi i h}+e^{\pi i k}+e^{\pi i l}+e^{\pi i(h+2 k)}+e^{\pi i(2 h+k)}+e^{\pi i(k+2 l)}+e^{\pi i(h+2 l)}+e^{\pi i(2 h+l)}+e^{\pi i(2 k+l)}\right. \end{aligned} \\] \\[ \begin{array}{l} \left.\quad+e^{\pi i(h+2 k+2 l)}+e^{\pi i(2 h+k+2 l)}+e^{\pi i(2 h+2 k+l)}\right]+f_{-} e^{\pi i(h+k+l)} \\ =f_{+}\left[1+(-1)^{h+k}+(-1)^{h+l}+(-1)^{k+l}\right] \\ \quad+f_{-}\left[(-1)^{h}+(-1)^{k}+(-1)^{l}+(-1)^{h+k+l}\right] \end{array} \\] Finally, show that \\[ F(h k l)=4\left(f_{+}+f_{-}\right) \\] if \(h, k,\) and \(l\) are all even; that \\[ F(h k l)=4\left(f_{+}-f_{-}\right) \\] if \(h, k,\) and \(l\) are all odd, and that \(F(h k l)=0\) otherwise.

What is the relation between the \(1 \overline{1}\) planes and the \(\overline{1}\) 1 planes of a two-dimensional square lattice?
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