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Chapter 3: Problem 48

Convert the (111) and (012) planes into the four-index Miller-Bravais scheme for hexagonal unit cells.

Short Answer

Expert verified
Question: Convert the given hexagonal unit cell planes in three-index Miller notation (111) and (012) to the four-index Miller-Bravais scheme. Answer: The four-index Miller-Bravais notation for the given planes are (1, 0, 1, 1) for (111) and (0, 1, 2, 1) for (012).

Step by step solution

01

Identify the three-index Miller notation

The three-index Miller notation of the given plane is (111).
02

Converting to four-index Miller-Bravais notation

For hexagonal unit cells, we can convert three-index notation into four-index notation using these formulas: 1. a* = (u, -u + v, i) 2. b* = (-a + u, v, j) 3. c* = (0, 0, 1) In this case, the three-index notation is (u, v, 1). Therefore, u = 1, v = 1, and i = 1.
03

Find a*, b*, and c* for Miller-Bravais notation

Now, plug the u, v, and i values into the equations: a* = (1, -1 + 1, 1) = (1, 0, 1) b* = (-1 + 1, 1, 1) = (0, 1, 1) c* = (0, 0, 1)
04

Write the converted Miller-Bravais scheme

The four-index notation for the (111) plane in hexagonal unit cells is [a*, b*, c*, i] = (1, 0, 1, 1). For the second plane (012), follow these steps:
05

Identify the three-index Miller notation for the second plane

The three-index Miller notation of the given plane is (012).
06

Converting to four-index Miller-Bravais notation

For hexagonal unit cells, we can convert three-index notation into four-index notation using the same formulas as before: In this case, the three-index notation is (u, v, 1). Therefore, u = 0, v = 1, and i = 2.
07

Find a*, b*, and c* for Miller-Bravais notation

Now, plug the u, v, and i values into the equations: a* = (0, -0 + 1, 2) = (0, 1, 2) b* = (-0 + 0, 1, 2) = (0, 1, 2) c* = (0, 0, 1)
08

Write the converted Miller-Bravais scheme

The four-index notation for the (012) plane in hexagonal unit cells is [a*, b*, c*, i] = (0, 1, 2, 1). So, the (111) and (012) planes in the four-index Miller-Bravais scheme are (1, 0, 1, 1) and (0, 1, 2, 1), respectively.

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Most popular questions from this chapter

Sketch within a cubic unit cell the following planes: (a) \((10 \overline{1})\) (b) \((2 \overline{1} 1)\) (c) (012) (d) \((3 \overline{1} 3)\) (e) \((\overline{1} 1 \overline{1})\) (f) \((\overline{2} 12)\) (g) \((3 \overline{1} 2)\) (h) (301)

Cite the indices of the direction that results from the intersection of each of the following pair of planes within a cubic crystal: (a) (110) and (111) planes, (b) (110) and \((1 \overline{1} 0)\) planes and (c) \((11 \overline{1})\) and (001) planes.

Within a cubic unit cell, sketch the following directions: (a) [101] (b) [211] (c) \([10 \overline{2}]\) (d) \([3 \overline{1} 3]\) (e) \([\overline{1} 1 \overline{1}]\) (f) \([\overline{2} 12]\) (g) \([3 \overline{1} 2]\) (h) [301]

The metal niobium has a BCC crystal structure. If the angle of diffraction for the (211) set of planes occurs at \(75.99^{\circ}\) (first-order reflection) when monochromatic x-radiation having a wavelength of \(0.1659 \mathrm{nm}\) is used, compute (a) the interplanar spacing for this set of planes and (b) the atomic radius for the niobium atom.

Convert the [110] and \([00 \overline{1}]\) directions into the four-index Miller-Bravais scheme for hexagonal unit cells.
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