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Chapter 3: Problem 40

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

Short Answer

Expert verified
Answer: The four-index Miller-Bravais representations for the given directions are [1 1 -2 0] for [110] and [0 0 0 -1] for [00-1].

Step by step solution

01

Write down the conversion rules for the Miller to Miller-Bravais scheme

Write down the conversion rules for moving from the Miller indices (h, k, l) to the Miller-Bravais indices (u, v, t, w). 1. \(u = h\) 2. \(v = k\) 3. \(t = -(h+k)\) 4. \(w = l\)
02

Convert the [110] direction to the Miller-Bravais scheme

Apply the conversion rules to the given direction. 1. For the \([110]\) direction, we have h=1, k=1, and l=0. 2. Using the conversion rules, we get: u = 1 v = 1 t = -(1 + 1) = -2 w = 0 3. Therefore, the \([110]\) direction in the four-index Miller-Bravais scheme is \([1\,1\,\overline{2}\,0]\).
03

Convert the [00-1] direction to the Miller-Bravais scheme

Apply the conversion rules to the given direction. 1. For the \([00\overline{1}]\) direction, we have h=0, k=0, and l=-1. 2. Using the conversion rules, we get: u = 0 v = 0 t = -(0 + 0) = 0 w = -1 3. Therefore, the \([00\overline{1}]\) direction in the four-index Miller-Bravais scheme is \([0\,0\,0\,\overline{1}]\).

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Most popular questions from this chapter

(a) What are the direction indices for a vector that passes from point \(\frac{1}{4} 0 \frac{1}{2}\) to point \(\frac{3}{4} \frac{1}{2}\) in a cubic unit cell? (b) Repeat part (a) for a monoclinic unit cell.

The metal rhodium (Rh) has an FCC crystal structure. If the angle of diffraction for the (311) set of planes occurs at \(36.12^{\circ}\) (first-order reflection) when monochromatic \(\mathrm{x}\)-radiation having a wavelength of \(0.0711 \mathrm{~nm}\) is used, compute the following: (a) The interplanar spacing for this set of planes (b) The atomic radius for a Rh atom

Show for the body-centered cubic crystal structure that the unit cell edge length \(a\) and the atomic radius \(R\) are related through \(a=4 R / \sqrt{3}\).

(a) Derive the planar density expression for the HCP (0001) plane in terms of the atomic radius \(R\). (b) Compute the planar density value for this same plane for titanium (Ti).

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