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Chapter 3: Problem 29

Sketch an orthorhombic unit cell, and within that cell indicate locations of the \(0 \frac{1}{2} 1\) and \(\frac{1}{3} \frac{1}{4}\) point coordinates.

Short Answer

Expert verified
Question: Sketch an orthorhombic unit cell and locate the points with coordinates (0, 1/2, 1) and (1/3, 1/4, 0) within the cell. Briefly describe how these points are located in the orthorhombic unit cell. Answer: An orthorhombic unit cell is a parallelepiped-shaped lattice with sides of different lengths and 90-degree angles. The point (0, 1/2, 1) is located at half the length of the b-axis and at the end of the c-axis, without moving along the a-axis. The point (1/3, 1/4, 0) is located at one-third of the length of the a-axis, one-fourth of the length of the b-axis, and on the same plane as the origin (0-coordinate) of the c-axis.

Step by step solution

01

Sketch the Orthorhombic Unit Cell

Begin by drawing a parallelepiped with three perpendicular axes: a, b, and c. Since the angles are all equal to 90 degrees, the lattice should look like a rectangular box with sides of different lengths. Label the corners of the unit cell with their corresponding coordinates (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).
02

Plot the \(0 \frac{1}{2} 1\) Point

The coordinates of the first given point are \(0, \frac{1}{2}, 1\). To plot this point in the orthorhombic unit cell, start from the origin (0, 0, 0), and then move along the b-axis half of its length (as given by the \(\frac{1}{2}\) coordinate). Finally, move along the c-axis until you reach the end (since the coordinate is 1). Mark the point with the coordinates \(0, \frac{1}{2}, 1\) in the orthorhombic unit cell.
03

Plot the \(\frac{1}{3} \frac{1}{4} 0\) Point

The coordinates of the second given point are \(\frac{1}{3}, \frac{1}{4}, 0\). To plot this point in the orthorhombic unit cell, start again from the origin (0, 0, 0). Move along the a-axis by a distance equal to one-third of its length (as given by the \(\frac{1}{3}\) coordinate). Then, move along the b-axis by a distance equal to one-fourth of its length (as given by the \(\frac{1}{4}\) coordinate), and stay on the same plane (0-coordinate) of the c-axis. Mark the point with the coordinates \(\frac{1}{3}, \frac{1}{4}, 0\) in the orthorhombic unit cell.
04

Label and Check the Points

Ensure that both points are clearly labeled with their respective coordinates. Double-check that the points are in the correct positions by comparing their distances to the origin and the nearby corners of the unit cell.

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Most popular questions from this chapter

Explain why the properties of polycrystalline materials are most often isotropic.

(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).

(a) Derive planar density expressions for FCC (100) and (111) planes in terms of the atomic radius \(R\). (b) Compute and compare planar density values for these same two planes for aluminum (Al).

Magnesium (Mg) has an HCP crystal structure and a density of \(1.74 \mathrm{~g} / \mathrm{cm}^{3}\). (a) What is the volume of its unit cell in cubic centimeters? (b) If the \(c / a\) ratio is \(1.624\), compute the values of \(c\) and \(a\).

Iron (Fe) undergoes an allotropic transformation at \(912^{\circ} \mathrm{C}:\) upon heating from a \(\mathrm{BCC}\) ( \(\alpha\) phase) to an FCC \((\gamma\) phase). Accompanying this transformation is a change in the atomic radius of \(\mathrm{Fe}-\) from \(R_{\mathrm{BCC}}=\) \(0.12584 \mathrm{~nm}\) to \(R_{\mathrm{FCC}}=0.12894 \mathrm{~nm}-\) and, in addition, a change in density (and volume). Compute the percentage volume change associated with this reaction. Does the volume increase or decrease?
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