A crystalline solid consists of atoms stacked up .

ABQADER

A crystalline solid consists of atoms stacked up in a repeating lattice structure.

A crystalline solid consists of atoms stacked up in a repeating lattice structure. Consider a crystal as shown in Figure P1.7a. The atoms reside at the corners of cubes of side L 5 0.200 nm. One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken. Suppose this crystal cleaves along a face diagonal as shown in Figure P1.7b. Calculate the spacing d between two adjacent atomic planes that separate when the crystal cleaves.

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Reference

Serway & Jewett's "Physics for Scientists and Engineers with Modern Physics 9th Edition" is a comprehensive textbook that covers fundamental physics concepts including mechanics, thermodynamics, electricity, magnetism, and modern physics. This problem comes from the crystallography section, which explores atomic arrangements in solid materials and their structural properties.

Question

A crystalline solid consists of atoms stacked up in a repeating lattice structure. Consider a crystal as shown in Figure P1.7a. The atoms reside at the corners of cubes of side L 5 0.200 nm. One piece of evidence for the regular arrangement of atoms comes from the flat surfaces along which a crystal separates, or cleaves, when it is broken. Suppose this crystal cleaves along a face diagonal as shown in Figure P1.7b. Calculate the spacing d between two adjacent atomic planes that separate when the crystal cleaves.

Explain Question

In this crystallography problem, we are examining a cubic crystal structure where atoms are positioned at the corners of cubes. The key elements we need to consider are the cube's side length L = 0.200 nanometers and the crystal's cleavage along a face diagonal. We need to determine the perpendicular distance (d) between adjacent atomic planes that form when the crystal cleaves along this diagonal.

Explain Solve

To solve this problem, we'll follow these detailed steps:

1. First, we need to understand that the spacing between diagonal planes (d) is half the distance between diagonally adjacent atoms on a flat plane. This relationship exists due to the geometric properties of the cubic structure.

2. To find the diagonal distance between atoms, we apply the Pythagorean theorem to a right triangle formed on the cube's face. The two legs of this triangle are equal to the cube's side length L.

3. Using the Pythagorean theorem: diagonal length = √(L² + L²) = L√2

4. Since the spacing d is half of this diagonal distance, we divide our result by 2: d = (L√2)/2

5. Plugging in L = 0.200 nm: d = (0.200 nm × √2)/2 = 0.141 nm

Summary

This problem demonstrates how geometric principles can be applied to understand atomic arrangements in crystals. The spacing between atomic planes is determined by considering the crystal's cubic structure and using the Pythagorean theorem to calculate diagonal distances, ultimately revealing how the crystal will cleave along specific geometric planes.

Solve

From the figure, we may see that the spacing between diagonal planes is half the distance between diagonally adjacent atoms on a flat plane. This diagonal distance may be obtained from the Pythagorean theorem, Ldiag = L2 + L2 . Thus, since the atoms are separated by a distance L = 0.200 nm, the diagonal planes are separated by 1/2√(L² + L²) = 0.141 nm

From the figure, we may see that the spacing between diagonal planes is half the distance between diagonally adjacent atoms on a flat plane. This diagonal distance may be obtained from the Pythagorean theorem, Ldiag = √(L² + L²). Thus, since the atoms are separated by a distance L = 0.200 nm, the diagonal planes are separated by 1/2√(L² + L²) = 0.141 nm.

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