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Chapter 12: Problem 22

Of the seven three-dimensional primitive lattices, which ones have a unit cell where all three lattice vectors are of the same length?

Short Answer

Expert verified
Among the seven three-dimensional primitive lattices, only the cubic (P) and rhombohedral (R) lattices have a unit cell where all three lattice vectors are of the same length.

Step by step solution

01

Analyze the Cubic Lattice

In a cubic lattice, all three lattice vectors (a, b, c) have the same length and the angles between them are 90 degrees. Thus, a cubic lattice has lattice vectors with equal lengths.
02

Analyze the Tetragonal Lattice

In a tetragonal lattice, two of the three lattice vectors (a, b) are equal in length, while the third vector (c) is different. The angles between them are 90 degrees. Thus, a tetragonal lattice does not have lattice vectors with equal lengths.
03

Analyze the Orthorhombic Lattice

In an orthorhombic lattice, all three lattice vectors (a, b, c) are of different lengths and the angles between them are 90 degrees. Hence, an orthorhombic lattice does not have lattice vectors with equal lengths.
04

Analyze the Rhombohedral Lattice

In a rhombohedral lattice, all three lattice vectors (a, b, c) have the same length but the angles between them are not 90 degrees. Thus, a rhombohedral lattice has lattice vectors with equal lengths.
05

Analyze the Monoclinic Lattice

In a monoclinic lattice, all three lattice vectors (a, b, c) are of different lengths, with one angle not equal to 90 degrees. Thus, a monoclinic lattice does not have lattice vectors with equal lengths.
06

Analyze the Triclinic Lattice

In a triclinic lattice, all three lattice vectors (a, b, c) are of different lengths and none of the angles are equal to 90 degrees. Thus, a triclinic lattice does not have lattice vectors with equal lengths.
07

Analyze the Hexagonal Lattice

In a hexagonal lattice, two of the three lattice vectors (a, b) are equal in length, while the third vector (c) is different and the angles between them are 120 degrees. Thus, a hexagonal lattice does not have lattice vectors with equal lengths.
08

Conclusion

Among the seven three-dimensional primitive lattices, only the cubic (P) and rhombohedral (R) lattices have a unit cell where all three lattice vectors are of the same length.

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Most popular questions from this chapter

True or false: (a) The band gap of a semiconductor decreases as the particle size decreases in the \(1-10-\mathrm{nm}\) range. (b) The light that is emitted from a semiconductor, upon external stimulation, becomes longer in wavelength as the particle size of the semiconductor decreases.

A white substance melts with some decomposition at \(730^{\circ} \mathrm{C}\). As a solid, it does not conduct electricity, but it dissolves in water to form a conducting solution. Which type of solid (molecular, metallic, covalent- network, or ionic) might the substance be?

A particular form of cinnabar (HgS) adopts the zinc blende structure, Figure \(12.26 .\) The length of the unit cell edge is \(5.852 \AA\). (a) Calculate the density of \(\mathrm{HgS}\) in this form. (b) The mineral tiemmanite (HgSe) also forms a solid phase with the zinc blende structure. The length of the unit cell edge in this mineral is \(6.085 \AA\). What accounts for the larger unit cell length in tiemmanite? (c) Which of the two substances has the higher density? How do you account for the difference in densities?

The semiconductor GaP has a band gap of \(2.2 \mathrm{eV}\). Green LEDs are made from pure GaP. What wavelength of light would be emitted from an LED made from GaP?

The electrical conductivity of titanium is approximately 2500 times greater than that of silicon. Titanium has a hexagonal close-packed structure, and silicon has the diamond structure. Explain how the structures relate to the relative electrical conductivities of the elements.
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