Page 30 - Modern physical chemistry

Problems 79

1. 14 The Quasicrystalline State
In an ideal crystal, a unit cell is repeated periodically in three independent direc-
tions. Such a system yields the Bragg diffraction results that we have described. It
also yields Laue diffraction patterns of sharp spots. A completely disordered struc-
ture, on the other hand, such as we have in a glass or liquid, yields a Laue pattern of
concentric circles.
A quasicrystal does not possess the periodicity in space of a crystal. Nevertheless, it
produces a Laue diffraction pattern along certain axes of sharp spots. Furthermore, these
indicate the presence of various "forbidden" symmetries.
For example, rapidly solidified alloys of Al-Mn, of Al-Li- Cu, and of Al-Fe-Cu exhibit
5-fold, 3-fold, and 2-fold symmetry axes. These are the symmetry axes of the icosahe-
dron. And the Laue spots are as sharp as those for a crystal.
Some have suggested that quasicrystals are merely glasses of icosahedral clusters.
Others that they are based on giant cell periodic structures. However, high quality single
grains of the Al-Fe-Cu alloy have been grown with dodecahedral faceting. This can be
described as a 3- dimensional slice of a 6-dimensional periodic structure. There are three
possible 6-dimensional icosahedral periodic arrangements, a primitive, a face-cantered,
and a body- centered lattice.

Questions

1.1 What limits exist on homogeneity?
1.2 Define the different states of matter.
1.3 Distinguish between a crystal, a quasicrystal, and a glass.
1.4 Distinguish between refraction and diffraction.
1.5 What are rays?
1.6 Define frequency, wavelength, phase velOCity.
1. 7 Distinguish a polychromatic wave from a monochromatic one.
1.8 Is a section of a sinusoidal wave strictly monochromatic? Explain.
1.9 Derive the law governing diffraction from a ruled grating.
1.10 How are X rays produced?
1.11 What does the reflection of X rays from a crystal indicate?
1.12 Derive the law governing Bragg diffraction.
1.13 Describe the possible symmetries of 3-dimensional crystal systems.
1.14 How are the Bravais lattices related to these?
1.15 Explain how Miller indices are defined.
1.16 What is the basis for a crystal?
1.17 What determines the structure of ( a) a covalent crystal, (b) a molecular crystal, (c) an ionic
crystal, (d) a metallic crystal?
1.18 How may Avogadro's number be determined from measurements on a crystal?
1.19 Why may the structure at the surface of a crystal differ from that in its interior?
1.20 How may this surface structure be determined?
1.21 What characterizes the quasicrystalline state?

Problems

1.1 When 1,540 A X rays from a copper target fell on a given crystal, the maximum in a certain
reflection was observed at e = 10° 27'. With a molybdenum target in the X-ray tube, the same
maximum was found at e = 4°48'. What is the wavelength of the latter radiation?
1.2 How many orders of Bragg reflection can occur from the (100) planes in a CsCl crystal if
an edge of a unit cube in the crystal is 4.12 A long and the wavelength of the incident radi-
ation is (a) 1.540 A, (b) 0.712 A?