ТОП просматриваемых книг сайта:
Solid State Physics. Philip Hofmann
Читать онлайн.Название Solid State Physics
Год выпуска 0
isbn 9783527837267
Автор произведения Philip Hofmann
Жанр Физика
Издательство John Wiley & Sons Limited
1.3.2 Other Methods for Structure Determination
While X‐ray diffraction is arguably the most widespread and most powerful method for structure determination, other techniques are used as well. Similar diffraction experiments can be carried out by making use of the wave character of neutrons or electrons. The former interact very weakly with matter because they carry no electric charge. They are also more difficult to produce than X‐rays. However, the use of neutrons has two distinct advantages over X‐rays: First, their interaction with light atoms is stronger than that of X‐rays, and second, they possess a magnetic moment, which means that they can interact with any magnetic moments in the solid, allowing one to determine its magnetic order. Electrons, on the other hand, have the advantages that they are easy to produce and that one can use electron‐optical imaging techniques, whereas making optical elements for X‐rays is very difficult. On the other hand, their very strong interaction with matter causes a breakdown of the kinematic approximation, that is, multiple scattering events have to be taken into account. Because of their strong interaction with matter, low‐energy electrons do not penetrate deeply into crystals. Therefore, they are more appropriate for surface structure determination.
1.3.3 Inelastic Scattering
Our discussion has been confined to the case of elastic scattering. In real experiments, however, the X‐rays or particles can also lose energy during the scattering events. This can be described formally by considering the diffraction from a structure that does not consist of atoms or ions at fixed positions but is time‐dependent, that is, which fluctuates with the frequencies of the atomic vibrations. We cannot go into the details of inelastic scattering processes here, but it is important to emphasize that inelastic scattering, in particular of neutrons, can be used to measure the vibrational properties of a lattice.
1.4 Further Reading
The concepts of lattice‐periodic solids, crystal structure, and X‐ray diffraction are discussed in all standard texts on solid state physics, for example:
Ashcroft, N.W. and Mermin, N.D. (1976). Solid State Physics. Holt‐Saunders.
Ibach, H. and Lüth, H. (2009). Solid State Physics, 4th edn. Springer.
Kittel, C. (2005). Introduction to Solid State Physics, 8th edn. John Wiley & Sons
Rosenberg, H.M. (1988). The Solid State. 3rd edn, Oxford University Press.
For a more detailed discussion of X‐ray diffraction, see, for example:
Als‐Nielsen, J. and McMorrow, D. (2011). Elements of Modern X‐Ray Physics, 2nd edn. John Wiley & Sons.
1.5 Discussion and Problems
Discussion
1 1.1 What mathematical concepts are used to describe the structure of any crystal?
2 1.2 What are the typical crystal structures of metals and why are they common?
3 1.3 Why do covalent crystals typically exhibit a much lower packing density than metallic crystals?
4 1.4 How can the reciprocal lattice conveniently be used to describe lattice‐periodic functions?
5 1.5 How can the structures of crystals be determined?
6 1.6 What is the difference between the Bragg and von Laue descriptions of X‐ray diffraction?
7 1.7 How can the reciprocal lattice of a crystal be used to predict the pattern of diffracted X‐rays?
Basic Concepts
1 1.1 Bravais lattice: Figure 1.13 shows four two‐dimensional lattices.Which of the following statements is true?All four lattices are Bravais lattices.Q is not a Bravais lattice.Q and R are not Bravais lattices.R is not a Bravais lattice.Draw the smallest possible unit cell of each lattice into the figure.Figure 1.13 Two‐dimensional lattices.
2 1.2 Basis:The left‐hand side of Figure 1.14 shows a two‐dimensional lattice with two types of atoms. We can think of the big white circles as nickel and the small grey ones as oxygen. When you describe this crystal by a two‐dimensional Bravais lattice and a basis, how many atoms are there in the basis?Figure 1.14 Left: two‐dimensional NiO crystal; Right: possible choices of the reciprocal lattice for this crystal.One.Two.Four.Nine.The right‐hand side of Figure 1.14 shows possible reciprocal lattices for this crystal. Which one is correct?
3 1.3 Unit cell of a lattice: Figure 9.6 shows a possible choice for the unit cell of barium titanate. The barium ions are located on the corners of the cube and the oxygen atoms on its faces. How many ions of the different types does this unit cell contain?Ba: 4, Ti: 1, O: 4.Ba: 8, Ti: 1, O: 6.Ba: 1, Ti: 1, O: 3.
4 1.4 The reciprocal lattice: Consider a real‐space Bravais lattice and the corresponding reciprocal lattice . Which of the following relations holds for all possible Bravais lattices? is parallel to . is perpendicular to the plane spanned by and . is perpendicular to the plane spanned by and . is perpendicular to .None of the above.
5 1.5 X‐ray diffraction: Which of the following can be determined from the positions of the spots in an X‐ray diffraction pattern?The reciprocal lattice.The Bravais lattice.Both A. and B.The position of the atoms in the unit cell.A., B., and D.
Problems
1 1.1 Fundamental concepts: For the two‐dimensional crystal in Figure 1.15, find (a) the Bravais lattice and a primitive unit cell, (b) a nonprimitive, rectangular unit cell, and (c) the basis.Figure 1.15 A two‐dimensional crystal.
2 1.2 Real crystal structures: Show that the packing of spheres in a simple cubic lattice fills 52% of the available space.
3 1.3 Real crystal structures: Figure 1.16 shows the structures of a two‐dimensional hexagonal packed layer of atoms, a hcp crystal, a two‐dimensional sheet of carbon atoms arranged in a honeycomb lattice (graphene), and three‐dimensional