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Description: it is the best number one book over the world.. Flag for Inappropriate Content. Download Now. For Later. Related titles. Carousel Previous Carousel Next. Krane Kenneth S. Solid State Physics Neil W. Ashcroft, N. David Mermin-Brooks Cole Jump to Page. Search inside document. Symmetry Operations 7 1.
Point Operations 8 1. The CsCl Structure 20 1. The Hydrogen Bond 37 2. Ionic Crystals 40 2. Elastic Compliance and Stiffness Constants 51 Bulk Modulus and Compressibility 54 2. Experimental Methods 82 3. Normal Modes of One-dimensional Diatomic Chain 4. Einstein Model 5. Phonon Density of States 5. The Free Electron Gas at 0K 6.
Wave Mechanical Interpretation of Energy Bands 7. Insulators, Semiconductors and Metals 7. L Representative Crystal Types of Ferroelectrics Theory of the Ferroelectric Displacive Transitions Antiferroelectricity The Heisenberg Model Spin Waves Law The details of this process are far from simple to picturize as this requires us to know mainly the nature of response of valence electrons in atoms to the presence of other atoms.
The solidification or freezing of matter may lead to the formation of an ordered or disordered state. The ordered state is commonly called the crystalline state whereas the disordered one is termed the amorphous state. The regularity in the arrangement or the periodicity extends over the whole volume of a crystalline matter or crystal. The whole volume of a crystal can be constructed by moving a building block of the smallest acceptable size along its edges.
This block consisting of atoms or a group of atoms is called a unit cell. Predictions turn out to be wrong as often as right.
Various crystal structures existing in nature can be understood by systematically exploring all the possible ways of arranging atoms in periodic arrays. This process involves the visualization of motional operations performed on atoms within a unit cell such that the operations leave the unit cell unchanged.
Every crystal structure satisfies the requirements of a specific group combination of certain symmetry operations. Different combinations of symmetry operations result in different crystal structures. Thus the knowledge of the symmetry operations helps us in making a complete list of possible crystal structures and in classifying crystals. We take up below the relationship between the geometrical symmetry and the constituent particles of a crystal.
This helps us in appreciating the meaning of unit cell, defined earlier. Further, the fact that the unit cell design is not always unique is very crucial in the theory of crystalline solids. In more general terms it is known as the space lattice. The arrangement of points defines the lattice symmetry. When an atom or an identical group of atoms is attached to every lattice point, we obtain a crystal structure.
The attached atom or the group of atoms, called basis, is identical for every lattice point in terms of composition, relative orientation and separation. These features are explained in Fig. The number of atoms in the basis of an inorganic crystal may approach whereas it is known to be as large as a few thousands in organic crystals like protein.
Since a unit cell is another name for the smallest structural unit that may be used as the building block of a crystal, a few words about its acceptable size are very much in order.
Examine the structure of the following two-dimensional crystal. The unit cell shown in Fig. This takes us closer to the introduction of symmetry and symmetry operations in crystals. Figure 1. In this structure the translation of the unit cell represented by a2 and by vectors is able to map the infinite pattern of the crystal whereas it is not possible with a; and b, as the translation by these vectors takes the smaller atom to the position of the bigger atom and vice versa.
Thus in the present case the size of the smallest structural unit representing the unit cell is given by ap, by Toe T FIG. These translations are actually the various combinations of a, b and c, naturally, with certain restrictions. These restrictions are closely linked with the structural aspects of a unit cell which are discussed below. A proper description of the subject may be carried out by referring to the three-dimensional space lattices. Draw a vector t connecting two lattice points P, and P, Fig.
Two lattice points P, and P, of a three-dimensional lattice connected by a translation vector t. The interpretation of 1. The primitive cell is also defined as the minimum volume unit cell having only one lattice point.
In Fig. We can also say that in a primitive cell, lattice points are located only at the corners of the cell. All the lattice points are equivalent in a primitive cell, whereas it is not so in general [Fig.
But we must not forget here that the basis of all the lattice points whether equivalent or non-equivalent remains identical in the whole volume of the crystal. There is one lattice point in the cell. All of these primitive cells are equally acceptable as each of them has the same volume.
This property can be easily demonstrated in a two-dimensional space lattice. Choices for the primitive cell of a square lattice. The use of a primitive cell or a non-primitive unit cell for the study of a crystalline material is purely a matter of convenience. There is no hard and fast rule in this respect. The space lattice of the unit cell shown in Fig. But the same structure has a rhombohedral symmetry for the design of its primitive cell Fig.
These form the edges of the primitive cell having the rhombohedral shape. An alternative primitive cell, known as Wigner-Seitz cell, is of historic importance in the theory of Solid State Physics. The first proper electron energy bands calculation was made by Wigner and Seitz using this design of the primitive cell.
The method of construction is as follows. A lattice point is joined to all the nearby lattice points with the help of lattice vectors. Then, a plane perpendicular to each of these vectors, connecting the central lattice point, is drawn at the mid- point of the vector.
The planes form a completely closed polyhedron which contains only one lattice point at its centre. The polyhedron serves as an ideal primitive cell and is named after Wigner and Seitz who are credited with having given the philosophy of its construction.
It should be noticed that the Wigner— Seitz cell of BCC lattice can in effect be constructed by chopping off all the comers of its lattice cube halfway along a diagonal from the centre to a corner point.
In this construction the lattice vectors connecting the lattice point at the centre and the next nearest lattice points are also involved.
J.P. Srivastava-Elements of Solid State Physics-Prentice-Hall of India (2006)
Elements of Solid State Physics