Diatomic Lattices

7 Key excerpts on "Diatomic Lattices"

• 

  eBook - PDF

  Quantum Field Theory Approach to Condensed Matter Physics

  • Eduardo C. Marino(Author)
  • 2017(Publication Date)
  • Cambridge University Press

    (Publisher)

  Surprisingly, for a vast amount of materials at room temperature and pressure, the most stable configurations would be such that the average position of the “atomic kernels” coincides precisely with the ideal points of a Bravais lattice, possibly with a certain base. The actual location of the solid’s basic constituents nevertheless oscillates around these equilibrium positions because of thermal and quantum fluctuations. Hence, since the physical proper- ties of crystal materials are strongly influenced by the symmetry of the underlying lattice structure, one should expect that such oscillations would have a profound impact on the physical properties of the solid. We shall consider in this chapter the dynamics of the oscillatory motion of the basic constituents of a material crystal, both from the classical and quantum- mechanical points of view. 2.1 The Harmonic Approximation For the sake of clarity, in this first approach we will consider crystal structures with a base containing just one kernel or, in other words, just a Bravais lattice. Then we can specify the position of each of the atomic kernels by X(R) = R + r(R). (2.1) Classically, r(R), the relative position with respect to the closest Bravais lattice point R, is a dynamical variable describing the position of an atomic kernel of 25 26 Vibrating Crystals mass M and momentum P(R) = M d r dt (R). Notice that we must have r = 0 so that the average position coincides with the Bravais lattice points, namely X = R. A pair of atomic kernels does interact through a potential energy that depends on the mutual separation X(R) − X(R  ). We assume only nearest-neighbors inter- actions will be relevant. Then, the total Hamiltonian describing the mechanical energy of the material lattice is therefore H =  R P 2 (R) 2 M +  RR   V ( R − R  + r(R) − r(R  ) ) .

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Thermal Energy at the Nanoscale

  • Timothy S Fisher(Author)
  • 2013(Publication Date)
  • WSPC

    (Publisher)

  Chapter 3 applies equally well to fluid phases. Within the array of solid-state materials, single-crystal structures are the most amenable for initial study, although even these structures become rather complex in three dimensions with various atomic arrangements such as face-centered cubic (fcc), body-centered cubic (bcc), and diamond configurations that are perhaps most familiar to readers. To minimize digression, here we refer the reader to the many excellent textbooks on solid-state physics (Ashcroft and Mermin, 1976; Kittel, 2007) and crystallography (De Graef and McHenry, 2012) for advanced treatment of 3D crystals.

  We will focus on one- and two-dimensional lattices for the sake of expediency and because the 2D graphene lattice has high contemporary scientific and technological importance. A simple 1D structure is obtained by repeating the diatomic arrangement of Fig. 1.2 indefinitely. Figure 1.5 shows the resulting configuration, with each atom of mass m connected to its neighbor by a bond with spring constant g. The equilibrium separation between atoms is represented by the lattice constant a. Somewhat surprisingly, this simple, idealized structure will enable us to develop almost all the essential tools for analysis of lattice vibrations and their quantum manifestation—called phonons.

  Fig. 1.4  (a) Two isolated, self-contained atoms and associated electron energy states. (b) Quantized energy states upon bond formation between the two isolated atoms. Energy levels are modified as electron orbitals become shared in a bond.

  Because an ideal crystal extends infinitely in all directions, we must find a way to concentrate the analysis on a smaller region. Fortunately, the regular order, or periodicity, of a crystal lattice makes this task straightforward. A primitive unit cell of a lattice is one that, if repeated throughout all space by well-defined translational vectors, would fill the space entirely and with no overlapping regions or void spaces. Figure 1.6 shows an example for a 2D monatomic rectangular lattice. Several possible shapes, positions, and orientations of the primitive unit cell exist for this lattice, as indicated by the shaded regions. The arrows denote basis vectors that define the periodic translation of the unit cells throughout the domain. The set of all possible translations by integer indexing of basis vectors forms a so-called Bravais lattice, whose discrete points are given by the lattice vector

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  The Crystal Lattice

  Phonons, Solitons, Dislocations, Superlattices

  • Arnold M. Kosevich(Author)
  • 2006(Publication Date)
  • Wiley-VCH

    (Publisher)

  Since the atoms of different type are distinguished not only by their chemical properties but also by their arrangement in the cell, even in a crystal of a pure element there can be more than one type of atom. If the unit cell consists of only one type of atom it is called monatomic, otherwise it is polyatomic. A monatomic lattice is also often called simple and a polyatomic lattice composite. Table salt (NaCl) containing atoms of two types is an example of a polyatomic crystal lattice (Fig. 0.2), and 2D lattice composed of atoms of two types is presented also in Fig. 0.3a. A polyatomic crystal lattice may also consist of atoms of the same chemical type. Figure 0.3b shows a highly symmetrical diatomic planar lattice whose atoms are located at the vertices of a hexagon. The differences between simple and composite lattices lead to different physical properties. For example, the vibrations of a diatomic lattice have some features that distinguish them from the vibrations of a monatomic lattice. We would like to emphasize that the unit cell of a crystal involves, by definition, all the elements of the translation symmetry of the crystal. By drawing the unit cell one can construct the whole crystal. However, the unit cell may not necessarily be symmetrical with respect to rotations and reflections as the crystal can be. This is clearly seen in Fig. 0.3 where the lattices have a six-fold symmetry axis, while the unit cells do not. 2) We note that the contribution to a cell of an atom positioned in a cell vertex equals 1/8, on an edge 1/4 and on a face 1/2. 0.2 Bravais Lattice 5 Fig. 0.2 NaCI crystal structure ( - Na, ● - Cl). Fig. 0.3 Hexagonal 2D diatomic lattice composed of atoms (a) of differ- ent types and (b) of the same type.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Structure of Materials

  An Introduction to Crystallography, Diffraction and Symmetry

  • Marc De Graef, Michael E. McHenry(Authors)
  • 2012(Publication Date)
  • Cambridge University Press

    (Publisher)

  This kind of atom motion is known as thermal motion or thermal vibration . It is present in every crystal structure and it is convenient to ignore it in a structural description of crystals. 1 The thermal motion of atoms only becomes important in the determination of the crystal structure by means of a suitable form of radiation (X-rays, electrons, neutrons) and can be adequately described by means of the so-called Debye–Waller factor , which will be introduced in Chapter 12 . From here on, we will always consider the average position to be the “real” position of the atom; this is an approximation, but it turns out to be a very convenient one because most mathematical relations to be derived in the remainder of this book become independent of time. The average position of an atom in a crystal structure does not change with time, so we can slightly revise our initial loose definition of a crystal structure to: a crystal structure is a time-invariant, 3-D arrangement of atoms or molecules on a lattice. We will take this statement as a starting point for this chapter. First, we need to define more precisely what we mean by the term “lattice.” 3.2 The space lattice • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • 3.2.1 Basis vectors and translation vectors The historical comments in Section 3.9 show how René Just Haüy built models of crystals by stacking rectangular blocks in such a way that the assembly resembled the external shape (or form ) of macroscopic crystals. By assuming the existence of a single shape, he 1 Thermal vibration is not limited to materials with a crystalline structure; it also occurs in liquids and gases, where there is no periodic structure. The vibrations are related to the curvature of the interatomic interaction potential introduced in the previous chapter.

  Sign up to read

  Learn more about book
• 

  eBook - ePub

  Modern Physical Metallurgy and Materials Engineering

  • R. E. Smallman, R J Bishop(Authors)
  • 1999(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  2.1b , one can appreciate why the structures of comparatively highly-ordered crystalline substances, such as chemical compounds, minerals and metals, have tended to be more amenable to scientific investigation than glasses.

  2.2 Crystal lattices and structures

  We can rationalize the geometry of the simple representation of a crystal structure shown in Figure 2.1a by adding a two-dimensional frame of reference, or space lattice, with line intersections at atom centres. Extending this process to three dimensions, we can construct a similar imaginary space lattice in which triple intersections of three families of parallel equidistant lines mark the positions of atoms (Figure 2.2a ). In this simple case, three reference axes (x, y, z ) are oriented at 90° to each other and atoms are ‘shrunk’, for convenience. The orthogonal lattice of Figure 2.2a defines eight unit cells, each having a shared atom at every corner. It follows from our recognition of the inherent order of the lattice that we can express the geometrical characteristics of the whole crystal, containing millions of atoms, in terms of the size, shape and atomic arrangement of the unit cell, the ultimate repeat unit of structure.2

  Figure 2.2 Principles of lattice construction.

  We can assign the lengths of the three cell parameters (a, b, c ) to the reference axes, using an internationally-accepted notation (Figure 2.2b ). Thus, for the simple cubic case portrayed in Figure 2.2a , x = y = z = 90°; a = b = c . Economizing in symbols, we only need to quote a single cell parameter (a ) for the cubic unit cell. By systematically changing the angles (α , β , γ ) between the reference axes, and the cell parameters (a, b, c ), and by four skewing operations, we derive the seven crystal systems (Figure 2.3 ). Any crystal, whether natural or synthetic, belongs to one or other of these systems. From the premise that each point of a space lattice should have identical surroundings, Bravais demonstrated that the maximum possible number of space lattices (and therefore unit cells) is 14. It is accordingly necessary to augment the seven primitive (P) cells shown in Figure 2.3 with seven more non-primitive cells which have additional face-centring, body-centring or end-centring lattice points. Thus the highly-symmetrical cubic system has three possible lattices: primitive (P), body-centred (I; from the German word innenzentrierte ) and face-centred (F). We will encounter the latter two again in Section 2.5.1 . True primitive space lattices, in which each lattice point has identical surroundings, can sometimes embody awkward angles. In such cases it is common practice to use a simpler orthogonal non-primitive lattice which will accommodate the atoms of the actual crystal structure.1

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Metals and Materials

  Science, Processes, Applications

  • R. E. Smallman, R J Bishop(Authors)
  • 2013(Publication Date)
  • Butterworth-Heinemann

    (Publisher)

  Chapter 2 Atomic arrangements in materials 2.1 The concept of ordering When attempting to classify a material it is useful to decide whether it is crystalline (conventional metals and alloys), non-crystalline (glasses) or a mixture of these two types of structure. The critical distinction between the crystalline and non-crystalline states of matter can be made by apply-ing the concept of ordering. Figure 2.1a shows a symmetrical two-dimensional arrangement of two different types of atom. A basic feature of this aggregate is the nesting of a small atom within the triangular group of three much larger atoms. This geometrical condition is called short-range order-ing. Furthermore, these triangular groups are regularly arranged relative to each other so that if the aggregate were to be extended, we could confi-dently predict the locations of any added atoms. In effect, we are taking advantage of the long-range ordering characteristic of this array. The array of (a) Figure 2.1 Atomic ordering in (a) crystals and (b) glasses of the same composition (from Kingery, Bowen and Uhlmann, 1976; by permission of Wûey-Interscience). 12 Metals and Materials (a) (b) Figure 2.2 Principles of lattice construction. Figure 2.1a exhibits both short- and long-range ordering and is typical of a single crystal. In the other array of Figure 2.1b, short-range order is discernible but long-range order is clearly absent. This second type of atomic arrangement is typical of the glassy state. 1 It is possible for certain substances to exist in either crystalline or glassy forms (e.g. silica). From Figure 2.1 we can deduce that, for such a substance, the glassy state will have the lower bulk density. Furthermore, in comparing the two degrees of order-ing of Figures 2.1a and 2.1b, one can appreciate why the structures of comparatively highly-ordered crystalline substances, such as chemical compounds, minerals and metals, have tended to be more amenable to scientific investigation than glasses.

  Sign up to read

  Learn more about book
• 

  eBook - PDF

  Materials Science for Engineers

  • J.C. Anderson, Keith D. Leaver, Rees D. Rawlings, Patrick S. Leevers(Authors)
  • 2004(Publication Date)
  • CRC Press

    (Publisher)

  Each corner site and face centre site is occupied by a covalent molecule I 2 , as shown in Fig. 6.16 (for clarity the molecular shape is not drawn to scale). Van der Waals forces that are nearly non-directional pull the molecules together, making it close-packed. Bromine and chlorine form similar crystals, which melt below room temperature. In the general case, each lattice point in any of the Bravais lattice shapes shown in Fig. 6.14 could be occupied by an identical molecule instead of an atom - provided, of course, that the intermolecular bonds are such as to stabilize the lattice. Each molecule may be in an identical orientation, or, as in crystalline iodine, they may have one of a set of different orientations. The requirement of long-range order means, however, that the structure must repeat itself at regular intervals. In crystals of simple molecules, the repeti-tion lengths (the lattice constants) are at most a few times the smallest intermolecular separation. However when large molecules form crystals the lattice constant may be anything from a few intermolecular distances to thousands of them. In the latter case, it may be difficult to distinguish the structure from one that is not ordered. The enormous variety of molecular solids, which includes most organic substances, means that only a tiny selection of simple crystal structures can be discussed here. We will concentrate on explaining a few of the principles governing them, and discuss three main types of molecular crystal, divided according to the types of attractive force discussed in Section 5.9.1. For convenience we summarize these again now. 6.14 MOLECULAR CRYSTALS 109 In the absence of charge transfer between molecules, electrical forces are dipolar in origin. These forces were classified in Section 5.9.1 as follows: • Permanent dipole-permanent dipole attractions, varying as 1/r 3 . • Induction forces, between a permanent dipole and an induced dipole, varying as 1/r 6 .

  Sign up to read

  Learn more about book