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Crystal Structures and Symmetry 1 Crystal Structures and Symmetry PHYS 624: Crystal Structures and Symmetry 1 Crystal Structures and Symmetry Introduction to Solid State Physics http://www.physics.udel.edu/∼bnikolic/teaching/phys624/phys624.html PHYS 624: Crystal Structures and Symmetry 2 Translational Invariance • The translationally invariant nature of the periodic solid and the fact that the core electrons are very tightly bound at each site (so we may ignore their dynamics) makes approximate solutions to many-body problem ≈ 1021 atoms/cm3 (essentially, a thermodynamic limit) possible. Figure 1: The simplest model of a solid is a periodic array of valance orbitals embedded in a matrix of atomic cores. Solving the problem in one of the irreducible elements of the periodic solid (e.g., one of the spheres in the Figure), is often equivalent to solving the whole system. PHYS 624: Crystal Structures and Symmetry 3 From atomic orbitals to solid-state bands • If two orbitals are far apart, each orbital has a Hamiltonian H0 = εn, where n is the orbital occupancy ⇐ Ignoring the effects of electronic corre- lations (which would contribute terms proportional to n↑n↓). + +++ ... = Band E Figure 2: If we bring many orbitals into proximity so that they may exchange electrons (hybridize), then a band is formed centered around the location of the isolated orbital, and with width proportional to the strength of the hybridization PHYS 624: Crystal Structures and Symmetry 4 From atomic orbitals to solid-state bands • Real Life: Solids are composed of elements with multiple orbitals that produce multiple bonds. Now imagine what happens if we have several orbitals on each site (ie s,p, etc.), as we reduce the separation between the orbitals and increase their overlap, these bonds increase in width and may eventually overlap, forming bands. ↓ • The valance orbitals, which generally have a greater spatial extent, will overlap more so their bands will broaden more. ↓ • Eventually we will stop gaining energy from bringing the atoms closer together, due to overlap of the cores ⇒ Once we have reached the optimal point we fill the states 2 particles per, until we run out of electrons. ↑ • Electronic correlations complicate this simple picture of band forma- tion since they strive to keep the orbitals from being multiply occupied. PHYS 624: Crystal Structures and Symmetry 5 Band developments and their filling quantum numbers nl elemental solid 1s H,He 2s Li,Be 2p B→Ne 3s Na,Mg 3p Al→Ar 4s K,Ca 3d transition metals Sc→Zn 4p Ga→Kr 5s Rb,Sr 4d transition metals Y→Cd 5p In-Xe 6s Cs,Ba 4f Rare Earths (Lanthanides) Ce→Lu 5d Transition metals La→Hg 6p Tl→Rn PHYS 624: Crystal Structures and Symmetry 6 • For large n, the orbitals do not fill up simply as a function of n as we would expect 2 4 E mZ e ⇒ Z → Z from a simple Hydrogenic model with n = 2¯h2n2 nl VPr) 5d z/r 4f atom 6s 5p 6spdf 4d d 5s 5spdf 4p 3d Ce Valence Shell 4s 4spdf 3p + 3s 3spd 6s 2p 5d 2s 2sp 4f s 1s 1s VPr) + C l Pl+1)/r2 Figure 3: Level crossings due to atomic screening. The potential felt by states with large l are screened since they cannot access the nucleus. Thus, orbitals of different principle quantum numbers can be close in energy. I.e., in elemental Ce, (4f 15d16s2) both the 5d and 4f orbitals may be considered to be in the valence shell, and form metallic bands. However, the 5d orbitals are much larger and of higher symmetry than the 4f ones. Thus, electrons tend to hybridize (move on or off) with the 5d orbitals more effectively. The Coulomb repulsion between electrons on the same 4f orbital will be strong, so these electrons on these orbitals tend to form magnetic moments. PHYS 624: Crystal Structures and Symmetry 7 Different Types of Chemical Bonds • The overlap of the orbitals is bonding: Bond Overlap Lattice constituents Ionic very small (<a) closest unfrustrated dissimilar packing Covalent small (∼ a) determined by the similar structure of the orbitals Metallic very large ( a) closest packed unfilled valence orbitals Table 1: The type of bond that forms between two orbitals is dictated largely by the amount that these orbitals overlap relative to their separation a. PHYS 624: Crystal Structures and Symmetry 8 Covalent Bonding • The pile-up of charge which is inherent to the covalent bond is important for the lattice symmetry. The reason is that the covalent bond is sensitive to the orientation of the orbitals. - P S S P + - + No bonding Bonding Figure 4: A bond between an S and a P orbital can only happen if the P- orbital is oriented with either its plus or minus lobe closer to the S-orbital. I.e., covalent bonds are directional! PHYS 624: Crystal Structures and Symmetry 9 Ionic Bonding • The ionic bond occurs by charge transfer between dissimilar atoms which initially have open electronic shells and closed shells afterwards. Bonding then occurs by Coulomb attraction between the ions. e- Na + 5.14 eV Na+ + e- Cl + Cl- + 3.61 eV + + Na + Cl- Na Cl- + 7.9 eV = 1.81 rCl = 0.97 rNa Figure 5: The energy per molecule of a crystal of sodium chloride is (7.9- 5.1+3.6) eV=6.4eV lower than the energy of the separated neutral atoms. The cohesive energy with respect to separated ions is 7.9eV per molecular unit. All values on the figure are experimental. PHYS 624: Crystal Structures and Symmetry 10 Metallic Bonding • Metallic bonding is characterized by at least some long ranged and non-directional bonds (typically between s-orbitals), closest packed lattice structures and partially filled valence bands. 2 2 3d x - y 4S Figure 6: In metallic Ni (FCC, 3d84s2), the 4s- and 3d-bands (orbitals) are almost degenerate and thus, both participate in the bonding. However, the 4s-orbitals are so large compared to the 3d-orbitals that they encompass many other lattice sites, forming non-directional bonds. In addition, they hybridize weakly with the d-orbitals (the different symmetries of the orbitals causes their overlap to almost cancel) which in turn hybridize weakly with each other. Thus, whereas the s-orbitals form a broad metallic band, the d-orbitals form a narrow one. PHYS 624: Crystal Structures and Symmetry 11 Discrete translations symmetry • Translational symmetry of the lattice: There exist a set of basis vec- tors (a,b,c) such that the atomic structure remains invariant under transla- tions through any vector rn = n1a+n2b+n3c where n1,n2,n3 are integers. a b Figure 7: One may go from any location in the lattice to an identical location by following path composed of integral multiples of the vectors a and b. • Note that basic building blocks of periodic structures can be more complicated than a single atom. For example in NaCl, the basic building block is composed of one Na and one Cl ion which is repeated in a cubic pattern to make the NaCl structure. PHYS 624: Crystal Structures and Symmetry 12 Lattice types and symmetry • A collection of points in which the neighborhood of each point is the same as the neighborhood of every other point under some translation is called Bravais lattice. • The primitive unit cell is the parallel piped (in 3D) formed by the prim- itive lattice vectors which are defined as the lattice vectors which produce the primitive cell with the smallest volume a · (b × c). • There are many different primitive unit cells—common features: each cell has the same volume and contains only one site of Bravais lattice (Wigner- Seitz cell → site is in the center of the cell). • Non-primitive unit cell: Minimal region (which can contain several particles) of a crystal that has the same Point Group symmetry as the crystal itself and that produces the full crystal upon repetition. • SPACE GROUP: The complete set of rigid body motions that take G T R θ, θˆ crystal into itself = rn + ( ) PHYS 624: Crystal Structures and Symmetry 13 Example: 2D Bravais lattices Square Rectangular b b a a |a| = |b|, = /2 |a| = |b|, = /2 Centered Hexangonal b a a b |a| = |b|, = /3 Figure 8: Two dimensional lattice types of higher symmetry. These have higher sym- metry since some are invariant under rotations of 2π/3,or2π/6,or2π/4,etc.The centered lattice is special since it may also be considered as lattice composed of a two- component basis, and a rectangular unit cell (shown with a dashed rectangle). PHYS 624: Crystal Structures and Symmetry 14 Example: 3D Bravais lattices • The situation in three-dimensional lattices can be more complicated: there are 14 lattice Bravais lattices (for example there are 3 cubic structures, shownintheFigure). Body Centered Face Centered Cubic Cubic Cubic c c b b a a a = x a = Px+y-z)/2 a = Px+y)/2 b = y b = P-x+y+z)/2 b = Px+z)/2 c = z c = Px-y+z)/2 c = Py+z)/2 Figure 9: Three-dimensional cubic lattices. Note that the primitive cells of the centered lattice is not the unit cell commonly drawn. PHYS 624: Crystal Structures and Symmetry 15 Lattice decorated with a basis • To account for more complex structures like molecular solids, salts, etc., one also allows each lattice point to have structure in the form of a basis. A good example of this in twodimensionsistheCuO2 planes which characterize the cuprate high temperature superconductors. Here the basis is composed of two oxygens and one copper atom laid down on a simple square lattice with the Cu atom centered on the lattice points.
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