Symmetry & Space Groups
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SYMMETRY & SPACE GROUPS
SYMMETRY & SPACE GROUPSPresented to: Prof. Dr. Mehmet Ali Glgn
Presented by: Muhammad Faisal JamilYelda Yorulmaz Amin Hodaei
MAT-509Sabanci University04 December, 2015
A Timeline of Symmetry in physics, chemistry and mathematics1890s, derivation of the 230 space groups in 3 dimensions (independently) by Arthur Moritz Schnflies, Evgraf Stepanovich Federov, and William Barlow1936, Frederick Seitz works out the representation theory of space groups, the symmetry groups of crystal lattices.
A. M. SchnfliesE. S. FederovW. BarlowF. SeitzSymmetry and crystalsThink, if you have to describe an infinite crystal with an infinite number of atoms or even a finite crystal, with some 1020 atoms.Sounds awful? Well, there is symmetry to help you out! Instead of an infinite number of atoms, you only need to describe the contents of one unit cell, the structural repeating motif Life could be even easier, if there are symmetry elements present inside the unit cell!Basic terminologies Atomic structures repeating itself in 3D is termed as Translational Symmetry Structures consist of simple groups of atoms that repeat periodically in space. This periodicity is called Lattice.
Crystal structure of the mineral cordierite (Mg2Al4Si5O18)Source: http://www.doitpoms.ac.uk/tlplib/crystallography3/intro.phpBasic terminologiesSet of identical points is the lattice, and each point within it is a lattice point.
Source: http://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.phpBasic terminologies Structure of a crystal can be seen to be composed of a repeated element in 3D, called Unit Cell. There are two types: Primitive Unit Cell: These cells have only one lattice point, which is made up from the lattice points at each of the corners. Non-Primitive Unit Cell: These contain additional lattice points, either on a face of the unit cell or within the unit cell, and so have more than one lattice point per unit cell.
Source: http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.phpLattice symmetry Lattice symmetry refers to unit cells size and shape. Without rules, there would be an infinite number of different unit cells to describe any given lattice.
Source: http://www.aps.anl.gov/Xray_Science_Division/Powder_Diffraction_Crystallography/Introduction_to_Crystallography/#03Lattice geometry The length of the unit cell along the x, y, and z direction are defined as a, b, and c. Alternatively, we can think of the sides of the unit cell in terms of vectors a, b, and c. The angles between the crystallographic axes are defined by: = the angle between b and c = the angle between a and c = the angle between a and b a, b, c, , , are collectively known as the lattice parameters or unit cell parameters or cell parameters.Source: http://www.doitpoms.ac.uk/tlplib/crystallography3/parameters.php
Lattice vectorsSource: http://www.doitpoms.ac.uk/tlplib/crystallography3/parameters.php
Common types of unit cellsSource: http://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.php
Crystal structure The structure of a crystal can be described by using the lattice type, the lattice parameters, and the motif. Lattice Type: Defines the location of the lattice points within the unit cell. Lattice Parameters: Defines the size and shape of the unit cell. Motif: List of the atoms associated with each lattice point, along with their fractional coordinates relative to the lattice point.For example, for ZnSCrystal structure: CubicLattice type: Face centeredMotif: Zn @ (0, 0, 0)S @ (, , )Source: http://www.doitpoms.ac.uk/tlplib/crystallography3/parameters.php
7 crystal systems
Source: http://126.96.36.199/wiki/images/JK-table1.png14 bravais latticesSource: http://learn.crystallography.org.uk/learn-crystallography/what-is-a-crystal/
Symmetry and symmetry elements Symmetry: In crystallization, the search for a minimum free energy and the regular packing of molecules, the crystal lattice often leads to a symmetric relationship between the molecules in addition to the unit cell translations in the crystal lattice.
Point symmetry-Rotation axis Symbol: n (e.g., 2, 3, 4, 6) An n-fold rotation axis will rotate the object by 360/n 2-fold: Diad; 3-fold: Triad; 4- fold: Tetrad; 6-fold: Hexad No change in handedness - referred to as proper symmetry operation The only rotational symmetries possible in a crystal lattice are 2, 3, 4 and 6, because it is not possible to fill space with other symmetries.
Point symmetry- Mirror planesSymbol: m A mirror plane changes the handedness of the object it is operating on. Cannot exist in crystals of an enantiomerically pure substance. Referred to as improper symmetry operation.
Point symmetry-inversion centersSymbol: i Turning an object inside out Equivalent to a point reflection through the inversion center. Similar to focal point of a lens. Changes handedness
translational symmetry-screw axisTranslation of an object by half a unit cell in the direction of the screw axis + 180 rotation. E.g., 2-fold screw 21 x, y, z = -x, -y, z + 1/2
translational symmetry-glide planesReflection and translation: a, b, c, n or d.Reflection across plane + translation of (usually) 1/2 a unit cell parallel to plane along a, b, c, face diagonal (n), or body diagonal (d)
Space groups Space groups are combination of point and translational symmetry operators. Translational symmetry operations yield 14 Bravais Lattices (3D) Point symmetry operations yield 32 Point Groups (3D) Combining simple point and translation symmetry elements together yields 73 Space Groups Two additional symmetry operations are possible Glides translation plus reflection by a mirror Screws translation plus rotation Combining these two operations yields 157 additional space groups Final total is 230 possible ways to repeat a motif in 3D or 230 Space GroupsSpace groupsHow to read (and understand) Volume A of International Tables for Crystallography: an introduction for nonspecialistsZ. Dauter et al.,J. Appl. Cryst. (2010). 43, 11501171Symmetry Properties and Their Relations to Electrical Properties Point groups as a factor for electrical propertiesNumber of rotational axes and reflection planesLack of center of symmetry
What is the center of symmetry?Point which crystal structure displays inversion symmetry.
Fig: Benzene is a centrosymmetric molecule having a center of symmetry at the centerhttps://en.wikipedia.org/wiki/Centrosymmetry#/media/File:Centrosymmetry-_Benzene.pngSymmetry Properties and Their Relations to Electrical PropertiesDielectric Polarization and Polar MaterialsAll materials form dipoles when there is an applied E Some of them has already dipoles without the effect of EHow do we decide if a material is polar or not?By looking its crystal structureIf it is symmetric anti-polarIf it has lack of center of symmetry Polar
http://hyperphysics.phy-astr.gsu.edu/hbase/electric/dielec.htmlFig: Dielectric polarizationElectric dipoles arise from charge (magnetics from spin)2332 Crystallographic Point Groups11 Centrosymmetric21 Non-centrosymmetric20 Piezoelectric1 Non-piezoelectric10 Pyroelectric10 Non-pyroelectricFerroelectricNon-ferroelectricClassification according to symmetry and polarization24Ex: BaTiO3
Above Tc=120C Cubic structure
CentrosymmetricNo spontaneous dipole momentDielectric (Anti-polar)
Below Tc=120C Tetragonal structure
Lack of center of symmetry Creates dipole momentFerroelectric
Critical temperature which the spontaneous polarization disappears is called as Tc: Curie Temperature.
http://www.gitam.edu/eresource/Engg_Phys/semester_2/dielec/BaTiO3.htmFig: (a) Tetragonal perovskite structure below Tc and the (b) Cubic structure above Tchttp://www.doitpoms.ac.uk/tlplib/ferroelectrics/phase_changes.php
25PiezoelectricityThe word piezo is Greek for push.
What is the definition?
Property of structure to form a polarization, P, under applied mechanical strain.Reversible Exhibiting mechanical strain, S, under applied electric field, E is possible.
Piezoelectricity EffectDirect piezoelectric effect the production of an electric polarization by a strain
Converse piezoelectric effect the production of a stress by an electric field
Fig: (a) Piezoelectric crystal w/o any applied stress or E (b) Direct piezoelectric effect (c) and (d) Conversed piezoelectricity (The dashed rectangle is the original sample size in )Principles of Electronic Materials and Devices, Second Edition, S.O. Kasap ( McGraw-Hill, 2002)Piezoelectricity EffectPolarization is directly proportional to the stress applied for direct piezoelectricity,Eqn 1:P=d wherePis polarization,dis piezoelectric coefficient,is stress.Strain is proportional to applied E for converse piezoelectricity,Eqn 2: =dE whereis strain,dis piezoelectric coefficient,Eis electric field.
Quartz (SiO2)SilkLead titanate (PbTiO2)*Barium titanate (BaTiO3) Lead zirconate (PbZrO3)Lithium niobate(LiNbO3)
Transducers such as ultrasoundSpeakersMicrophoneEarphoneSensors
http://www.doitpoms.ac.uk/tlplib/piezoelectrics/applications.phpAirbag patlamas29Pyroelectricity The word pyro is Greek for fire.What is the definition?Property of structure to make a spontaneous change in polarization, due to the thermal effects.It is a bridge between piezoelectric and ferroelectric.
Fig: Relationship of piezoelectric, pyroelectric and ferroelectricPyroelectricity
T increase creates spontaneou