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://chemistrytextbookcrawl.blogspot.com.tr/2012/08/chapter-3.html

Source: http://202.141.40.218/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.

Ex: Benzene

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.

http://www.doitpoms.ac.uk/tlplib/piezoelectrics/polarisation.phpPiezoelectricityExamples

Quartz (SiO2)SilkLead titanate (PbTiO2)*Barium titanate (BaTiO3) Lead zirconate (PbZrO3)Lithium niobate(LiNbO3)

Applications

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.

http://www.doitpoms.ac.uk/tlplib/piezoelectrics/spontaneous_polar.php

Fig: Relationship of piezoelectric, pyroelectric and ferroelectricPyroelectricity

T increase creates spontaneous polarization.

ExamplesLithium tantalate(LiTaO3)Polyvnyl fluoridesApplicationsThermal imaging (IR detection)Pollutant control

http://www.slac.stanford.edu/grp/arb/tn/arbvol5/AARD459.pdfFig: Pyroelectric crystal with Intrinsic dipole moment and its reaction to increase in T

Figure 1. If a pyroelectric crystal with an intrinsic dipole moment (top) is fashioned into a circuit with electrodes attached on each surface (middle), an increase in temperature T prompts the spontaneous polarization PS to decrease as the dipole moments, on average, diminish in magnitude. The horizontal tilting of the dipoles, pictured at bottom, signifies the effect. A current flows to compensate for the change in bound charge that accumulates on the crystal edges.

http://www.doitpoms.ac.uk/tlplib/pyroelectricity/infrared.php

31Ferroelectricity The prefix ferro means iron (but not most do not contain iron).

What is the definition?Property of structure to have a spontaneous dipole moment (polarization) which can be reversed by an applied electric field.Switchable polarization

Subset of both piezoelectric and pyroelectric materialsNet permanent dipole moment

Hysteresis Loop:Assume we have fully polarized material.Aligned dipole moments are affected by reversed E.Direction of dipole moments changes in time. Polarization is reversed at the end (until applied E is removed).http://www.doitpoms.ac.uk/tlplib/ferroelectrics/switching2.php

Fig: Hysteresis loop

http://www.ceramicindustry.com/articles/92711-the-world-of-ferroelectric-ceramicsHysteresis Loop:Temperature dependenceWhen T increases, hysteresis loop becomes sharper and thinner.

ReasonsDisorder which increases with T and decreases net dipole momentPhase transformations which can create new dipole moments

Fig: Hysteresis loop for different T

http://www.doitpoms.ac.uk/tlplib/ferroelectrics/temp_dependence.phpFerroelectricityExamples

Barium titanate (BaTiO3) bt. 298-393 KRochelle salt bt 255-296 K

Applications

Computer memory elementsCapacitors

http://www.doitpoms.ac.uk/tlplib/ferroelectrics/why.php

Applications and explanation of ferroelectricity, pyroelectricity and piezoelectricity Ferroelectric Materials ApplicationsFerroelectric capacitorsMedical ultrasound machinesHigh quality IR cameras

Ferroelectric RAMSensorsFire sensorsVibration sensorsSonar Sensors

J.F. Scott (2000). Ferroelectric Memories. Springer. ISBN3-540-66387-8.

https://www.youtube.com/watch?v=-aFm8_nE6bM&list=PLC3582BD2E81221C9

Ferroelectricity: zoomed in a little bit!Pyroelectric materials applicationsPyroelectricity can be visualized as one side of a triangle, where each corner represents energy states in the crystal:kinetic, electrical and thermal energy.

The side between electrical and thermal corners represents the pyroelectric effect and produces no kinetic energy. The side between kinetic and electrical corners represents the piezoelectric effect and produces no thermal energy.

Buchanan, Relva C. (2004). Ceramic Materials for Electronics: Third Edition, Revised and Expanded (Third ed.). Cincinnati, Ohio: Marcel Dekker, Inc. p. 217. ISBN 0-8247-4028-9.Pyroelectric materials applicationsApplications: IR sensors (heat sensors)Heat engines (e. g. lowering the operating temperatures, using less bulky equipment, and having fewer moving parts) can be used in an Ericsson cycle to reach higher levels of Carnot efficiency

AdvantagesWide thermal and electromagnetic sensitivityFast response (0 to 10 Hz) Low-costWork at ambient temperature

Kouchachvili, L; Ikura, M (2007). "Pyroelectric conversionEffects of P(VDFTrFE) preconditioning on power conversion". Journal of Electrostatics 65: 182. doi:10.1016/j.elstat.2006.07.014.

Piezoelectric materials applicationsIn piezoelectric materials, electrical energy is converted into mechanical energy and/or vice versa. So, piezoelectricity is used in:

SpeakersMicrophonesSolar devices ActuatorsHigh precision positioning with electrical polarizationIgniters

MAT 408 lecture notes, Prof. Dr. Mehmet Ali Gulgun, Faculty of Engineering and Natural Sciences, Sabanci University, Fall 2015.Piezoelectric materials applications:grand mothers part

Perovskite Structure: ABO3

Perovskite structure has a general formula of ABO3 Perovskite is one of the widely found structures in the materials (mostly ceramics) exhibiting ferroelectricity, pyroelectricity and piezoelectricity or a combination of them.

Perovskites are composed of two different cations and an anion, for which the sizes of cations differ considerably, like in CaTiO3 (perovskite) perovskite is also the name of CaTiO3 mineral

Where Ca+2 and O-2 form a closed-packed cubic structure, with the smaller cation (Ti+4) in the octahedral interstitial site at the center of the cube.

MAT 408 lecture notes, Prof. Dr. Mehmet Ali Gulgun, Faculty of Engineering and Natural Sciences, Sabanci University, Fall 2015.

Perovskite Structure: ABO3

Fig. 1. Perovskite Structure.MAT 408 lecture notes, Prof. Dr. Mehmet Ali Gulgun, Faculty of Engineering and Natural Sciences, Sabanci University, Fall 2015.Goldschmidt tolerance factor

Goldschmidt's tolerance factor indicates the relation of the chemical composition of a perovskite to its stabilitythe tolerance factor can be used to calculate the compatibility of an ion with a crystal structureThe first description of the tolerance factor for perovskite was made by Victor Moritz Goldschmidt in 1926rA is the radius of the A-cation. rB is the radius of the B-cation. rO is the radius of the anion.

Goldschmidt tolerance factor

A practical example of using piezoelectric materials in new technologies

Referenceshttp://www.doitpoms.ac.uk/tlplib/crystallography3/intro.phphttp://www.doitpoms.ac.uk/tlplib/crystallography3/lattice.phphttp://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.phphttp://www.aps.anl.gov/Xray_Science_Division/Powder_Diffraction_Crystallography/Introduction_to_Crystallography/#03http://www.doitpoms.ac.uk/tlplib/crystallography3/parameters.phphttp://www.doitpoms.ac.uk/tlplib/crystallography3/parameters.phphttp://www.doitpoms.ac.uk/tlplib/crystallography3/unit_cell.phphttp://www.doitpoms.ac.uk/tlplib/crystallography3/parameters.phphttp://chemistrytextbookcrawl.blogspot.com.tr/2012/08/chapter-3.htmlhttp://202.141.40.218/wiki/images/JK-table1.pnghttp://learn.crystallography.org.uk/learn-crystallography/what-is-a-crystal/Hummel, R. E. (2011). Electronic Porperties of Materials. New York: Springer.Moulson, A. J., & Herbert, J. M. (1990). Electroceramics. Cambridge: Chapman&Hall.Glgn, M.A..(2014). An Introductory Course to Ceramics. https://www.ucl.ac.uk/qsd/people/teaching/EOPM-Part3.pdfReferenceshttp://www.doitpoms.ac.uk/tlplib/ferroelectrics/phase_changes.phphttp://www.doitpoms.ac.uk/tlplib/ferroelectrics/why.phphttp://www.gitam.edu/eresource/Engg_Phys/semester_2/dielec/BaTiO3.htmhttp://www.nanomotion.com/piezo-ceramic-motor-technology/piezoelectric-effect/http://www.doitpoms.ac.uk/tlplib/piezoelectrics/polarisation.phphttp://www.slac.stanford.edu/grp/arb/tn/arbvol5/AARD459.pdfhttp://www.ceramicindustry.com/articles/92711-the-world-of-ferroelectric-ceramicshttp://www.doitpoms.ac.uk/tlplib/ferroelectrics/switching2.phphttp://academic.uprm.edu/pcaceres/Courses/Smart/SMD-5A.pdfJ. F. Scott (2000). Ferroelectric Memories. Springer. ISBN3-540-66387-8https://www.youtube.com/watch?v=-aFm8_nE6bM&list=PLC3582BD2E81221C9Buchanan, Relva C. (2004). Ceramic Materials for Electronics: Third Edition, Revised and Expanded (Third ed.). Cincinnati, Ohio: Marcel Dekker, Inc. p. 217. ISBN 0-8247-4028-9Kouchachvili, L; Ikura, M (2007). "Pyroelectric conversionEffects of P(VDFTrFE) preconditioning on power conversion". Journal of Electrostatics 65: 182. doi:10.1016/j.elstat.2006.07.014MAT 408 lecture notes, Prof. Dr. Mehmet Ali Gulgun, Faculty of Engineering and Natural Sciences, Sabanci University, Fall 2015