Group theory is the study of symmetry (more than just proofs)

Group theory is the study of symmetry (more than just proofs)
Symmetric = anything invariant under transformations Examples: circle invariant under rotation x2 + y2 + z2 (invariant under rearrangement of x, y & z) sin t and cos t are invariant when t → t+2π Laws of physics are unchanging in time (invariance in time) → conservation of energy Invariance under "translation" in space → conservation of momentum Invariance under rotations → conservation of angular momentum Predicted many elementary particles. The structure and behavior of molecules and crystals depends on their different symmetries. GT often not taught in typical graduate education. And when taught, typically focuses on proofs (at least early on). But doing proofs is not really necessary to utilize GT. (good since many people don’t like proofs). Group theory can help with the analysis of a symmetric object. Group theory predicted the existence of many elementary particles before they were found experimentally.

Behavior of Crystals Many measurable quantities depend on material symmetry. Ex: The 3x3 dielectric tensor 𝜀ij consists of 6 independent components. Crystal symmetry can reduce the number of independent quantities. Like other 3x3 matrices we’ve studied, it has only 6 independent components as xy is the same as yx.

Reminder: Related to Stress and Strain (Section 3.5)
Compliance tensor Cijkl represents elastic constants. In triclinic systems, it has 21 independent components, which are reduced to 3 in cubic materials. Reminder: Related to Stress and Strain (Section 3.5) Following the same logic as the 3x3 dielectric tensor with equal off diagonal components, we were able to limit four by four tensors to only 21 possible independent constants. Compliance, stiffness tensor or the elasticity tensor Related to stress and strain tensors x=(a-b)/2

Group Theory can be used to solve the Rubik’s Cube
2 I put a reference on the class website for more details about this.

Introduction to Group Theory
Outline for today: Start easy: Discuss/identify types of symmetry Combine to Assign Point Groups Introduce Some Notation of Point Groups (If time) Explore Character Tables Extra slides on today’s powerpoint

Point Symmetry Defined as operation on a point that leaves structure the same Each Bravais lattice has one or more types of symmetry Point symmetry also occurs in molecules Triclinic is inversion symmetric (more on that soon)

Symmetry and the Hamiltonian
No difference if symmetry operation occurs before or after Hamiltonian (aka, it commutes with H) Meaning: rotate/flip/etc and get same energy Commuting operators have common eigenstates Thus, possible to classify eigenvalues of the Hamiltonian with respect to symmetry operations So? is also an eigenstate of H

Examples of Symmetry Ease
Start with the Basics Calculating eigenfunctions (i.e., diagonalization of a Hamiltonian) can be hard. Easier if symmetry restricts your choice of wavefunctions. Example? Example 1: Atom invariant under rotations, Lz and L2, leads to spherical harmonics! Example 2: Simplify vibrational problem, invariant under inversion means 1 eigenfunctions Without group theory: First you would need to identify the equations of motion (similar to what we did for lattice dynamics in Chapter 3), then find eigenvalues, then use these to find eigenfunctions. With group theory: not mirror symmetry means eigenfunctions should have mirror symmetry.

What is a line of symmetry?
A line on which an object can be folded so that both sides match exactly. Trace half a heart onto folded paper, Have them make a snowflake. then cut it out. The two halves make a whole heart. The two halves are symmetrical.

In 3D: Reflection (Mirror) Plane
A plane such that, when a mirror reflection across this plane is performed (e.g., x’=-x, y’=y, z’=z), the object looks identical. Mirror plane indicated by symbol m Butterfly: vertical mirror plane. In the plane? Maybe if ignore feet and antenna. m Approximately

Matrix Representation of Mirror Planes
(1,1,0) (-1,1,0) (1,1,0) m m (1,-1,0)

Identify a mirror plane in water
Water molecule has two mirror symmetries

Group: How many lines of symmetry do these polygons have?
equilateral 5 4 3 Do you see a pattern? 6 8

Mirror Planes in 3D [001] [100] [010] [010] [001] [100] 3+2*3=9 Group: How many mirror planes are in a cube? Note: can re-label the axes but leave the object unchanged.

Some Symmetry Definitions
Motif: the part of a symmetric design that, when repeated, creates the whole pattern (in a crystal, motif = basis) Operation: some act that reproduces the motif to create the pattern Element: an operation located at a particular point in space

Practice: 2-D Symmetry Symmetry Elements 1. Reflection (m)
Reflection across a “mirror plane” reproduces a motif = symbol for a mirror plane

Practice: 2-D Symmetry Rotation through an angle about a certain axis 2a. Two-fold rotation = 360o/2 rotation to reproduce a motif in a symmetrical pattern A Symmetrical Pattern 6 6

6 6 Practice: 2-D Symmetry a. Two-fold rotation = 360o/2 rotation
Operation a. Two-fold rotation = 360o/2 rotation to reproduce a motif in a symmetrical pattern = the symbol for a two-fold rotation 6 Motif Element 6

6 6 2-D Symmetry a. Two-fold rotation = 360o/2 rotation
to reproduce a motif in a symmetrical pattern = the symbol for a two-fold rotation 6 first operation step second operation step 6

6 6 6 2-D Symmetry b. Three-fold rotation = 360o/3 rotation
to reproduce a motif in a symmetrical pattern 6 6 6

6 6 6 2-D Symmetry b. Three-fold rotation = 360o/3 rotation
to reproduce a motif in a symmetrical pattern 6 step 1 6 step 3 6 step 2

Rotational Symmetry Link
Rotation Axes Trivial case is 360o rotation Order of rotation: 2-, 3-, 4-, and 6- correspond to 180o, 120o, 90o, and 60o. These are only symmetry rotations allowed in crystals with long-range order Small aggregates (short-range order) or molecules can also have 5-, 7-, etc. fold rotational symmetry Rotational Symmetry Link

Cut out a snowflake Identify the different symmetry elements in your snowflake. Discuss with your neighbor.

Identify the mirror planes and rotation for the pattern on my shirt.
Group Exercise: Analogy to straining a crystal Identify the mirror planes and rotation for the pattern on my shirt. How does the pattern sym. change if I stretch?

Rotation Representation What about the sign?
Rotate about x by 180 degrees Cos(180°)=-1 Sine(180°)=0 Sign depends on direction of rotation (clockwise or counterclockwise) Compare to reflection matrix

Inversion Center (x,y,z) --> (-x,-y,-z)
Center of symmetry: A point at the center of a molecule. (x,y,z) --> (-x,-y,-z) It is not necessary to have an atom in the center (benzene, ethane). Tetrahedral, triangles, pentagons do not have a center of inversion symmetry. All Bravais lattices are inversion symmetric. Once you add the basis, this symmetry may be lost. (SHG) Mo(CO)6 Draw triangle on board. Then draw upside down; not the same. Discuss how BaTiO3 perovskite not inversion symmetry. Ti atom is moved off center. All piezoelectrics are not inversion symmetric.

6 6 2-D Symmetry Symmetry Elements 3. Inversion (i)
inversion through a center to reproduce a motif = symbol for an inversion center Inversion is identical to 2-fold rotation in 2-D, but is unique in 3-D (try it with your hands) X → - X Y → -Y (2D) 6 Two fingers (x,y) goes to (-x,-y) with rotation by 180 degrees (do with left hand because easier to turn and point along positive x and y initially) Three fingers with rotation goes to (-x,-y,z). Rotation does not equal inversion in 3D. 6

Using the Rubik’s Cube to Introduce Group Theory
2

Crystal Symmetry We now have 6 unique 2-D symmetry operations: m (inversion not unique) Combinations of symmetry elements are also possible The group of operations that can map a crystal into itself defines the crystal symmetry Groups with lots of operations have high symmetry Let’s start with 2-D examples

Try combining a 2-fold rotation axis with a mirror
2-D Symmetry Try combining a 2-fold rotation axis with a mirror

2-D Symmetry Try combining a 2-fold rotation axis with a mirror
Step 1: reflect (could do either step first)

Step 1: reflect Step 2: rotate (everything) Are those the only operations? No! A second mirror is required

The result is Point Group 2mm “2mm” indicates 2 mirrors The mirrors are different (not equivalent by symmetry)

Now try combining a 4-fold rotation axis with a mirror
2-D Symmetry Now try combining a 4-fold rotation axis with a mirror

Now try combining a 4-fold rotation axis with a mirror Step 1: reflect
2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Step 1: reflect

2-D Symmetry Now try combining a 4-fold rotation axis with a mirror
Step 1: reflect Step 2: rotate 1

2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements?

2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors

2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name??

2-D Symmetry Now try combining a 4-fold rotation axis with a mirror Any other elements? Yes, two more mirrors Point group name?? 4mm Why not 4mmmm?

3-fold rotation axis with a mirror
2-D Symmetry 3-fold rotation axis with a mirror Guess the point group. Point group 3m Why not 3mmm?

6-fold rotation axis with a mirror creates ???
2-D Symmetry 6-fold rotation axis with a mirror creates ???

6-fold rotation axis with a mirror creates ??? point group 6mm
2-D Symmetry 6-fold rotation axis with a mirror creates ??? point group 6mm

2-D Symmetry The original 6 elements plus the 4 combinations creates 10 possible 2-D Point Groups: m 2mm 3m 4mm 6mm Any 2-D pattern of objects surrounding a point must conform to one of these groups Surfaces or interfaces sometimes have a different symmetry than the bulk.

Back to your snowflake Identify the point group of your snowflake. Discuss with your neighbor.

Group Exercise (if time) 2D Analogy for strained vs unstrained crystal
4mm left, 2mm right Identify the point groups of the unstretched and stretched patterns.

2D vs 3D We now have 6 unique 2-D symmetry operations: m (inversion not unique) 2D has 10 point groups: m 2mm 3m 4mm 6mm In 3D, inversion is unique Now 7 symmetry operations Every crystal described by a combination of point-symmetry elements

Rotation-inversion Axes
Combination of simultaneous rotation and inversion results in new symmetry element International symbol representation: This operation involves a rotation by (360/3) ° followed by an inversion through the center of the object.

3-D Symmetry New Symmetry Elements 4. Rotoinversion
b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Note: this is a temporary step, the intermediate motif element does not exist in the final pattern

b. 2-fold rotoinversion ( 2 ) Step 1: rotate 360/2 Step 2: invert

b. 2-fold rotoinversion ( 2 ) The result:

b. 2-fold rotoinversion ( 2 ) This is the same as m, so not a new operation

3-D Symmetry New 3-D Symmetry Elements 4. Rotoinversion
a. 1-fold rotoinversion ( 1 )

a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity)

a. 1-fold rotoinversion ( 1 ) Step 1: rotate 360/1 (identity) Step 2: invert (x,y,z) -> (-x,-y,-z) This is the same as i, so also not a new operation

We now have 10 unique 3-D symmetry operations: 1 2 3 4 6 i m 3 4 6
Examples of others at end of PPT (no time) Combinations of these elements are also possible A complete analysis of symmetry about a point in space requires that we try all possible combinations of these symmetry elements

3-D Symmetry 3-D symmetry element combinations
a. Rotation axis parallel to a mirror Same as 2-D 2 || m = 2mm 3 || m = 3m, also 4mm, 6mm b. Rotation axis  mirror 2  m = 2/m 3  m = 3/m, also 4/m, 6/m c. Most other rotations + m are impossible 2-fold axis at odd angle to mirror? Some cases at 45o or 30o are possible Examples of cyclic point groups. (From left to right and top to bottom) 2, 2mm, 3, 3m, 4, 4mm, 6, and 6mm (From Bernal, pp. 45, and 47-53)

3-D Symmetry 3-D symmetry element combinations
d. Combinations of rotations 2 + 2 at 90o  222 (third 2 required from combination) 4 + 2 at 90o  422 ( “ “ “ ) 6 + 2 at 90o  622 ( “ “ “ )

3-D Symmetry As in 2-D, the number of possible combinations is limited only by incompatibility and redundancy There are only 22 possible unique 3-D combinations, when combined with the 10 original 3-D elements yields the 32 3-D Point Groups

Symmetry operations Various Terminology E Cn Sn i σ σh σv σd
The identity transformation Cn Rotation (clockwise) through an angle of 2π/n radians, where n is an integer. Sn An improper rotation (clockwise) through an angle of 2π/n radians. Improper rotations are regular rotations followed by a reflection in the plane perpendicular to the axis of rotation. Also known as alternating axis of symmetry and rotation-reflection axis. i The inversion operator (the same as S2). In Cartesian coordinates, (x, y, z)→(−x, −y, −z). Irreducible representations that are even under this symmetry operation are usually denoted with the subscript g, and those that are odd are denoted with the subscript u. σ A mirror plane σh Horizontal reflection plane σv Vertical reflection plane σd Diagonal reflection in a plane through the origin. Element Schoenflies Hermann-Mauguin Operation Rotation axis Cn n n-fold rotation (360º/n) Identity E 1 nothing Plane of symmetry  m Reflection Center of symmetry i -1 Inversion Improper rotation axis Sn - n-fold rotation + reflection Rotary Inversion axis -n or “n bar” n-fold rotation + inversion

Two Notations for Point Groups
Rules: Each component refers to a different direction The position of an “m” in a symbol indicates the direction of the normal to the mirror plane. e.g. “mm2” indicates mirror planes perpendicular to x & y; the “2” indicates a 2-fold rotation along z. (mm2 = m2m = 2mm; just renaming the axes.) The terms 2/m (read ‘two over m”), 4/m. 6/m combine as 1 component and refer to only 1 direction. e.g. 4/m ≡ there is a mirror plane perpendicular to the 4-fold rotation axis. Element Sc Hermann-Mauguin Operation Rotation axis Cn n n-fold rotation Identity E 1 nothing Plane of symmetry  m Reflection Center of symmetry i -1 Inversion Improper rotation axis Sn - n-fold rotation + reflection Rotary Inversion axis -n or “n bar” n-fold rotation + inversion

Identify the symmetry operations.
Diamond Structure 23 might be tempting but more symmetry than just that. Same for m3 432 won’t work, no four fold axis.

Simple Crystal Structures Face-centered Cubic
12 nearest neighbors (coordination number) Close-packed planes are perpendicular to cube diagonal Close-packed planes are 6-fold symmetric (hexagons) Stacking (ABCAB…) reduces symmetry to three-fold Four 3-fold rotation axes + mirror plane, therefore Oh point group (octahedral symmetry) Examples: Cu, Ag, Au, Ni, Pd, Pt, Al

Schoenflies Terminology
The identity transformation Cn Rotation through an angle of 2π/n radians. Sn Improper rotations are regular rotations followed by a reflection in the plane perpendicular to the axis of rotation. i The inversion operator (x, y, z)→(−x, −y, −z). σ A mirror plane σh Horizontal reflection plane σv Vertical reflection plane σd Diagonal reflection in a plane through the origin. d d Point group C2v=mm2 What operations? I showed an example last week. Let’s try this the other way around. Give you point group, and identify symmetry. h Can you think of an example? v v

Representations of Point Groups
e.g. H2O, CH2, NH2 Representations of Point Groups The C2v Point Group consists of the following elements: E C2 σ(xz) σ(yz) Choose set of x,y,z axes z is usually the Cn (rotation) axis xz plane is usually the plane of the molecule Examine what happens after the molecule undergoes each symmetry operation in the point group (E, C2, 2s) σ(xz) σ(yz) What single number might represent this transformation?

With a partner: Write down the 4 matrices for these operations
E C2 σ(xz) σ(yz) C2 Transformation Matrix x,y,z  -x, -y, z Correct matrix is: sv(xz) Transformation Matrix x,y,z  x,-y,z sv(yz) Transformation Matrix x,y,z  -x,y,z σ(xz) σ(yz)

These 4 matrices are the “Matrix Representation” of the C2v point group
All point group properties transfer to the matrices as well Example: A * B = C C2 * sv(xz) =? A * B = C A*B = B*A Associative: (A*B)*C = A*B*C Identity element E: A*E = A Inverse: A-1 *A = E C2 * sv(xz) = sv(yz)

Representations σ(xz) σ(yz)
The C2v Point Group elements: E C2 σ(xz) σ(yz) What is the effect of C2 rotation on a px orbital? Quantitate the description of symmetry by using numbers to represent symmetry operation; these numbers are called Representations. Again, what single number might represent this change? C2 px = -1px Character of others? σ(xz) σ(xz) px = +1px σ(yz) σ(yz) px = -1px E E px = +1px z z — → x C2 x ← — Why it’s called B1 we will get into later. C2v E C2 σ(xz) σ(yz) character x Character table selection

Defining Representations
C2v E C2 σ(xz) σ(yz) B x Representations are subsets of the complete point group – they indicate the effect of the symmetry operations on different kinds of mathematical functions. (like the x axis) Representations are orthogonal to one another. Thus, what other functions might we have? Or just x (instead of x axis)

Representations What set of numbers represent the effect of the operations on a py? How about the z axis? Since they are spherical and the most highly symmetric, s orbitals always belong to the first symmetry class of any point group. C2v E C2 σ(xz) σ(yz) character y y C2v E C2 σ(xz) σ(yz) character z σ(xz) σ(yz)

Reducible and Irreducible Representations
Each matrix in the C2v matrix representation can be block diagonalized To block diagonalize, make each nonzero element into a 1x1 matrix When you do this, the x,y, and z axes can be treated independently Positions 1,1 always describe x-axis Positions 2,2 always describe y-axis Positions 3,3 always describe z-axis Generate a partial character table from this treatment E C2 sv(xz) sv(yz) Axis used E C2 sv(xz) sv(yz) x 1 -1 y z G 3 Irreducible Representations Reducible Repr.

rotations about an axis
Representations z σ(xz) σ(yz) The full set of representations is included in the Character Table of the group. The numbers in this table formally called The Characters of the Irreducible Representations. [NOT irreproducible!] s orbital is totally symmetric and always belongs to the A1 representation. rotations about an axis Axis used E C2 sv(xz) sv(yz) x 1 -1 y z G 3 Irreducible Representations Reducible Repr.

Character Table Representations
1. A representations indicate that the functions are symmetric with respect to rotation about the principal axis of rotation (z). 2. B representations are asymmetric with respect to rotation about the principal axis. 3. E representations are doubly degenerate. 4. T representations are triply degenerate. 5. Subscripts u and g indicate asymmetric (ungerade) or symmetric (gerade) with respect to a center of inversion.

Application for C2v Consider that an atom with a single electron in a p orbital (B or Al) is placed at a site in a crystal with C2v symmetry. The character table tells us that in general, the px, py, and pz states will all have different energies. On the other hand if the symmetry were that of a square (D4h), px and py would be degenerate, but pz might be different, and in the symmetry of an octahedron (Oh) or tetrahedron (Td), the three p states will be degenerate.

Why Else Should I Care? Character tables of point groups are used to classify molecular and crystal vibrations according to their symmetry and to predict whether a transition between two states is forbidden for symmetry reasons.

Motivation: Orbital parity
We could determine whether orbitals are even = gerade (g) or odd=ungerade (u) (in German). An orbital is g if it has a center of inversion, and u if it does not. Whether absorption of a photon to produce an electronic transition can occur is determined by whether the two orbitals involved are g or u. According to selection rules, transitions from gu and ug are allowed, but gg and uu are forbidden. Color is wave function phase One more motivation for symmetry and group theory to add onto what we’ve already said. not a center of inversion a ≠ b a a center of inversion a = b a b b b s-orbital p-orbital d-orbital gerade (g) ungerade (u) gerade (g)

Infrared Absorption and Raman
For IR spectra, a vibrational mode is active in the infrared only if it corresponds to a representation with the same symmetry as the x,y, or z axis. Raman requires relation to x2 or xy. Is c2v centrosymmetric? (inversion) No.

Example: Raman of Water
Example: Raman of Water Hydrogen-bonded 5-molecule tetrahedral nearest-neighbor structure commonly found in liquid water Hydrogen bond bending at 65 cm-1 involves O–O–O bending., whereas the hydrogen bond stretching at 162 cm-1 involves O–O stretching along the O–H…O or hydrogen bond direction--or transverse and longitudinal acoustic phonons

Simple Crystal Structures Hexagonal Close-packed
Similar to fcc, but stacking is ABAB… Smallest possible unit cell contains two atoms This leads to 6-fold symmetry Three 2-fold rotation axes perp. to 3-fold axis & close-packed layer lies in mirror plane Point group is therefore D3h E.G. Zn, Cd, Be, Mg, Re, Ru, Os points of primitive cell 120o

Double Degeneracy The C4 rotation turns px into py.
These orbitals do NOT have separate identities in this point group ... there exists a symmetry operation in a D4h molecule that turns one into the other. There is no symmetry operation that turns pz into px or py so this one remains separate. Note the d orbitals here that are doubly degenerate.

Triple Degeneracy In a pure octahedron, there are symmetry elements that turn any of the p orbitals into any others. A triply degenerate set of orbitals. By the way, these are the labels for the on-axis d orbitals ... Note the “long form” here for dz2.

Summary of Character Tables
Each point group has a set of possible symmetry operations that are conveniently listed as a matrix known as a Character Table. Point Group Label Symmetry Operations – The Order is the total number of operations In C2v the order is 4: 1 E, 1 C2, 1 v and 1 ’v C2V E C2 v (xz) ’v (yz) A1 1 A2 -1 B1 B2 Character Representation of B2 Symmetry Representation Labels Representations are subsets of the complete point group – they indicate the effect of the symmetry operations on different kinds of mathematical functions. Representations are orthogonal. Character: integer that indicates the effect of operation in a representation.

Symmetry of orbitals and functions
A pz orbital has the same symmetry as an arrow pointing along the z-axis. z z  y y E C2 v (xz) ’v (yz) x x No change  symmetric  1’s in table C2V E C2 v (xz) ’v (yz) A1 1 z x2,y2,z2 A2 -1 Rz xy B1 x, Ry xz B2 y, Rx yz

A px orbital has the same symmetry as an arrow pointing along the x-axis. z z y x y  E v (xz) No change  symmetric  1’s in table x z z y y C2 ’v (yz) Opposite  anti-symmetric  -1’s in table x x C2V E C2 v (xz) ’v (yz) A1 1 z x2,y2,z2 A2 -1 Rz xy B1 x, Ry xz B2 y, Rx yz

A py orbital has the same symmetry as an arrow pointing along the y-axis. z z y y E ’v (yz) No change  symmetric  1’s in table  x x z z y y C2 v (xz) Opposite  anti-symmetric  -1’s in table x x C2V E C2 v (xz) ’v (yz) A1 1 z x2,y2,z2 A2 -1 Rz xy B1 x, Ry xz B2 y, Rx yz

Rotation about the n axis, Rn, can be treated in a similar way. y y The z axis is pointing out of the screen! If the rotation is still in the same direction (e.g. counter clock-wise), then the result is considered symmetric. If the rotation is in the opposite direction (i.e. clock-wise), then the result is considered anti-symmetric. x x E C2 No change  symmetric  1’s in table y y x x v (xz) ’v (yz) Opposite  anti-symmetric  -1’s in table C2V E C2 v (xz) ’v (yz) A1 1 z x2,y2,z2 A2 -1 Rz xy B1 x, Ry xz B2 y, Rx yz

d orbital functions can also be treated in a similar way y y The z axis is pointing out of the screen! x x E C2 No change  symmetric  1’s in table y y x x v (xz) ’v (yz) Opposite  anti-symmetric  -1’s in table C2V E C2 v (xz) ’v (yz) A1 1 z x2,y2,z2 A2 -1 Rz xy B1 x, Ry xz B2 y, Rx yz

d orbital functions can also be treated in a similar way y y The z axis is pointing out of the screen! So these are representations of the view of the dz2 orbital and dx2-y2 orbital down the z-axis. x x No change  symmetric  1’s in table y y E C2 v (xz) ’v (yz) x x C2V E C2 v (xz) ’v (yz) A1 1 z x2,y2,z2 A2 -1 Rz xy B1 x, Ry xz B2 y, Rx yz

Symmetry Operations Rotation Matrix:
This 3D representation can be simplified into 1D changes and therefore is reducible 1D coordinate transformations are always irreducible: x y 3

Symmetry Operations What is the matrix for this rotation?
In this case, there are two coordinate changes; irreducible representation for 3-, 4-, 6-fold rotation axes are 2D x y Note: in general, ccw rotation about angle q represented by

Character Table for C4V σd σv What symmetry operations do you expect?
All members of a class have the same character. Top view A1,B1,B2,E For E, just look at C4, and the mirror planes. Discussing C2 would be confusing. A2 horizontal planes

Energy Degeneracies For operators with 2D irreducible representations result in intrinsic degeneracies Operator can produce new state (e.g., C3y ) New state must also be eigenstate of H State can be same as original state (C3y =y ) Or it is a new state; in this case, new state and unchanged state must be degenerate For diamond, fcc, bcc lattices 3D irreducible representations apply Td (e.g., diamond) and Oh point symmetry groups Result in 3-fold degeneracies 3 fold degeneracy of t2g state is lifted with a tetragonal distortion, like strain

Leaving 3N – 6=3 “vibrations”
Vibrational Modes Bond stretching and bending involves moving of the atoms in water along their individual x, y, and z axes. “Degrees of Freedom” Since there are 3 atoms and each can go 3 different directions, there are 9 (3N) different motions to keep track of. - 3 are “translational” - 3 are “rotational” Leaving 3N – 6=3 “vibrations”

(Chiral) Raman requires relation to x2 or xy.

Stereographic Projections
Represents 3-D object on 2-D surface Frequently project up, down or through equator Frequently done with the Earth’s map

Representation of Crystal Symmetry in Stereographic Projection
CRYSTAL PROJECTIONS – represents 3-D crystal on 2-D surface Unit cell axis with highest symmetry is usually selected as the polar axis. Rotation axes not in the equatorial plane are drawn with the symbol representing the rotation order of the axis at the projection point on the equatorial plane. Mirror planes are drawn as thickened lines. Inversion centers are drawn as open circles (o) in the center of the polar axis, inversion center. Rotation (2, 3, 4, 6): Rotation-inversion: Reference:

Stereographic projections
Plane of half the spherical projection (N hemisphere) Plane of projection -equatorial plane of sphere Primitive Circle outlining projection -equator Relationship between spherical & stereographic projection Stereographic projection of an isometric crystal with the faces identified by  vectors

Rotoinversion 1 (= i), 2 (= m), 3, 4, 6
Stereographic projections of symmetry groups Types of pure rotation symmetry 001 Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) symmetry elements equivalent points

Stereographic projections of symmetry groups Types of pure rotation symmetry 010 Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) symmetry elements equivalent points

Stereographic projections of symmetry groups Types of pure rotation symmetry Rotation 1, 2, 3, 4, 6 Rotoinversion 1 (= i), 2 (= m), 3, 4, 6 Draw point group diagrams (stereographic projections) symmetry elements equivalent points

Stereographic projections of symmetry groups More than one rotation axis - point group 222
symmetry elements equivalent points

Stereographic projections of symmetry groups More than one rotation axis - point group 222
symmetry elements equivalent points orthorhombic Orthorhombic a ≠ b ≠ c, a = b = g = 90º

Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm
[001] symmetry elements equivalent points

Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm
symmetry elements equivalent points

(by definition c is the unique axis)
Stereographic projections of symmetry groups Rotation + mirrors - point group 4mm symmetry elements equivalent points tetragonal Tetragonal a = b ≠ c, a = b = g = 90º (by definition c is the unique axis)

Group Exercise In groups, plot the rotations on the stereographic projection of a cube.

Stereographic projections of symmetry groups Rotation + mirrors - point group 2/m
[010]

Stereographic projections of symmetry groups Rotation + mirrors - point group 2/m
symmetry elements equivalent points monoclinic

There are a total of 32 independent point groups

Fun Motivation: Gaming
Escher Mobile runs on phones and PCs create symmetrical images to “effortlessly master the main crystallographic concepts”. “Not only does this application make the fundamentals of crystallography a cinch, but it’s been designed to actually be a lot of fun” Check out the P6mm rotationally symmetrical “Duck Crystal”

Point Groups There are a total of 32 point groups that conform to this rule (the “crystallographic point groups”) and the combination of crystallographically acceptable point groups with the 14 Bravais lattices gives rise to space groups.

3-D Symmetry The 32 3-D Point Groups
Every 3-D pattern must conform to one of them. This includes every crystal, and every point within a crystal Table 5.1 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

3-D Symmetry The 32 3-D Point Groups Regrouped by Crystal System
(more later when we consider translations) Table 5.3 of Klein (2002) Manual of Mineral Science, John Wiley and Sons

Determination of Symmetry Point Groups (Group Theory)
Many papers have this information about the materials.

Determining Point Groups
Find the highest order rotation axis = n. Are there n C2 axes perpendicular to this principle axis? Is there a mirror plane perpendicular to the principle axis? Are there dihedral mirror planes? Are there vertical mirror planes? yes no Dn set Cn set yes no Dnh yes no Cnh yes no Dnd Dn yes no Cnv 5. Is there a S2n? yes no S2n Cn

Draw and describe the symmetry elements of all Bravais lattices.
2.4

Group Example In groups, determine the point group (Schoenflies notation) of an equilateral triangle. Then define the matrices that leave the triangle invariant.

Matrix representation of the rotation point groups
- Matrix representation of the rotation point groups What is a group? A group is a set of objects that form a closed set: if you combine any two of them together, the result is simply a different member of that same group of objects. Rotations in a given point group form closed sets - try it for yourself! Complicated slide with much data. Illustrates the different matrices used. Note that the coefficients for the 6-fold rotation axes do not have unit values. Note: the 3rd matrix in the 1st column (x-diad) is missing a “-” on the 33 element; this is corrected in this slide. Also, in the 2nd from the bottom, last column: the 33 element should be +1, not -1. In some versions of the book, in the last matrix (bottom right corner) the 33 element is incorrectly given as -1; here the +1 is correct. Kocks: Ch. 1 Table II

Symmetry Representations: Mulliken Symbols and their Meaning

Character Table Representations
1. Characters of +1 indicate that the basis function is unchanged by the symmetry operation. 2. Characters of -1 indicate that the basis function is reversed by the symmetry operation. 3. Characters of 0 indicate that the basis function undergoes a more complicated change.

C3V What symmetry operations do you expect?

Stretches always have a symmetric mode (doesn’t change the symmetry)
Also flattening doesn’t change the symmetry (until completely flat)

Symmetry Product Table

Graphene with and without Strain
Without strain Graphene is in space group 191 which has a factor group isomorphic to D6h 2 atoms/unit cell  3 optical phonon modes E2g and B2g modes exist F Tuinstra and JL Koenig. Jour of Chem Phy, 55 3, 1126 (1970)

Only one mode Raman active (first order)
G band, doubly degenerate

Strain Under uni-axial strain the symmetry group is broken along with the degeneracy Huang et al. PNAS April 21, 2009, 106 (16)

Identify the cubic groups.
Which one is diamond? The O-group has no inversion symmetry Gold is Oh Applets: (Gold)

Which of these flags have a line of symmetry?
United States of America Canada No Maryland England No

Invariance to transformation as an indicator of facial symmetry:
Mirror image Symmetry frequently considered more pleasing.

Examples (Groups) Determine the number of mirror planes. Triclinic has no mirror planes. Monoclinic has one plane midway between and parallel to the bases.

Group Exercise Below are two “crystals” and a polygon.
Identify the point group symmetry operations of the three objects (assume the crystals are of infinite size). Show that the point group of the two crystals are different and that one of them has the equivalent point group to the polygon. M1.1, A and C have eight symmetry elements (4mm): identity, 90, 180, 270 rotation, four mirrors In 2D, inversion does not count as it is the same as 180 rotation. B does not have the mirror planes, and thus cannot be the same point group. a b c

Lattice goes into itself through Symmetry without translation
Operation Element Inversion Point Reflection Plane Rotation Axis Rotoinversion Axes The system of symmetry operations The point group symbol Or the Schonflies symbol (popular for group theory or spectroscopy)

Which crystal lattice has higher symmetry?
High Symmetry Means Many Operations recreate the same crystal (after operation it is impossible to tell that an operation occurred) Ortho Which crystal lattice has higher symmetry?

Symmetry Element Properties (also true for 2D)
Must satisfy conditions: Two successive symmetry operations result in a further symmetry element: A * B = C Associative rule: (A*B)*C = A*B*C Identity element E: A*E = A Every symmetry element possesses inverse: A-1 *A = E A*B = B*A This leads to 32 distinct crystallographic groups If you were also to allow translations, 7 --> 14, allowing 230 combinations, known as space groups

New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 )
3-D Symmetry New Symmetry Elements 4. Rotoinversion c. 3-fold rotoinversion ( 3 )

c. 3-fold rotoinversion ( 3 ) Step 1: rotate 360o/3 Again, this is a temporary step, the intermediate motif element does not exist in the final pattern 1

c. 3-fold rotoinversion ( 3 ) Step 2: invert through center

c. 3-fold rotoinversion ( 3 ) Completion of the first sequence 1 2

c. 3-fold rotoinversion ( 3 ) Rotate another 360/3

c. 3-fold rotoinversion ( 3 ) Invert through center

c. 3-fold rotoinversion ( 3 ) Complete second step to create face 3 3 1 2

c. 3-fold rotoinversion ( 3 ) Third step creates face 4 (3  (1)  4) 3 1 4 2

c. 3-fold rotoinversion ( 3 ) Fourth step creates face 5 (4  (2)  5) 5 1 2

c. 3-fold rotoinversion ( 3 ) Fifth step creates face 6 (5  (3)  6) Sixth step returns to face 1 5 1 6

3-D Symmetry New Symmetry Elements 4. Rotoinversion This is unique
c. 3-fold rotoinversion ( 3 ) This is unique 3 5 1 4 2 6

New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 )
3-D Symmetry New Symmetry Elements 4. Rotoinversion d. 4-fold rotoinversion ( 4 )

d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4

d. 4-fold rotoinversion ( 4 ) 1: Rotate 360/4 2: Invert

d. 4-fold rotoinversion ( 4 ) This is also a unique operation

e. 6-fold rotoinversion ( 6 ) Begin with this framework:

New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 )
3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 1

3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 2

3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 2

3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 2 4

3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 5 2 4

3-D Symmetry New Symmetry Elements 4. Rotoinversion e. 6-fold rotoinversion ( 6 ) 1 3 5 2 6 4

e. 6-fold rotoinversion ( 6 ) Note: this is the same as a 3-fold rotation axis perpendicular to a mirror plane (combinations of elements follows) Top View

Hermann-Mauguin Symbols
To begin, write a number representing each of the unique rotation axes present. A unique rotation axis is one that exists by itself and is not produced by another symmetry operation. In this case, all three 2-fold axes are unique, because each is perpendicular to a different shaped face, so we write a 2 (for 2-fold) for each axis, A: 2 2 2

Next we write an "m" for each unique mirror plane. Again, a unique mirror plane is one that is not produced by any other symmetry operation. In this example, we can tell that each mirror is unique because each one cuts a different looking face. So, we write: 2 m 2 m 2 m

If any of the axes are perpendicular to a mirror plane we put a slash (/) between the symbol for the axis and the symbol for the mirror plane. In this case, each of the 2-fold axes are perpendicular to mirror planes, so our symbol becomes: 2/m 2/m 2/m The Dnh Groups: Briefly know as, D2h =mmm, D4h =4/mmm, D6h =6/mmm,

What happens if we stretch the cube along one direction?
Let’s try another… Each have E, 8C3, 3C2, 6s4, 6sigmad 6 comes from 2 edges along each part S4’s also where 6C2’s are (hard to visualize, so not including on slide) What happens if we stretch the cube along one direction?

Group theory is the study of symmetry (more than just proofs)

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Group theory is the study of symmetry (more than just proofs)

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