applications of representations theory in solid-state...
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University of Hamburg, Department of Earth Sciences
Applications of representations theory in Applications of representations theory in solidsolid--state physics and chemistrystate physics and chemistry
Boriana Mihailova
Vibrations in molecules and solidsVibrations in molecules and solids
GoalsGoals
Group theory analysis of vibrational spectra
“User-friendly” approach to group-theory analysis
fun easy absolutely necessary!
without forgetting thatwithout forgetting that
Human beingHuman being Clicking species
= = Thinking species
OutlineOutline
2. Brillouin zone centre spectroscopy. Raman and Infrared the most commonly used methods to study atomic dynamics
1. Atomic dynamics. Phonons
3. Group theory analysis (GTA) :- selection rules for conventional IR and Raman spectra
“by hand” and using on-line tools
- “deviations” from the GTA predictions
- second order phonon processes; compatibility relations;hyper-Raman scattering, a note about resonance Raman spectra
4. Electron states. Selection rules for optical transitions
ReferencesReferences
http://reference.iucr.org/dictionary/
http://www.cryst.ehu.es
http://www.staff.ncl.ac.uk/j.p.goss/symmetry/index.html
http://minerals.gps.caltech.edu/
http://riodb01.ibase.aist.go.jp/sdbs/cgi-bin/cre_index.cgi?lang=eng
Rousseau, Bauman & Porto, Normal mode determination in crystals.J. Raman Spectrosc. 10, (1981) 253-290
Fateley, McDevitt & Bentley, Infrared and Raman selection rules for lattice vibrations: the Correlation method. Appl. Spectrosc. 25, (1971) 155-173
M. Dresselhaus, G. Dresselhaus, A. Jorio, Group Theory. Application to the Physics of Condensed Matter. Springer 2008
Atomic dynamics in crystals Atomic dynamics in crystals
Visualization: UNISOFTVisualization: UNISOFT, Prof. G. Eckold et al., University of Göttingen
KLiSOKLiSO44, hexagonal, hexagonal
6
Crystal normal modes (Crystal normal modes (eigenmodeseigenmodes))
Atomic vibrations in crystals = Superposition of normal modes (eigenmodes)
a mode involving mainly S-Ot bond stretchinge.g.,
a mode involving SO4 translations and Li motions vs K atoms
PhononsPhonons
Atomic vibrations in a periodicperiodic solid
standing elastic waves normal modes (S, {ui}s )
crystals : N atoms in the primitive unit cell vibrating in the 3D space 3N degrees of freedom finite number of normal states
quantization of crystal vibrational energy
N atoms 3 dimensions 3N phonons phonons
phononphonon quantum of crystal vibrational energy
phonons: quasi-particles (elementary excitations in solids)- En = (n+1/2)ħ, - m0 = 0, p = ħK (quasi-momentum), K q RL- integer spin
Bose-Einstein statistics: n(,T)= 1/[exp(ħ/kBT)-1] (equilibrium population of phonons at temperature T)
Harmonic oscillator
2
n=0
n=1
n=2
n=3
Phonon frequencies and atom vector displacementsPhonon frequencies and atom vector displacements
phonon S, {ui}s eigenvalues and eigenvectors of D = f (mi, K({ri}), {ri})
K m2m1
a
Hooke’s low : Kxxm Atomic bonds elastic springs
Equation of motion for a 3D crystal with N atoms in the primitive unit cell :
in a matrix form: qq wqDw )(2
(3N1) (3N1)(3N3N)
)(1)( ''
''' qq ss
ssss mm
D dynamical matrix
second derivativesof the crystal potential
= 1,2,3 i = 1,..., N
atomic vector displacements
''
,''',',2 )(
iiiii wDw qq q
qq ,,
1 i
ii u
mw
0)( 2 qwδqD
mK
phonon S, {ui}s carry essential structural information !
Brillouin zone
chain in 1D
Types of phononsTypes of phonons
phonon dispersion: ac(q) op(q),
diatomic chain
AcousticAcoustic phonon: u1, u2, in-phaseOpticalOptical phonon: u1, u2, out-of-phase
for q 0, op > ac
1 Longitudinal1 Longitudinal: wave polarization (u) || wave propagation (q)22 TransverseTransverse: wave polarization (u) wave propagation (q)
LALA
TATA
LOLO
TOTO
3D crystal with N atoms per cell : 3 acoustic and 3N – 3 optical phonons
chain in 3D
induced dipole moment interact with light
Phonon (Raman and IR) spectroscopyPhonon (Raman and IR) spectroscopy
Infrared absorption: Infrared absorption:
Raman scattering Raman scattering inelastic light scattering from optical phonons
electromagnetic wave as a probe radiation (photon – opt. phonon interaction):
, kground state
excited state
s, k s, ki
Stokes anti-Stokes is is
)()( phononGS
phononESphoton EE
Kkk is Kkk is
anti-Stokes Stokes
Phonon (Raman and IR) spectroscopyPhonon (Raman and IR) spectroscopy
• only optical phonons near the BZ centre are involved Kkk si ikkK 2max (e.g. Raman, 180-scattering geometry)
a
K max
0Kphoton-phonon interaction only for
(a ~ 10 Å)i (IR, vis, UV) ~ 103 – 105 Å ki ~ 10-5 – 10-3 Å Kmax
10 [cm-1] 1.24 [meV] 10 [cm-1] 0.30 [THz] [Å].[cm-1] = 108
cm-1 E= ħck = ħc(2/) = hc(1/)• spectroscopic units:
• IR and Raman spectra are different for the same crystal
different interaction phenomena different selection rules !
Raman and IR intensitiesRaman and IR intensities
IR activityIR activity: induced dipole moment due to the change in the atomic positions
Qk – configurational coordinate
0, IR activity
),,( zyx μ
IR: “asymmetrical”, “one-directional”
Raman: “symmetrical”, “two-directional”
N.B.! simultaneous IR and Raman activity – only in non-centrosymmetric structures
...)(0
0 kk
Q μμμ
2
0
k
IR QI μ
Raman activityRaman activity: induced dipole moment due to deformation of the e- shell
Polarizability tensor:
0, Raman activity
zzyzxz
yzyyxy
xzxyxx
α P = .E
induced polarization (dipole moment per unit cell)
...)(0
0 kk
Q ααα
2
0i
kSRaman Q
I eαe
Raman and IR activity in moleculesRaman and IR activity in molecules
sym. stretching
bending
anti-sym. stretching
H2O
IRIR : change in RamanRaman : change in pol. ellips. size (Tr()), shape () and/or orientation (ij, ij)
1
2
3
Raman and IR activity in crystalsRaman and IR activity in crystals
Pb2
Pb1
Op
Ot
P
ba
c
Pb2
Pb1
Op
Ot
P
ba
c
Raman-active
Pb2
Pb1
Op
Ot
P
ba
c
IR-active
Raman-active IR-active
Pb2
Pb1
Op
Ot
P
ba
c
Isolated TO4 group
IR-active
Raman-active
R3mCrystal: Pb3(PO4)2,
Methods for normal phonon mode determinationMethods for normal phonon mode determination
N.B.! Tabulated information for:first-order, linear-response, non-resonance interaction processes
(one phonon only) (one photon only) (ħi < EESelectron-EGS
electron)
Three techniques of selection rule determination at the Brillouin zone centre:
• Factor group analysis (Bhagavantum & Venkatarayudu)
• Molecular site group analysis (Correlation method)
the effect of each symmetry operation in the factor group on each type of atom in the unit cell
Rousseau, Bauman & Porto, J. Raman Spectrosc. 10, (1981) 253-290
Bilbao Server, SAM, Bilbao Server, SAM, http://www.cryst.ehu.es/rep/sam.html
symmetry analysis of the ionic group (molecule) site symmetry of the central atom + factor group symmetry
• Nuclear site group analysisNuclear site group analysis
site symmetry analysis is carried out on every atom in the primitive unit cell set of tables ensuring a great ease in selection rule determinationpreliminary info required: space group and occupied Wyckoff positions
Spectroscopy and Group theory. The PhilosophySpectroscopy and Group theory. The Philosophy
normal phonon modes irreducible representations
Lattice (molecular) dynamics
obey the symmetry restrictions from all symmetry operations in the group
= normal phonon modes
symmetry-specific transformation upon different symmetry operations
Crystals: Space groups specificpoint symmetry at each k
Raman and IR k = 0 (-point)Crystals, molecules: Point group Irreducible representations
n3
4
3
2
1
Dynamical matrix in a diagonal formDynamical matrix in a diagonal form
jiij uu
UD
2
ii wu s 2
nnzznxz
xxxzxyxx
xzzzyzxz
xyyzyyxy
nnxzxxxzxyxx
DD
DDDDDDDDDDDD
DDDDD
1
22121212
12111111
12111111
12111111
41
41
41
31
31
21
11
43
43
43
43
42
42
42
42
41
41
41
41
32
32
32
32
31
31
31
31
2111
22111
2111
111222
1111
zyxzyxzyx
nznynxzyxzyx
uuuuuuuuuuuuuuuuuu
(eigenvalues)(eigenvectors)
singlesingle
doublydegenerate
triplydegenerate
atomic displacement vectors for a given normal mode (symmetry of irrep)
2 symmetry-equivalent sets of a.d.v.
3 symmetry-equivalent sets of a.d.v.
1 symmetry-allowed set of a.d.v.another symmetry-allowed set
Dynamical matrix and Group TheoryDynamical matrix and Group Theory
D commutates with all matrix forms of the mechanical representation m
m D = mDm block-diag. form via irreps, i.e. S: S-1mS = m = S S-1
D S S-1 = D S S-1 S-1()S
S-1DS = S-1DS
block-diagonalized form of D !
D has two parts “force-field” F and “geometry” G
F peak positions (exact values of eigenfrequencies) and intensities (exact values of the atomic displacement amplitudes)
G mode degeneracy (relative values of eigen frequencies)mode symmetry (sets of relative atomic amplitudes)
S-1DS = S-1DSS-1S S-1S
Character tablesCharacter tables
normal phonon modes irreducible representations
Symmetry element: matrix representation A
C3v (3m)
001100010
r3
r2
r1
C3 (3)
v (m)
Point group
1 3 m
3:
Character: i
iiA)(Tr A
Symmetry elements
100010001
1:
100010001
m:
010100001
reducible irreducible (block-diagonal)
3 0 1reducible
characters
21
230
23
210
001
100010001
irreducible A1
E1 1 1
2 -1 0
A1 + E 3 0 1
Mulliken symbols
Reminder: (1 231)
(232)
MullikenMulliken symbolssymbols
A, B : 1D representations non-degenerate (single) modeonly one set of atom vector displacements (u1, u2,…,uN) for a given wavenumber
A: symmetric with respect to the principle rotation axis n (Cn) B: anti-symmetric with respect to the principle rotation axis n (Cn) E: 2D representation doubly degenerate mode
two sets of atom vector displacements (u1, u2,…,uN) for a given wavenumber
T (F): 3D representation triply degenerate modethree sets of atom vector displacements (u1, u2,…,uN) for a given wavenumber
E mode
2D system
X
Y
T mode
3D system
X
Y
Z
subscripts g, u (Xg, Xu) : symmetric or anti-symmetric to inversion 1superscripts ’,” (X’, X”) : symmetric or anti-symmetric to a mirror plane msubscripts 1,2 (X1, X2) : symmetric or anti-symmetric to add. m or n
• biaxial crystals (tricl, monocl, orth) : only A , B• uniaxial crystals (tetra, rhombo, hexa): A,B, E• cubic: A, E, T
Symbols and notationsSymbols and notations
Triclinic Monoclinic Trigonal (Rhombohedral)
Tetragonal Hexagonal Cubic
C1 1 C2 2 C3 3 C4 4 C6 6 T 23 Ci 1 CS m C3i 3 S4 4 C3h 6 C2h 2/m C4h 4/m C6h 6/m Th 3m C2v mm2 C3v 3m C4v 4mm C6v 6mm D3d m3 D2d 42m D3h 6m2 Td m34 D2 222 D3 32 D4 422 D6 622 O 432 D2h mmm D4h 4/mmm D6h 6/mmm Oh mm3
Dn: E, Cn; nC2 to Cn; T: tetrahedral symmetry; O: octahedral (cubic) symmetry
Point groups:
Symmetry element Schönflies notation International (Hermann-Mauguin)
Identity E 1 Rotation axes Cn n = 1, 2, 3, 4, 6 Mirror planes m to n-fold axis || to n-fold axis bisecting (2,2)
hvd
m, mz mv, md, m’
Inversion I 1 Rotoinversion axes Sn 6,4,3,2,1n Translation tn tn Screw axes k
nC nk Glide planes g a, b, c, n, d
(on the Bilbao Server: POINT)
Spectroscopy and Group theory. General outlinesSpectroscopy and Group theory. General outlines
Identify the symmetry operations defining the point group G in equilibrium
Find the irrep decomposition of equivalence rep eq (atoms remaining invariant under the sym. operations of G); (eq) for each SO is the number of inv. atoms vibrations transformation properties of a radial vector r (x,y,z)
mol.vibr. = eq vec – trans – rot
latt.vibr. = eq vec,
different degrees of freedom
trans: simple translation
rot : simple rotation
about the centre of mass
irreps of polar vector, e.g. r (x,y,z)irreps of axial vector, e.g. r p
)1(iC
)1(iC
Infrared activity: transformation properties of a vector (1st-rank tensor) Gvec irrep symtotally initialfinal x,y,z remain positive
• GTA of the point group the corresponding k-point (N.B. ac(k0) 0)• compatibility relations between neighbouring k-points to form a phonon branch
Lattice vibrations for k 0:
opt.latt.vibr. = eq vec – acous (ac(k=0) = 0)
Raman activity: transformation properties of a 2nd-rank symmetric tensor
x2, y2, z2, xy, zy, zx remain positive finalinitial Ramanirrep'
Working example for simple point groups, Cs (m),
Constructing character tablesConstructing character tables
X
Z
Y ),,( zyx
),,( zyx
Cs (m)
sym. operations
E (1) h (m) functions
Point group selection rules
irreps(symmetry types of modes)
1
1
Two single irreps: totally sym A’, antisym to reflection A’’
A’
A”
1
1
x, y
z
change in sign
)0,,()'( yxA μ
),0,0()''( zA μIR di
pole
momen
ts
, x2, y2, z2, xy
, xz, yz small rot. about Z does not change the symmetry
, Jz
, Jx, Jy
Raman
tens
ors
c
bdda
A )'(α
fe
fe
A )''(α
ExerciseExercise: construct the character table for D2 (222)
Constructing character tablesConstructing character tables
D2 (222)
sym. operations
functions
Point group selection rules
irreps(modes)
A
B2
B1
B3
1
1
1
1 1 1 1
11
1
1
1
11
1
1
E (1) C2x (2x) C2z (2z)C2y (2y)
C2x (2x)
),,( zyx
C2z (2z)
C2y (2y)
),,( zyx ),,( zyx
),,( zyx
x2, y2, z2
x, Jx, yzy, Jy, xz
z, Jz, xy
),0,0()( 1 zB μ )0,,0()( 2 yB μ )0,0,()( 3 xB μ
c
ba
A)(α
dd
B )( 1α
e
eB )( 2α
ffB )( 3α
A : IR-inactive
Dipole moments
Polarizability tensors
),,( zyx ),,( zyx ),,( zyx ),,( zyx
1:2x:2y:2z:
ExerciseExercise: construct the character table for H2O
Normal modes in molecules. HNormal modes in molecules. H22OO
sym. operationsPoint group
selection rules
C2 (mm2) E (1) C2 (2) y (my) x (mx) functions
A1
B1
A2
B2
1
11
1
1 11
11
11
1 11
1 1irreps
Jz, xyx, Jy, xz
y, Jx, yz
z, x2, y2, z2
mol.vibr. = eq vec – trans – rot
x (mx)
y (my)C2 (2)
),,( zyx ),,( zyx ),,( zyx ),,( zyx
E (1):C2 (2):y (my):x (mx):
Z
XY
Characters for atomic site transformations1 2 my mx
eq(H2O) 3
atomsremaining invariant
OHH
O OHH
O
11 3 2A1 + B2
either 2A1 + B2or A1 + 2B2
only 2A1 + B2
mol.vibr. = (2A1+B2) (A1+B1+B2) –(A1+B1+B2) –(A2+B1+B2) =
= 2A1+2B1 +2B2+B2+A2+A1–– A1 – B1 – B2 –A2 – B1 – B2
H2O = 2A1 (Raman, IR)+B2 (Raman, IR)
3 irreps
Correlation method. Example HCorrelation method. Example H22OO
x (mx)
y (my)C2 (2)
Z
XY
Using tabulated info: Character tables Correlation tables
O: site symmetry C2
H: site symmetry CS = C1h
H2O: Point group C2
e.g. http://www.staff.ncl.ac.uk/j.p.goss/symmetry/index.html
Correlation method. Example HCorrelation method. Example H22OO
x (mx)
y (my)C2 (2)
Z
XY
O: site symmetry C2
H: site symmetry CS = C1h
H2O: Point group C2
Oxygen
Hydrogen
Identify the translation modes: (vibration translation)
O (C2) : A1 + B1 + B2
2H (Cs) : 2(2A’ + A”)= 4A’ + 2A”
Correlation method. Example HCorrelation method. Example H22O O (C2)
Correlate the site group irreps with the point group irrep
O (C2) : A1 + B1 + B2 the same
2H (Cs) : 4A’ + 2A” 2(A1 + B2) + A2 + B1
H2O = A1+B1+B2 + 2A1+A2+B1+ 2B2 – (A1+B1+B2) – (A2+B1+B2) = 2A1 + B2
O atom H atoms translation rotation
Normal modes in molecules. Bilbao Server: SAMNormal modes in molecules. Bilbao Server: SAM
O: (1a) (0,0,z)H: (2g) (0,y,z) (0,-y,z) x (mx)
y (my)C2 (2)
Normal modes in molecules. Bilbao Server: SAMNormal modes in molecules. Bilbao Server: SAM
O: (1a) (0,0,z)H: (2g) (0,y,z) (0,-y,z)
H2O C2 (mm2)
Pmm2
Normal modes in molecules. Bilbao Server: SAMNormal modes in molecules. Bilbao Server: SAM
H2O C2 (mm2)O: (1a) (0,0,z)H: (2g) (0,y,z) (0,-y,z)
OH
Point group
normal modes
symmetry operations
characters
selection rules
H2O : A1+B1+B2 + 2A1+A2+1B1+2B2 – (A1+B1+B2) – (A2+B1+B2) = 2A1 + B2
O atom H atoms H2O translation H2O rotation
sym. stretching
bending
anti-sym. stretching
A1 A1 B2
Working example:
CaCO3, calcitecalcite, R3c (167) D3d6
Ca: (6b) 0 0 0C : (6a) 0 0 0.25O : (18e) 0.25682 0 0.25
(0,0,0)+ (2/3,1/3,1/3)+ (1/3,2/3,2/3)+
Structural information onhttps://icsd.fiz-karlsruhe.de/icsd/
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
Ca: (6b) 0 0 0C : (6a) 0 0 0.25O : (18e) 0.25682 0 0.25
or input, thenclick
CalciteCalcite
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
CalciteCalcite
rotation (inactive)
Ca: (6b) : C : (6a) : O : (18a) :
A1u + A2u + 2EuA2g + A2u + Eg + EuA1g + A1u + 2A2g + 2A2u + 3Eg + 3Eu
IR-active z 0x , y 0 + acoustic
(N = 6:3 +6:3+18:3 = 10)Total: 10A + 10E = 30 3N = 30 opt = A1g(R) + 2A1u(ina) + 3A2g(ina) + 3A2u(IR)+ 4Eg(R) + 5Eu(IR)
5 Raman peaks and 8 IR peaks are expected
xx = -yy xyxzyz
Raman-activenon-zero componentsxx = yy zz
acoustic: A2u+Eu (the heaviest atom)
silent (inactive)
number of operation of each class
Point group
normal modes
symmetry operations
characters
selection rules
Spectra fromSpectra from
134
5
2
Practical exercisePractical exercise: number of expected Raman and IR peaks of aragonite
CaCO3, aragonite, Pnma (62) D2h16
Ca : (4c) 0.24046 0.25 0.4150C : (4c) 0.08518 0.25 0.76211O1 : (4c) 0.09557 0.25 0.92224O2 : (8d) 0.08726 0.47347 0.68065
SolutionSolution: opt = 9Ag(R) + 6Au(ina) + 6B1g (R) + 8B1u (IR) + 9B2g (R) + 5B2u (IR) + 6B3g (R) + 8B3u (IR)
30 Raman peaks and 21 IR peaks are expected
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
Spectra of CaCOSpectra of CaCO33
Calcite: Calcite: Raman-active = A1g + 4Eg
Aragonite:Aragonite: Raman-active = 9Ag + 6B1g + 9B2g + 6B3g
22 observed
Normal modes in Crystals. Normal modes in Crystals. BhagavantamBhagavantam--VenkatarayuduVenkatarayudu MethodMethod
The perovskite structure type ABO3
mPm3 1hO
A: (1b): 0.5 0.5 0.5B: (1a): 0 0 0O: (3d): 0.5 0 0
Working exampleWorking example::
1000cossin0sincos
R+ for proper rotation at = 360/n- for improper rotation (rotation + reflection in h)
Magic formula
– number of modes for each symmetry species
)(1 )()( Rgg
n crystalRR
R
R
Rgg – order of the point group G
gR – number of elements for each class R
)(R – the character for R and irreducible representation
– character for a certain crystal
wR – number of atoms remaining invariant under operation R
)cos21()( Rcrystal wR
(E is proper rot. at 0, I improper rot at 180, h improper rot. at 0)
Normal modes in Crystals. Normal modes in Crystals. BhagavantamBhagavantam--VenkatarayuduVenkatarayudu MethodMethod
1. Determine wR (number of atoms invariant under R) and crystal(R)
• Number of atoms per unit cell: A: 1; B: 8 1/8 = 1; O: 12 1/4 = 3; total = 5; Z = 1
ABO3• Identity E (1), gR = 1 :
wR = 5, cr= 5(+1+2cos0)=5(1+2)=53= 15
• four-fold axis C4 (4), gR = 6 :1C4
2C43C4
4C4 5C4N.B. we consider only one axis from all 6 equivalent as well as all axes parallel to this certain axis, which go through the unit cell
wR = 1 1A + 4(2 1/8 B + 1 1/4 O) = 3
wR = 1 1A + 2 1/8B + 6 1/8 B = 1 + 1 =2
• three-fold axis C3 (3), gR = 8 :1C3
2C33C3
5C3
4C3
6C3
7C3
cr= 3(+1+2cos90)=3(1+20)= 31 = 3
cr= 2(+1+2cos120)=3(1-21/2)= 30 = 0
Normal modes in Crystals. Normal modes in Crystals. BhagavantamBhagavantam--VenkatarayuduVenkatarayudu MethodMethod
• two-fold axis C2 (2), gR = 3 :
wR = 1 1A + 4(21/8B+11/4O) + 4( 2 1/4O) = 1+2+2=5
1C22C2
3C24C2
5C2
6C2
7C28C29C2
wR = (1 1A + 21/4O) + 2( 2 1/8B) + +4(11/8B) + 2(11/4O) = 1+1/2+1/2+1/2+1/2=3
• two-fold axis C2’ (2’), gR = 8 :
1C2’
2C2’
3C2’
4C2’5C2’
6C2’
7C2’
8C2’
9C2’
• centre of inversion I ( ), gR = 1 :1wR = 5 (the same as for identity operation)
cr= 5(+1+2cos180)=5(1-21)= 5(-1) = -5
cr= 3(+1+2cos180)=3(1-21)= 3(-1) = -3
cr= 5(-1+2cos180)=5(-1-2)= 5(-3) = -15
Normal modes in Crystals. Normal modes in Crystals. BhagavantamBhagavantam--VenkatarayuduVenkatarayudu MethodMethod
wR = 3 (the same as for proper four-fold rotation C4)• improper four-fold axis S4 ( ), gR = 6 :4
wR = 2 (the same as for proper three-fold rotation C3) • improper three-fold axis S6 ( ), gR = 8 :3
wR = 3(11A+81/8B+121/4O) = 5 (all atoms invariant) 2h
1h
3h
• mirror plane h (m), gR = 3 :
wR = 11A+81/8B+41/4O = 3• mirror plane d (m’), gR = 6 :
1d
2d3d
cr= 3(-1+2cos90)=3(-1+0)= 3(-1) = -3
cr= 2(-1+2cos60)=2(-1+1)= 20 = 0
cr= 5(-1+2cos0)=5(-1+2)= 51 = 5
cr= 3(-1+2cos0)=3(-1+2)= 31 = 3
Normal modes in Crystals. Normal modes in Crystals. BhagavantamBhagavantam--VenkatarayuduVenkatarayudu MethodMethod
Oh ( mm3 )
E (1)
C4 (4)
C2 (2)
C3 (3)
C2’ (2’)
I ( 1 )
S4 ( 4 )
S6 ( 3 )
h (m)
d (m’)
gR 1 6 3 8 6 1 6 8 3 6
A1g 1 1 1 1 1 1 1 1 1 1 A1u 1 1 1 1 1 -1 -1 -1 -1 -1 A2g 1 -1 1 1 -1 1 -1 1 1 -1 A2u 1 -1 1 1 -1 -1 1 -1 -1 1 Eg 2 0 2 -1 0 2 0 2 -1 0 Eu 2 0 2 -1 0 -2 0 -2 1 0 T1g 3 1 -1 0 -1 3 1 -1 0 -1 T1u 3 1 -1 0 -1 -3 -1 1 0 1 T2g 3 -1 -1 0 1 3 -1 -1 0 1 T2u 3 -1 -1 0 1 -3 1 1 0 -1
cr(R) 15 3 -5 0 -3 -15 -3 0 5 3
)(1 )()( Rg
gn crystalR
RR
g = 48
ABO3
A1g: n = (1511+613-15+0-18-15-18+0+15+18)/48 = 0 no such modeA1u: n = (1511+613-15+0-18+15+18+0-15-18)/48 = 0 no such mode
likewise A2g, A2u, Eg, Eu, T1g, T2u: n() = 0 no such modes T1u: n = 4 T2u: n = 1
latt.vibr.(ABO3) = 4T1u + T2u opt.vibr.(ABO3) = 3T1u + T2u
2. Apply the magic formula
PerovskitePerovskite--type structure type structure ABABOO33
A: (1b): 0.5 0.5 0.5B’/B’’: (1a): 0 0 0
O: (3d): 0.5 0 0
A : (8c) : 0.25 0.25 0.25B’ : (4a) : 0 0 0B”: (4b) : 0.5 0 0O : (24e): 0.255 0 0
(0,0,0)+ (0,1/2,1/2)+ (1/2,0,1/2)+ (1/2,1/2,0)+
single perovskite-type
chemical B-site disorder
double perovskite-type
chemical 1:1 B-site order
mPm3 1O h(221)A(B’,B”)O3 mFm3AB’0.5B”0.5O3 5O h(225)
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
A: (1b):B: (1a): O: (3d):
T1u
T1u
2T1u + T2u
Total: 5T = 15 3N (5 atoms)opt = 3T1u(IR) + T2u(ina)
acoustic
xyz
BO6 stretching
mPm3 1O h(221)A(B’,B”)O3
BO6 bending (3 sets)(3 sets)
B-c. transl. (3 sets)
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
A : (8c):B’: (4a): B”: (4b): O: (24e):
T1u + T2g
T1u
T1uA1g + Eg + T1g + 2T1u + T2g + T2u
Total: 1A+1E+ 9T = 30 3N (N=8:4+4:4+4:4+24:4)
opt = A1g + Eg + T1g(ina) + 3T1u(IR) + T2u(ina)
T1u acoustic
ExerciseExercise: determine the atom vector displacements for A1g, Eg, T2g, and add. T1u
mFm3AB’0.5B”0.5O3 5O h(225)
A1g Eg
xy
z
T2g
T2g
T1u
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
Ba: (1a): 0.0 0.0 0.0Ti: (1b): 0.5 0.5 0.595
O1: (1b): 0.5 0.5 -0.025O2: (2c): 0.5 0.0 0.489
mPm3 1O h(221)BaTiO3
PerovskitePerovskite--type structure type structure ABABOO33 : ferroelectric phases: ferroelectric phases
orthorhombic
cubic
rhombohedraltetragonal
Ba: (1a): 0.0 0.0 0.0Ti: (1b): 0.5 0.5 0.595O: (3c): 0.5 0 0.5
P4mm (99) 14vC
Ba: (2a): 0.0 0.0 0.0Ti: (2b): 0.5 0.5 0.515
O1: (2b): 0.5 0.5 0.009O2: (4c): 0.5 0.264 0.248
Ba: (1a): 0.0 0.0 0.0Ti: (1a): 0.4853 0.4853 0.4853O: (1b): 0.5088 0.5088 0.0185
Amm2 (38) 142vC
(0,0,0)+ (0,1/2,1/2)+R3m (160) 5
3vC
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
Ba:Ti:O:
BaTiO3
PerovskitePerovskite--type structure type structure ABABOO33 : ferroelectric phases: ferroelectric phases
mPm3 P4mm Amm2 R3m
A1 + E A1 + E A1 + EA1 + EB1 + E
T1u
T1u
T1u
T1u
T2u
Ba:Ti:
O1:O2:
A1 + B1 + B2
A1 + B1 + B2
A1 + B1 + B2
A1 + B1 + B2
A1 + A2 + B2
Ba:Ti:
O1:O2:
A1 + E A1 + E A1 + EA1 + EA2 + E
Ba:Ti:O:
Polar modesPolar modes:simultaneously Raman and IR active
xxz, yy
z, zzz
mode polarization along (~ u)
LO: q || TO: q
Normal modes in Crystals. Bilbao Server: SAMNormal modes in Crystals. Bilbao Server: SAM
Experimental geometryExperimental geometry
IIR 2 Iinc Itrans Imeas
z
or
y
Infrared transmission (only TO are detectible)
polarizerX
Y
Z
IRaman 2
back-scatteringgeometry
Porto’s notation: A(BC)DA, D - directions of the propagation of incident (ki) and scattered (ks) light, B, C – directions of the polarization of the incident (Ei) and scattered (Es) light
ki
ks
ki
ks
(qx,qy,0)
(qx,0,0)
yy
Raman scattering
right-anglegeometry
Ei Es
zzn = x LOn = y, z TO
yz
yyn zzn yzn
Ei Es Ei Es
xy xz zy
xyn xzn zyn
zz
zzn n = x,y LO+TOn = z TO
X
Y
Z
X
Y
Z
XYYX )( XZZX )( XYZX )(
(ki = ks+q , E is always to k) XXYY )( XXZY )( XZYY )( XZZY )(
Experimental geometryExperimental geometry
cubic system
Ei EsA1, E
T2(LO)Ei Es
ZYYZ )(ki
ks
Z
X
Y
)q,0,0( zqki = ks + q μq ||
ki
ks
)q,0,q( zxq
Z
X
Y Ei Es ZZYX )(
Ei Es
ZZXX )(
xyz
)0,0,μ( xμ
yxz
)0,μ,0( yμ
μq ||x
μq zT2(LO+TO)
μq T2(TO)
yy
ZYXZ )(
e.g., Td ( ) m34
xy
)μ,0,0( zμ
z
non-cubic systeme.g., rhombohedral, C3v (3m)
ki
ks
Z
X
Y z
yy )μ,0,0( zμ
)q,0,0( zq ZYYZ )(Ei Es
Ei Es
ZYXZ )( E(TO)
YZ
X ki
ksA1(TO) z
zzEi Es
YZZY )( )μ,0,0( zμ)0,q,0( yq
E(LO)
)q,0,0( zq )0,μ,0( yμ
Experimental geometryExperimental geometry
yxy
(hexagonal setting)
Ei Es
Y’ Y Z
X ki
ks XYYX )''(
contribution from
xyy
xyy )0,0,μ( xμ
A1(LO)
E(TO)
zyy )μ,0,0( zμ A1(TO)
)0,0,μ( xμ )0,0,q( xq
LOLO--TO splitting TO splitting
Cubic systemsCubic systems: LO-TO splitting of T modes: T(LO) + T(TO)
NonNon--cubic systemscubic systems: {A(LO) ,A(TO)}; {B(LO),B(TO)}; {E(LO),E(TO)}
)()( TOLO general rule: (the potential for LO: U+E; for TO: U)
Cubic crystalsCubic crystals: LO-TO splitting covalency of atomic bonding LO-TO: larger in ionic crystals, smaller in covalent crystals
More peaks than predicted by GTA may be observedMore peaks than predicted by GTA may be observed (info is tabulated)
UniaxialUniaxial crystals:crystals:if short-range forces dominate:
Raman shift (cm-1)
TO TOLO LO
EA
if long-range forces dominate :
Raman shift (cm-1)
A AE ELOTO
under certain propagation and polarization conditions quasi-LO and quasi-TO phonons of mixedmixed A-E character
LOLO--TO splitting TO splitting
LO-TO splitting: sensitive to local polarization fields induced by point defects
a change in IILOLO//IITOTO depending on the type and concentration of dopant
BiBi1212SiOSiO2020:X:X
Raman shift / cm-1
0.094
0.476
0.8991018 cm-3
undoped
Raman shift / cm-1
BiBi44GeGe33OO1212:Mn:MnI23 324IBi(24f), Si(2a),O1(24f),O2(8c), O3(8c)
Bi(16c),Ge(12a),O(48e)
(Td6)(T0
3)
OneOne--mode / twomode / two--mode behaviour in solid solutions mode behaviour in solid solutions
different types of atoms in the same crystallogr. position,
covalent character of chemical bonding : short correlation length relatively large difference in f(B’/B”-O) and/or m(B’/B”)
• two peaks corresponding to “pure” B’-O and B”-O phonon modes • intensity ratio I(B’-O )/I(B”-O ) depends on x
e.g. (B’1-xB”x)Oy
• one peak corresponding to the mixed B’-O/B”-O phonon mode • ~ lineal dependence of the peak position on dopant concentration x
ionic character of chemical bonding : long correlation length similarity in ri(B’/B”), f(B’/B”-O) and m(B’/B”)
twotwo--mode behaviour:mode behaviour:
oneone--mode behaviour:mode behaviour:
intermediate classesintermediate classes of materials: two-mode over x (0, xm) and one-mode over x (xm, 1)
OneOne--mode / twomode / two--mode behaviour in solid solutions mode behaviour in solid solutions
one-mode behaviour
PbSc0.5(Ta1-xNbx)0.5O3
mf
two-mode behaviour
Pb3[(P1-xAsx)O4]2
T > TC
NonNon--centrosymmetriccentrosymmetric crystals with compositional disordercrystals with compositional disorder
LOLO--TOTO splitting + one-mode/twotwo--modemode behaviour: we may observe fourfour peaks instead of oneone !
aa
a
000000
bb
b
2000000
000003030
bb
022
200
200
dd
d
d
022
200
200
dd
d
d
0000000
dd
aa
a
000000
bb
b
2000000
0000000
dd
00000
00
d
d
0000
000
dd
000030003
bb
X
Y
ZX’
Y’
Z
rotZ(
’ = UTU
100 200 300 400 500 600 700 800 9000.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
Inte
nsity
/ a.
u.
Raman shift / cm-1
PSTcpp 0 deg, p-pol. PSTdpp 45 deg, p-pol.
100 200 300 400 500 600 700 800 9000.00
0.05
0.10
0.15
0.20
Inte
nsity
/ a.
u.
Raman shift / cm-1
PSTccp 0 deg, c-pol. PSTdcp 45 deg, c-pol.
Transformation of Transformation of polarizabilitypolarizability tensors tensors
A1g(O) + Eg(O)
I(F2g )
F2g(O) + F2g(Pb)
I(Eg )
How GTA helps in complex systems How GTA helps in complex systems
chemical B-site order/disorder ferro-/paraelectric
+
B’
B’’
(A’,A’’)(B’,B’’)O3
Perovskite-type relaxor ferroelectrics
Diffraction methods:
dynamic PNR
mPm3
average structure over
time and space
bold: Ei || [100]thin: Ei || [110]
PSTPST
PSNPSN
p-pol c-pol
Pb-O bo
nd st
r.
Pb-O bo
nd st
r.
F2uF1uF1u
BO 3tra
nsl. v
s Pb
BO 3tra
nsl. v
s PbF1u
B-catio
n locali
sed
B-loca
lised
F1u
BO 6str
etchin
g
BO 6str
.
A1gEg
BO 6be
nding
BO 6be
nding
F2g
Pb-loc
alise
d
Pb-loc
alise
d
F2g
prototype structuremFm3
Pb : F2g (R)
O : A1g(R) + Eg (R) + F1g(s) + 2F1u (IR) + F2g(R) + F2u(s)
B-cations : F1u(IR) + F1u(IR)
A-siteB-siteO
GTA of GTA of RelaxorsRelaxors. Peak assignment. Peak assignment
m(Ta) > m(Nb)
f(Ta-O) > f(Nb-O)
X-ray diffraction Raman scattering
hk0
F1u
F1u F2u
F2g
phonon modes
GTA of GTA of RelaxorsRelaxors. Temperature as variable . Temperature as variable
BB--localised mode ~ 240 cmlocalised mode ~ 240 cm--11
TmTa
PbPb--localised mode ~ 53 cmlocalised mode ~ 53 cm--11
allowed
anomalous
TB T*
“intra-cluster” “inter-cluster”
T* Tm
GTA of GTA of RelaxorsRelaxors. Temperature as variable . Temperature as variable
X-ray diffractionRaman scattering
F1u
F2uF1u
phonon modes
F2g
006 046
GTA of GTA of RelaxorsRelaxors. Pressure as variable . Pressure as variable
SecondSecond--order phonon processes order phonon processes
, kground state
excited state
one photon ↔ two phonons
low probability low intensity
• overtones: 21, 22, 23, …
• combinational modes: 1 2, 1 3, 2 3, …slight -deviation due to mode-mode interactions
General rules:General rules: the symmetry of the II-order state = direct product of the I-order states
overtones: i i combinations: i i
(high-order states : further multiplication) modes away from the BZ-centre may contribute to II-order spectra
e.g. phonon(-k) + phonon(+k) = state (k=0)N.B.! acoustic(k0) 0, i.e. non- acoustic modes may also participate
Benefits:Benefits: inactive -point modes may be observed in IR and Raman spectra non--point modes may be observed in IR and Raman spectra
SecondSecond--order phonon processes order phonon processes
Reminder about the direct product rules: Reminder about the direct product rules:
Product tables:
SecondSecond--order phonon processes. Bilbao Serverorder phonon processes. Bilbao Server
e.g. A1 A2 = A2 inactive; EE = A1+A2+E Raman-active; IR-inactive
SecondSecond--order phonon processes. Bilbao Server order phonon processes. Bilbao Server
Dresselhaus et al, Springer, 2008
Td (43m)
RamanRaman
Raman, IRRaman, IR
SecondSecond--order phonon processes. k order phonon processes. k 00
• Determine the point group at the corresponding k-point • Find the corresponding normal modes and overtones and combinations
e.g. overtones and combinations at R are the same as at
SecondSecond--order phonon processes. k order phonon processes. k 00
In some cases, the determination of non- overtones and combination modes may be far from straightforward, this holds especially for non-symmorphic groups (having glide plane or screw axis)
(Special Thanks to Alexander Litvinchuk, University of Houston)Bilbao Server, DIRPRO can help
SecondSecond--order phonon processes. Bilbao Server order phonon processes. Bilbao Server
SecondSecond--order phonon processes. Bilbao Server order phonon processes. Bilbao Server
• Scroll down to the end of the resulting file (the most important info)
• Different nomenclature is confusing consider the matrices of S to recognize the symmetrical type, compare with matrix representation in POINT/SAM; be patientbe patient!
The phase
Compatibility relationsCompatibility relations
(the associated characters cannot “jump”)
Going away from Going away from --point (applied to phonon and electron states):point (applied to phonon and electron states):
• Symmetry of the wave-vector group may change
- fewer elements of symmetry at k 0 (if no glide planes or screw axes )- more elements of symmetry at k 0 (glide planes or screw axes )
• The effect of a given symmetry element on the wave function must be continuous between the centre and the boundary of the Brillouin zone
• k-branches must follow the compatibility relation principle
Pb
Kuzmany, Springer 1998
Compatibility relationsCompatibility relations
Working example:
Consider the compatibility relations along the line X for Pm3m
1. Compare the character tables of (m-3m) and (4mm)
2. Compare the character tables of (4mm) and X (4/mmm)
3. Find which modes can develop along the X-direction
Compatibility relationsCompatibility relations
Find the normal modes having the same characters for the corresponding symmetry elements in the two point groups
m-3m 4mmA1g A1
Fully compatible modesFully compatible modes:
A1u A2A2g B1A1g B2
For degenerate modes possible combinations (sums of characters) must be considered
e.g. E in 4mm is not compatible with modes in m-3m ((2) for 4mm for m-3m)
Compatibility relationsCompatibility relations
Fully compatible modesFully compatible modes:
X
4/mmm4mmA1
A2B1
B2
Eg and EuE
A1g and A2u
A2g and A1u
B1g and B2u
B2g and B1u
Compatibility relationsCompatibility relations
Dresselhaus et al, Springer, 2008
mPm3
Some modes split following the compatibility relations along the k-line
When phonons (e- levels) of differentsymmetry approach one another: • crossing: retain their original symmetry after crossing
• anticrossing: admixed wave functions in an appropriate linear combination phonon-phonon (e--e-) interaction
When phonons (e- levels) of the samesymmetry approach one another:
HyperHyper--Raman ScatteringRaman Scattering
NonNon--linear processlinear process
...61
21 lkjijklkjijkjiji EEEEEEP
ijk - 3rd-rank tensor
...)(0
0 kk
Q βββ
0, hyper-Raman activity
ijk= ikj
HR spectra of a perovskite-type relaxor
Hehlen et al, PRB 2007
If Ej || Ek || crystallographic axis
ijk= ijj 33 non-symmetric matrix
zzzzyyzxx
yzzyyyyxx
xzzxyyxxx
ijj
hyperpolarizability tensor
s, k s, ki)
Stokes anti-Stokes is 2 Kkk is 2
is 2 Kkk is 2
HyperHyper--Raman ScatteringRaman Scattering
Selection rulesSelection rules: according to the transformation properties of a 3rd-rank tensor
Outlines:Outlines: IR-active modes are always HR-active, no matter of the crystal symmetry In centrosymmetric systems only the odd modes (‘u’) may be HR-active Raman-active modes are not hyper-Raman-active and vice versa in non-centrosymmtrical systems some types of modes may be both Raman and HR-active modes inactive in both Raman and IR spectra may be HR-active the nonsymmetrical part of the HP tensor which is symmetrical in two indices gives rise to additional HR-active modes as compared to the completely symmetrical tensor
Denisov et al., Phys.Rep. 151 (1987) 1-92
HyperHyper--Raman ScatteringRaman Scattering
V.N.Denisov , B.N. Mavrin, V.P. Podobedov, Phys.Rep. 151 (1987) 1-92
HyperHyper--Raman ScatteringRaman Scattering
V.N.Denisov , B.N. Mavrin, V.P. Podobedov, Phys.Rep. 151 (1987) 1-92
HyperHyper--Raman ScatteringRaman Scattering
V.N.Denisov , B.N. Mavrin, V.P. Podobedov, Phys.Rep. 151 (1987) 1-92
Resonance Raman ScatteringResonance Raman Scattering
s, k s, ki
Stokes anti-Stokes is is Kkk is Kkk is
e- ES
• Selection rules for both e- and phonon states ()• Fundamental and II-order harmonics may be enhanced• LO are strongly enhanced
200 400 600 800 10000
1000
2000
parallel-polarised
Inte
nsity
/ a.
u.Raman shift / cm-1
457 nm 413 nm 350 nm 334 nm
2.0 2.5 3.0 3.5 4.0 4.50
2
4
6
8
Eg(PST)
RRS
3.20eV(387.5nm)
2.71eV(457nm)
3.00eV(413nm)
3.55eV(350nm)
3.71eV(334nm)
diel
ectri
c co
nsta
nt
Energy / eV
r
i
valence
conductionEg
Electron statesElectron states
GTA conceptGTA concept : the same for phonon and electron states s
2 ↔ Enus ↔ n
Atomic Atomic orbitalsorbitals: : imm
lmlm
l ePNY )(cos),( ,spherical harmonics:
Normalization coefficient Legendre polynomial
),( ml
lYr linear combination of simple Cartesian coordinate functions
l = 0 : 1;l = 1 : x, y, z;l = 2 : xy, yz, zx, x2-y2, 3z2-r2, …
s-likep-liked-like, …
1s 2p
3d
4f
D ↔
Electron statesElectron states
Cubic crystal field environment for a paramagnetic transition Cubic crystal field environment for a paramagnetic transition cationcation
Crystal field theory:Crystal field theory: how the atomic surroundings affects the electron energy levels
3d dx2-y2
dxy
dz2
dxz dyz ),(2 mY
five-fold level
three-fold level
two-fold level
Electron statesElectron states
o: c: d: t = 1 : -8/9 : -1/2 : -4/9
octahedron (6-fold polyhedron): t2g – stabilized, eg – destabilized;cube (8-fold), dodecahedron (12-fold), tetrahedron (4-fold): opposite
dx2-y2
dxy
dz2
dxz dyz
the electron lobes are toward the anions add. repulsive forces destabilization
vice versa
Electron statesElectron states
basic functions for e- states ↔ irreducible representationsCharacter tables:
e.g.
• splitting of levels: according to transformation rules of irreps
Optical absorption processes
, k
• electric dipole transitions ↔ IR selection rulestransformation properties of a polar vector
Gvec irrep symtotally initialfinal
dipole operator
Optical transitions. Symmetry selection rulesOptical transitions. Symmetry selection rules
Working example: Determine the allowed electric dipole transitions in Oh from T2g
• check for each irrep if contains A1g
• irrep for the dipole operator: T1u
T1u T2g = A2u + Eu + T1u + T2u (see product tables)
A1g (A2u+Eu+T1u+T2u) = (A2u+Eu+T1u+T2u) no
A1u (A2u+Eu+T1u+T2u) = (A2g+Eg+T1g+T2g) noA2g (A2u+Eu+T1u+T2u) = (A1u+Eg+T2u+T1u) no
A2u (A2u+Eu+T1u+T2u) = (A1g+Eg+T2g+T1g) yes
Eg (A2u+Eu+T1u+T2u) = (Eu+A1u+A2u+Eu+T2u+T1u+T2u+T1u) noEu (A2u+Eu+T1u+T2u) = (Eg+A1g+A2g+Eg+T2g+T1g+T2g+T1g) yes
T1g (A2u+Eu+T1u+T2u) = (T2u+T1u+T2u+A1u+Eu+T2u+T1u+A2u+Eu+T2u+T1u) no
T2g (A2u+Eu+T1u+T2u) = (T1u+T1u+T2u+A2u+Eu+T2u+T1u+A1u+Eu+T2u+T1u) noT1u (A2u+Eu+T1u+T2u) = (T2g+T1g+T2g+A1g+Eg+T2g+T1g+A2g+Eg+T2g+T1g) yes
T2u (A2u+Eu+T1u+T2u) = (T1g+T1g+T2u+A2g+Eg+T2g+T1g+A1g+Eg+T2g+T1g) yes
Tabulated info on transition products
Optical transitions. Symmetry selection rulesOptical transitions. Symmetry selection rules
Group theory:Group theory: predicts the number of expected IR and Raman peaksone needs to know: crystal space symmetry + occupied Wyckoff positions
Deviations Deviations from the predictions of the groupfrom the predictions of the group--theory analysis: theory analysis:
• LO-TO splitting if no centre of inversion (info included in the tables)• one-mode – two-mode behaviour in solid solutions
• local structural distortions (length scale ~ 2-3 nm, time scale ~ 10-12 s)
• Experimental difficulties (low-intensity peaks, hardly resolved peaks)
ConclusionsConclusions
What should we do before performing a Raman or IR experiment?What should we do before performing a Raman or IR experiment?
Do groupDo group--theory analysistheory analysis