Pružný rozptyl, luminiscence, Ramanův rozptyl
v. 1c
16.12.2024
1
Josef Humlíček, humlicek@physics.muni.cz
F8150, 2024
• pružný a nepružný rozptyl fotonu
• objev Ramanova jevu
• fenomenologický popis, kvantové přechody
• ukázky spekter
Pružný (Rayleighův) rozptyl
Pružný (Rayleighův) rozptyl – záření dipólu
Pružný (Rayleighův) rozptyl - experiment s čistým vzduchem
Pružný (Rayleighův) rozptyl
Nepružný rozptyl rtg záření (Comptonův rozptyl)
A. Compton 1923: částicové vlastnosti vlnění, inspirace pro analogii v optickém oboru
Compton23a.pdf
A. Compton 1923:
částicové
vlastnosti vlnění
Compton23a.pdf
A. Compton 1923: částicové vlastnosti vlnění
Compton23a.pdf
Nepružný rozptyl světla na nehomogenitách (např. akustických fononech)
předpověď Brillouin 1922
Snadercock 1978 (Si, Ge)
Sandercock 1978 (kovy, polovodiče)
Nepružný rozptyl viditelného světla - Raman & Krishnan , Nature1928
1930 Nobelova cena (Raman)
Nepružný rozptyl viditelného světla - Raman & Krishnan , Nature1928
Nepružný rozptyl viditelného světla - Raman & Krishnan , Nature1928
Ramanův „spektrograf“ (Hilger), Hg výbojka + oko nebo fotografická deska
Předpověď: Adolf Smekal
Současný objev „kombinačního rozptylu“
QM teorie: Georg Placzek
QM teorie: Georg Placzek
QM teorie: Georg Placzek
Georg Placzek
symetrie vlastních kmitů
Rozptylový proces
posouvá se s energií excitujícího fotonu
1 4 eV
(8000 32000 cm
-1
)
initial/final
state 2
state 1
Rayleigh Raman
(elastic) Stokes) (Antistokes)
energy
10 4000 cm
-1
(1 500 meV)
intermediate
(virtual) state
21
Ramanův rozptyl
odlišnost od nekoherentního luminiscenčního (fluorescenčního) procesu (ten je
nezávislý na excitační vlnové délce, není v antistokesově větvi)
22
thermalization
(incoherent)
initial/final
state
excitation
energy
luminescence
excited
states
Normal vibrational modes
A molecule consisting of N atoms has 3N degrees of freedom;
classical equations of motion with the restoring forces proportional to the
displacements ukl of the k-th atom in the l-th direction (k=1,...,N, l=x,y,z) can
be transformed into 3N algebraic equations with a real (dynamical) matrix, assuming
harmonic, in-phase motion of the atoms.
These equations can be solved via finding the eigenvectors ekl
(r) of the dynamical
matrix, and the corresponding eigenvalues (frequencies) w(r).
The actual displacements can be expanded in terms of the eigenvectors,
23
( )1
,  r
kl r kl
rk
u Q e
m
where mk is the mass of the k-th atom. The expansion coefficients are called
normal coordinates.
Normal vibrational modes – restrictions due to the symmetry
The 3N–dimensional vectors of the displacements form a basis for a (reducible)
representation of the symmetry group, that can be decomposed in terms of the
irreducible representations.
The infrared activity of a given mode requires the presence of a dipole moment,
which transforms as a three-dimensional vector, forming a basis for a (reducible)
representation. If an irreducible component of this representation is among the
components forming the normal modes, the corresponding normal vibration is
infrared active. Otherwise is the excitation of the normal mode by infrared
waves forbidden by symmetry.
Listing he basis functions in the character tables is of help, because the dipole
moment transforms as the vector (x,y,z).
24
Normal vibrational modes – restrictions due to the symmetry
The Raman activity of a given mode requires the presence of a nonzero derivative
of the induced dipole moment with respect to the vibrational displacements;
these transform as a symmetric tensor of rank 2, forming a basis for a (reducible)
representation. If an irreducible component of this representation is among the
components forming the normal modes, the corresponding normal vibration is
Raman active. Otherwise is the Raman scattering involving the normal mode
forbidden by symmetry.
Listing he basis functions in the character tables is of help, because the functions
{x2, y2, z2, xy, yz, xz} form the basis for the tensorial representation.
25
Vibrational modes of the water molecule
C2v symmetry group:
E (identity), C2 (two-fold axis z), mirror plane sy (xz) , mirror plane sx (yz)
each of the elements forms a class → 4 one-dimensional irreducible representations
9 degrees of freedom, 9-6=3 vibrational ones
Let us first count the number of atoms NR which are not moved by the symmetry
operation R (only these atoms are relevant for the characters of representations,
since they contribute non-vanishing diagonal elements):
C2v E C2 sy sx
NR 3 1 3 1 26
Vibrational modes of the water molecule
Next, a (proper) rotation about the z axis through an angle a transforms the
displacements from equilibrium positions (actually, any vector) according to
E C6
C4 C3 C2 i S6 S4 S3 S2s
3 2 1 0 -1 -3 -2 -1 0 1
'
'
'
cos sin 0
sin cos 0 .
0 0 1
x x
y y
z z
u u
u u
u u
a a
a a
    
        
        
Its contribution to the character, independent of the orientation of the axis, is
( ) 1 2cos .vect Ra a 
Improper rotations iRa, consisting of proper rotations and inversion i, contribute
( ) 1 2cos .vect iRa a  
Using the allowed angle for the crystallographic point groups, we obtain the
contributions (for the water molecule, we need the colored entries only):
27
Vibrational modes of the water molecule
All the atoms left unmoved by the symmetry operation R contribute the same value,
vect(R), to the character of the 3N-dimensional representation based on the
3N-dimensional vector of displacements:
We arrive at the sought (the last row) character table:
( ) ( ).R vectR N R 
C2v E C2 sy sx
NR 3 1 3 1
vect(R) 3 -1 1 1
(R) 9 -1 3 1
Not to forget: the corresponding representation contains three translation and
rotation modes of the molecule as a whole.
28
Vibrational modes of the water molecule
The coordinates of the translation form a basis of the 3-dimensional representation
with the character trans(R)=vect(R). The rotation is described by an axial
vector (no change of sign on inversion); consequently, the character is 1+2cos(a)
for both proper and improper rotations by the angle a. We can subtract the two
irrelevant motions from the character  to obtain the wanted description of
vibrations:
C2v E C2 sy sx
(R) 9 -1 3 1
trans(R) 3 -1 1 1
rot(R) 3 -1 -1 -1
vibr(R) 3 1 3 1
The 3-dimensional representation vibr is reducible.
29
Vibrational modes of the water molecule
The irreducible components are easily found.
1. One of them must be A1, we subtract it:
C2v E C2 sy sx
vibr(R) 3 1 3 1
A1 1 1 1 1
vibr(R)-A1 2 0 2 0
2. The projection of the rest on A1 is nonzero; the latter is contained once more,
we subtract it:
C2v E C2 sy sx
vibr(R)-A1 2 0 2 0
A1 1 1 1 1
vibr(R)-2A1 1 -1 1 -1
The rest is B1 and we arrive at the decomposition of 2A1 + B1.
30
Vibrational modes of the water molecule
The symmetry analysis classifies the three normal modes:
(a) 3651.7 cm-1, symmetric stretching, monotonic dependence of the polarizability
on the normal coordinate  nonzero slope at equilibrium (Raman active), dipole
moment along z (IR active);
(b) 1595 cm-1, bending, low frequency due to small restoring forces, monotonic
dependence of the polarizability on the normal coordinate  nonzero slope at
equilibrium (Raman active), dipole moment along z (IR active);
(a) 3755.8 cm-1, asymmetric stretching, monotonic dependence of the polarizability
on the normal coordinate  nonzero slope at equilibrium (Raman active), dipole
moment along z (IR active). 31
Vibrational modes of the H2O molecule - overview Herzberg
32
Examples of Raman spectra - (liquid) water
(Ph. Valee et al., Journal of Molecular Structure, 2003, 651-653: 371-379)
33
Examples of Raman spectra - water
(Ph. Valee et al., Journal of Molecular Structure, 2003, 651-653: 371-379)
34
Examples of Raman spectra -
Calcite
(Porto et al., Phys. Rev. 147, 608 (1966))
Note:
the 1088 cm-1 A1g mode corresponds
probably to the “l1= 9.1 m Trabant”
observed by Landsberg&Mandelstam
35
Examples of Raman spectra - intrinsic silicon
(T.R. Hart et al., Phys. Rev. B1, 638 (1970))
36
Examples of Raman spectra - heavily doped p-type silicon
(M.V. Klein, Light Scatt. Sol. I, p. 173)
37
Raman spectra of “topological insulators” Bi2Te3, Bi2Se3
ÚFKL 2012-13, epilayers on BaF2 (JKU Linz)
defects, steps on the substrates
MBE2805 backside 5x
MBE2805 backside 20x
Crystalline Bi2Te3, 3 quintuples
Te(1) - Bi, 0.173 nm
Te(2) - Bi, 0.203 nm
projection on (110)
Bi2Te3 stacking sequence: 3x[Te(1)-Bi-Te(2)-Bi-Te(1)]
5
5
5
c=3.03 nm Te(2) center
of symmetry
Te(1) - Te(1), 0.260 nm
van der Waals
Raman spectra, backscattering geometry, unpolarized
633 and 514 nm excitation, two possibilities
BaF2 substrate (~1 mm thick) transparent for exciting and scattered light
negligible overlap of the spectra of film and substrate
"back side""front side"
film BaF2
BaF2
lens
Renishaw In-Via
Raman spectra, 3 narrow Gaussian profiles on a flat background
(when measured at the interface), very slight asymmetry
60 80 100 120 140
0.0
0.5
1.0
133 / 10
(134 / 10)
100.8 / 4.0
(102.3 / 6.0)
60.6 / 3.1
(62.0 / 5.0)
position / FWHM (cm
-1
) / (bulk, Kullmann et al.)
AgEg
Bi2
Te3
(830 nm) / BaF2
, 633 nm, interface
RAMANSIGNAL(arb.u.)
STOKES SHIFT (cm
-1
)
Ag
BiTe_Raman1_Bi2Te3_Ram1a
Identification of normal modes (gerade)
very weak 61 101 133 cm
-1
Chis et al. (2012) 42.1 64.2 112.3 139.2
first-principles
DFT & DFPT
(perturbation th.)
Bi2Te3 Raman modes
A1g
(2)A1g
(1)A1g
(1) Eg
(2)Eg
(1)
Jenkins et al., 1972, 9 adjustable parameters, fitting measured elastic moduli
50.6 71.1 118.5 128.3 cm
-1
BiTe - three doublets:
50 100 150 200
0.0
0.5
1.0
131
123.7101.7
92.7
60.4
58.5
film
Bi1
Te1
MBE2882
BiTe23-11_Comp23-11d
RAMANSIGNAL(arb.u.)
STOKES SHIFT (cm
-1
)
subs
633 nm
The lower symmetry of BiTe quintuplets
(the interlayer spacings in Angstroems):
2.598
1.729
2.032
2.032
1.729
Bi
Te1
Te2
Bi1
Te1
2.040
2.650
1.656
2.177
BiTe
Te1
Te1
Te3
Te2
Te3
Bi1
Bi2
Bi3
Bi2
Bi3
Te2
Bi
Te1
1.870
1.131
1.822
Bi2Te3
7 3
different spacings
Several weaker bands seen in the detailed comparison, found also
in the spectra of thick films taken from the surface;
deviations from perfect crystal stronger at the surface;
the growth need not end with a complete cell of Bi2Te3 (3.03 nm)
40 60 80 100 120 140
0.0
0.2
0.4
interface
48
71
75
91
94
633 nm, surface BiTe_RamanThin_B830-22sdd
Bi2
Te3
MBE2780 830 nm
Bi2
Te3
MBE2918 22 nm
RAMANSIGNAL(arb.u.)
STOKES SHIFT (cm
-1
)
82
three Bi2Se3 bands
about sqrt(127.6/79)=1.27 times higher in frequencies than Bi2Te3
0 50 100 150 200 250
0.0
0.5
1.0
133
175131
10161
Bi2Se3 2836Bi2Te3 2883
BiSe_2013_04_30_TeSe
RAMANSIGNAL(arb.u.)
STOKES SHIFT (cm
-1
)
subs
72