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Uni Siegen
Electronic Transitions
Jablonski
Diagram:
ISC
T1
S1
S0
QM Description of Electronic Transitions:
Ø
Ø
Ø
Selection rules
Transition strength
Franck-Condon-Factor
10.12.2004
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we need:
time-dependent Schroedinger equation
perturbation theory
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Physikalische Chemie
Uni Siegen
QM Description of Electronic Transitions
Schroedinger equation of the system is solved:
H 0ϕ n = Enϕ n
H0
ϕn
En
Hamilton operator of the system
eigenfunctions and
eigenvalues of the n-th eigenstate are known
Incident light perturbs the system
Hamilton operator of the perturbation
Hp
Perturbation is time-dependent (oscillating electromagnetic field)
⇒ time-dependent Schroedinger equation has to be solved
& = − i (H + H )Ψ
Ψ
0
P
h
wavefunction (state of the system) is changed
in time
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because of Hamliton operators
act on system
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Physikalische Chemie
Uni Siegen
QM Description of Electronic Transitions
1. Without perturbation, H0 time-independent
& =−i H Ψ
Ψ
0
h
for t=0 system is in state n
Ψn0 (t = 0 ) = ϕ n
solution for time-dependent
 i

0
Ψ
(
t
)
=
ϕ
exp
−
E
t

n
n
n 
Schroedinger equation
h


Ψn0 time-dependent eigenfunctions of the unperturbed system
for probability density we get
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2
2
2
i

 i

Ψn0 (t ) = ϕ n exp Ent  exp − Ent  = ϕ n
h

 h

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Physikalische Chemie
Uni Siegen
QM Description of Electronic Transitions
Idea: If the perturbation is weak, the wavefunctions of the perturbed system
are similar to those of the unperturbed system
the new wavefunctions can be constructed by summing over the eigenfunctions
of the unperturbed system with the weighting factors cn.
2. Ansatz for the wavefunctions of the perturbed system:
Ψ = ∑ cn (t )Ψn0
n
cn
2
c2
probability that state n is populated
c1
c0
changes in cn describe transitions
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Physikalische Chemie
2
2
2
Uni Siegen
QM Description of Electronic Transitions
2. Ansatz for the wavefunctions of the perturbed system:
Ψ = ∑ cn (t )Ψn0
n
cn
2
c2
probability that state n is populated
c1
for t=0, system is in the state ground state n=0
c0
Ψ (t = 0 ) = c0 (t = 0 )Ψ00 (t = 0 ) = ϕ 0
c0(t=0)=1 and cn(t=0)=0 for all other n
(without perturbation the system would always stay in the ground state n=0)
at t=0, perturbation is switched on, with time c0 decreases and cn increase
⇒ system undergoes transitions between states
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Physikalische Chemie
2
2
2
Uni Siegen
QM Description of Electronic Transitions
now we have to determine cn(t)
we put our ansatz Ψ = ∑ cn (t )Ψn0 into the time-dependent Schroedinger-Eqn.
n
& = − i (H + H )Ψ
Ψ
0
P
h
i
i
0
0
0
&
&
c
Ψ
+
c
Ψ
=
−
c
H
Ψ
−
∑n n n ∑n n n h ∑n n 0 n h ∑n cn H P Ψn0
i
&
Ψ
=
−
H 0Ψ
using
h
i
0
&
c
Ψ
=
−
∑n n n h ∑n cn H P Ψn0
we get
∑ c&n Ψm0*Ψn0 = −
10.12.2004
n
i
cn Ψm0* H P Ψn0
∑
h n
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multiplied with
* Ψm0*
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Physikalische Chemie
Uni Siegen
QM Description of Electronic Transitions
0* 0
&
c
Ψ
Ψn dV = − ∫
∑
n
m
∫
integration
on both sides
V
we use that
eigenfunctions
are orthogonal
n
V
i
cn Ψm0* H´ P Ψn0dV
∑
h n
0*
0
Ψ
Ψ
∫ m n dV = 1
for m=n
0*
0
dV = 0
Ψ
Ψ
m
n
∫
for m≠n
V: Volume of molecule
V
V
c&m = −
i
cn ∫ Ψm0* H P Ψn0dV
∑
h n V
transition matrix element
c&m = −
i
P
cn H mn
∑
h n
P
H mn
= ∫ Ψm0* H P Ψn0dV
…
. .c
al
cu
lat
ion
s
V
with
and
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P
P
H mn
= H mn
(0) exp(ω mnt )
ω mn = 1 (Em − En )
h
Light at frequency ωmn
Transition energy hωmn
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Physikalische Chemie
Uni Siegen
Probability of Electronic Transitions
2
4 sin 2 (ω mnt / 2 ) P
2
probability to find system in state m
cm =
H
0
(
)
mn
2
h 2 (ω mn )
at time t
2
if it was in state n at t=0
d cm
wmn =
dt
transition probability n→m
Result:
wmn
2 sin (ω mnt ) mn 2
=
H P (0 )
h 2ω mn
transition probability depends on the transition matrix element
P
H mn
= ∫ Ψm0* H P Ψn0dV
V
wavefunction of the
final state
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Hamilton operator
of the perturbation
general form
- can be applied to any system
wavefunction
of the initial state
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Physikalische Chemie
Uni Siegen
QM Description of Electronic Transitions
In our case: Perturbation by light
⇒ light interacts with dipole moment
rr
rr
H P = Ep = eEx
r
r
p = ex
dipole moment
r r
r i [kr⋅ xr −ωt ] light as harmonic
E ( x , t ) = E0 e
em plane wave
transition matrix
element
(
)
r i [krxr ] r 0
H (0 ) = e ∫ Ψ E0e x Ψn dV
P
mn
0*
m
V
is extremely difficult to calculate…..
Idea: Size of the molecule x (<0.5 nm) is much smaller than the wavelength λ(~500 nm)
x << λ ⇒ kx = 2π λ x ≈ 0
(fails for extended
systems, e.g. crystals)
rr
ik x
Electric
dipole
approximation:
e
≈1
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Physikalische Chemie
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Uni Siegen
Probabilty of Electronic Transitions
Using the Electric dipole approximation we get
r
r
P
H mn (0 ) = eE ∫ Ψm0* xΨn0dV
V
⇒
P
H mn
(0) is called “transition
dipole moment”
r
because of ex
r2
wmn ∝ E ∝ I
⇒ transition probability (absorption) depends
linearly on the intensity of the light
Photon picture:
Excitation probability depends linearly on the
number of photons
wmn ∝ I = nhω
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10
Physikalische Chemie
Uni Siegen
Electronic Transitions in H-Atoms
y
Application of the model:
S
x
Hydrogen atom: lowest state 1S
optical transition between 1S and 2S?
both states are symmetric:
(angular momentum l=0)
r
r
Ψ10S (− x ) = Ψ10S ( x )
r
r
Ψ20S (− x ) = Ψ20S ( x )
∞
0
r 0
r
0* r
0
H (0 ) = ∫ Ψ xΨ1 dV = ∫ Ψ2 xΨ1 dV + ∫ Ψ20* xΨ10 dV
−∞
0
−∞
integral split in two
P
21
(skipped
Ee)
∞
0*
2
∞
∞
r r
r
r r
r
H 21P (0 ) = ∫ Ψ20* ( x )x Ψ10 ( x )dV + ∫ Ψ20* (− x )(− x )Ψ10 (− x )dV
0
0
∞
∞
r r
r
r r
r
H 21P (0 ) = ∫ Ψ20* ( x )x Ψ10 ( x )dV − ∫ Ψ20* ( x )xΨ10 ( x )dV = 0
0
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0
⇒ no electronic transition between 1S and 2S !!!
11
Physikalische Chemie
Uni Siegen
Electronic Transitions in H-Atoms
y
Application of the model:
-
Hydrogen atom: lowest state 1S
optical transition between 1S and 2P?
Px
+
x
r
r
Ψ20P (− x ) = −Ψ20P ( x )
2P is asymmetric:
(angular momentum l=1)
H
H
P
21
P
21
∞
(0) = ∫0
∞
(0) = ∫0
∞
r r 0 r
r r
r
Ψ ( x )x Ψ1 ( x )dV + ∫ Ψ20* (− x )(− x )Ψ10 (− x )dV
0*
2
0
∞
r r 0 r
r r
r
Ψ ( x )x Ψ1 ( x )dV + ∫ Ψ20* ( x )xΨ10 ( x )dV ≠ 0
0*
2
0
⇒ transition 1S-2P is possible
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Physikalische Chemie
Uni Siegen
Selection Rules
r
r
Ψ (− x ) = Ψ ( x )
r
r
Ψ (− x ) = −Ψ ( x )
A function with
A function with
is called gerade (symmetric)
is called ungerade (asymmetric)
The operator of the electric field is ungerade
r
r
− (x ) = − x
∆l=0
Transition between two gerade (g) functions
H
P
21
∞
∞
(0) = ∫−∞ gugdV = ∫−u∞dV = 0
“forbidden”
Transition between gerade (g) and ungerade (u) function
∞
∞
−∞
−∞
H 21P (0 ) = ∫ uugdV = ∫ gdV ≠ 0
∆l=1
“allowed”
Selection rule for electronic transitions: ∆l=1
and ∆n≠0
10.12.2004
because energy of the system is changed by photon energy
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Physikalische Chemie
Uni Siegen
Electronic Transitions: Particle in a box
ground state: g-symmetry
1st exited state: u-symmetry
Ψ00 (− x ) = Ψ00 ( x )
Ψ10 (− x ) = − Ψ10 ( x )
0
x
x
0
=> optical transition between ground and 1st excited state is allowed
2nd exited state: g-symmetry
Ψ20 (− x ) = Ψ20 ( x )
x
0
=> optical transition between ground and 2nd excited state is forbidden
⇒ selection rule ∆n = ± 1
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Physikalische Chemie
Uni Siegen
Electronic Transitions
Jablonski
Diagram:
ISC
S1
Absorption:
T1
d
sample
light
source
I(d)
detector
Fluorescence / Raman
S0
sample
light
source
I detector
How do can we study these transitions?
we need
Ø
Ø
Ø
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1. light source
2. detector
3. spectral resolution
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Physikalische Chemie
1. Light sources
characterization of a light source:
source
wavelength λ (energy per photon)
E = hc/ λ = h ν = h / 2 π ω = h / 2 π k
light
intensity: energy / time / area
(=power density)
cε 0 r 2
I = n E / (t A) or I =
E
T
2
n number of photons
area A
time average
? source emits more than one wavelength (e.g. lamp)
n(λ)
n dλ= n(λ) dλ
(number of photons detected within wavelength interval dλ)
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dλ
λ
Spectral radiation density :
I(λ) = n(λ) E(λ) / (t A)
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Lamps
continous output
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Lamps
continous output
S1
OD(λ)=log(I0 λ) /I (λ))
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used for absorption measurements
S0
18
Lamps
line ouput
S1
S0
spectral calibration
fluorescence
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19
Lamps
FIR (Far Infra Red):
Nernst-Stift (rare earth oxides)
Globar (SiC)
NIR / Vis:
Gas discharge lamps: (Hg, Xe, D2..)
Heated wires (CrNi, W, …)
UV:
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Gas discharge lamps: (Hg, Xe, D2..)
20
Lamps
Black body radiation (“heat radiation”)
spectral radiant exitance:
intensity per wavelength
interval
(number of photons
within ∆λ)
Planck’s law:
−1
ch


W (λ , T ) = 8ωhcλ−5  e kλT − 1

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
output spectrum depends on temperature
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21
Laser: Stimulated emission
(Light amplification by stimulated emission of radiation)
Spontaneous emission:
Stimulated emission:
excited state
excited state
ground state
ground state
excited state relaxation
one photon is emitted
incident photon stimulates emission
of second photon => amplification
process is instantaneous
emission is isotropic
(“no preferred direction”)
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emission is directed
two photons out of one: amplification!22
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Uni Siegen
Probabilty of Electronic Transitions
r
r
H (0 ) = eE ∫ Ψm0* x Ψn0 dV
P
mn
Transition dipole moment:
V
general rule for all states n and m of the system
stimulated emission:
H
P
mn
absorption:
excited state n
excited state m
ground state m
ground state n
(0)
2
= H
P
nm
(0)
2
⇒ probabilities for stimulated emission and absorption are the same
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Physikalische Chemie
Uni Siegen
Laser: Inversion
competition between absorption and stimulated emission
N1,N2 number of atoms in ground state, excited state
excited state (N2)
excited state
ground state (N1)
ground state
N1>>N2
absorption more likely
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N1<<N2
stimulated emission more likely
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Physikalische Chemie
Uni Siegen
Laser: Inversion
Normal population:
E
N2 / N1 = g2 / g1 exp(-(E2-E1)/kT)
N2
N1
N(E)
(BoltzmannDistribution)
(e.g. E2-E1 =2 eV, T=300K, g2=g1 N2/N1 = 3.6*10-34)
(N2)
N1 >> N2
(N1)
needed: N1 < N2 !
Population inversion
(N2)
(N1)
energy pump
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Physikalische Chemie
Uni Siegen
Laser: 2, 3 and 4-level systems
2-level
N2
3-level
N2
N2
pump
pump
N1
4-level
N1
inversion not
possible*
pump
N1
fast non-radiative relaxation
N2 should have long lifetime
N1 ≈ 0
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Physikalische Chemie
Uni Siegen
Laser: Elements
d
active
medium
mirror
(high reflector)
RHR
mirror
ROC
(output coupler)
energy pump
essential elements:
1. laser active gain medium (3-, 4-level system), amplifies incident radiation
2. energy pump, produces population inversion
3. optical resonator, stores energy in particular resonator modes
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Physikalische Chemie
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feature:
Gain medium:
Laser: Types
semiconductor material
dye molecules
gas
doped crystals, Ruby laser
……
(1960)
Pumping process: light
electric current
Resonator cavity: linear
ring
folded
……
Ouput:
(1960)
(1964)
(1962)
cw (continuous wave), pulsed (Q-switching, mode-locking..)
fixed frequency, tunable
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Physikalische Chemie
Uni Siegen
Gain medium
example: fluorescent dye molecule Rhodamin 6G
4-level
N2
absorption
lasing
fluorescence
pump
λ
N1
vibrational relaxation
pump
lasing
amplification (gain) only within fluorescence band
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Physikalische Chemie
Uni Siegen
Laser: Longitudinal resonator modes
L
metallic mirror,
electric field in a
metal is zero
λ= 2L
λ=L
λ=L2/3
standing waves
Mode spectrum:
λ= 2 L/m
ν=m c/2L
m=1, 2, 3……
mode spacing:
∆ν=c/2L
lasing not for arbitrary wavelength !
only for resonator modes
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Physikalische Chemie
Uni Siegen
Laser: Longitudinal resonator modes
Gain
threshold
1
∆ν=c/2L
ν
1
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ν
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Physikalische Chemie
Uni Siegen
Laser vs. Lamps
Adavantages of lasers:
Adavantages of lamps:
- high output powers
- collimated output beams
- narrow spectral linewidth
- short output pulses (~4 fs)
- coherence
……
- high output powers
- broad spectral output
- cheap
……
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