CHM 5175: Part 2.7
Emission Quantum Yield
Source
Detector
hn
F(em) =
# of photons out
# of photons in
Sample
Ken Hanson
MWF 9:00 – 9:50 am
Office Hours MWF 10:00-11:00
1
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Emission Quantum Yield
Source
Emission Quantum Yield (F)
Detector
hn
F=
# of photons emitted
# of photons absorbed
Sample
Ground State
(S0)
hn
Singlet Excited
State (S1)
hn
Excited State Decay
Radiative Decay
Excitation
F=
# of photons emitted
# of photons absorbed
Non-emissive Decay
Non-radiative Decay
Excited State Decay
Reaction Kinetics
S1
Energy
kA
S0
kA knr
kr
S0
kA = excitation rate
kr = radiative rate
knr = non-radiative rate
S1
kr + knr
S0
If it is assumed that all processes are first order
with respect to number densities of S0 and S1
(nS0 and nS1 in molecules per cm3)
Then the rate Equation:
dnS1
dt
 k A nS0  (k r  k nr )nS1
Excited State Decay
S1
Rate equation:
dnS1
Energy
kA knr
kr
S0
dt
 k A nS0  (k r  k nr )nS1
Sample is illuminated with photons of constant
intensity, a steady-state concentration of S1 is
rapidly achieved. dnS1/dt = 0
nS1 
kA = excitation rate
kr = radiative rate
knr = non-radiative rate
nS0 k A
k F  knr
Substitution for photon flux and the relationship
between kA and kr then (math happens):
kr
F 
knr  kr
F = Emission Quantum Yield
Quantum Yield
kr
F 
knr  kr
=
# of photons emitted
# of photons absorbed
kr
knr
Non-radiative Rates
kr
FF =
kr + knr
kr
FF =
kr + kchem + kdec + kET + ket + kpt + ktict + kic + kisc …
Rate constants:
kr
= radiative
kchem = photochemistry
kdec = decomposition
kET = energy transfer
ket = electron transfer
ktict = proton transfer
ktict = twisted-intramolecular charge transfer
kic = internal conversion
kisc = intersystem crossing
Emission Quantum Yield
Quantum Yield:
kr
F
k r  kisc  k nr
Emission Quantum Yield
Emission Quantum Yield
F=
# of photons emitted
# of photons absorbed
= 0 to 1
Fluorescence Quantum Yield
kr
FF =
kr + kchem + kdec + kET + ket + kpt + ktict + kic + kisc …
1) Internal conversion (kic)
-non radiative loss via collisions with solvent or via internal vibrations.
2) Quenching
-interaction with solute molecules capable of quenching excited state
(kchem, kdec, kET, ket )
3) Intersystem Crossing Rate
4) Temperature
- Increasing the temperature will increase of dynamic quenching
5) Solvent
- viscosity, polarity, and hydrogen bonding characteristics of the solvent
-Increased viscosity reduces the rate of bimolecular collisions
6) pH
- protonated or unprotonated form of the acid or base may be fluorescent
7) Energy Gap Law
Intersystem Crossing
S1
T2
T1
E
S0
Excitation
Fluorescence
Intersystem Crossing
Phosphorescence
Atom Size
ISC
Increase in strength of spin-orbit interaction
FFluorescence
FPhosphorescence
Temperature Dependence
Measuring Quantum Yield
Source
Detector
hn
Sample
kr
F 
knr  kr
=
# of photons emitted
# of photons absorbed
We don’t get to directly measure F, kr or knr!
We do measure transmittance and emission intensity.
Measuring Quantum Yield
Relative Quantum Yield
“Absolute” Quantum Yield
Relative Quantum Yield
kr
F 
knr  kr
=
# of photons emitted
# of photons absorbed
If is proportional to the amount of the radiation from the excitation
source that is absorbed and Ff .
If = Ff I0 (1-10-A)
If = emission intensity
Ff = quantum yield
I0 = incident light intensity
A = absorbance
I f ∝ Ff
If ∝ A
Relative Quantum Yield
I f = Ff I 0
(1-10-A)
Ff ∝ I f
If ∝ A
Compare sample (S) fluorescence to reference (R).
Reported
(1-10-AR)
IS
FS
x
=
x
-A
IR
FR
(1-10 S)
Measured
F
I
A
n
= quantum yield
= emission intensity
= absorbance
= refractive index
nS2
nR2 Known
Relative Quantum Yield
(1-10-AR)
IS
FS
x
=
x
-A
IR
FR
(1-10 S)
Absorption
Emission
Reference
Emission
AR
AS
nS2
nR2
I
Sample
Emission
Same instrument settings:
excitation wavelength, slit widths, photomultiplier voltage…
Relative Quantum Yield
(1-10-AR)
IS
FS
x
=
x
-A
IR
FR
(1-10 S)
nS2
nR2 ?
Excitation
Snell’s Law:
Sinqi
ni
=
no
Sinqo
Detector
Sinqi2
nS2
=
2
nR
Sinqo2
Emission
References
Relative Quantum Yield
Emission
Detector
Source
Absorption
Detector
IF
P0
P
Sample
Reflectance
Scatter
(1-10-AR)
IS
FS
x
=
x
-A
IR
FR
(1-10 S)
nS2
nR2
Relative Quantum Yield
[Ru(bpy)3] 2Cl
in H2O
F = 0.042-0.063
[Ru(bpy)3] 2PF6
in MeCN
F = 0.062-0.095
Relative Quantum Yield
(1-10-AR)
IS
FS
x
=
x
-A
IR
FR
(1-10 S)
Minimizing Error:
• Sample/Reference with similar:
- Emission range
- Quantum yield
• Same Solvent
• Known Standard
• Same Instrument Settings
- Excitation wavelength
- Slit widths
- PMT voltage
nS2
nR2
Absolute Quantum Yield
F=
# of photons emitted
# of photons absorbed
Source
Detector
Integrating Sphere
Absolute Quantum Yield
Hamamatsu: C9920-02
(99% reflectance for wavelengths from
350 to 1650 nm and over 96%
reflectance for wavelengths from 250
to 350 nm)
Fig. 2. Schematic diagram of integrating sphere (IS) instrument for measuring absolute
fluorescence quantum yields. MC1, MC2: monochromators, OF: optical fiber, SC:
sample cell, B: buffle, BT-CCD: back-thinned CCD, PC: personal computer.
Absolute Quantum Yield
Absolute Quantum Yield
Instrumentation
Hamamatsu: C9920-02G
Absolute quantum yield measurement system
Absolute Quantum Yield
Horiba QY Accessory
Data Acquisition
1) Set excitation l
2) Insert reference
- holder + solvent
3) Irradiate Reference
4) Detect output across
- excitation and emission
5) Insert Sample
Holder + solvent + sample
6) Repeat 3 and 4
Absolute Quantum Yield
Absolute Quantum Yield
Self Absorption/Filter Effect
Anthracene
Fluorescence intensity
Self Absorption/Filter Effect
Inner filter
effect
If = Ff I0 (1-10-A)
Concentration (M)
(1-10-AR)
IS
FS
x
=
x
-A
S
IR
FR
(1-10 )
nS2
nR2
Single Crystal Measurements
Quantum Yield and Lifetime
Substitution for photon flux and the relationship
between kA and kr then (math happens):
kr
F 
knr  kr
Intrinsic or natural lifetime (tn):
lifetime of the fluorophore in the absence of non-radiative processes
1
kr
tn =
Radiative Rate and Extinction Coefficient:
Extinction Coefficient
1 n n

 n dn
kr =
8   
t no 3.42 10
Radiative Rate
2
max
2
Relationship between absorption intensity and fluorescence lifetime
Strickler and Berg “Relationship between Absorption Intensity and Fluorescence Lifetime of
Molecules” J. Chem. Phys. 1962, 37, 814.
Strickler-Berg relation
The relation of the radiative lifetime of the molecule and the absorption
coefficient over the spectrum [ref. 5]
2
 n2
1 n max

 n dn
kr =
8   
t no 3.42 10
n: refractive index of medium
n: position of the absorption maxima in wavenumbers [cm-1]
 : absorption coefficient
Relationship between Einstein A and B coefficients
Suppose a large number of molecules, immersed in a nonabsorbing medium with
refractive index n, to be within a cavity in some material at temperature T,
The radiation density within the medium is given by Planck’s blackbody radiation law,
8 hn 3n3
 (n ) 
c3
hn


exp(
)

1


kT
1
-(1)
Blackbody Radiation
By the definition of the Einstein transition probability coefficients,
the rate of molecules going from lower state 1 to upper state 2 by absorption of radiation,
N1a B1a2b  (n1a2b )
-(2)
N1a : number of molecules in state 1a
v 1a 2b : frequency of the transition
Einstein transition
The rate at which molecules undergo this downward transitionprobability
is given bycoefficients
N2b [ A2b1a  B2b1a  (n 2b1a )]
-(3)
spontaneous emission probability
induced emission probability
At equilibrium the two rates must be equal, so by equating (2) and (3),
Relationship between Einstein A and B coefficients
A2b1a
N
 [ 1a  1] (n 2b1a )
B2b1a
N 2b
-(4)
According to the Boltzmann distribution law, the numbers of molecules in the two
states at equilibrium are related by
N 2b
hn
 exp[ 2b1a ]
N1a
kT
-(5)
Ratio of molecules in the
ground and excited state
Substitution of Eqs. (1) and (5) into (4) results in Einstein’s relation,
A2b1a  8 hn 2b1a n3c3 B2b1a
3
-(6)
Relationship B coefficient to Absorption coefficient
The radiation density in the light beam after it has passed a distance x cm through
the sample, the molar extinction coefficient  (n ) can be defined by
 (n , x)
 10 (n )Cx  e 2.303 (n )Cx
 (n , 0)
Radiation Density
Photons per distance
C: concentration in moles per liter
If a short distance dx is considered, the change in radiation density may be
d  (n )  2.303 (n )  (n , 0)Cdx
-(7)
For simplicity, all the molecules will be assumed to be in the ground vibronic state, 10
Cdx  1000N10 N A
1
-(8)
The number of molecules excited per second with energy hv is given by
N (n )  
c d  (n )
n hn
-(9)
Excitations/second
Relationship B coefficient to Absorption coefficient
Combining Eqs. (7), (8), and (9), it is found
N (n )
 [2303c (n ) / hn nN A ] (n , 0)
N10
-(10)
Probability of a single
excitation
transition
the probability that a molecule in state 10 will absorb of energy hv and go
to some excited
state
To obtain the probability of going to the state 2b, it must be realized that this can occur
with a finite range of frequencies, and Eq. (10) must be integrated over this range. Then
N102b
2303c
[
 (n )d lnn ] (n 102b )
N10
hnN A 
-(11)
If the molecules are randomly oriented, the average probability of absorption for
N1a B1a2b  (n1a2b )
a large number of molecules, Eq (2) give a similar relation to (11)
-(2)
2303c
 B102b 
-(12)
 d lnn
hnN A

for all
The probability coefficient for all transitions to the electronic state Probability
2
excitation transitions
B102b   B102b 
b
2303c
 d lnn

hnN A
-(13)
Lifetime relationship for molecules
The wavefunctions of vibronic states are functions of both the electronic coordinates x
and the nuclear coordinates Q,
-(14)
1a ( x, Q)  1 ( x, Q)F1a (Q)
If M(x) is the electric dipole operator for the electrons, the probability for induced
Absorption or emission between two states is proportional to the square of the
Matrix element of M(x) between two states
B1a 2b  B2b1a  K |  1a ( x, Q) M ( x) 2b ( x, Q)dxdQ |2
*
-(15)
Using (14), the integral in this expression can be
dQ
Excitation
 1a ( x, Q)M ( x) 2b ( x, Q)dxdQ   1a (Q)M 12 (Q) 2b (Q)Relating
*
*

where M 12 (Q)  1 ( x, Q) M ( x) 2 ( x, Q)dx
*
Probability to Relaxation
Probability of atoms
Electronic transition moment integral for the transition
Assuming the nuclei to be fixed in a position Q
Lifetime relationship for molecules
It can be expand in a power series in the normal coordinates of the molecule
M12 (Q)  M12 (0)   (M12 / Qr )0 Qr 
-(16)
r
For strongly allowed transitions in a molecule, the zeroth-order term is dominant
Then (15) reduces to
B1a2b  B2b1a  K M12 (0)
2
F
*
1a
F2b dQ
2
-(17)
Expanding the model to
molecules
Taking the appropriate sums, we find
B102   B102b  K M 12 (0)
b
F 2b comprise
 F
*
1a
F 2b dQ
2
b
B102  K M 12 (0)
Since the
2
2
a complete orthonormal set in Q space
-(18)
Lifetime relationship for molecules
The rate constant for emission from the lowest vibrational level of electronic state 2
to all vibrational levels of state 1, A201 , can be written by using Eqs. (6) and (17)
A201   A201a (8 hn / c ) K M12 (0)   v201a
3
2
3
a
F
3
*
1a
a
It is desirable to be able to evaluate the term
v
201a
a
3
F
*
1a
F 2b dQ
2
-(19)
2
F 2b dQ experimentally .
If the fluorescence band is narrow, v3 can be considered a constant and removed from
the summation, the remaining sum being equal to unity
Relating relaxation
2
*
probabilities to lifetime
By dividing by  a  F1a F20 dQ  1
for a single transition
 v201a
a
3
*
F
1
 a F20 dQ
 F
*
1a
a
F 20 dQ
2
2
Lifetime relationship for molecules
The sums over all vibronic bands can be replaced by integrals over the fluorescence
spectrum, so the expression reduces to
 I (n )dn
n
3
I (n )dn
 n f
3
 Av
1
Expanding to all
transitions
Now, by combining Eqs. (13), (18), and (19), we obtain
Finally a relationship
-(20)
between
lifetime and
extinction coefficient
8  2303 n2
3
1
A201 

n

 d lnn
f
Av
2

c NA
It is convenient to write this equation in terms of the more common units
1
t0
1
 A201  8  2303 cn2 N A  n f
Natural Lifetime
 2.880 109 n2  n f
3
3
 Av
 Av
1
1
Extinction Coefficient
g1
 d lnn

g2
g1
 d lnn

g2
g1 and g2 : degeneracies of the 1, 2 states
-(21)
Quantum Yield and Lifetime
Math happens:
kr
F 
knr  kr
Intrinsic or natural lifetime (tn):
lifetime in the absence of nonradiative processes (F = 1)
=
# of photons absorbed
Experimental lifetime (t):
lifetime with radiative and nonradiative processes
1
kr
tn =
F
=
# of photons emitted
t
tn
1
t =
kr + knr
Algebra happens:
kr = F/t
knr = (1 − F)/t
F and t from experiment, calculate kr and knr
Radiative vs Non-radiative
kr
F 
knr  kr
kr = Φ/τ
X = Br
X= I
knr = (1 − Φ)/τ
kr
2.0 x 108 s-1
2.1 x 108 s-1
knr
1.1 x 108 s-1
1.4 x 109 s-1
• Same extinction coefficient
• Same radiative rate
• knr larger with I
Quantum Yield and Lifetime
Unquenched Emission
kr
F=
kr + kic
t = 1/(kr + kic)
With an acceptor molecule
kr
F=
kr + kic + kET
t = 1/(kr + kic + kET)
kET
knr
t
F
Quantum Yield and Lifetime
Unquenched Emission
kr
F=
kr + kic
t = 1/(kr + kic)
With a quenching molecule
kr
F=
kr + kic + kq[C]
t = 1/(kr + kic + kq[C])
[C]
knr
t
F
Side Note: Energy Gap Law
kr
F=
kr + knr
Large Gap
Small Gap
Higher energy
absorption/emission
E1
Lower energy
absorption/emission
knr
E
E
E1
knr
E0
E0
Energy Gap Law
Energy Gap (E)
knr
Poor overlap
Strong overlap
S1
S0
S1
S0
S1
S0
Energy Gap Law
knr (1011 s-1)
Energy Gap (E)
knr
kr
F 
knr  kr
833
667
555
476
416
cm-1
nm
F
Energy Gap
knr = 1013e-aE (sec-1)
If you red shift emission
efficiency goes down!
Implications of Energy Gap Law
Organic Solar Cells
Charge
Separation
Excitation
hn
A
A
C
C*
kcs
knr
A
C*
kcs
h ∝
kcs + knr
h  Solar cell efficiency
A-
C+
Implications of Energy Gap Law
kcs
kcs
h ∝
kcs + knr
knr
Irradiance (W M-2 nm-1)
knr
h
h Solar cell efficiency
C*
A
Energy Gap (E)
AM1.5 (Global tilt)
1.5
To increase efficiency:
1.0
Decrease E
Reduce knr
0.5
Increase kcs
0.0
500
750
1000 1250 1500 1750 2000
Wavelength (nm)
Implications of Energy Gap Law
Organic Light Emitting Diodes
F = 50%
N
N
lmax = 650 nm
t
= 90 ms
Pt
N
N
500Å Ag
1000Å Mg:Ag
100Å Alq3
400Å PtOEP : Alq3
350Å NPD
60Å CuPc
ITO
Red OLED >20% efficiency
Implications of Energy Gap Law
Organic Light Emitting Diodes
F = 40%
lmax = 765 nm
t
= 50 ms
Visible Image 550 nm
EL Intensity (a.u.)
LiF/Al
AlQ3
AlQ3:Pt(TBP), 6%
NPD
300 400 500 600 700 800 900
Wavelength (nm)
nIR OLED <7 % efficiency
ITO
IR monocular which replaces infrared
emission (800 nm) with green
Side Note: Photochemical Yield
Source
hn
Sample
F=
# of reacted molecules
# of photons absorbed
Actinometry
Reference
hn
F =
# of events
# of absorbed photons
hn
100% yield
# of absorbed photons = # of events
From NMR/UV-Vis/MS
of the photoproduct…
Variables
Photons/second/area
Wavelength
Absorption
Extinction Coefficient
Sample
hn
hn
?% yield
From NMR/UV-Vis/MS…
# of events
=
F =
# of absorbed photons
3
5
= 60% yield
From Actinometer
Quantum Yield End
Any Questions?