Transcript Slide 1

Some of this weeks seminars:
Dynamical Studies of the Photodissociation of Ozone: From the
Near IR to the VUV
February 12 | 4-5 p.m. | Pitzer Auditorium, 120 Latimer Hall
Dr. Reinhard Schinke, Max-Planck-Institut fuer Dynamik und
Selbstorganisation, Goettingen, Germany
Engineering Organic-to-Semiconductor Heterojunctions
February 13 | 4-5 p.m. | Pitzer Auditorium, 120 Latimer Hall
Professor Thomas F. Kuech, Dept. of Chemical & Biological
Engineering, University of Wisconsin - Madison
Using Supported Lipid Bilayers as a Separation Matrix
February 15 | 4-5 p.m. | 775A Tan Hall
Professor Paul Cremer, Dept. of Chemistry, Texas A & M University
Absorption/Emission
Connections between the rates of stimulated and spontaneous emission:
Case a)
Thermal equilibrium in a cavity
E2,N2,g2
A21
E1,N1,g1
B12W(w)
B21W(w)
W(w), the energy
density and A,B
the Einstein A and B
coefficients which are
the rate constants (per
molecule), excepting the
energy density for the
transition probability,
Wif. Also N large so we
need not consider
statistics.
At equilibrium dN1/dt = -dN2/dt = 0 = N2A21-N1B12W(w)+N2B21W(w)
Connect A and B to Golden Rule
2
1
Wif 
if   fi 
2
6 0
N1Wif  N1 Bif   fi 
Bif 
A fi 
if
2
6 0 2
gi
gf
8h 3
if
c3
B
2
A fi 
if
gi
8h 3
g f 6  2 c 3
0
rate / m olecule
A21
W (w )  N1
B  B21
N 2 12
 E1 / kT
g
e
N1
1
w / kT


g
/
g
e


1
2
N2
 E2 / kT
g 2e

W w   A21 / g1 / g 2 ew / kT B12  B21

w 3
dw
Planck's law W w dw  2 3 w / kT
 c e
1
The two expressions are equal at all T only if:
g1
g2
B12  B21
 B  A
w 3
 2c 3
21
21
Comparing again to Planck’s Law
A21 / B21W w   e
T=300K
w / kT  1
w / kT
1
at l=50m, 6THz, 200 cm-1
Longer wavelengths stimulated exceeds spontaneous rate
Shorter stimulated emission is slow compared to spontaneous rate
Case b)
A light source, Now W is not thermal energy density but the energy density of
the light source (assumed to be large enough that we can neglect the thermal
field).
At
BW / A  1
the stimulated and spontaneous rates are equal.
Consider visible light of frequency 5x1014 Hz, 3x10-19J
 w 3 
W dw   A / B dw   2 3 dw  1014 J / m3
 c 
Intensity obtained by ,multiplying by c is 3x10-6 dw W/m2
dw for an ordinary spectroscopic light source is ~1011 Hz
The intensity required to equalize spontaneous and stimulated
emission rates is ~105W/m2
Some light sources
Strong Hg Lamp
I (W/m2)
104
E(V/m) n/V(m-3)
103
1014
Photons/mode
10-2
cw laser
105
104
1015
1010
pulsed laser (ns)
1013
108
1023
1018
Footnote: Derivation of relations between A, B, assumed thermal radiation. The
relations hold so long as either the radiation field or the molecules are randomly
oriented in space—not necessarily for solids interacting with lasers.
E2,N2,g2
N2+N1=N
A21
B12W(w)
B21W(w)
absorption
negligible
E1,N1,g1
dN1/dt = -dN2/dt = N2A21+ (N2-N1) B12W(w)

NBW
 A 2 BW t 
N 2 t  
1 e
A  2 BW

For short time
s : A  2BW t 1
Expanding the exponential as
1-(A+2BW)t
N 2 t   NBW t
For long times: A  2 BW t  1
NBW
N 2 t  
A  2 BW
N2/N
Why this behavior?
At short times
N 2 t   NBW t
At long times
NBW
N 2 t  
A  2 BW
time
N2/N
Steady state value
What is the limiting ratio?
1
BW/A
4
How does this connect to Beer’s
Law?
Recall
I=I0e-Nsl=I02.30310-Cl
dI  
 I 0  N 2  N1 BW / N    I 0  Ns   I 0  2.303C
dx
Ac2 g  
c 2 g  
BW / N 

s
2
3
2
3
8n h
8n h  rad
g() a normalized lineshape function
n index of refraction
The square of the transition dipole can be
expressed as an integral (over all
frequencies) of the absorption crosssection (or molar absorptivity).
What happens
when we
increase [ ] vs. I?
(Discussion)
Breakdown of Beer’s Law.
BW/A>>1
Intensity falls off linearly with distance in absorber
not exponentially and is independent of I0
I  I 0   NA
Oscillator Strength (CH4.4.3)
f if 

2me if wif
2
3e 
f if  1
f
fif=1; 1 electron allowed transition
>1 multiple transitions
=0.001-0.01; forbidden transitions