Download Ch543Spr07-NotesIpart3

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I. Part 3 -- Transitions
Left off at transition probability – for absorbance: choose  = mk
Pm (t ) 
1
2
  
 mk 2
4 2

0
2
 mk

sin 2
( mk   ) t
[ ( mk   ) 2 ]
f  2t  (  )
for t  
last term is band shape  f (t,  )
Long time:
Note:
Transition Rate:
rkm 
mk

2
2
where  = mk – 
~ 1 for conditions where  = 1
Pm ( t )
t


 mk  

4 2
2
 (  km ) -  (   km )
absorption
stimulated emission
Probability linear in time → longer expose sample to light
the higher probability of a transition
Rate is what we measure experimentally – flux of light
stimulate an absorbance (loss of flux  rate abs)
Uncertainty – lifetime
f(, t) has a width:   ~ 2 t
t → lifetime of state or duration of a pulse (especially femto-sec)
       (2 t )  t     t    1
Ch 4.3 Book does nice relationship of density of photon states and the rate of transition.
Development not central to our course
Ch 4.4 Then it has a detailed discussion of polarizability. We will put this off until we
address scattering. Now focus on dipolar transitions - absorption and emission
Ch. 3
Frequency dependent polarizability – recall start from classical: W = 1/2 E..E
Note: complex due to relaxation, here  is lifetime of state,  is rate of decay  ~
 ( ) 
e2
m
 
fj
j
2
 j0
 2  i 

1


express  quantum mechanically
n0 0  n n 0 0

2 
0n 
n02  2
n0
this picture misses line widths → relaxation → can insert 1 i ~ i into denom.
2

Allows quantum mechanical definition of oscillator strength
2
2m 0n n0
2
f0n 
like probability, f0n ~  0n
2
3e 
convenient method of categorizing transition (sum rule):  fnm  1
m


1
I-
How is this evidenced in matter?
Aside: refraction:
speed light in vacuum – c (const)
speed light in material – v  1 
refractive index
n  cv

 0 0
(4 x 10-7 N/amp2) ?
relative permitivity
=
most non-magnetic  ~ 0
n ~  0 ~ r
actually complex – real part → dispersion (refraction)
imaginary part → absorption
since index normally n > 1 (n = 1 vacuum), refraction
– will cause deviation from path when material has charge in n
– will be greater at frequencies that correspond to an absorbance
Absorbance – sample attenuate intensity in the beam: A   log 10
I  I 0 e  x  I 0 10 - bc
 
I
I0
b = x (path length)
 = 2.3  c, so  ~  ~ k  m
2
at km
thus absorbance relates to probability of charge state
McHale,Ch. 3 Polarizability is response of material to electric field
induced dipole moment: ind   ,  also ~er (charge x distance)
if model response of e to force as Hook's Law → for
harmonic oscillator → F   m2 x = -eE
0
2
e2
ind  e
 so by analogy:   m  0 2
m 02

2
for multiple electrons:   e
m
see text – time dependent:

j
fj
 j2

where fj is oscillator strength
 ( ) 
e2

fj
m

2
j0
  2  i 


relaxation: 

real part respond (), disp
()  ()  i() 
imag - out of phase - absorb
 s () ds
Interchange - Kramers Kronig: ()  2 p 
0

s2  2
 () ds
()  - 2 p 
0

s2  2
2
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3
2rk k  r r  k
rk 2  2
Note see resonance when  matches energy diff.  ~ rk   r   k , result --> big 
QM polarizability:
k   1 
f kn ~ k  r
Now see oscillator strength:
2
relative permittivity - response of medium to field
 r   0  1
– apparent field / E0 – applied
i.e. dielectric constant – factor of reduction for Coulombic force in a medium
Applied field induces polarization (P) in medium to oppose it

 
  0 - P  0



P   0 ( r  1)    e 0
Electric susceptibility:  = r - 1
isotropic medium, field dependent - non-linear terms
Parallels polarizability discussion above:
if using frequency
r ()  r ()  ir ()
n()  nr ()  i ()
same with refraction
since:
n2() = r()
plug in prop.vector
 r ( )  n 2 ( )   2 ( )
 r()  2n n ()   ()
k  2n with k || x

into
  0 exp [ i(kx  t ) ]
x

   0 exp [ i (2n r x/   t ) ] exp  -2




nr  mod ify in medium
i.e.  same but v c n
Now:
I ~ 
damp out
the amplitude
2
I ~ I 0 exp (- x)  I 0 exp  - 4kx 
  
()
()
()  4k  4 


c 2nr
c nr
( )
for solution:   2.3c    2.3c
A = bc ~ cxs where xs - path, c - conc.
so absorption coefficient:
I-
Now want to relate this to dipole expressions
McHale, Ch 6; Struve, Ch 8; Bernath, Ch 1
Einstein relationships are phenormalized expressions of rates of ???
up
r12 = N1 B12 ()
N1 – population lower state
B12 – stimulated rate constant at 
down r21 = N2 (B21 () + A21)
() – energy density
A21 – spontaneous rate
note: only interested in  = 12 → resonant frequency
dN2
simple kinetics – no light
r21  N2 A 21 
dt
N2  N02 e -A21t
radiative lifetime:
 1
A 21
1st order decay
if light on a long time system comes to equilibrium
N1 B12  = N2(B21  + A21) and
solve for
( ) 
(relating () to kinetics)

A 21N2
(N1B 21 - N2B 21)
N2 g2 - 

e
N1 g1
kt
 = h
A 21N2
h


 B g1

kt
e

B
21 
 12 g2


if let () be a black body light source (also equilibrium)
1
3 h kt
8

h

( ) 
e
 1  g1 B21  g2 B21 (gives denomination term)
3
c
3
A 21
 8h
B 21
c3
see that A21 depends strongly on 3 → probability of spontaneous emission increase
as go to the uv
Two emission processes
A 21
~  eh kt  1
Compare rates
B21  

high  → uv – spontaneous dominance
low  → IR – stimulated dominance
Important →
→
spontaneous (fluorescence) – incoherent
stimulated (e.g. laser) – same properties as

incident photoreduction ̂ and polarization A
So how do lasers work in vis-uv (note kT ~ 200 cm-1 – for IR)
non-equilibrium devices → population inversion

 
4
I-
must make N2 > N1 (non ???)
Recall Lambert Law:
dI = -I dx
I = I0 e-x
positive   absorption  loss of intensity
negative
emission (stimulated)  amplification
 
dI  (N  N ) r ( ) h
2
1 12
dx
assume B12 = B21
relate power/volume to energy density/time:
relate to Einstein:
r12 
A 21C3( )
8n3h3
c 2 I ( ) g ( )

8n2h3
Lambert Law:
(correct c for n) non degenerate B12()
() → nI()g()/c






2 g ( )

(
N

N
)
c
dI  2 1
I ( ) relative pop (N 2
dx

8 n 2  2 rad


now back to macroscopic: r  1   e
  x cn
 r   e
c3g()(N1  N2 )
 
162n3rad
Relate back to Golden Rule:
 N1 ) determine abs - cm
complex part
of induced polarization
~ 2n() k()
2
12
B12 
6 0  2
2 3 3
3
g2 16 21 n 
A12 
 1
g1
3
rad
3 0 h c
rethink Lambert Law:
assume electronic/no stimulated emission
 dI  N [ (h) dx ] r21
I in increment dx (cross section: 1-unit) (N2 ~ 0)
Beers Law: -dI = 2.303 a()cIdx
I = I0e-acx
a() = e()
(c-conc. M, a(n) – malar absorption (per cm) molec abs
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r21 
Combine to get rate:
( I  cu( )
2.303 a()CI
2.303 ca() u( )

Nh 
NA h  n
u() energy density C  1000 N
n
NA
N  mole )
cm3
( ) d 
d

() a 
r21  2303 C 
d
NAh n

To account for bandwidth u( )  
but normally () constant over bandwidth – take out and compare to relationship for B12
a()
B21  2303 C 
d
NAh n

so if measure spectra, integrate, connect for path and concentration
can determine B12 experimentally also
(602 ) 2303 C a()
12 
  d
NAh n
a( )
 0.92  10  2 
d

2
in D2
(devices for homework!)
McHale 6.6? Line shapes
Homogenous → “all affect same way” → typical lifetime
  ~ 1  time and freq complementary variable (inverse)

t
Fourier Transform relate then: A( t )  A 0 e  (on correlation function)


I max  1 
 2
 4 
Lorentzian: IL ( ) 


(   0 ) 2   1 
 2
 4 
(very long tails)
FWHM 
This concept for single state transition →
for electronic –vibration (mix) or (rotation-vibration) the
distribution of states (if unresolved)  shape
12
 
6
I-
Bernath, p31
Fig 1-22
Inhomogenous broadening → collection of molecules has a distribution
of resonant freq 12
 - (   )2 
0 
Gaussian dist
IG ()  Imax exp 
 22 
(broaden FWHM but tails drop fast)
1 2  2 (2n2)1 2 6
G = standar deviation:
(often use  )
gas phase – velocity distribution means random with r/t detected
shift frequency due to Doppler effect
   0 1   
c

12
 2k T n2 

B

straight on (+) or away (-)
d1 2  2


mc


Voigt profile
I voigt () 
 I 0 () I L (  ) d
convolutes both allows “resolution” homogeneous and in homogeneous (numerical)
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