5.33 Lecture Notes,Frequency- and Time-Domain Spectroscopy
We just showed that you could characterize a system by taking an absorption spectrum,We
select a frequency component using a grating or prism,irradiate the sample,and measure how
much gets absorbed we just change frequency and repeat,
(1) Absorption Spectrum? Frequency domain
Vary frequency of driving field,
P
av
(ω)
ω
0
γ
Disperse color → measure power absorbed
ω
The variables ω
0
and γ characterize the time-dependent behavior of our H.O,
So indeed we could measure these variables also if we applied a short driving force and watched
the coordinate directly,It would oscillate and relax to equilibrium,and we would
characterize ω
0
+ γ that way,
(2) Watch coordinate? Time domain
Q
2π
t
eA
γ?
short driving force → watch relaxation/oscillation
0
This is the basis for time-resolved spectroscopy using short pulses of light to exert an impulse
reponse on the system and watch chemical processes happen,
Also,all modern NMR instruments work this way,applying a burst of RF radiation to sample in
field and watch relax,
The information content is essentially the same in either domain,Why use time?
1) All data collected in single observation—faster than collecting one frequency point at a
time!
2) Resolving power between peaks is often better,
In practice,different methods work better for different spectroscopic or different types of
measurements/information,
Fourier Transform Relations
In fact,there is a formal relationship between the time domain and frequency domain
→ Fourier Transformation,
Joseph Fourier showed that any periodic function can be expressed as an expansion in cosines +
sines,
∞
Ft() =
∑
[
a
n
cos(nω′t)+ b
n
sin(nω′t)
]
n=1
So we can either represent data as the time response,or the value of all of the expansion
coefficients in freq,→ the spectrum!
Time domain St? Frequency domain S(ω)( )
These are the sine
transforms,More
These are related through the Fourier Transform integrals,
generally the term
sinωt is written as e
iωt
,
and we look at the real
+∞
ω ω
2π
∫
∞
() sin ωt dt the integral.S(t) =
2π
1
∫
∞
S () sin ωt dω? S ()=
1
+∞
S t
or imaginary part of
t( this gives S () if we know S (ω) ) ( this gives S (ω) if we know S ( t ) )
For our harmonic oscillator we find,
γ
S(t) ∝ e
γt
sin ω
0
t? S ()
(ω?ω
0
)
2
+γ
2
ω ∝
Parameters in the time and frequency domains,
Parameter Time Domain S(t)
Large ω
0
Fast oscillations
Small ω
0
Slow oscillations
Large γ Fast decay
Small γ Slow decay
Frequency Domain S(ω)
High frequency
Low frequency
Broad linewidth
Narrow linewidth
5.33 Lecture Notes,Time and Frequency Domains Page 2
Examples of Fourier Transforms
1) C(t) = e
γt
sin(ωt) with ω/2π = ν = 0.1
(The units of time or frequency are
arbitrary – lets call it seconds,The time
period for oscillation 1/ν = 10 sec and the
time scale for relaxation is 1/γ = 100 sec)
S is the Fourier transform of C(t),
and γ = 0.01
1
Ct() 0
1
0 5 10 15 20 25 30 35 40
ν
.
t
2
S
j
0
0 0.05 0.1 0.15 0.2
ν
j
If we have two superimposed oscillations,C(t) = e
γt
sin(ω
1
t) + e
γt
sin(ω
2
t)
with ω
1
/2π = ν
1
= 0.1; ω
2
/2π = ν
2
= 0.11; and γ = 0.01
then we expect two peaks in the spectrum at the two frequencies,each with linewidth 2γ,
In time,this manifests itself as two beat frequencies,One with a time period corresponding to
the mean frequency (ν
1
+ν
2
)/2,and the other to the frequency splitting (ν
1
ν
2
)/2,
2
Ct() 0
2
0 5 10 15 20 25 30 35 40
2/(ν
1
+ν
2
)
2/(ν
1
ν
2
)
ν
.
t
2
S
j
0
0 0.05 0.1 0.15 0.2
ν
j
ν
2
ν
1
5.33 Lecture Notes,Time and Frequency Domains Page 3
2) Other examples,
Two superimposed oscillations,C(t) = A
1
exp(?γ
1
t)sin(ω
1
t) + A
2
exp(?γ
2
t)sin(ω
2
t)
with ω
1
/2π = ν
1
= 0.1; ω
2
/2π = ν
2
= 0.125; γ
1
= 1/50; γ
2
= 1/10; A
1
= 3; A
2
= 5,
Ct()
A
1
.
exp γ
1
t
,
0
A
2
.
exp γ
2
t
,
0 5 10 15 20
ν
1
ν
2
.
t
2
4
S
j
2
0
0 0.05 0.1 0.15 0.2
ν
j
10
A sum of harmonics in ω,C(t) =
∑
exp(?γt)sin(n ωt) with ω/2π = ν = 0.03 and γ = 0.01,
n=1
These are similar to what you will see in rotational and rotational-vibrational spectra,A regularly
spaced set of absorption features represent a behavior in time,where all frequencies constructively
interfere at periodic multiples of the fundamental period (recurrences),
10
Ct() 0
10
0 2 4 6 8 10 12 14
ν
.
t
1
S
j
0
0 0.1 0.2 0.3 0.4
ν
j
5.33 Lecture Notes,Time and Frequency Domains Page 4
DAMPING
Damping is any process that influences the amplitude or phase of an oscillating
system,
Let’s look more carefully at this,
For a single harmonic oscillator,the decay is due to loss of energy → friction
Molecule,1) interaction with environment (heat – nonradiative)
2) emission of radiation (radiative)
This is known as lifetime broadening → T
1
We make measurements on large numbers of molecules:,ensembles.”
Properties of the collection can influence observed relaxation effects,For N molecules,
()Qt
N N
=
∑
Q
i
()=
∑
A
i
e
γ
i
t
sin (?
i
t )t
i1 i1= =
What happens if the frequencies are a bit different?
Time Domain Frequency Domain
Q1
Q2
Q3
Q4
Q?
t γ?
1
ω
2
ω
3
ω
4
ω
inhom
2γ
e
inhom
There are tricks for removing this,single-molecule spectroscopy / echo experiments
5.33 Lecture Notes,Time and Frequency Domains Page 5
What about phase?
We use a light field that drives all oscillators in
phase,but the oscillator phase can vary in time →
for instance,collisions,
Destructive interference → pure dephasing
*
Pure dephasing time,T
2
Q?
*
2eff
T t t
ee
=
γ
There are other damping processes,
For instance,in your NMR experiment you measure the rates of exchange between pyruvic
acid and dihydroxypropanoic acid by measuring linewidths,This is dephasing due to
exchange between two species with distinct frequencies,
molecule 1
molecule 2
Q?
molecule 3
τ t
e
When averaged over all molecules,the destructive interference due to hopping between sites
leads to a decay that reflects the mean residence time,τ,
Also,rotational motion is seen in linewidth,
All processes that influence the amplitude or phase of the oscillations are observed in a
spectrum,
5.33 Lecture Notes,Time and Frequency Domains Page 6
The damping rates of a collection of oscillators is a sum over all relaxation rates,
γ= γ
lifetime
+γ
dephasing
+γ
inhom
+ "
1 1 1
T
2
=
T
1
+
T
2
*
+ "
γ 2
How can you separate these in a spectrum? You can’t without additional measurement,
in gas—organics mainly lifetime,,,
in room temperature solution—mainly dephasing (IR and electronic)
in low T systems (liquids/crystals) —γ
inhom
5.33 Lecture Notes,Time and Frequency Domains Page 7
We just showed that you could characterize a system by taking an absorption spectrum,We
select a frequency component using a grating or prism,irradiate the sample,and measure how
much gets absorbed we just change frequency and repeat,
(1) Absorption Spectrum? Frequency domain
Vary frequency of driving field,
P
av
(ω)
ω
0
γ
Disperse color → measure power absorbed
ω
The variables ω
0
and γ characterize the time-dependent behavior of our H.O,
So indeed we could measure these variables also if we applied a short driving force and watched
the coordinate directly,It would oscillate and relax to equilibrium,and we would
characterize ω
0
+ γ that way,
(2) Watch coordinate? Time domain
Q
2π
t
eA
γ?
short driving force → watch relaxation/oscillation
0
This is the basis for time-resolved spectroscopy using short pulses of light to exert an impulse
reponse on the system and watch chemical processes happen,
Also,all modern NMR instruments work this way,applying a burst of RF radiation to sample in
field and watch relax,
The information content is essentially the same in either domain,Why use time?
1) All data collected in single observation—faster than collecting one frequency point at a
time!
2) Resolving power between peaks is often better,
In practice,different methods work better for different spectroscopic or different types of
measurements/information,
Fourier Transform Relations
In fact,there is a formal relationship between the time domain and frequency domain
→ Fourier Transformation,
Joseph Fourier showed that any periodic function can be expressed as an expansion in cosines +
sines,
∞
Ft() =
∑
[
a
n
cos(nω′t)+ b
n
sin(nω′t)
]
n=1
So we can either represent data as the time response,or the value of all of the expansion
coefficients in freq,→ the spectrum!
Time domain St? Frequency domain S(ω)( )
These are the sine
transforms,More
These are related through the Fourier Transform integrals,
generally the term
sinωt is written as e
iωt
,
and we look at the real
+∞
ω ω
2π
∫
∞
() sin ωt dt the integral.S(t) =
2π
1
∫
∞
S () sin ωt dω? S ()=
1
+∞
S t
or imaginary part of
t( this gives S () if we know S (ω) ) ( this gives S (ω) if we know S ( t ) )
For our harmonic oscillator we find,
γ
S(t) ∝ e
γt
sin ω
0
t? S ()
(ω?ω
0
)
2
+γ
2
ω ∝
Parameters in the time and frequency domains,
Parameter Time Domain S(t)
Large ω
0
Fast oscillations
Small ω
0
Slow oscillations
Large γ Fast decay
Small γ Slow decay
Frequency Domain S(ω)
High frequency
Low frequency
Broad linewidth
Narrow linewidth
5.33 Lecture Notes,Time and Frequency Domains Page 2
Examples of Fourier Transforms
1) C(t) = e
γt
sin(ωt) with ω/2π = ν = 0.1
(The units of time or frequency are
arbitrary – lets call it seconds,The time
period for oscillation 1/ν = 10 sec and the
time scale for relaxation is 1/γ = 100 sec)
S is the Fourier transform of C(t),
and γ = 0.01
1
Ct() 0
1
0 5 10 15 20 25 30 35 40
ν
.
t
2
S
j
0
0 0.05 0.1 0.15 0.2
ν
j
If we have two superimposed oscillations,C(t) = e
γt
sin(ω
1
t) + e
γt
sin(ω
2
t)
with ω
1
/2π = ν
1
= 0.1; ω
2
/2π = ν
2
= 0.11; and γ = 0.01
then we expect two peaks in the spectrum at the two frequencies,each with linewidth 2γ,
In time,this manifests itself as two beat frequencies,One with a time period corresponding to
the mean frequency (ν
1
+ν
2
)/2,and the other to the frequency splitting (ν
1
ν
2
)/2,
2
Ct() 0
2
0 5 10 15 20 25 30 35 40
2/(ν
1
+ν
2
)
2/(ν
1
ν
2
)
ν
.
t
2
S
j
0
0 0.05 0.1 0.15 0.2
ν
j
ν
2
ν
1
5.33 Lecture Notes,Time and Frequency Domains Page 3
2) Other examples,
Two superimposed oscillations,C(t) = A
1
exp(?γ
1
t)sin(ω
1
t) + A
2
exp(?γ
2
t)sin(ω
2
t)
with ω
1
/2π = ν
1
= 0.1; ω
2
/2π = ν
2
= 0.125; γ
1
= 1/50; γ
2
= 1/10; A
1
= 3; A
2
= 5,
Ct()
A
1
.
exp γ
1
t
,
0
A
2
.
exp γ
2
t
,
0 5 10 15 20
ν
1
ν
2
.
t
2
4
S
j
2
0
0 0.05 0.1 0.15 0.2
ν
j
10
A sum of harmonics in ω,C(t) =
∑
exp(?γt)sin(n ωt) with ω/2π = ν = 0.03 and γ = 0.01,
n=1
These are similar to what you will see in rotational and rotational-vibrational spectra,A regularly
spaced set of absorption features represent a behavior in time,where all frequencies constructively
interfere at periodic multiples of the fundamental period (recurrences),
10
Ct() 0
10
0 2 4 6 8 10 12 14
ν
.
t
1
S
j
0
0 0.1 0.2 0.3 0.4
ν
j
5.33 Lecture Notes,Time and Frequency Domains Page 4
DAMPING
Damping is any process that influences the amplitude or phase of an oscillating
system,
Let’s look more carefully at this,
For a single harmonic oscillator,the decay is due to loss of energy → friction
Molecule,1) interaction with environment (heat – nonradiative)
2) emission of radiation (radiative)
This is known as lifetime broadening → T
1
We make measurements on large numbers of molecules:,ensembles.”
Properties of the collection can influence observed relaxation effects,For N molecules,
()Qt
N N
=
∑
Q
i
()=
∑
A
i
e
γ
i
t
sin (?
i
t )t
i1 i1= =
What happens if the frequencies are a bit different?
Time Domain Frequency Domain
Q1
Q2
Q3
Q4
Q?
t γ?
1
ω
2
ω
3
ω
4
ω
inhom
2γ
e
inhom
There are tricks for removing this,single-molecule spectroscopy / echo experiments
5.33 Lecture Notes,Time and Frequency Domains Page 5
What about phase?
We use a light field that drives all oscillators in
phase,but the oscillator phase can vary in time →
for instance,collisions,
Destructive interference → pure dephasing
*
Pure dephasing time,T
2
Q?
*
2eff
T t t
ee
=
γ
There are other damping processes,
For instance,in your NMR experiment you measure the rates of exchange between pyruvic
acid and dihydroxypropanoic acid by measuring linewidths,This is dephasing due to
exchange between two species with distinct frequencies,
molecule 1
molecule 2
Q?
molecule 3
τ t
e
When averaged over all molecules,the destructive interference due to hopping between sites
leads to a decay that reflects the mean residence time,τ,
Also,rotational motion is seen in linewidth,
All processes that influence the amplitude or phase of the oscillations are observed in a
spectrum,
5.33 Lecture Notes,Time and Frequency Domains Page 6
The damping rates of a collection of oscillators is a sum over all relaxation rates,
γ= γ
lifetime
+γ
dephasing
+γ
inhom
+ "
1 1 1
T
2
=
T
1
+
T
2
*
+ "
γ 2
How can you separate these in a spectrum? You can’t without additional measurement,
in gas—organics mainly lifetime,,,
in room temperature solution—mainly dephasing (IR and electronic)
in low T systems (liquids/crystals) —γ
inhom
5.33 Lecture Notes,Time and Frequency Domains Page 7
