cnclosed are two
𝐷
2
Q
-
branch spectra that are puzzling. Spectrum A is a "Conventional" CARs spectrum in which an
𝜔
3
anti
-
Stokes frequency is observed when
𝜔
2
is tuned such that the difference
𝜔
1
-
𝜔
2
=
𝑄
𝐽
for the
𝑉
=
0
→
1
vibrational transition as depicted below.
v
v
a
.

𝜔
2
frequenei forJ
0

1
)
Q Q transition i
)
i
)
for
𝐷
2
.
𝜔
𝑒
=
3
1
1
5
.
5
0
,
𝜔
𝑒
𝑥
𝑒
=
6
1
.
8
2
,
𝐵
𝑒
=
3
0
.
4
4
3
6
,
𝛼
𝑒
=
0
.
4
1
9
8
b
.
Measure the actual
𝜆
2
values form spectrum
𝛬
and convert to
𝑐
𝑚
-

1

.
Use these and your calculated
𝑄
𝑗
values from part a to
deduce the
(
unspecified
)
value of the fixed frequency source
𝜔
1
in
𝑐
𝑚
-
1
.
The above is preliminary to the intriguing question of the source of the much weaker
(
1
0
-
9
)
Q
-
branch spectrum
𝐵
which is observed when onc tuncs
𝜔
2
to a frequency resion where there should be no CARS resonances. One suggestion is that the spectrum results from a new Kind of
6
-
wave mixing process the might be depicted as follows
𝑄
𝐽
=
2
(
𝜔
1
-
𝜔
2
)
If this picture is true, it suggests that this mothod might be a good way to measure higher overtone energy levels, something that is relatively difficult to do in many moleculos.
C
.
Check whether spectrum
𝐵
is consistent with this
6
-
wave mixing picture. Mcasure
𝜆
2
values for the
𝑞
!

transitions on spectrum
𝐷
and convert to
𝜔
2
values. Using
𝜔
1
from part
𝑏
,
calculatc
2
(
𝜔
1
-
𝜔
2
)
and compare with calculated
𝑣
=
0
→
2
𝑄
𝑗
values using Herzberg's constants. How does
𝜔
1
-
𝜔
2
compare with the
𝑣
=
1
-
2
transitions that could be expected if there is any appreciable thermal population in the
𝑣
=
1
statc?

Question

cnclosed are two 
𝐷
2
 ﻿Q
-
branch spectra that are puzzling. Spectrum A is a "Conventional" CARs spectrum in which an 
𝜔
3
 ﻿anti
-
Stokes frequency is observed when 
𝜔
2
 ﻿is tuned such that the difference 
𝜔
1
-
𝜔
2
=
𝑄
𝐽
 ﻿for the 
𝑉
=
0
→
1
 ﻿vibrational transition as depicted below.
v
v
a
.
 
𝜔
2
 ﻿frequenei forJ 
0
 
1
)
 ﻿Q Q transition i
)
 ﻿i
)
 ﻿for 
𝐷
2
.
𝜔
𝑒
=
3
1
1
5
.
5
0
,
𝜔
𝑒
𝑥
𝑒
=
6
1
.
8
2
,
𝐵
𝑒
=
3
0
.
4
4
3
6
,
𝛼
𝑒
=
0
.
4
1
9
8
b
.
 ﻿Measure the actual 
𝜆
2
 ﻿values form spectrum 
𝛬
 ﻿and convert to 
𝑐
𝑚
-
 
1
 
.
 ﻿Use these and your calculated 
𝑄
𝑗
 ﻿values from part a to
deduce the 
(
unspecified
)
 ﻿value of the fixed frequency source 
𝜔
1
 ﻿in 
𝑐
𝑚
-
1
.
The above is preliminary to the intriguing question of the source of the much weaker 
(
1
0
-
9
)
 ﻿Q
-
branch spectrum 
𝐵
 ﻿which is observed when onc tuncs 
𝜔
2
 ﻿to a frequency resion where there should be no CARS resonances. One suggestion is that the spectrum results from a new Kind of 
6
-
wave mixing process the might be depicted as follows
𝑄
𝐽
=
2
(
𝜔
1
-
𝜔
2
)
If this picture is true, it suggests that this mothod might be a good way to measure higher overtone energy levels, something that is relatively difficult to do in many moleculos.
C
.
 ﻿Check whether spectrum 
𝐵
 ﻿is consistent with this 
6
-
wave mixing picture. Mcasure 
𝜆
2
 ﻿values for the 
𝑞
!
 
 ﻿transitions on spectrum 
𝐷
 ﻿and convert to 
𝜔
2
 ﻿values. Using 
𝜔
1
 ﻿from part 
𝑏
,
 ﻿calculatc 
2
(
𝜔
1
-
𝜔
2
)
 ﻿and compare with calculated 
𝑣
=
0
→
2
𝑄
𝑗
 ﻿values using Herzberg's constants. How does 
𝜔
1
-
𝜔
2
 ﻿compare with the 
𝑣
=
1
-
2
 ﻿transitions that could be expected if there is any appreciable thermal population in the 
𝑣
=
1
 ﻿statc?

Accepted Answer

∻Solution∻ ‖ 1 ⋮ **Measure $\lambda_2$ values from spectrum A**: To convert the measured wavelengths ($\lambda_2$) to wavenumbers ($ ilde{ u}$), use the relation $ ilde{ u} = \frac{1}{\lambda}$. ‖ ‖ 2 ⋮ **Calculate $ ilde{ u}_2$ and deduce $ ilde{ u}_1$**: Use the given Herzberg constants to calculate the $Q_J$ values for the $v=0 ightarrow 1$ transition. Then, use the relation $ ilde{ u}_1 - ilde{ u}_2 = Q_J$ to find the fixed frequency source $ ilde{ u}_1$. ‖ ‖ 3 ⋮ **Check consistency with 6-wave mixing**: For spectrum B, measure $\lambda_2$ for the $q_1$ transitions and convert to $ ilde{ u}_2$ values. Calculate $2( ilde{ u}_1 - ilde{ u}_2)$ and compare with the calculated $v=0 ightarrow 2$ $Q_J$ values using Herzberg's constants. ‖ ∻Answer∻ ⚹ [Insert final answer here after performing the calculations based on the measured $\lambda_2$ values and Herzberg's constants] ⚹ ∻Key Concept∻ ⚹ Conversion of wavelength to wavenumber and the use of Herzberg's constants to analyze vibrational transitions ⚹ ∻Explanation∻ ⚹ Wavenumber is the reciprocal of wavelength and is used in spectroscopy to characterize transitions. Herzberg's constants allow for the calculation of energy levels and transitions in diatomic molecules. ⚹◊How can the 6-wave mixing process be used to measure higher overtone energy levels in molecules?⍭ Generate me a similar question◊

Sia