Faculty of
Natural Sciences
CO Project
Submitted by:
Sophie Stolte
sophie.stolte@uni-ulm.de
Teacher:
Smita Omkarnath Ganguly
2022
Contents
1 Exercise 8
1
2 Exercise 9
5
3 Exercise 10: Valence electron spectrum
7
4 Exercise 10: Core electron spectrum
14
5 Exercise 11
17
Bibliography
19
1 Exercise 8
The following equation should be demonstrated:
Γ21 + Γ22 = Γ2c
(1.1)
Γ presents the fwhm of a gaussian density function. It is important, to only convolute two gaussian functions. Otherwise the equation does not hold. Two gaussian
functions were created in python. According to the hint given in the manual one of
these were defined as main and the other as broadening. A function to determine
the fwhm of a function was implemented. To test this, even from the starting functions the fwhm is determined. Different examples are chosen, for each of them the
equation holds. However, there are sometimes small deviations because the determined fwhm is not completely exact. One of the reason is the linespace chosen
for x. The resolution is not perfect. Fig. 1.1, 1.2, 1.3, show different examples.
The chosen values for Γ1 and Γ2 are written in the caption. Using the code in the
appendix allows to generate further examples.
1
1 Exercise 8
Figure 1.1: Γ1 = 10, Γ2 = 7, Γ2c = 148.84
2
1 Exercise 8
Figure 1.2: Γ1 = 1, Γ2 = 2.52, Γ2c = 7.18
3
1 Exercise 8
Figure 1.3: Γ1 = 3, Γ2 = 4, Γ2c = 25
4
2 Exercise 9
The exercise dealt with the effects of the lineshape of the convolution if the fwhm
of the lorentzian or gaussian are changed. For the lorentzian density function a
function in python was implemented. The formula given at the manual was used
(further details in the appended code). Representativly three cases are considered.
In fig. 2.1 the fwhm of the gaussian equals the lorentzian. In fig. 2.2 ΓG = 0.005 and
in fig. 2.3 ΓG = 0.1. The lineshape of the voigt function differs for all three cases.
For equal fwhm values, the lineshape is a combination of the two distinct density
function appearances. For the other two cases the lineshape always becomes more
similar to the lineshape of the density function with the greater fwhm. Thus, for
ΓG < ΓL the lineshape of the voigt function is more like the lorentzian. And the
other way around.
Figure 2.1: ΓG = 0.05, ΓL = 0.05
5
2 Exercise 9
Figure 2.2: ΓG = 0.005, ΓL = 0.05
Figure 2.3: ΓG = 0.1, ΓL = 0.05
6
3 Exercise 10: Valence electron
spectrum
We received a data file for a CO molecule. It was differentiate for valence and
core electrons. This chapter deals with analyzing the valence electron spectrum.
The dataset contains the binding energy and the counts, intensity respectively. For
the fitting the peaks were found in the first place. Therefore, the python function,
find peaks was used. Related energies are also saved. For the first guess, 17
peaks were used. A clear separation of three different region can be seen. These
electronic states are defined by a 5σ , 1π and 4σ orbitals. Each of them have got
distinct vibrational states, which are also recorded. For further analysis the fwhm
is needed. As a first guess, the value given in the manual was used for ΓG . The
lorentzian fwhm was set on the base of some calculations. For the first guess for
each electronic state the peak with the highest intensity was used. The fwhm was
afterwards calculated. As it holds, that Γv = ΓL , it was useful for the first guess.
The first guess fit is shown in fig. 3.1.
7
3 Exercise 10: Valence electron spectrum
Figure 3.1: First guess of the spectra of the valence electrons.
This was used for further optimization of the fit, resulting in fig. 3.2.
8
3 Exercise 10: Valence electron spectrum
Figure 3.2: Resulting fit of the spectra of the valence electrons.
For this fit all parameters were displayed by python, which are used for further
analysis. They are given as:
9
3 Exercise 10: Valence electron spectrum
Peak number
Binding energy [eV]
Intensity
ΓL
ΓG
1
14.3641
0.7591
0.0068
0.0421
2
14.6344
0.0388
0
0.043
3
16.8854
0.2034
0.005
0.0468
4
17.0752
0.4582
0.0046
0.0468
5
17.2629
0.5863
0.0072
0.0464
6
17.445
0.5184
0.0056
0.0462
7
17.6249
0.3772
0.0055
0.0462
8
17.8014
0.2493
0.0094
0.0465
9
17.9748
0.1406
0.0083
0.0448
10
18.1447
0.0764
-0.0000
0.0476
11
18.3103
0.0416
0.0142
0.0416
12
18.4748
0.0198
0.0049
0.0458
13
18.6344
0.0102
0.0443
0.0162
14
20.0123
1.0041
0.0078
0.0430
15
20.2187
0.4027
0.0081
0.0425
16
20.4200
0.0894
0.0106
0.0403
17
20.6146
0.0150
0.0138
0.0374
Table 3.1: Table peaks, corresponding energy and the two fwhm.
For each of the three electronic states it is possible to calculate the lifetime τ . Therefore the following equation is used
τ=
1
.
2πΓL
(3.1)
Until now ΓL is given in Hertz. It can be used, that
E = hν
(3.2)
holds.
Calculating the Huang-Rhys parameter can be of interest in the case of considering
the Franck-Condon factor. It considers emitted photons during electronic transitions
[1]. According to the results of exercise 2 it can be determined by the relation of two
consecutive intensities of two vibrational levels.
10
3 Exercise 10: Valence electron spectrum
It holds that
δ2
2
S=
(3.3)
(cf. manual). Therefore and on base of task 3, the change in bond length can be
calculated via
√
2Sℏ
.
ωµ
∆re
(3.4)
The reduced mass µ can be found with the help of values given in a periodic table.
It results in µ = 1.138 · 10−26 kg. In the manual a value for the wavenumber k is
given. Therefore, ω can be written as ω = kc, with c as speed of light. Hence for
further calculation the following formula was used
√
∆re
2Sℏ
.
kcµ
(3.5)
The resulting lifetimes, Huang-Rhys parameters and ∆re are presented in the following table.
State
τ in s
S
∆re in m
5σ
9.69 · 10−14
1.315 · 10−13
8.45 · 10−14
0.051
9.55 · 10−12
6.34 · 10−11
2.678 · 10−11
1π
4σ
2.25
0.401
Table 3.2: State and corresponding lifetime, S values and ∆re
Last but not least, the morse potential can be determined, using the spectra. It
holds that
(
E=
(
(cf. manual). If now ν +
1
2
)
1
ν+
2
)
(
1
ℏωe − ν +
2
)2
ℏωe xe
(3.6)
is substituted by x, a quadratic fit y = ax2 + bx + c can
be made by python. A coefficient comparison delivers values for the factors. Out of
them ωe and xe can be determined. For fitting the binding energy in eV is plotted
against the vibrational quantum number. Of course it has to be differentiate for all
11
3 Exercise 10: Valence electron spectrum
three states. Also not all peaks, which where used for the first guess where also
used here. For the second state they differ, because the intensity of a few of them
is really low. However, using more peaks for the first guess, gives the possibility for
a better fit, which is why they were used there and not in case of determining the
morse potential. The resulting plot and corresponding fit can be seen in fig 3.3.
Figure 3.3: Plot of the energy against the vibrational quantum number
For the first state no fit can be made, as the number of points is not sufficient. For
the second and third one, the resulting function are given as
E2 (x) = −1.79 · 10−3 x2 + 1.92 · 10−1 x + 1.68892857 · 101
(3.7)
E3 (x) = −5 · 10−3 x2 + 2.15 · 10−1 x + 2.001 · 101 .
(3.8)
Hence, b = ℏωe and a = −ℏωe xe . Calculating ωe , xe is crucial to determine De and
α. The used formulas are given in the manual. Furthermore ωe as the vibrational
frequency is also of general interest.
The following table summarize all the values for the second and third state.
12
3 Exercise 10: Valence electron spectrum
State
ωe in Hz
xe
De
α
1π
2.92 · 1014
2.92 · 1014
9.32 · 10−3
9.32 · 10−3
5.16
2.34
60.92
101.256
4σ
Table 3.3: State and corresponding ωe , xe , De , α
Afterwards, the morse potential can be drawn. It can be seen in figure 3.4. At some
point there has to be a error in the calculation, because the order of magnitude of
the energy does not make sense. However, the general lineshape does not look too
bad.
Figure 3.4: Resulting morse potential, for the second and third state.
13
4 Exercise 10: Core electron
spectrum
The same steps, as described before, were also conducted for the core electron
spectrum. As the steps are accomplished in the same way, the description is rather
short and minimal here.
For the first guess, the 3 shown peaks were used. Each of them represent a vibrational state. For further analysis the fwhm is needed. The first guess fit is shown in
fig. 4.1.
Figure 4.1: First guess of the spectra of the valence electrons.
This was used for further optimization of the fit, resulting in fig. 4.2.
14
4 Exercise 10: Core electron spectrum
Figure 4.2: Resulting fit of the spectra of the valence electrons.
For this fit all parameters were displayed by python, which are used for further
analysis. They are given as:
Peak number
Binding energy [eV]
Intensity
ΓL
ΓG
1
295.6726
0.9335
0.0654
0.1452
2
295.9727
0.6239
0.1247
0.1142
3
296.2741
0.1603
0.1841
0.0024
Table 4.1: Table peaks, corresponding energy and the two fwhm.
Also the lifetime τ , the Huang-Rhys parameter and ∆re was calculated for this spectrum. They are presented in the following table.
τ in s
S
∆re in m
1.004 · 10−14
0.668
1.379 · 10−11
Table 4.2: State and corresponding lifetime, S values and ∆re
15
4 Exercise 10: Core electron spectrum
Last but not least, the morse potential can be determined, using the spectra, too. It
also gives the vibrational frequency. The following table summarize all the values.
The way to calculate them is described in detail for the valence spectrum. The steps
are analog.
ωe in Hz
xe
De
α
4.74 · 1014
0.042
1.859
164.75
Table 4.3: ωe , xe , De , α
Afterwards, the morse potential can be drawn. It can be seen in fig. 4.3. At some
point there has to be a error in the calculation here as well, because the order of
magnitude of the energy does not make sense. However, the general lineshape
does not look too bad.
Figure 4.3: Resulting morse potential for the core hole spectrum.
16
5 Exercise 11
The last part considers some comparisons between founded results. Especially the
differences whether a core electron or a valence electron is considered are of interest.
The binding energy of the core electron is a lot higher than EB of the valence electrons. This is in accordance to literature. Of course more energy is needed when
an electron closely attached to the nucleus should be taken out.
Furthermore, it can be observed by looking at the spectra, that the valence electron spectrum is more narrowed than the core-hole spectrum. This is acually really
broad, indicating the already mentioned stronger binding. Looking at the lifetime
unfolds that the lifetime of the core state is the shortest. Also this can be explained
and approved by literature. For an atom it is even more unfavorable to have got a
hole near the nucleus.
The vibrational frequency of the core hole state is significantly larger than the vibrational frequency of the valence states.
Attached to this file, another file with some handwritten notes can be found. Furthermore, the code for the analysis. Two notebooks were saved, just because some
modifications were made for the two distinct spectra. However, the code does not
really differ for both spectra.
17
List of Figures
1.1 Γ1 = 10, Γ2 = 7, Γ2c = 148.84 . . . . . . . . . . . . . . . . . . . . . .
1.2 Γ1 = 1, Γ2 = 2.52,
Γ2c
2
= 7.18 . . . . . . . . . . . . . . . . . . . . . .
1.3 Γ1 = 3, Γ2 = 4, Γ2c = 25 . . . . . . . . . . . . . . . . . . . . . . . . .
3
2.1 ΓG = 0.05, ΓL = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2 ΓG = 0.005, ΓL = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . .
6
2.3 ΓG = 0.1, ΓL = 0.05 . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
3.1 First guess of the spectra of the valence electrons. . . . . . . . . . .
8
3.2 Resulting fit of the spectra of the valence electrons. . . . . . . . . . .
9
3.3 Plot of the energy against the vibrational quantum number . . . . . .
12
3.4 Resulting morse potential, for the second and third state. . . . . . . .
13
4.1 First guess of the spectra of the valence electrons. . . . . . . . . . .
14
4.2 Resulting fit of the spectra of the valence electrons. . . . . . . . . . .
15
4.3 Resulting morse potential for the core hole spectrum. . . . . . . . . .
16
18
4
Bibliography
[1]
Wikipedia. ‘Huang-rhys-faktor.’ (2022), [Online]. Available: https://de.wiki
pedia.org/wiki/Huang-Rhys-Faktor (visited on 12/02/2022).
19