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