Improvement of Energy Levels in Praseodymium-I by a Line Combination Approach

Bachelor Thesis by Martin Nuss - 0630923 TU Graz - Institute of Experimental Physics

under the supervision of Univ.-Prof. Dr. L. Windholz

April, 2009 Contents

1 Abstract 3

2 Introduction 4

3 Fundamentals 5 3.1 Praseodymium - P r59 ...... 5 3.2 Atomicmodel...... 6 3.3 Atomic fine structure ...... 8 3.4 Atomic hyperfine structure ...... 14 3.5 Selection Rules ...... 19 3.6 Complex Spectra ...... 20 3.7 Experimental Basis ...... 26

4 Improvement of Energy Levels 30 4.1 Energy Level correction approach ...... 30 4.2 Working Process ...... 32

5 Results 41 5.1 Results of the corrections ...... 41 5.2 Prospectives ...... 57

6 Appendix 58 6.1 Example of a data sheet ...... 58 6.2 Sourcecode ...... 65

Bibliography 68

List of figures 70

2 1 Abstract

In this Bachelor’s thesis the energy values of currently known Praseodymium-I energy levels in a range of 2000cm−1 from 9400cm−1 to 11400cm−1 were examined and corrected (improved). The goal when all work will be finished in the future is to reach a final uncertainty in level energy of about ±0.003cm−1 which is about 1 − 2 orders of magnitude lower than today. In the inspected energy region 29 even levels were examined and corrected as ’lower’ levels using an averaging method. Resulting from the correction of these lower levels 114 odd ’upper’ levels were corrected using well identifyable and assignable spectral lines. In addition several A-factors of magnetic hyperfine structure were improved. The correction process was done using high resolu- tion Fourier Spectrographic data as well as data from Laser Induced Fluorescence measurements. Furthermore a table of all known or suspected energy levels of Praseodymium-I was available. All this information was brought together in a program developed by Prof. L Windholz to handle this enourmous amount of entities in an easy way.

In dieser Bakkelaureatsarbeit wurden die bis jetzt bekannten Energiewerte von Praseodym-I Energieniveaus in einem Bereich uber¨ 2000cm−1 von 9400cm−1 bis 11400cm−1 untersucht und korrigiert (verbessert). Das Ziel, wenn alle Arbeiten an den Energieniveaus abeschlossen sein wer- den, ist es, eine Unsicherheit der Energie von ±0.003cm−1 zu erreichen. Das ist ungef¨ahr 1 − 2 Gr¨ossenordnungen kleiner als die Unsicherheit heute ist. In dem untersuchten Energiebereich wurden 29 gerade Niveaus untersucht und mit einer Mittel- wertmethode als ’untere’ korrigiert. Daraus resultierend wurden 114 ungerade ’obere’ Niveaus auf- grund von gut identifizierbaren und zuordenbaren Spektrallinien korrigiert. Ebenso wurden einige A-Faktoren der magnetischen Hyperfeinstruktur verbessert. Die notwendigen Daten der Spek- trallinien stammen aus Messungen von Fouriertransformationsspektrographen und Laserinduzierter Fluoreszenz. Dazu kommt eine Tabelle mit daraus errechneten, bekannten oder vermuteten En- ergieniveaus von Praseodym-I. Die gesamte Information wurde in einem von Prof. L Windholz entwickelten Programm zusammengefuhrt,¨ welches die einfache Handhabung der riesigen Daten- mengen erm¨oglicht.

3 2 Introduction

This work consists of a theoretical part giving an overview of the necessary fundamentals of atomic physics and a second part explaining the correction process in detail. While the first part may be of general interest, the second part may be of special interest for anybody who is going to work on the task of improving Pr-I energies. This may concern many students, because the work on the Pr-I energy levels will take a long time (there are simply too many).

Abbreviations, phrases and conventions used throughout this thesis are:

• hfs - hyperfine structure • FTS - Fourier Transformation Spectroscopy • LIF - Laser Induced Fluorescence • lower level - an energy level chosen to be the lower level of a transition is called lower level • upper level - an energy level chosen to be the upper level of a transition is called upper level • good spectral line - a spectral line which is clearly classified (i.e. matches perfectly in hfs and energy and may additionally be seen in fluorescence) • Pr-I - Prasodymium in its natural valence configuration. (Pr-II would be singly ionized Pr, and so on.) • The Energy is given in form of wave numbers (in units cm−1 (1cm−1= 1 Kayser = 1000mK)).

4 3 Fundamentals

3.1 Praseodymium - P r59 In this section the most important and relevant data and chracteristics of the element Praseodymium are stated. The data given here has been acquired from [11] and [1].

Praseodymium (Pr) is a (under normal conditions NC) soft, paramag- netic metal in solid phase (see fig. 3.1). It is easily oxidised at air and develops a greenish oxide layer. It’s crystal structure is hcp (hexagonal close packed) (αPr). Praseodymium’s 59 protons assign it to the Lan- thanoid group (rare earth metals) in the periodic table of the elements g (see fig. 3.2). The relative atomic mass Ar is 140, 907 mol at the only sta- ble isotope (see below). Furthermore Praseodymium has a mass density g of 6.48 cm3 , a melting point of 1204K and a boiling point of 3212K. On our planet earth Pr is very scarce with a mass fraction of 10−4 − 10−5% (in the form of minerals: Monazite, Cerite, ...).

Figure 3.1: Praseodymium sample (from [11])

Figure 3.2: Periodic table of the elements - Praseodymium in Lanthanoid group (from http://www.dayah.com/periodic/ 07.04.2009)

For the classification of spectral lines using hyperfine structure the nuclear spin quantum num- 5 ber I of 2 is very important.

141 Pr is a anisotope element, which means it occurs naturally in one stable isotope only: P r59 . There are 38 radioactive artificial isotopes but all of them (except for two) have half lives in the magnitude of seconds to minutes only. This fact simplifies the hyperfinstructure analysis signifi- cantly because the isotopic shift (see 3.4 A) does not occur.

5 kJ Praseodymium’s first energy of ionization is 527 mol . The electronic configuration of the ground state is:

[Xe]4f 36s2 [1s22s22p63s23p63d104s24p64d105s25p6]4f 36s2

The electronic configuration of the ground state in terms of electrons per principal quantum num- ber is:

K(2) − L(8) − M(18) − N(21) − O(8) − P (2) Praseodymium like all Lanthanoids has an extraordinarily complex and line-rich spectrum due to its valence configuration.

Praseodymium is used in combination with Cerium or Neodymium for the manufacturing of catalysts. Furthermore in alloys with Magnesium for the construction of high-strength met- als for aircraft engines and in alloys with Cobalt for permanent magnets. Another application of Praseodymium is in lighting industry where it is used in carbon arc- and projector lights. Praseodymium compounds give glasses a yellow color. Moreover Prasedymium is a component of didymium glass, which is used to make special welder’s and glass blower’s goggles. Like all rare earth metals, Praseodymium is of low toxicity. Praseodymium has no known biological role.

3.2 Atomic model

In the early 20th century the nowadays used atomic model was established. An early atomic model was Thomson’s plum pudding model which thought of atoms as lumps of positive charge with small negative point particles mounted on springs (see fig. 3.3 (left)). Lenard’s Dynamiden model accounted for the transition of radiation through matter and described atoms as empty space with randomly placed small positively and negatively charged particles (see fig. 3.3 (right)).

Figure 3.3: Evolution of the atomic model: (left) Thomson’s Plum Pudding model, (right) Lenards Dynamiden model

It was Rutherford’s scattering experiment of α particles at thin gold foils which revealed the existence of a positively charged very small nucleus. He also introduced the model of electrons revoluting in circles around the nucleus (see fig.3.4 (left)). The enourmous difficulties the Rutherford atomic model had in explaining how electrons can rev- olute in stable orbits without radiating energy (and collapsing into the nucleus) as proposed by classical electrodynamics where taken on by Bohr. He used also Balmer’s formula for the stable energies in the newly introduced Energy level scheme. Combining his postulates with Balmer’s

6 formula he ended up with the Bohr atom model (the first semiclassical description of atoms) (see fig.3.4 (right)):

1. Electrons revolute on stable spherical orbits around the nucleus with enegy En. Electron orbits are only stable if the orbital angular momentum is a multiple of Planck’s constant.

p = n~ In contradiction to classical electrodynamics the electrons in such orbits do not radiate and thereby loose energy. Apart from these orbits classical electrodynamics is valid (coulomb force = centrifugal force).

2. The radius of revolution of an electron cannot be changed continously. It can only absorb a photon and climb up one energy level (increased radius) or emit a photon and climb one down.

En − En0 = hν

Figure 3.4: Evolution of the atomic model: (left) Rutherford’s atomic model, (right) Bohr’s atomic model

Sommerfeld extended the Bohr theory to elliptical orbits and also introduced a second quantum number k accounting for the different axial ratios (see fig. 3.5 (left)). For Hydrogen the Bohr-Sommerfeld semi-classical theory reaches good agreement with the exper- iment up to fine-structure, but it is mathematically inconsistent and also lacks conviction in a physical understanding altought it is very vivid. The first mathematically consistent theory was developed by Schr¨odinger, Heisenberg, etc. and gives a totally different picture of atoms. In quantum mechanics every electron is described by a complex probability amplitude (see fig. 3.5 (right)). It should be remarked that the results of the Bohr-Sommerfeld theory agree with the results of the Schr¨odinger equation up to fine-structure for Hydrogen. In the 20th century the quantum theory was refined in several steps. Dirac for example achieved a fully relativistically correct quantum mechanical description for the Hydrogen Atom. Further quantum effects were added in second quantization (QED).

7 Figure 3.5: Evolution of the atomic model: (left) Sommerfeld’s atomic model, (right) Quantum mechanical atomic model

The semi-classical and quantum mechanical description of the nature of atoms supported the idea of descrete energy levels. The at the beginning of the 20th century with the help of experimen- tal examinations created Term scheme became a standard tool for visualizing energy circumstances in atomic (or generally all quantized) systems. Spectrographers are trying to build very accurate maps of atoms in energy scale since then. Al- though the serious work on atomic spectra to determine the energy schemes was started more than 100 years ago up to today only a fraction of the spectra and term schemes of all elements are known with high precision.

3.3 Atomic fine structure

From experimental data [7] one can see that the simple dependence of the Energy levels

Z2 E = −R · (3.1) n n2 with R the Rydberg-constant, Z the atomic number, and n the main quantum number, calcualted either by the classic Bohr-Sommerfeld theory or by the non-relativistic Schr¨odinger equation for the Hydrogen atom or Hydrogen like ions is only approximatively correct. For non-Hydogen like systems the electron-electron interaction is a mayor contribution to the interaction energy. It is very complicated to solve problems for more electrons revoluting around the nucleus including electron-electron interaction and most of the time only appimations can be calculated. The electron-electron interaction accounts for the splitting of the spectral lines (see chapter 3.6). If you examine spectra more carefully (i.e. with spectrographs with higher resolution) you will see the En levels shifted a little bit to lower energy (depending on Z, n and l) and you will also experience a splitting of the levels in dublets, triplets,... (depending on J). A fully relativistically correct solution to the Hydrogen problem is given by the Dirac equation which will not be discussed here. I have decided to rather give a summary of the effects using a semi-classical and a non-relativistic quantum theory (i.e. Schroedinger equation and perturbation theory) which are much more demonstrative and mathematically easier to understand.

In this Ansatz it is comfortable and common to split the origins of the fine structure into three parts: • A) Relativistic electron mass correction • B) Spin-orbit interaction • C) Darwin correction

8 A) Relativistic electron mass correction

The calculations in this section follow ([2], p143 ff.) closely. Visualizing an electron as a point particle circling around the nucleus in elliptic orbits, its motion can be described as a relativistic Kepler problem. The electron picks up and loses velocity during one revolution because its angular momentum is sustained.

~l = m(~r × ~v) for (~r⊥~v) ⇒ mvρ = const. (3.2) So the electron is slowest in the aphelion (point farthest away from the nucleus) and fastest in the perihelion (point nearest to the nucleus). Relativity theory tells us that the faster electron has a higher mass and therefore has a smaller radius of revolution. 1 m(v) = m0 · (3.3) q v2 1 − ( c2 ) using 3.2 and 3.3 the now reduced radius is r const. v2 ρ = · 1 − ( 2 ) (3.4) m0v c r v2 ρ = ρ · 1 − ( ) (3.5) 0 c2 This leads to a perihel rotation (see fig. 3.6) of the orbit and a lowering of the energy level.

Figure 3.6: Visualization of a relativistic electron in an elliptic orbit with perihel rotation due to it’s relativistc mass

Using the relativistic energy of the electron q (rel) 2 2 2 2 E = c · m0c + p − m0c + U (3.6)

q p2 and expanding in 1 + 2 2 m0c

s 2 (rel) p 2 E = c · m0c · 1 + 2 2 − m0c + U (3.7) m0c 2 4 (rel) 2 1 p 1 p p 6 2 E = m0c · (1 + · 2 2 − · 4 4 + O(( ) )) − m0c + U (3.8) 2 m0c 8 m0c m0c 2 4 (rel) p 1 p 2 p 6 E = + U + (− · 3 2 + m0c · O(( ) )) (3.9) 2m0 8 m0c m0c p E(rel) = E + ∆E(rel) + O(( )6) (3.10) m0c

9 we obtain 4 (rel) 1 p ∆E = − · 3 2 (3.11) 8 m0c as a relativistic correction for the energy levels.

Compared to the level energy this correction is small so it can be included in the Schr¨odinger equation as a perturbation 1 ˆ (rel) ˆ4 H = 2 P (3.12) 8m0c 4 (rel) ~ 0 ˆ4 0 ∆E = 2 < ψ |∇ |ψ > (3.13) 8m0c The following calculations are done for simplicity only for the Hydrogen atom to get an idea of what the solution may look like in simple systems. Using the Hydrogen wavefunction ψ the energy shift can be evaluated

2 2 (rel) Z e 2 3 n ∆E = En · 2 · ( ) · ( − 1 ) (3.14) n 4π0~c 4 l + 2 2 2 (rel) Z α 3 n ∆E = En · 2 · ( − 1 ) (3.15) n 4 l + 2

2 with the fine structure constant α = e . 4π0~c As one can see from 3.15 the level energy from now on is not only depending on the principle quantum number n but also on the azimuthal quantum number l. This fine structure correction is large for low l and low n (i.e. l = s, n = K) and decreases with increasing n and l.

B) Spin orbit interaction

The spin orbit interaction examination follows [2] and [14]. The relativistic correction done above shifts the energy levels to better values, but does not account for the observed splitting of spectral lines. This splitting is founded in so called spin-orbit interaction.

Figure 3.7: Spin orbit interaction

10 The gyrating electron (see fig. 3.7) represents a current e I = − (3.16) e τ −ev I = (3.17) e 2πρ This steady current accounts for the orbital magnetic moment of the electron

~µl = I · A~ (3.18) −ev ~µ = · ρ2πnˆ (3.19) l 2πρ −e ~µ = · ρ2vnˆ (3.20) l 2 which is connected to the orbital angular momentum given by

L~l = ~ρ × m~v (3.21)

L~l = m|~ρ||~v|nˆ for ~ρ⊥~v (3.22)

So we end up with the orbital magnetic moment of the electron −e ~µl = · L~l (3.23) 2me e~ with the Bohr magneton µB = (3.24) 2me µB ~µl = − · L~l (3.25) ~

(In these formulas the Lande factors gl are neglected because they are equal to one.) Viewed from the rest frame of the electron the nucleus orbits the electron with a given frequency νn, this representing a current again causes a magnetic induction at the position of the electron according to Biot-Savart’s law given by µ Ze B~ = 0 · (~v × −~ρ) (3.26) l 4πρ3 µ Ze B~ = 0 · L~ (3.27) l 4πρ3 l

The potential energy of a magnetic dipole El,s (in this case caused by the electron spin ~µs) in an external magnetic field B~l (by the orbiting nucleus) is given by

El,s = − ~µs · B~l (3.28) µB µ0Ze El,s = −(−gs · L~s) · ( · L~l) (3.29) ~ 4πρ3 After the transformation to the rest frame of the nucleus a further correction (Thomas factor) 1 occurs due to relativistic effects which is exactly one half of the potential energy. So a factor of 2 has to be added. 1 µB µ0Ze El,s = −( ) · (−gs · L~s) · ( · L~l) (3.30) 2 ~ 4πρ3 Simplifying this result further using the gyromagnetic ratio γ (the ratio of magnetic moment to angular momentum) which can for example be deduced from the Einstein-DeHaas effect (see [14]).

e |~µl| µB γl = gl · = = (3.31) 2me |L~l| ~ where gL the Lande factor of the orbital angular momentum is 1 (the Lande factor for spin gs is approximately 2). Now the potential energy can be written as

11 2 µ0Ze 1 ~ ~ El,s = 2 · 3 · (Ls · Ll) (3.32) 8πme ρ From this result one can see that the splitting of the energy level is depending on the orientation of L~s to L~l. Introducing the total (hull) angular momentum L~j = L~l + L~s the scalar product can be rewritten using the law of cosines

2 2 2 2 L~j = (L~j + L~s) = L~j + L~s − 2|L~j||L~s| · cos(l, s) (3.33) 1 |L~ ||L~ | · cos(l, s) = (L~ · L~ ) = · (L~ − L~ − L~ ) (3.34) j s s l 2 j l s 1 |L~ ||L~ | · cos(l, s) = · 2 · (j(j + 1) − l(l + 1) − s(s + 1)) (3.35) j s 2 ~

Putting everything together one obtains for the total energy En,l,j including the unperturbed terms of the Schroedinger equation and finestructure spin orbit interaction < a > E = E + · (j(j + 1) − l(l + 1) − s(s + 1)) (3.36) n,l,s n 2 with the spin orbit coupling constant

2 2 µ0Ze ~ 1 < a >= 2 · < Ψ| 3 |Ψ > (3.37) 8πme ρˆ The following calculations are done for simplicity only for the Hydrogen atom to get an idea of what the solution may look like in simple systems. Using the Hydrogen wavefunction ψ the spin orbit coupling constant a can be evaluated

Z2α2 < a >= En · 1 (3.38) n · l(l + 2 )(l + 1) with the fine structure constant α (see 3.15). The fine structure splitting is therefore proportional to Z4 ∆E ∝ (3.39) l,s n3l(l + 1)

Z2 (The additional n2 is coming from En.)

C) Darwin correction

The Darwin Term is explained at [8]. The Darwin Term is a further correction of the energy levels contributing to the fine structure. The term may be obtained from the Dirac equation or be accounted for as perturbation in the Schroedinger equation. It accounts in a nonrelativistic approximation for the non-locality of the electric electron-nucleus interaction which also depends on a small area around the electron (Zit- terbewegung). The term arising replaces the term l = 0 in the spin-orbit correction (which should be zero) making the formula correct.

2 ˆ ~ HDarwin = 2 2 · 4πeρnucleus (3.40) 8mec 2 2 ˆ ~ πZe 3 HDarwin = 2 2 · δ (~r) (3.41) 2mec with the nuclear charge density ρnucleus.

12 summary of fine structure components

The energy levels including fine structure can be calculated using the three perturbations A, B and C given above. For no external magnetic induction or electric field the single electron Hamiltonians look like this:

Hˆatom = Hˆ0 + Hˆee + Hˆrel + Hˆso + HˆD (3.42) with:

• The unperturbated Hamiltonian for an electric central force field

~2 Ze2 1 Hˆ0 = − ∆ˆi + · (3.43) 2m 4π0 ρˆi • The electron-electron interaction (major contribution - no perturbation)

Ze2 Hˆee = i 6= j (3.44) ρij • The relativistic correction term

ˆ 1 ˆ4 Hrel = 2 Pi (3.45) 8m0c • The spin-orbit interaction term

2 ˆ µ0Ze 1 ~ ~ Hso = 2 · 3 · (Ls · Ll) (3.46) 8πm ρˆi • The Darwin term

2 2 ˆ ~ πZe 3 HD = 2 2 · δ (~ri) (3.47) 2mec The energy levels may be obtained by solving the time independent Schroedinger equation

ˆ 0 0 Hatom|Ψn,l,s >= En,l,s|Ψn,l,s > (3.48) and using perturbation theory (given here are corrections for the eigenenergy and the eigenvectors to first order)

1 0 ˆ 0 En = < Ψn|Hi|Ψn > (3.49) 0 ˆ 0 X < Ψ |H1|Ψ > |Ψ1 > = |Ψ0 > m n (3.50) n m E0 − E0 n6=m n m

One more correction arising from the second quantization the Lamb Shift (see [14]) has to be added to the above corrections.

All of the calculations may also be done in a semiclassical calculation (see above) or using rela- tivistic quantum mechanics.

13 3.4 Atomic hyperfine structure

The contents of this section are primarily based on [14], [2] and [7]. The interactions of the nucleus with the electron hull which are not based on the isotropic coulomb potential are responsible for the hyperfine structure of atomic systems. These effects are three orders lower in energetic magnitude compared to the fine structure (electron-electron interaction, spin orbit coupling). In general the hyperfine structure is visible for doppler free high precision spectroscopic methods only (for example FTS, LIF, ... (see 3.7)).

Generally we differ three effects of the core-hull interaction:

• A) isotopic shift • B) magnetic hyperfine structure • C) electric quadrupole hyperfine structure

Of these three effects the latter two are accounted as actual hyperfinstructure with C being the dominant effect usually while A is a accompaniment for elements occuring in isotopic mixtures.

In account of Praseodymium’s anisotopy the isotopic shift does not occur. This affects the obtained spectra in a way that they are much clearer than they would otherwise be. Further it should be mentioned that also the electric quadrupole interaction is very low because of the spherical symmetry of the potential of the nucleus (a nearly perfect Coulomb potential). This helps on assignment of spectral lines to upper and lower energy levels because the B factor (see below D) can be set equal to zero.

A) isotopic shift

Naturally most elements occur as a mixture of stable (or longliving) isotopes. When using spectroscopic methods you will observe the spectral lines of all these isotopes at once. One can split the isotopic shift into three contributions:

• α) normal mass shift • β) specific mass shift • γ) volume effect

The normal mass shift causes a blurring of the hfs due to the different masses of the different isotopes. The cause of this effect is that the electrons also move the nucleus while their movement around it (because the nucleus is not infinitely heavy). After a mathematical transformation in the rest frame of the nucleus, this fact has to be accounted for by using the effective mass

me µ(mn) = (3.51) 1 + me mn Due to that different isotopes have different wavelengths of the same spectral lines: The dependency of the effective mass on the mass of the nucleus mn results in a dependency of the Rydberg-constant of the mass on the nucleus

µ(mn) RM (mn) = R∞ · (3.52) me This dependency causes the one for the spectral lines

2 1 1 ν¯ 0 (m ) = R¯(m ) · Z · ( − ) (3.53) n,n n n n2 n02

14 For light elements the isotopic shift is large (Z <= 10).

In figure 3.8 the effect of isotopic shift is illustrated for Lyman-α lines in Hydrogen, Deuterium and Tritium. The wavelength is calculated using Balmer’s formula 3.53. The natural linewidth 1 was calculated by δν = 2∗π∗τ and is 10 times magnified in this figure. The lifetime of the upper level was estimated to τ = 1 · 10−9s.

Figure 3.8: Visualization of isotopic shift (normal mass shift only) for Lyman-α lines in Hydrogen, Deuterium and Tritium

The specific mass shift is due to correlations between electrons. The described (see obove) movement of the nucleus with the electrons is of course very sensitive on the positions of the electrons. There are constellations where the effect is almost canceled out and others where it is amplified. For heavy elements this correction is large compard to the normal mass shift.

The volume effect accounts for the different volume of different isotopes. The coulomb potential of the nucleus gets more and more distorted from its initial form of the central force field of a point charge when nucleons are added. This influences especially electrons near (in) the nucleus. At spectroscopic analysis of isotopic mixtures this effect also causes a weak blurring of the spectral lines because of the different volumes of different isotopes.

B) magnetic hyperfine structure

Beforehand it should be mentioned that the actual calculation of the later mentioned A factors is a very difficult theoretical subject and only a shortened and simplified approach is given here. Atomic nuclei have a mechanical angular momentum called nuclear spin L~ I . This nuclear spin can be described by the nuclear spin quantum number I which is either integer or half integer (ranging 15 from 0 to 2 ) depending on the number of nucleons. I describes the maximum component of the absolute square of L~ I . The projection of the nuclear spin onto the designated direction (i.e. direction of external magnetic induction B~ ext) can only take on quantized values:

~ |LI,Z | = mI · ~ with − I ≤ mI ≤ +I ∆ = 1 (3.54)

15 5 In figure 3.9 the quantization of the nuclear spin of Pr (I = 2 ) is represented in a vector model. One can see that the z-component can only take on certain values.

5 Figure 3.9: Scheme of nuclear spin quantization of Pr (I = 2 )

The effect of magnetic hyperfine structure is based on the coupling of the angular momentum of the nuclear spin L~ I and the total angular momentum of the hull L~ j to yield the overall angular momentum L~ F in terms of the vector model.

L~ F = L~ J + L~ I (3.55) ~ p |LF | = F (F + 1) · ~ with |I − J| ≤ F ≤ |I + J| ∆ = 1 (3.56) ~ |LF,Z | = mF · ~ with − |F | ≤ mF ≤ +|F | ∆ = 1 (3.57) As one can see in formula 3.56 the coupling results in a quatization of the orientation of the nuclear angular momentum with respect to the total hull angular momentum. The number of hyperfine levels n is

n = 2I + 1 if I ≤ J (3.58) n = 2J + 1 if I > J (3.59)

Otherwise stated this results in a magnetic dipole interaction of the magnetic moment of the nucleus ~µI with the magnetic induction B~ J (0) generated by the hull electrons at the nucleus.

Like in spin orbit interaction (see 3.3) the gyrating electrons (magnetic moments) interact with a magnetic moment but this time with the magnetic moment of the nucleus. From the magnitude of the magnetic moment of the nucleus ~µI

µn ~µI = gI · L~ I (3.60) ~ me with µn = µB · the nuclear magneton (3.61) mp one can immediately see that the arising energies will be me ≈ 1000 times lower in magnitude mp than those of fine structure. Attention should also be payed to the opposite sign of the magnetic moment - angular momentum relation in contradiction to the case of fine stucture.

16 The constants of proportionality between the magnetic moment and the angular momentum of the nucleus are: the nuclear magneton µn and the nuclear Lande factor gI .

The additional magnetic interaction energy Ehfs,m arising is the scalar product of the nuclear magnetic moment ~µI and the magnetic induction driven by the gyrating electron cloud B~ J (0) (the <> brackets indicating a time average (expectation value)).

Ehfs,m = ~µI · < B~ J (0) > (3.62)

Ehfs,m = |~µI | · | < B~ J (0) > | · cos(L~ J , L~ I ) (3.63)

L~ J · L~ I Ehfs,m = |~µI | · | < B~ J (0) > | · (3.64) |L~ J | · |L~ I | taking the overall angular momentum L~ F into account it follows using the law of cosines

~ ~ ~ ~ ~ ~ ~ 2 ~ 2 ~ ~ LF · LF = (LJ + LI ) · (LJ + LI ) = LJ + LI + 2 · Lj · LI (3.65) ~ 2 ~ 2 ~ 2 1 LF − Lj − LI Ehfs,m = |~µI | · | < B~ J (0) > | · (3.66) 2 |L~ j| · |L~ I | 2 ~ 1 ~ (F (F + 1) − J(J + 1) − I(I + 1)) Ehfs,m = |~µI | · | < BJ (0) > | · p p (3.67) 2 ~ J(J + 1) · ~ I(I + 1) we end up with the additional energy coming from nuclear orbit coupling

Ehfs,m = hα(F )A (3.68) with the geometric factor (Casimir Factor) α(F )

F (F + 1) − J(J + 1) − I(I + 1) α(F ) = (3.69) 2 and the hyperfine magnetic coupling constant A using 3.61 and 3.56

|~µ | · | < B~ (0) > | A = I J (3.70) hpJ(J + 1) · pI(I + 1) p ~ (gI µn~ I(I + 1)) · | < BJ (0) > | = p p (3.71) ~h J(J + 1) · I(I + 1) g µ · | < B~ (0) > | = I n J (3.72) hpJ(J + 1)

Therefore the hyperfine splitting arises from the different orientations of L~ I and L~ j.

C) electric quadrupole hyperfine structure

In some nuclei the charge distribution is not spherically symmetric (i.e. the protons are not spherically arranged). This distortion of the coulomb potential gives rise to a quadrupole mo- jk ment of the nucleus Qn . The gradient of the electric field produced by the gyrating electrons < ∂zzΦe− (0) > at the nucleus couples to this quadrupole moment resulting in one more energy shift of the spectral terms. The resulting energy correction is

17 ik 3α(F )(2α(F ) + 1) − 2I(I + 1)J(J + 1) E = e(Q < ∂ Φ − (0) >) · (3.73) hfs,e n zz e 4I(2I − 1)J(2J − 1)

Ehfs,e = hβ(F )B (3.74) with the geometric factor (Casimir Factor) β(F )

3α(F )(2α(F ) + 1) − 2I(I + 1)J(J + 1) β(F ) = (3.75) 4I(2I − 1)J(2J − 1) and the hyperfine quadrupole coupling constant B (in MHz)

ik e(Q < ∂ Φ − (0) >) B = n zz e (3.76) h summary of hyperfine structure components

The total energy of a hyperfine structure term can be stated by using 3.68 and 3.74

Ehfs = Ec + Ehfs,m + Ehfs,e (3.77) with the center of gravity level energy Ec (i.e. finestructure level energy) and the two contributions from magnetic and electric hyperfine structure:

Ehfs = Ec + hα(F )A + hβ(F )B (3.78)

νhfs = νc + α(F )A + β(F )B (3.79)

The radiation frequency of a hyperfine transition can therefore be written as

νFu,Fl = ν00 + (αuAu + βuBu) − (αlAl + βlBl) (3.80) with ν00 the center of gravity frequency.

To be able to determine Au,Al,Bu,Bl, ν00 from a hyperfine spectrum the situation of five hfs components including the corresponding quantum numbers need to be known. If more are known it is possible to determine the constants by a Least-Squares fit (i.e. Gauss-Newton-Marquard numerically).

Because Pr’s B factors are almost zero all quadrupole terms can be neglected in further medi- tations.

When B = 0 the Lande interval rule states that the spacing of the hfs levels is proportional to the greater quantum numbers of the levels. 17 As an example the spacing of the hyperfine components of an energy level with J = 2 and a 5 nuclear spin of I = 2 is plottet in figure 3.10.

18 17 5 Figure 3.10: Lande interval rule applied to J = 2 , I = 2

It is also possible that the hfs of a term is inverted (negative A factor) meaning the highest F value lies deepest (contrary to fig. 3.10). The nucleus-orbit interaction tends to produce normal 1 hfs. The nucleus-spin interaction tends to invert the term j = l + 2 .

3.5 Selection Rules

The selection rules for dipole transitions used for the complex spectra of Pr are: The change of Parity P :

∆P = 1 (3.81) The Parity is the sum of the orbital angular quantum numbers X P = li (3.82) i The change of the total (hull) angular momentum J:

∆J = −1, 0, +1 (3.83) The transition from J = 0 to J 0 = 0 is forbidden generally.

The change of the overall angular momentum F :

∆F = −1, 0, +1 (3.84) The transition from F = 0 to F 0 = 0 is forbidden generally.

19 3.6 Complex Spectra

For reference see [14] and [7]. Praseodymium is an element with an extraordinary complex spectrum including an awfully great amount of spectral lines. In this section the basics of complex spectra shall be treated and a short abstract of the Pr spectrum features will be given on the following topics:

• A) Complex coupling • B) Complex Pr spectrum - high field quantum numbers • C) Term scheme nomenclature

A) Complex coupling

The coupling of angular momenta in complex atomic systems is very hard to describe theoretically. The two most common limiting cases considered for the description of coupling are

• LS (Russel-Sounders) -coupling • jj -coupling Both are no good approximations for the coupling in Pr-I. Nevertheless a quick overview of these two schemes shall be given here. The LS -coupling scheme is essential for the understanding of the nomenclature of spectral terms, which in the case of Pr can be labeled in LS nomenclature. Every term in Pr will be a mixture of many LS terms. The most dominant one is given usually as a label for the level.

Simple systems can be characterised by neglecting parts of the interaction of angular momenta shown in figure 3.11. Depending on which interactions are strong in comparison with the others the two coupling schemes arise.

Figure 3.11: Possible interactions of angular momenta for two electrons

20 LS (Russel-Sounders) -coupling

Figure 3.12: Vector model for perfect LS -coupling in a two electron system

If the coupling between the orbital angular momenta and the coupling between the spins of the different electrons is stronger than the spin orbit interaction of the individual electrons the coupling can be approximated by LS -coupling (see fig. 3.11: purple lines dominate orange lines, green lines totally neglected). The arising total orbital angular momentum is the vectorial sum of all orbital angular momenta which is then quantized as follows (see fig.3.12):

X L~ L = L~ l,i (3.85) i p |L~ L| = L · (L + 1) (3.86) X X L = min(| ±L~ l,i|), ..., max(| ±L~ l,i|), ∆ = 1 (3.87) i i The arising total spin is also the vectorial sum of all electron spins which is then quantized as follows:

X L~ S = L~ s,i (3.88) i p |L~ S| = J · (S + 1) (3.89) X X S = min(| ±L~ s,i|), ..., max(| ±L~ s,i|), ∆ = 1 (3.90) i i The resultant total (hull) angular momentum is

L~ J = L~ L + L~ S (3.91) J = |L − S|, ..., |L + S|, ∆ = 1 (3.92)

The Terms arising in complex spectra can be characterized as a mixture of many LS terms. The notation for LS coupled terms is:

r n LJ with n ... the principal quantum number with r ... the multiplicity (r = 2S + 1) with L ... orbital quantum number with J ... the total (hull) quantum number

21 This scheme applies quite well to atoms with low atomic number.

jj -coupling

Figure 3.13: Vector model for perfect jj -coupling in a two electron system

Here the interaction between spin and orbit of the same electrons is stronger (see fig. 3.11: orange lines dominate purple lines, green lines totally neglected). This results in total angular momenta for every electron (see fig. 3.13).

L~ j,i = L~ l,i + L~ s,i (3.93) These individual angular momenta sum up to the total angular momentum of the electron cloud

X L~ J = L~ j,i (3.94) i p |L~ J | = J · (J + 1) (3.95) X X J = min(| ±L~ j,i|), ..., max(| ±L~ j,i|), ∆ = 1 (3.96) i i The notation for jj coupled terms is:

(ji, ..., jk)J with jp ... the total quantum numbers of the single electrons with J ... the total (hull) quantum number

This scheme applies quite well to atoms with high atomic number (because spin-orbit interac- tion strength is increasing with atomic number).

B) Complex Pr spectrum - high field quantum numbers

To get a feeling of the arising spectral terms although the coupling scheme for Pr can be described neither by LS nor by jj coupling, we have a look on the so called high field quantum numbers. In high external magnetic fields every coupling between angular momenta in the atomic ensemble is destroyed (Full-Paschen-Back effect). This allows us to investigate all possible arising terms

22 according to Pauli’s principle. Unexcited Praseodymium has three valent f electrons and two s electrons. According to the alternation law for multiplicities (see fig. 3.14) Pr should behave like CrI in the figure and build Triplets, Quintets and Septets. This is not the case because the alternation law does not account for the special constellation of Pr where the two 6s electrons lie energetically higher than the three 5f electrons. Therefore we will see that Pr acts as if it had three valence electrons.

Figure 3.14: Alternation law of multiplicities (from [7])

We now try to compute the spectral terms arising from the three unsaturated f electrons. We 1 have angular momenta l of 3 and spins of s of 2 . We enumerate all possible one electron states which are listed in table 3.1.

Table 3.1: Possible one electron states for 5f 3 valence electrons of Pr. ml ... magnetic orbital quantum number ms ... magnetic spin quantum number id ... letter assigned to identify the one electron state

ml ms id 1 -3 2 a 1 -2 2 b 1 -1 2 c 1 0 2 d 1 1 2 e 1 2 2 f 1 3 2 g 1 -3 − 2 h 1 -2 − 2 i 1 -1 − 2 j 1 0 − 2 k 1 1 − 2 l 1 2 − 2 m 1 3 − 2 n

By building now all possible combinations of three single electron states the arising composite states can be determined. To fulfill Pauli’s principle it is sufficient to calculate only the combina- tions without putting back and neglecting the order. This is done in table 3.2.

The Pauli exclusion principle states that no two equivalent fermions may occupy the same quantum state (i.e. the total wave function has to be anti-symmetric). Under circumstances con- sidered here (i.e. single atomic systems) the principle could be expressed by the fact that no two equivalent electrons can have the same quantum numbers.

23 Table 3.2: Allowed combinations of single electron states for 5f 3 valence electrons of Pr. Ml ... sum of magnetic orbital quantum numbers ms Ms ... sum of magnetic spin quantum numbers ml ids ... combination of single electron states applied

Ml Ms ids -8 -0.5 ahi -8 0.5 abh -7 -0.5 ahj bhi -7 0.5 abi ach -6 -1.5 hij -6 -0.5 ahk aij bhj chi -6 0.5 abj aci adh bch -6 1.5 abc -5 -1.5 hik -5 -0.5 ahl aik bhk bij chj dhi -5 0.5 abk acj adi aeh bci bdh -5 1.5 abd -4 -1.5 hil hjk -4 -0.5 ahm ail ajk bhl bik chk cij dhj ehi -4 0.5 abl ack adj aei afh bcj bdi beh cdh -4 1.5 abe acd -3 -1.5 him hjl ijk -3 -0.5 ahn aim ajl bhm bil bjk chl cik dhk dij ehj fhi -3 0.5 abm acl adk aej afi agh bck bdj bei bfh cdi ceh -3 1.5 abf ace bcd -2 -1.5 hin hjm hkl ijl -2 -0.5 ain ajm akl bhn bim bjl chm cil cjk dhl dik ehk eij fhj ghi -2 0.5 abn acm adl aek afj agi bcl bdk bej bfi bgh cdj cei cfh deh -2 1.5 abg acf ade bce -1 -1.5 hjn hkm ijm ikl -1 -0.5 ajn akm bin bjm bkl chn cim cjl dhm dil djk ehl eik fhk fij ghj -1 0.5 acn adm ael afk agj bcm bdl bek bfj bgi cdk cej cfi cgh dei dfh -1 1.5 acg adf bcf bde 0 -1.5 hkn hlm ijn ikm jkl 0 -0.5 akn alm bjn bkm cin cjm ckl dhn dim djl ehm eil ejk fhl fik ghk gij 0 0.5 adn aem afl agk bcn bdm bel bfk bgj cdl cek cfj cgi dej dfi dgh efh 0 1.5 adg aef bcg bdf cde 1 -1.5 hln ikn ilm jkm 1 -0.5 aln bkn blm cjn ckm din djm dkl ehn eim ejl fhm fil fjk ghl gik 1 0.5 aen afm agl bdn bem bfl bgk cdm cel cfk cgj dek dfj dgi efi egh 1 1.5 aeg bdg bef cdf 2 -1.5 hmn iln jkn jlm 2 -0.5 amn bln ckn clm djn dkm ein ejm ekl fhn fim fjl ghm gil gjk 2 0.5 afn agm ben bfm bgl cdn cem cfl cgk del dfk dgj efj egi fgh 2 1.5 afg beg cdg cef 3 -1.5 imn jln klm 3 -0.5 bmn cln dkn dlm ejn ekm fin fjm fkl ghn gim gjl 3 0.5 agn bfn bgm cen cfm cgl dem dfl dgk efk egj fgi 3 1.5 bfg ceg def 4 -1.5 jmn kln 4 -0.5 cmn dln ekn elm fjn fkm gin gjm gkl 4 0.5 bgn cfn cgm den dfm dgl efl egk fgj 4 1.5 cfg deg 5 -1.5 kmn 5 -0.5 dmn eln fkn flm gjn gkm 5 0.5 cgn dfn dgm efm egl fgk 5 1.5 dfg 6 -1.5 lmn 6 -0.5 emn fln gkn glm 6 0.5 dgn efn egm fgl will be continued on next page

24 Table 3.2 – continued Ml Ms ids 6 1.5 efg 7 -0.5 fmn gln 7 0.5 egn fgm 8 -0.5 gmn 8 0.5 fgn

From these allowed combinations it is possible to build the arising spectral terms by a clever scheme (see [7]). Beginning at the highest available L (here 8) one has to pick every mL (from L to -L) out of the list. The possible combinations given in table 3.1 represent the number of how often this state is available. This whole series represent a spectral term in table 3.3. The procedure is repeated until every state is used as often as it is available altering L every run. A good way to check if the calculation is right is to see if every state has been used exactly as often as available.

Table 3.3: Possible spectral terms for 5f 3 valence electrons of Pr. r ... multiplicity r = 2s + 1 term ... arising spectral term

r term

2 J8.5,7.5 2 K7.5,6.5 4 I7.5,6.5,5.5,4.5 2 I6.5,5.5 2 H5.5,4.5 2 H5.5,4.5 4 G5.5,4.5,3.5,2.5 2 G4.5,3.5 2 G4.5,3.5 4 F4.5,3.5,2.5,1.5 2 F3.5,2.5 2 F3.5,2.5 4 D3.5,2.5,1.5,0.5 2 D2.5,1.5 2 D2.5,1.5 2 P1.5,0.5 4 S1.5,0.5

The spectral terms in table 3.3 are the theoretical values for Pr-I in the groundstate obeying the Pauli exclusion principle. But they are not correct in a way that practically a term can never have more multiplicity than its angular quantum number plus one. So S terms can maximally be single, P terms double, D terms triple and so on.

To be able to characterize the groundstate for the given electron configuration we have to make use of Hund’s rule which states: 1. The term with maximum multiplicity r has the lowest energy (r = 2S + 1, so maximum r equals maximum S). 2. For the chosen maximum multiplicity r the term with the largest value of L has the lowest energy. 3. The value for the total (hull) angular momentum J is J = |L − S| when the shell is less than half full and is J = |L + S| when it is more than half filled. For exactly half filled shells L = 0 and so J = |S|.

25 4 The ground state could be identified according to Hund’s rule as I4.5. This is done as follows:

4 1. From all entries in table 3.3 pick those with maximum r (here rmax = 4): I7.5,6.5,5.5,4.5, 4 4 4 4 G5.5,4.5,3.5,2.5, F4.5,3.5,2.5,1.5,( D3.5,2.5,1.5,0.5, S1.5,0.5 are not really 4 (see above))

4 2. From all selected entries choose the one with maximum L (here Lmax = I): I7.5,6.5,5.5,4.5

3. The f shell the three electrons reside in provides space for 14 electrons so it is less than half filled (here J = |L − S|) so J = |I − 1.5| = 4.5

All data provided in this section (tables 3.1, 3.2 and 3.3) was calculated using a calculation scheme given in [7] which I programmed in matlab. The program is capable of dealing with dif- ferent number of valence electrons in any given subshell and the source code is provided in the appendix.

C) Term scheme nomenclature

A Termsymbol represents a certain wavefunction. It is based on a coupling scheme for angular momenta (LS- or jj- coupling). Because of the complexity of the Pr spectrum every term is a mixture of many LS terms, of which the dominant one is stated. 5 In addition to the nuclear spin which is I = 2 for Pr, for every energy level the following information is needed for unique identification • E ... the level Energy (usually in K (Kayser) i.e. cm−1)

P • P ... the level Parity (the sum of all orbital angular momenta i li)

• J ... the total (hull) angular momentum

• A ... the magnetic hyperfine structure coupling constant (usually in MHz)

• B ... the electric quadrupole hyperfinstructure coupling constant (usually in MHz)

• g ... the Land´e g factor

3.7 Experimental Basis

Altough my work did not include any experimental aspects it relys heavily on spectroscopic data. In this section a quick overview of the experimental methods used to obtain the spectral data is given. The most important two were:

• A) Fourier Transformation Spectroscopy (FTS) • B) Laser Induced Fluorescence (LIF)

For the selection of spectral lines a Fourier spectrum was used. In addition to this spectrum the data of Laser Induced Fluorescence measurements was available.

26 A) Fourier Transformation Spectroscopy (FTS)

The information of this section is based on [5] and [3]. FTS is a special type of emission spectroscopy which is based upon a Michelson interferometer replacing the dispersive elements (i.e. gratings or prisms). The main benefits of this technology are that it is very fast (scanning all wavelengths at the same time), it has a very high resolution (hyperfine structure), also intensity weak components are visible in the spectrum and the wave- lengths can be determined very accurately. A schematic drawing of a Fourier Transformation Spectroscope is given in figure 3.15.

Figure 3.15: Schematic drawing of a Fourier Transformation Spectroscope (from [5]), 1: light source, 3: aperture, 5: beam splitter, 10: specimen, 12: detector, (13,14): HeNe Laser, 17: Laser detector, (2,4,6,7,8,11,15,16): mirrors, 9: specimen chamber

The most important aspect of FTS is that the whole spectrum is imaged at once (polychromatic light source). This can be done using a Michelson interferometer and Fourier transformation. Behind the Michelson interferometer one gets interference of the two splitted light beams. This causes the intensity F (x) to be a function of the optical path difference x. Considering the two splitted electromagnetic waves pointing in z direction and calculating the square of the amplitudes which is the intensity I(δ) we get:

p i(kxx−ωt) Ex,1 = I1 · e (3.97)

p i(kxx−ωt+δ) Ex,2 = I2 · e (3.98) p I(δ) = I1 + I2 + 2 I1I2 cos (δ) (3.99)

I(δ) = 2I0(1 + cos (δ)) (3.100)

For the last modification of I(δ) a perfect (50%/50%) beam splitter is assumed. To see that the interferometer is recording the intensities as a function of optical path difference we can use the diffraction conditions

2nd cos (θ) = l · λ (3.101) δ = l · 2π (3.102)

27 and simplify by assuming vacuum conditions (n = 1), the beams perfectly aligned on the optical axis (θ = 0rad) and setting the optical path difference x = 2d:

F (x) = 2I0(1 + cos (2πνx)) (3.103) To end up with the interferogram I(x) (intensity as a function of optical path difference) one has to substract an offset term 1 F (0) = I (3.104) 2 0 From the recorded intensity data as a function of optical path difference I(x) one can now obtain the wanted intensity as a function of frequency Ie(ν) by Fourier transformation 1 Z ∞ I(x) = F (x) − F (0) = 2 Ie(ν) cos (2πνx)dν (3.105) 2 0 1 Z ∞ I(x) = Ie(ν)e−i2πνxdν (3.106) N −∞ The two most important parameters of an FTS device are the maximum displacement of the Michelson interferometer mirror dmax which determines the frequency resolution and the minimum stepping of the mirror ∆xmin which determines the lower and upper bounds of the frequency range.

The spectra used were taken with a Bruker IFS 120 HR at Hannover. For the selection of spec- tral lines which can be used for correcting energy levels these Fourier spectra were the main tool to find the center of gravity wavelength of the transitions and to determine whether the hyperfine structure of the classified spectral line matches the calculation or not (see section 4.1).

B) Laser Induced Fluorescence (LIF)

For references see [6] and [3]. LIF uses the effects occuring when laser light is absorbed by an ensemble of particles for example in a discharge. The laser (for example a dye laser tuneable over a wide wavelength range) is tuned to the energy of the transition to be examined. The particles in the system get excited by the laser. One way for the system to relax is by emission of light (fluorescence). The additional intensity arising from laser excitation is very weak in discharge samples because the atoms are highly randomly excited anyway and radiating broadband high intensity. One way to get a signal is to intensity-modulate the laser by chopping the beam and detecting the signal using a lock in amplifier. There are three types of observeable effects (see fig.3.16):

• positive fluorescence (see fig.3.16 red) Tuning the laser to the resonance frequency of transition E1 → E2 (for example the strongest component of one hyperfine multiplett) it excites the atomic ensemble to E2. This level then decays to levels E3 and E4, emitting the corresponding wavelengths. When the laser is in operation the signal of λ23 and λ24 is increased.

• negative fluorescence (see fig.3.16 green) If E1 is a high lying level and has a high decay probability to level E5 negative fluorescence is observed. The laser excitation depletes the population of level E1 and so the signal of the transition E1 → E5 will become weaker when the laser is on.

• impact/collisional coupling (see fig.3.16 blue) The third effect occurs when two or more of the excited levels lie close to each other. The population of E2 gets transferred to level E6 by collision processes. Afterwards of course E6

28 decays according to dipole selection rules.

Figure 3.16: Energy scheme for the LIF process

This method is mainly used for the confirmation of new energy levels. For example by using positive fluorescence to determine the upper level E2 the excitation frequency has to be combined with the two fluorescence lines.

For this work the fluorescence measurements are of importance because in the selection process of lines useable for doing level corrections those lines which fit not perfectly in hfs can be identified by the fluorescence data of the involved levels (see section 4.1).

29 4 Improvement of Energy Levels

4.1 Energy Level correction approach

The correction of energy levels in Praseodymium-I is based on high resolution spectral data from FTS and LIF. Because the theoretical predictions are only good within about ±50cm−1 they are no help in determining high accuracy level energies. The correction process (based on the Rydberg-Ritz combination principle, which states that the spectral lines of any element include frequencies that are either the sum or the difference of the frequencies of other lines) uses the well identifyable transitions (i.e. spectral lines from FTS) between energy levels for a correction procedure.

The correction process is further based on the assumption, that the Fourier spectra are abso- lutely wavelength calibrated. This can be checked by viewing the center of gravity wavelengths of Ar-I or Ar-II lines in the spectra. The center of gravity of the selected lines can be determined. For doing this the hyperfine struc- ture constants of the upper and lower level involved need to be known.

−1 13 The lowest even level (4432.240cm , 2 , e) (see fig. 4.1 black) was chosen as a fixed basis for the corrections. It is assumed that the level energy is correct, at least at the moment. Every transition of the level was examined and those transitions definitely assignable to known upper odd levels used to correct the upper level energies.

From here on every even level is picked as a lower level for transitions (see fig.4.1 red, blue, purple). All spectral lines classified for this level are examined and those which are good (i.e. definitely assignable in terms of energy, hfs and fluorescence) are used to build a mean deviation in wave number by summing over all good spectral lines to already corrected upper levels, calcu- lating the difference of the formerly presumed level energy difference and the difference proposed by the transition to the known upper level:

1 X < ∆¯ν >= · ∆level energy proposed by transitionν¯ number of good spectral lines formerly assumed level energy all good spectral lines (4.1) After this the picked lower level is corrected by the mean difference in wave number. Here attention must be paid to the fact that a negative correction makes in fact the energy of the lower level more positive:

Elower,after = Elower,before− < ∆¯ν > (4.2) Now the energy of the lower level is assumed to be correct and all good transitions to formerly not corrected upper levels can be used to correct the level energies of these by:

Eupper,after = Eupper,before + ∆¯νselected good transition (4.3)

30 Figure 4.1: Scheme of the correction process, black: arbitrarily fixed level, green: true ground state, red&blue&purple: levels involved in correction process; (The given level data is from Pr-I levels but the transitions are chosen fictitiously.)

This process is repeated for every known even energy level as a lower level. By doing so the even levels get more correct relatively to each other by averaging and also a pool of corrected upper odd levels is established.

After going through all even levels the whole process should be repeated now using the odd levels as lower levels and doing the correction vice versa. In principle it is possible to do the whole cycle again and again to iteratively improve the energy levels further and further until they all are below a certain uncertainty limit.

31 The whole correction will take very long time because there are about 6000 energy levels and about 23000 spectral lines currently in the Pr-I database (which is steadily growing) and every spectral line needs special attention in the correction process which can only be done by hand.

A deeper insight in the necessary steps of the procedure will be given in the next chapter.

It should be mentioned that a certain very valuable point will be reached when enough good transitions from corrected levels to the groundstate will be known (see fig.4.1 green, purple), mean- −1 13 ing the formerly presumed correct level (4432.240cm , 2 , e) can be replaced by the groundstate and every corrected level can be shifted by a certain amount of energy to be not only correct relatively to other levels but absolutely correct in energy (of course with a certain uncertainty expected to be about ±0.003A).˚

4.2 Working Process

The main purpose of this work was to correct the experimentally constructed energy levels of Praseodymium-I. Every recorded spectral-line (in our case the latest data comes from FTS and LIF) is identified by its hyperfine structure. So the line has a fingerprint depending on the upper and the lower level of the transition it came from. This makes an assignment to energy levels possible because for every two hypothetical levels the "hyperfine-fingerprint" can be calculated using the total angular momentum J and the hyperfine constants A and B of the two involved levels. By means of that, it is possible to gradually build a level-scheme for the element (which can be compared to theoretical data only roughly because the experimental data is much more accurate (∆ ≈ 50cm−1)). The work on energy levels of Pr is not completed by now, so new levels are found regularly, but there is a set of confirmed energy levels which were identified by the process described above. We now aim to improve the previously set energy levels by an iterative process of comparing energy differences between levels. This process works as follows:

• A) choosing a lower level to be corrected • B) stepping through all known spectral lines having been identified to have the chosen level as lower level • C) for each line the classification is updated with the latest data • D) for each line the latest Fourier spectrum is again compared with the calculated hyperfine structure • E) each line is recorded on paper and it is assessed whether the classification is definite or not (categorizing the recorded lines) • F) averaging over the good deltas and correction of the lower level • G) scanning all spectral lines again to determine the new wave number difference • H) correcting upper levels which have been categorized for correction in F • I) correcting the data of all spectral lines involved in any corrected level • J) A factor correction (optional)

The whole work was done using a computer-program developed by Prof. L Windholz called Classification of Spectral Lines by means of their Hyperfine Structure (see fig. 4.2). For further information on the tool see [13].

32 Figure 4.2: Main Screen of the used Program: Classification of Spectral Lines by means of their Hyperfine Structure

A) choosing a lower level to be corrected

The process of level correction was started at the lowest even level in Pr-I E = 4432.24cm−1, 9 J = 2 , e, A = 929(1), B = −92, 7. This level was fixed to be correct in energy and is now used as a reference. Hence all corrections done are done in relation to this level and not absolutely. When in high lying levels transitions to zero energy arise all corrected levels can be shifted by a constant but are correct with respect to each other. Starting from this level every even level is used as a lower level (increasing in energy). My work 13 began at level E = 9464.45cm−1, J = 2 , e. It should be mentioned here that once the whole correction process is completed with even lower levels it should be repeated starting from the lowest lying odd level using all odd levels as lower levels (see fig.4.3).

33 Figure 4.3: Extract of the even levels of Pr-I in the database.

B) stepping through all known spectral lines having been identified to have the chosen level as lower level

Having selected a lower level, the program Class lw allg10 offers a comfortable way to step through all spectral lines associated with it. This is done by Select → Lines with Level Energy. Now the main screen shows a spectral line in the left part and proposes alternatives for classifica- tion in the right part of the screen (see 4.4). It is now possible to examine the hyperfine structure of all these lines.

Figure 4.4: Mainscreen showing the currently selected spectral line and the calculated suggestions for the hfs

34 C) for each line the classification is updated with the latest data

As many people are actively working on the classification of Pr-I lines (either by adding new lines, changing classifications or correcting lines,... ) the data shown for the level (left part of fig.4.4) may not be up to date. It is necessary to compare the data again with the proposals given for this spec- tral line. (Usually the energy of the upper level has changed by a magnitude of 10−2 −10−3cm−1.) The updated information in general includes all known data for the lower and upper level of the transition (i.e. energies, parity, angular momentum, A&B factors and comments). The wavelength of the line is not changed in this process (only the calculated wavelength which is calculated using the energies of the upper and lower level can change according to changes in one of these energies).

D) for each line the latest Fourier spectrum is again compared with the calculated hyperfine structure

Having the latest information on the currently examined spectral line one can examine the hyper- fine structure again to ensure the quality of the classification. This is one of the most important parts in the whole process because comparing the spectra is

• essential to find the right center of gravity wavelength (see fig. 4.5),

• often quite difficult (ambiguous proposals, blend situations, ...) and

• it is crucial for the whole correction process to identify lines which are good for correction.

Every line which is clearly and definitely identifiable is in principal good for doing corrections of energy levels. These corrections can be of the lower or the upper level depending on other criteria which will follow. The hyperfine structure seen in the Fourier spectrum is compared with the calculated spectrum. Here it is possible to shift the centre of gravity wavelength of the spectral line to achieve a better match (see fig. 4.5). If the match of the hyperfine structure is not clear the centre of gravity wavelength is not changed. (In case of new lines it is set to the calculated value.)

Figure 4.5: Example of shifting the center of gravity wavelength

35 E) each line is recorded on paper and it is assessed whether the classification is definite or not (categorizing the recorded lines)

Following data of every spectral line associated with the current lower level is recorded on paper:

• Wavelength of the spectral line before corrections of energy levels are applied λold

• Wavelength of the spectral line after the center of gravity wavelength has been shiftet to fit the calculated hfs λnew

• Energy of the upper level of the transition Eupper

• Difference from calculated wave number to transition wave number before the correction of the lower level ∆old

• Comment describing the quality of the spectral line (see below)

• Difference from calculated wave number to transition wave number after the correction of the lower level ∆old

• Number of spectral lines already used to correct upper level noL

As an example one working-sheet is given in the appendix. The rest of the data can be viewed in the office.

Following the shifting of the center of gravity wavelength the most important part for the cor- rection is done. Every line is classified whether to be a basis for corrections or not. Following categories have been used in this work:

• g = GOOD The hyperfine structure of the line in the Fourier spectrum and the hyperfine structure cal- culated from known A and B factors match exactly or very well (see fig. 4.6). These spectral lines will be used for correction of either the upper or lower level depending on whether the upper level has been corrected before or not. In addition to that it can be checked if the line has been observed as a fluorescence line when exciting the upper level with laser light to ensure the classification of the spectral line (see fig.4.7).

Figure 4.6: Examples for good match of Fourier spectrum to calculated hfs spectrum

36 Figure 4.7: Information on laser-induced fluorescence in the comment concerning the spectral line

• a = CLASSIFICATION DOUBTFUL The match of the hyperfine structure is not very good (see fig. 4.8). In addition to the hfs information also the fluorescence information may be viewed where available to ensure the decision. These spectral lines are recorded but not dealt with any longer.

Figure 4.8: Examples for bad match of Fourier spectrum to calculated hfs spectrum

• b = NO MATCH IN HFS The match of the hyperfine structure is very bad (see fig. 4.9). These spectral lines are recorded but not dealt with any longer.

Figure 4.9: Examples for no match of Fourier spectrum to calculated hfs spectrum

37 • c = BAD SIGNAL/NOISE RATIO The S/N ratio of the Fourier spectrum is so bad (see fig. 4.10) that no conclusion of the hyperfine structure match can be found or there is no Fourier spectrum available for this part of the frequency domain. These spectral lines are recorded but not dealt with any longer.

Figure 4.10: Examples for a situation in the Fourier spectrum

• d = UPPER LEVEL ALREADY CORRECTED This category does not concern the quality of the classification. It includes spectral lines which upper level has been already corrected before (at least once). These spectral lines will be used for correction of the lower level by averaging if they are g = GOOD.

• e = BLEND TOO STRONG Here the situation in the Fourier spectrum shows more then one spectral line very close together, this makes qualitative predictions very difficult (see fig. 4.11). For safety these spectral lines are recorded but not dealt with any longer.

38 Figure 4.11: Examples for a blend situation in the Fourier spectrum

To sum up there are several possibilities for a spectral line to be not clearly enough classified to participate in the correction process. Only lines which hyperfine structure is definitely fitting the calculated hyperfine structure of the transition are used for correcting energy levels.

F) averaging over the good deltas and correction of the lower level

Every line which is GOOD and has an upper level which has been corrected at least one time is used for correcting the lower level. This is done by averaging all deltas of these lines and then subtracting the mean from the initial lower level. (See formula 4.2.)

G) scanning all spectral lines again to determine the new wave number difference

Every spectral line associated with the current lower level is viewed again and the new data for the lower level energy is adopted. This results in a new wave number difference, which is recorded again.

39 H) correcting upper levels which have been categorized for correction in F

Now every GOOD spectral line (which is not D) serves as basis for the correction of upper energy levels which

• α) have not been corrected before or

• β) have been corrected before but only on a basis of 1 or 2 lines and have a greater wave number difference than 0.005.

Process α is straight forward because every not corrected level is simply corrected by the wave number difference of the GOOD line. (See formula 4.3.) Process β is tougher because it involves viewing the lines used for the previous correction of the line again and deciding which line is a better basis for correcting the level.

I) correcting the data of all spectral lines involved in any corrected level

At the end of the correction process every spectral line of every corrected level has to be updated again to give the latest data of the involved energy levels.

J) A factor correction (optional)

For some energy levels it was necessary to update the A factor. This is only done in strongly suspicious cases and the new A factor is checked by viewing different transitions from the level and seeing if the new A factor fits to them also. The effects of a different A factor can be comfortably viewed in the used program (see fig. 4.12). The corrected A factors have to be identified with an s for simulation (for example 560s) to be able to distinquish them from fitted or otherwise generated A factors.

Figure 4.12: Correction of an A factor

40 5 Results

5.1 Results of the corrections

The following even levels have been used as lower levels for the correction procedure: 13 • A) 9464.450 2 e 11 • B) 9483.540 2 e 3 • C) 9650.060 2 e 11 • D) 9675.040 2 e 7 • E) 9704.750 2 e 5 • F) 9710.640 2 e 15 • G) 9745.420 2 e 17 • H) 9770.330 2 e 7 • I) 9918.170 2 e 15 • J) 9951.823 2 e 17 • K) 10157.090 2 e 7 • L) 10194.740 2 e 13 • M) 10266.510 2 e 9 • N) 10356.710 2 e 13 • O) 10423.680 2 e 15 • P) 10466.730 2 e 13 • Q) 10470.300 2 e 17 • R) 10532.001 2 e 5 • S) 10828.988 2 e 11 • T) 10841.482 2 e 11 • U) 10904.070 2 e 9 • V) 10920.380 2 e 1 • W) 10956.833 2 e 5 • X) 11107.690 2 e 19 • Y) 11151.490 2 e 9 • Z) 11184.410 2 e 7 • AA) 11274.180 2 e 11 • BB) 11282.870 2 e 3 • CC) 11361.817 2 e

As an example one full correction table is given in the appendix. All others are available in the office.

13 A) 9464.450 2 e

−1 13 The level 9464.450cm , 2 , e, A = 1056.3(1)MHz, B = −14.8(52)MHz has been corrected on a basis of 57 lines average difference in wave number. The calculated correction wave number difference was −0.005cm−1. This results in a new level energy of 9464.455cm−1. Due to the correction of this level 5 upper odd levels could be corrected which are listed in table 5.1.

41 −1 13 Table 5.1: Corrections of upper odd levels based on transitions from 9464.455cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4904.465 29848.320 29848.346 o 5.5 710(3) - 1 0.026 0.000 4 5395.932 27991.773 27991.768 o 6.5 775 - 3 -0.007 -0.002 7 5753.070 26841.653 26841.658 o 7.5 467 - 3 0.005 0.000 1 5266.028 28448.810 28448.815 o 6.5 785.9(3) -52(23) 1 0.005 0.000 7 5182.584 28754.380 28754.461 o 7.5 639(2) - 1 0.096 0.000 7

11 B) 9483.540 2 e

−1 11 The level 9483.540cm , 2 , e, A = 731.3(2)MHz, B = −15.4(80)MHz has been corrected on a basis of 31 lines average difference in wave number. The calculated correction wave number difference was +0.007cm−1. This results in a new level energy of 9483.533cm−1. Due to the correction of this level 8 upper odd levels could be corrected which are listed in table 5.2.

−1 11 Table 5.2: Corrections of upper odd levels based on transitions from 9483.533cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4714.622 30688.196 30688.208 o 5.5 575.19 -45 1 0.012 0.000 5 4766.964 30455.358 30455.380 o 6.5 610.0v - 1 0.022 0.000 4 4832.704 30170.090 30170.101 o 5.5 667(1) -103(1) 1 0.011 0.000 3 4891.421 29921.790 29921.811 o 4.5 752.58 - 1 0.021 0.000 4 4983.731 29543.242 29543.224 o 6.5 602.4 -21 1 -0.018 0.000 3 5099.060 29089.519 29089.526 o 5.5 595.83 - 1 0.007 0.000 2 5613.504 27292.780 27292.774 o 4.5 767.86 - 1 -0.006 0.000 3 6140.074 25765.476 25765.475 o 6.5 581.04 - 1 -0.001 0.000 7

42 3 C) 9650.060 2 e

−1 3 The level 9650.060cm , 2 , e, A = 1554.9vMHz, B = −15.4(80)MHz was examined but no transitions to already corrected upper levels existed so it could not be corrected.

11 D) 9675.040 2 e

−1 11 The level 9675.040cm , 2 , e, A = 683.2(5)MHz has been corrected on a basis of 26 lines average difference in wave number. The calculated correction wave number difference was −0.004cm−1. This results in a new level energy of 9675.044cm−1. Due to the correction of this level 4 upper odd levels could be corrected which are listed in table 5.3.

−1 11 Table 5.3: Corrections of upper odd levels based on transitions from 9675.044cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4683.080 31022.550 31022.537 o 5.5 670 - 1 -0.013 0.000 4 4836.005 30347.573 30347.492 o 5.5 700v - 1 -0.081 0.000 5 4850.767 30284.595 30284.582 o 6.5 681.4 - 1 -0.013 0.000 5 5201.247 28895.858 28895.850 o 5.5 641.77 - 1 0.002 0.000 13

7 E) 9704.750 2 e

−1 7 The level 9704.750cm , 2 , e, A = 779.1(6)MHz, B = −52(6) has been corrected on a basis of 4 lines average difference in wave number. The calculated correction wave number difference was −0.009cm−1. This results in a new level energy of 9704.759cm−1. Due to the correction of this level 6 upper odd levels could be corrected which are listed in table 5.4.

43 −1 7 Table 5.4: Corrections of upper odd levels based on transitions from 9704.759cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4564.063 31604.099 31604.123 o 3.5 817.4 - 1 0.024 0.000 1 4605.311 31412.715 31412.737 o 2.5 998(12) - 1 0.022 0.000 2 4766.274 30679.611 30679.642 o 2.5 932.8 - 1 0.031 0.000 1 5005.949 29675.409 29675.420 o 3.5 865.5 - 1 0.011 0.000 2 5029.989 29579.880 29579.975 o 3.5 720 - 1 0.095 0.000 5 5055.780 29478.545 29478.587 o 3.5 776.18 - 1 0.042 0.000 2

5 F) 9710.640 2 e

−1 7 The level 9710.640cm , 2 , e, A = 164.5(10)MHz has been corrected on a basis of 2 lines average difference in wave number. The calculated correction wave number difference was +0.025cm−1. This results in a new level energy of 9710.615cm−1. Due to the correction of this level 8 upper odd levels could be corrected which are listed in table 5.5.

−1 5 Table 5.5: Corrections of upper odd levels based on transitions from 9710.615cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4927.178 30000.578 30000.543 o 1.5 983.92 - 1 -0.035 0.000 2 5204.093 28920.810 28920.909 o 2.5 1012 - 1 0.099 0.000 3 5371.233 28323.110 28323.138 o 3.5 891 - 1 0.028 0.000 11 5388.090 28264.931 28264.928 o 3.5 720.36 - 1 -0.003 0.000 5 9084.199 20715.680 20715.719 o 1.5 979v - 1 0.039 0.000 2 9162.490 20621.720 20621.684 o 2.5 356.7(6) - 1 -0.036 0.000 4 9293.861 20467.480 20467.454 o 2.5 800 - 1 -0.026 0.000 2 9663.405 20056.197 20056.146 o 2.5 644v - 1 -0.051 0.000 1

44 15 G) 9745.420 2 e

−1 7 The level 9745.420cm , 2 , e, A = 539.6MHz has been corrected on a basis of 15 lines average difference in wave number. The calculated correction wave number difference was +0.029cm−1. This results in a new level energy of 9745.391cm−1. Due to the correction of this level 15 upper odd levels could be corrected which are listed in table 5.6.

−1 15 Table 5.6: Corrections of upper odd levels based on transitions from 9745.391cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4297.518 33008.119 33008.091 o 7.5 508.3 - 1 -0.028 0.000 2 4468.213 32119.424 32119.423 o 6.5 637.7v - 1 -0.001 0.000 2 4511.075 31906.863 31906.840 o 6.5 560.89 - 1 -0.023 0.000 4 4719.636 30927.551 30927.539 o 6.5 615v - 1 -0.012 0.000 3 4738.567 30842.933 30842.915 o 6.5 580.3 -56.6 1 -0.018 0.000 7 4991.212 29774.970 29775.016 o 6.5 592 - 1 0.046 0.000 2 5019.250 29663.117 29663.130 o 6.5 651 - 1 0.013 0.000 13 5093.196 29373.997 29373.957 o 8.5 784.5 - 1 -0.040 0.000 7 5117.290 29281.511 29281.540 o 8.5 503v -24v 1 0.029 0.000 3 5151.922 29150.250 29150.216 o 8.5 574v -33v 1 -0.034 0.000 19 5810.516 24424.544 24424.515 o 8.5 1072(1) - 1 -0.029 0.000 3 7464.946 23137.680 23137.646 o 8.5 1095.9 - 1 -0.034 0.000 11 5029.392 29623.001 29622.966 o 8.5 500.9 14 1 -0.035 0.000 4 4689.722 31062.660 31062.650 o 7.5 567.7(6) - 1 -0.010 0.000 3 5075.069 29444.080 29444.065 o 6.5 906.7v - 1 -0.015 0.000 4

17 H) 9770.330 2 e

−1 17 The level 9770.330cm , 2 , e, A = 905.498MHz, B = −40.819MHz has been corrected on a basis of 22 lines average difference in wave number. The calculated correction wave number difference was +0.042cm−1. This results in a new level energy of 9770.288cm−1. Due to the correction of this level 11 upper odd levels could be corrected which are listed in table 5.7.

45 −1 17 Table 5.7: Corrections of upper odd levels based on transitions from 9770.288cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4798.392 30604.830 30604.778 o 8.5 525.6v - 1 -0.052 0.000 3 4986.048 29820.680 29820.657 o 7.5 560(4) - 1 -0.023 0.000 5 4991.069 29800.480 29800.487 o 7.5 672.4v - 1 0.007 0.000 3 5024.276 29668.136 29668.103 o 8.5 542.7v - 1 -0.033 0.000 3 5250.884 28809.430 28809.400 o 9.5 566(3) - 1 -0.030 0.000 8 5315.657 28577.408 28577.405 o 7.5 781.78 - 1 -0.003 0.000 5 5583.703 27674.590 27674.578 o 9.5 471.35 - 1 -0.012 0.000 3 5779.287 27068.720 27068.663 o 9.5 578.1(2) -39(8) 1 -0.057 0.000 3 5792.949 27027.920 27027.867 o 9.5 793.1 - 1 -0.053 0.000 5 7889.311 22442.190 22442.180 o 7.5 961 - 1 -0.010 0.000 8 6746.189 24589.400 24589.381 o 8.5 433(3) - 1 -0.019 0.000 9

7 I) 9918.170 2 e

−1 7 The level 9918.170cm , 2 , e, A = 1057.4(5)MHz, B = 2(6)MHz has been corrected on a basis of 10 lines average difference in wave number. The calculated correction wave number difference was −0.035cm−1. This results in a new level energy of 9918.205cm−1. Due to the correction of this level 2 upper odd levels could be corrected which are listed in table 5.8.

−1 7 Table 5.8: Corrections of upper odd levels based on transitions from 9918.205cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5294.612 28800.010 28800.076 o 2.5 1057 - 1 0.066 0.000 4 6480.685 25344.399 25344.409 o 2.5 692(2) -1(6) 1 0.010 0.000 3

46 15 J) 9951.823 2 e

−1 15 The level 9951.823cm , 2 , e, A = 291.67MHz was examined but no lines at all transit from this level. It could not be corrected and is doubtful in existence.

17 K) 10157.090 2 e

−1 17 The level 10157.090cm , 2 , e, A = 488.7MHz was examined but no good transitions were found therefore it could not be corrected.

7 L) 10194.740 2 e

−1 7 The level 10194.740cm , 2 , e, A = 854.9(4)MHz has been corrected on a basis of 7 lines average difference in wave number. The calculated correction wave number difference was −0.040cm−1. This results in a new level energy of 10194.780cm−1. Due to the correction of this level 5 upper odd levels could be corrected which are listed in table 5.9.

−1 7 Table 5.9: Corrections of upper odd levels based on transitions from 10194.780cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4925.138 30493.058 30493.112 o 4.5 717v - 1 0.054 0.000 3 5160.433 29567.550 29567.602 o 4.5 731.1(20) - 1 0.052 0.000 3 5385.649 28757.493 27757.482 o 4.5 748.2v - 1 -0.011 0.000 6 5294.835 29075.830 29075.855 o 2.5 829 - 1 0.025 0.000 6 5649.665 27890.028 27890.032 o 3.5 882.60v - 2 0.008 0.004 6

47 13 M) 10266.510 2 e

−1 13 The level 10266.510cm , 2 , e, A = 972.3MHz, B = −7MHz has been corrected on a basis of 35 lines average difference in wave number. The calculated correction wave number difference was −0.006cm−1. This results in a new level energy of 10266.516cm−1. Due to the correction of this level 6 upper odd levels could be corrected which are listed in table 5.10.

−1 13 Table 5.10: Corrections of upper odd levels based on transitions from 10266.516cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5130.965 29750.596 29750.598 o 5.5 610.1v 31.5v 1 0.002 0.000 3 5311.402 29088.660 29088.699 o 6.5 591.4v - 1 0.039 0.000 6 5800.910 27500.414 27500.414 o 6.5 883.34 -35.53 2 -0.002 -0.001 3 5899.660 27212.011 27211.960 o 7.5 607.7(1) -20(16) 1 -0.051 0.000 4 6339.440 26036.419 26036.419 o 6.5 651.34 - 2 0.000 0.000 0 8210.847 22442.180 22442.180 o 7.5 961 - 2 0.000 0.000 0

−1 13 Moreover the upper level 26708.776cm , 2 , o, A = 762.3MHz was A factor corrected on basis of transition 6080.202A.˚ The new A factor is 745s MHz.

9 N) 10356.710 2 e

−1 9 The level 10356.710cm , 2 , e, A = 1406(1)MHz has been corrected on a basis of 12 lines average difference in wave number. The calculated correction wave number difference was −0.042cm−1. This results in a new level energy of 10356.752cm−1. Due to the correction of this level 3 upper odd levels could be corrected which are listed in table 5.11.

48 −1 9 Table 5.11: Corrections of upper odd levels based on transitions from 10356.752cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5042.225 30183.743 30183.737 o 3.5 604.86 - 1 -0.006 0.000 1 5634.928 28098.267 28098.283 o 5.5 1025v - 1 0.016 0.000 4 7479.183 23723.610 23723.515 o 3.5 830(18) - 1 -0.095 0.000 1

−1 9 In addition to that the lower Level 10356.710cm , 2 , e, A = 1420MHz was A factor corrected on basis of the transition from 9652.901cm−1 to 0.0cm−1. The new A factor is 1406(1)f MHz as stated above.

13 O) 10423.680 2 e

−1 13 The level 10423.680cm , 2 , e, A = 868.8MHz, B = 25MHz has been corrected on a basis of 54 lines average difference in wave number. The calculated correction wave number difference was +0.011cm−1. This results in a new level energy of 10423.669cm−1. Due to the correction of this level 5 upper odd levels could be corrected which are listed in table 5.12.

−1 13 Table 5.12: Corrections of upper odd levels based on transitions from 10423.680cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5077.984 30110.987 30111.035 o 5.5 785.9 - 1 0.048 0.000 3 5353.809 29096.754 29096.765 o 5.5 714.93 - 1 0.011 0.000 4 5736.700 27850.475 27850.489 o 6.5 635.0(3) - 1 0.014 0.000 3 6100.929 26810.076 26810.078 o 6.5 860.29 6.91 2 0.003 0.001 7 7440.860 23859.280 23859.275 o 5.5 635.5v - 1 -0.005 0.000 4

49 15 P) 10466.730 2 e

−1 15 The level 10466.730cm , 2 , e, A = 1041.7(15)MHz, B = −19(10)MHz has been corrected on a basis of 34 lines average difference in wave number. The calculated correction wave number difference was +0.026cm−1. This results in a new level energy of 10466.704cm−1. Due to the correction of this level 3 upper odd levels could be corrected which are listed in table 5.13.

−1 15 Table 5.13: Corrections of upper odd levels based on transitions from 10466.704cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4867.414 31005.746 31005.778 o 6.5 588 - 2 0.032 0.000 18 5012.308 30411.982 30412.029 o 6.5 658 - 1 0.047 0.000 12 5643.163 28182.321 28182.345 o 8.5 615.52 - 1 0.024 0.000 2

13 Q) 10470.300 2 e

−1 13 The level 10470.300cm , 2 , e, A = 927.9vMHz has been corrected on a basis of 7 lines average difference in wave number. The calculated correction wave number difference was −0.044cm−1. This results in a new level energy of 10470.344cm−1. Due to the correction of this level 1 upper odd level could be corrected which is listed in table 5.14.

50 −1 13 Table 5.14: Corrections of upper odd levels based on transitions from 10470.300cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4991.430 30498.989 30499.094 o 5.5 708 - 1 0.105 0.000 3

−1 17 In addition to that the upper Level 28429.490cm , 2 , o, A = 546.3MHz was A factor cor- rected on basis of transition 5565.525A.˚ The new A factor is 559s MHz as stated above.

17 R) 10532.001 2 e

−1 17 The level 10532.001cm , 2 , e, A = 546v MHz has been corrected on a basis of 7 lines average difference in wave number. The calculated correction wave number difference was +0.035cm−1. This results in a new level energy of 10531.966cm−1. Due to the correction of this level 6 upper odd levels could be corrected which are listed in table 5.15.

−1 17 Table 5.15: Corrections of upper odd levels based on transitions from 10531.966cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4660.252 31984.040 31984.027 o 8.5 543.4(20) - 1 -0.013 0.000 11 4796.799 31373.394 31373.375 o 7.5 604.92 -19.48 1 -0.019 0.000 2 5161.717 29899.984 29899.969 o 8.5 529(1) - 1 -0.015 0.000 8 5267.660 29510.473 29510.445 o 9.5 531.6(2) -38(8) 1 -0.028 0.000 4 6737.304 25370.600 25370.601 o 7.5 814.54 - 2 0.002 0.001 6 7505.565 23851.803 23851.743 o 8.5 1091v -4 1 -0.060 0.000 1

51 5 S) 10828.988 2 e

−1 5 The level 10828.988cm , 2 , e, A = 979.7MHz, B = −14MHz was examined but could not be corrected because there were no good spectral lines to already corrected upper levels.

11 T) 10841.482 2 e

−1 11 The level 10841.482cm , 2 , e, A = 530(3)MHz has been corrected on a basis of 4 lines average difference in wave number. The calculated correction wave number difference was +0.065cm−1. This results in a new level energy of 10841.417cm−1. Due to the correction of this level 1 upper odd level could be corrected which is listed in table 5.16.

−1 11 Table 5.16: Corrections of upper odd levels based on transitions from 10841.482cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5035.613 30694.484 30694.435 o 6.5 728.8v - 1 -0.049 0.000 5

11 U) 10904.070 2 e

−1 11 The level 10904.070cm , 2 , e, A = 301.1(1)MHz, B = −22(2)MHz has been corrected on a basis of 24 lines average difference in wave number. The calculated correction wave number difference was +0.021cm−1. This results in a new level energy of 10904.049cm−1. Due to the correction of this level 2 upper odd levels could be corrected which are listed in table 5.17.

52 −1 11 Table 5.17: Corrections of upper odd levels based on transitions from 10904.049cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5239.264 29985.524 29985.387 o 5.5 775(2) - 1 -0.137 0.000 12 5297.618 29775.180 29775.206 o 5.5 803 - 1 0.026 0.000 1

9 V) 10920.380 2 e

−1 9 The level 10920.380cm , 2 , e, A = 632vMHz has been corrected on a basis of 3 lines average difference in wave number. The calculated correction wave number difference was +0.002cm−1. This results in a new level energy of 10920.378cm−1. Due to the correction of this level 1 upper odd level could be corrected which is listed in table 5.18.

−1 9 Table 5.18: Corrections of upper odd levels based on transitions from 10920.378cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4828.477 31625.080 31625.051 o 5.5 602.7(10) - 1 -0.029 0.000 5

1 W) 10956.833 2 e

−1 1 The level 10956.833cm , 2 , e, A = 1560(1)MHz was examined but it is not involved in any known transitions. This makes the existence of this level doubtful.

53 5 X) 11107.690 2 e

−1 5 The level 11107.690cm , 2 , e, A = 658MHz has been corrected on a basis of 1 line difference in wave number. The calculated correction wave number difference was −0.015cm−1. This results in a new level energy of 11107.705cm−1. Due to the correction of this level 1 upper odd level could be corrected which is listed in table 5.19.

−1 5 Table 5.19: Corrections of upper odd levels based on transitions from 11107.705cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 7883.290 23789.247 23789.203 o 1.5 -137v - 1 -44.000 0.000 3

19 Y) 11151.490 2 e

−1 19 The level 11151.490cm , 2 , e, A = 876.1(3)MHz, B = −31(12)MHz has been corrected on a basis of 5 lines average difference in wave number. The calculated correction wave number difference was +0.042cm−1. This results in a new level energy of 11151.448cm−1. Due to the correction of this level 14 upper odd levels could be corrected which are listed in table 5.20.

−1 19 Table 5.20: Corrections of upper odd levels based on transitions from 11151.448cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4720.773 32328.452 32328.494 o 8.5 494.6v - 1 0.042 0.000 3 4763.268 32139.580 32139.568 o 8.5 715.1v - 1 -0.012 0.000 2 will be continued on next page

54 Table 5.20 – continued λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 4850.633 31761.614 31761.555 o 8.5 513.47 - 1 -0.059 0.000 3 4868.231 31687.069 31687.054 o 8.5 593.0(6) - 1 -0.015 0.000 3 4917.616 31480.870 31480.828 o 9.5 471.3(5) - 1 -0.042 0.000 4 4968.936 31270.870 31270.866 o 8.5 512 - 1 -0.004 0.000 9 5022.311 31057.076 31057.048 o 9.5 402.8v - 1 -0.028 0.000 2 5033.165 31014.149 31014.122 o 8.5 674.1v - 1 -0.027 0.000 3 5208.303 30346.230 30346.214 o 8.5 518 - 1 -0.016 0.000 3 5231.533 30261.013 30260.983 o 9.5 539.02 - 1 -0.030 0.000 3 5280.717 30083.020 30083.001 o 8.5 524.9v - 1 -0.019 0.000 5 5403.645 29652.280 29652.331 o 8.5 432 - 1 0.051 0.000 6 6229.454 27199.787 27199.779 o 8.5 378.6v - 1 -0.008 0.000 3 6852.761 25740.100 25740.080 o 8.5 440s - 1 -0.020 0.000 8

−1 19 In addition to that the upper Level 25740.080cm , 2 , o, A = 553MHz was A factor corrected on basis of transition 6852.761A.˚ The new A factor is 440sMHz as stated above.

9 Z) 11184.410 2 e

−1 9 The level 11184.410cm , 2 , e, A = 692.1(4)MHz, B = 14(28)MHz needn’t be corrected. 8 lines average difference in wave number showed exactly a deviation of ±0.000cm−1. Due to the affirmation of this level 2 upper odd levels could be corrected which are listed in table 5.21.

−1 9 Table 5.21: Corrections of upper odd levels based on transitions from 11184.410cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5051.337 30975.574 30975.622 o 3.5 734 - 1 0.048 0.000 3 5061.203 30937.040 30937.058 o 3.5 710.8(20) - 1 0.018 0.000 1

7 AA) 11274.180 2 e

−1 7 The level 11274.180cm , 2 , e, A = 1285.6(6)MHz, B = −7(3)MHz has been corrected on a ba- sis of 4 lines average difference in wave number. The calculated correction wave number difference

55 was −0.064cm−1. This results in a new level energy of 11274.244cm−1. Due to the correction of this level 1 upper odd level could be corrected which is listed in table 5.22.

−1 7 Table 5.22: Corrections of upper odd levels based on transitions from 11274.244cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5262.719 30270.493 30270.541 o 2.5 842.02 - 1 0.048 0.000 2

11 BB) 11282.870 2 e

−1 11 The level 11282.870cm , 2 , e, A = 1048.9vMHz has been corrected on a basis of 16 lines average difference in wave number. The calculated correction wave number difference was −0.010cm−1. This results in a new level energy of 11282.880cm−1. Due to the correction of this level 1 upper odd level could be corrected which is listed in table 5.23.

−1 11 Table 5.23: Corrections of upper odd levels based on transitions from 11282.870cm , 2 , e λ ... wavelength of spectral line used for correction of upper level Eupper,before ... energy of upper level before correction Eupper,after ... energy of upper level after correction P ... parity of upper level (odd:o, even:e) J ... total (hull) angular momentum of upper level A ... A-factor of upper level B ... B-factor of upper level nol ... overall number of lines used for correction of this level ∆before ... difference in wave number before correction ∆after ... difference in wave number after correction nofc ... number of follow up corrections

λ Eupper,before Eupper,after P J A B nol ∆before ∆after nofc A˚ cm−1 cm−1 MHz MHz cm−1 cm−1 5185.335 30562.700 30562.667 o 6.5 536.6(8) - 1 -0.033 0.000 3

56 3 CC) 11361.817 2 e

−1 3 The level 11361.817cm , 2 , e, A = 53.8MHz was examined but no good transitions to upper already corrected levels could be identified. The level was not corrected.

5.2 Prospectives

−1 9 In the investigated energy region 6 good transitions to the ground state (0.0cm , 2 , o) where found. They all agree fairly good in their wave number difference to the proposed level energies (see table 5.24).

−1 9 Table 5.24: Good transitions to the groundstate of Pr-I 10.0cm , 2 , o Eupper,new ... confirmed energy of upper level J ... total (hull) angular momentum of upper level λ ... wavelength of spectral line to groundstate ∆before ... difference in wave number before correction

Eupper,new J λ ∆before cm−1 A˚ cm−1 9704.759 3.5 10301.420 -0.008 10194.780 3.5 9806.265 -0.013 10356.752 4.5 9652.901 -0.013 10920.378 5.5 9154.679 -0.012 11184.410 4.5 8938.574 -0.014 11282.880 5.5 8860.564 -0.015

This result suggests a correction of all already corrected energy levels by −0.013cm−1 to make them correct on an absolute energy scale.

57 6 Appendix

6.1 Example of a data sheet

On the following six pages an example of a full working dataset is given.

58

6.2 Sourcecode

%∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗% % name: spectralTerms % author: Martin Nuss % date: 19.03.2009 % use: calculates the spectral terms arising from % given l,s,eanz using " high field quantum % numbers in a totally decoupled approach % in : % − l: orbital quantum number (0,1,2,3,4,5, ...) % − s: spin quantum number (usually 1/2) % − eanz: number of equivalent electrons in this orbital % out : % − stat: status is set to 0 if all element in magnetic % quantum number matrices have been used correctly % is set to number of not used or more often used % numbers , % i t i s −1 if number of maximum electrons per orbital is exceeded % f i l e −out : % − ’ isdmqn l s eanz.dat’: calculated combinations of decoupled % magneticquantumnumbers % − ’ cdmqn l s eanz.dat’: initial set of decoupled magnetic % quantumnumbers % − ’spectralTerms l s eanz.dat’: calculated spectral terms %∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗% % This program was checked with data given by E.White: " Introduction to % Atomic Spectra " (1934) page 254. % All spectral terms for p and d levels given there were calculated % correctly so I assume the program works for any configuration! %∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗% % Version 1.0 19.03.2009 % Version 1.1 21.03.2009 %∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗% % know bugs : % − none %∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗% function [stat] = spectralTerms(l ,s,eanz)

%check input i f ( eanz > 2∗(2∗ l +1)) s t a t = −1; e l s e

%build ml and ms ml = [− l : 1 : l ] ; ms = [ s :−1:− s ] ;

%build working ml mlw = repmat(ml,1 ,numel(ms));

%build working ms msw = ones(1,numel(ml)); msw(:) = ms(1); for(k = [2:1:numel(ms)]) msw = [msw,ones(1,numel(ml))]; msw(msw == 1) = ms( k ) ; end

65 %calculate principally possible structure (no putting back, not ordered) classf = [1:1:numel(mlw)]; combo =nchoosek(classf ,eanz);

%append calculated L and S to combo combo = [combo,zeros(size(combo,1) ,2)]; combo(: ,eanz+1) = mlw(combo(: ,1)); combo(: ,eanz+2) = msw(combo(: ,1)); for(k=[2:1:eanz]) combo(: ,eanz+1) = combo(: ,eanz+1) + mlw(combo(: ,k)) ’; combo(: ,eanz+2) = combo(: ,eanz+2) + msw(combo(: ,k)) ’; end

%sortrows after ML then MS combo = sortrows(combo,[eanz+1, eanz+2]);

%save to file : calculated combinations of decoupled magnetic quantum numbers fname = [ ’ cdmqn ’,num2str(l),’ ’,num2str(s),’ ’,num2str(eanz) , ’.dat ’]; fid = fopen(fname,’w’); l e t = ’ 0 ’ ; draw = 1 ; %this whole proc is to print the data MLMS once with all combinations besides %them and not every combination with ML MS for(k = [1:1:size(combo,1)]) if(draw == 1) fprintf(fid ,[num2str(combo(k,eanz+1),’%1.0f ’), ’ ’]); fprintf(fid ,[num2str(combo(k,eanz+2),’%1.1f ’), ’ ’]); end for(j = [1:1:eanz]) fprintf(fid ,num2let(combo(k,j ))); end i f ( k < size(combo,1)) if ((combo(k+1,eanz+1) == combo(k,eanz+1))&&(combo(k+1,eanz+2)... == combo(k,eanz+2))) %ML unchanged MS unchanged fprintf(fid,’ ’); draw = 0 ; elseif ((combo(k+1,eanz+1) ˜= combo(k,eanz+1))&&(combo(k+1,eanz+2)... == combo(k,eanz+2))) %ML changed MS unchanged fprintf(fid ,’ \ n ’ ) ; draw = 1 ; elseif ((combo(k+1,eanz+1) == combo(k,eanz+1))&&(combo(k+1,eanz+2)... ˜= combo(k,eanz+2))) %ML unchanged MS changed fprintf(fid ,’ \ n ’ ) ; draw = 1 ; elseif ((combo(k+1,eanz+1) ˜= combo(k,eanz+1))&&(combo(k+1,eanz+2)... ˜= combo(k,eanz+2))) %ML changed MS changed fprintf(fid ,’ \ n ’ ) ; draw = 1 ; e l s e fprintf(fid ,’! \ n ’ ) ; draw = 1 ; end e l s e fprintf(fid ,’ \ n ’ ) ; end end fclose(fid);

66 %save to file: initial set of decoupled magnetic quantum numbers ll = [1:1:numel(mlw)]; letl = zeros(1,numel(mlw)); for(k = [1:1:numel(ll )]) letl(k) = num2let(ll(k)); end b = zeros(1,numel(mlw)); b ( : ) = ’ ’ ; tab = [num2str(mlw’,’%1.0f ’) ,b’,num2str(msw’,’%1.1f ’) ,b’,letl ’]; fname = [ ’isdmqn ’,num2str(l),’ ’,num2str(s),’ ’,num2str(eanz) , ’.dat ’]; fid = fopen(fname,’w’); for(k = [1:1:numel(ll )]) fprintf(fid ,[tab(k,:),’ \ n ’ ] ) ; end fclose(fid);

%transform combo to combomat where number of systems per configuration is %given ch = 1 ; cnt = 1 ; combomat = zeros(size(combo,1) ,3); j = 1 ; for(k = [1:1:size(combo,1)]) i f ( k < size(combo,1)) if ((combo(k+1,eanz+1) == combo(k,eanz+1))&&(combo(k+1,eanz+2)... == combo(k,eanz+2))) %ML unchanged MS unchanged ch = 0 ; cnt = cnt + 1 ; elseif ((combo(k+1,eanz+1) ˜= combo(k,eanz+1))&&(combo(k+1,eanz+2)... == combo(k,eanz+2))) %ML changed MS unchanged ch = 1 ; elseif ((combo(k+1,eanz+1) == combo(k,eanz+1))&&(combo(k+1,eanz+2)... ˜= combo(k,eanz+2))) %ML unchanged MS changed ch = 1 ; elseif ((combo(k+1,eanz+1) ˜= combo(k,eanz+1))&&(combo(k+1,eanz+2)... ˜= combo(k,eanz+2))) %ML changed MS changed ch = 1 ; e l s e ch = 1 ; end

i f ( ch == 1) combomat(j ,1) = combo(k,eanz+1); combomat(j ,2) = combo(k,eanz+2); combomat(j ,3) = cnt; j = j + 1 ; cnt = 1 ; end else %last elem if (size(combo,1) >1) if ((combo(k−1,eanz+1) == combo(k, eanz+1))&&(combo(k−1,eanz+2)... == combo(k,eanz+2))) combomat ( j −1,3) = combomat(j −1 ,3) + 1 ; e l s e combomat(j ,1) = combo(k,eanz+1); combomat(j ,2) = combo(k,eanz+2);

67 combomat(j ,3) = 1; end e l s e combomat(j ,1) = combo(k,eanz+1); combomat(j ,2) = combo(k,eanz+2); combomat(j ,3) = 1; end end end %cut zero filled lines! (line number is not determineable beforehand a = sum(abs(combomat) ,2); a ( a˜=0) = 1 ; cutanz = sum(a(:)); combomat = combomat([1:cutanz ] ,:);

%seperate terms fname = [ ’spectralTerms ’,num2str(l),’ ’,num2str(s),’ ’,... num2str(eanz) , ’.dat ’]; fid = fopen(fname,’w’); k = 1 ; while ( k <=size (combomat,1)) if (combomat(k,3) == 0) k = k + 1 ; e l s e s v a l s = [−abs(combomat(k,2)):1: abs(combomat(k,2))]; pv = zeros(size(svals)); pv(:) = abs(combomat(k,1)); pv = pv−s v a l s ; %p r i n t term lorbit = num2orbit(abs(combomat(k,1))); fprintf(fid ,[num2str(abs(2∗ combomat(k,2))+1),’ ’ ,... l o r b i t , ’ ’ ,num2str(pv(pv>=0),’%1.1f ’) , ’ \ n ’ ] ) ; %take out for(j =[k:1: size(combomat,1) −k+1]) for(p = [1:1:numel(svals)]) if(combomat(j ,2) == svals(p)) combomat(j ,3) = combomat(j ,3) −1; end end end end end fclose(fid);

%check if all quantum numbers were used correctly stat = sum(abs(combomat(: ,3))); end end

68 Bibliography

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[13] L Windholz. Classification Program for Atomic and Ionic Spectral Line Manual. Institut fur¨ Experimentalphysik, TU Graz, 2007. [14] L Windholz. Skriptum zur Experimentalphysik 3 (Atom- und Kernphysik), volume III of Vorlesungsskripten Experimentalphysik TU Graz. Institut fur¨ Experimentalphysik, TU Graz, 2007.

[15] L Windholz. Skriptum zur Experimentalphysik 4 (Molekulphysik,¨ Spektroskopische Methoden, Quantenmesstechnik), volume IV of Vorlesungsskripten Experimentalphysik TU Graz. Institut fur¨ Experimentalphysik, TU Graz, 2008.

69 List of Figures

3.1 Praseodymium sample (from [11]) ...... 5 3.2 Periodic table of the elements - Praseodymium in Lanthanoid group (from http://www.dayah.com/periodic/ 07.04.2009) ...... 5 3.3 Evolution of the atomic model: (left) Thomson’s Plum Pudding model, (right) Lenards Dynamiden model ...... 6 3.4 Evolution of the atomic model: (left) Rutherford’s atomic model, (right) Bohr’s atomic model ...... 7 3.5 Evolution of the atomic model: (left) Sommerfeld’s atomic model, (right) Quantum mechanical atomic model ...... 8 3.6 Visualization of a relativistic electron in an elliptic orbit with perihel rotation due to it’s relativistc mass ...... 9 3.7 Spin orbit interaction ...... 10 3.8 Visualization of isotopic shift (normal mass shift only) for Lyman-α lines in Hydro- gen, Deuterium and Tritium ...... 15 5 3.9 Scheme of nuclear spin quantization of Pr (I = 2 )...... 16 17 5 3.10 Lande interval rule applied to J = 2 , I = 2 ...... 19 3.11 Possible interactions of angular momenta for two electrons ...... 20 3.12 Vector model for perfect LS -coupling in a two electron system ...... 21 3.13 Vector model for perfect jj -coupling in a two electron system ...... 22 3.14 Alternation law of multiplicities (from [7]) ...... 23 3.15 Schematic drawing of a Fourier Transformation Spectroscope (from [5]), 1: light source, 3: aperture, 5: beam splitter, 10: specimen, 12: detector, (13,14): HeNe Laser, 17: Laser detector, (2,4,6,7,8,11,15,16): mirrors, 9: specimen chamber . . . . 27 3.16 Energy scheme for the LIF process ...... 29

4.1 Scheme of the correction process, black: arbitrarily fixed level, green: true ground state, red&blue&purple: levels involved in correction process; (The given level data is from Pr-I levels but the transitions are chosen fictitiously.) ...... 31 4.2 Main Screen of the used Program: Classification of Spectral Lines by means of their Hyperfine Structure ...... 33 4.3 Extract of the even levels of Pr-I in the database...... 34 4.4 Mainscreen showing the currently selected spectral line and the calculated sugges- tions for the hfs ...... 34 4.5 Example of shifting the center of gravity wavelength ...... 35 4.6 Examples for good match of Fourier spectrum to calculated hfs spectrum . . . . . 36 4.7 Information on laser-induced fluorescence in the comment concerning the spectral line...... 37 4.8 Examples for bad match of Fourier spectrum to calculated hfs spectrum ...... 37 4.9 Examples for no match of Fourier spectrum to calculated hfs spectrum ...... 37 4.10 Examples for a situation in the Fourier spectrum ...... 38 4.11 Examples for a blend situation in the Fourier spectrum ...... 39 4.12 Correction of an A factor ...... 40

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