Coherence effects of diatomic homonuclear molecules and sequential two-photon processes of noble gases in the photoionization

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Coherence effects of diatomic homonuclear

molecules and sequential two-photon processes of

noble gases in the photoionization

vorgelegt von Diplom-Physiker Gregor Hartmann

aus Berlin

von der Fakultät II -Mathematik und Naturwissenschaften der Technischen Universität Berlin

zur Erlangung des akademischen Grades Doktor der Naturwissenschaften

-Dr. rer. nat.-genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. rer. nat. Mario Dähne

Gutachter: Prof. Dr. rer. nat. Thomas Möller

Gutachter: Dr.-Ing. Jens Viefhaus

Tag der wissenschaftlichen Aussprache: 10.04.2014 Berlin 2014

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Abstract

Atomic and molecular photoionization produces electrons and ions which can be analyzed with respect to cross sections and angular distributions of photoemission applying time-of-flight spec-troscopy. The electron angular distributions of both ionization steps in sequential two-photon double ionization of noble gases are analyzed in a photon energy range of 38 - 91 eV. This angular dependence of emitted photoelectrons is determined by the transfer of the angular momentum from the ionizing photon to the ejected electron. In the case of two-photon double ionization the total angular momentum transfer gives rise to a Legendre polynomial of fourth order. This

so-called β4 term has a weighting factor given by the alignment of the ionic core left by the

first ionization step. This additional term does not only affect the electrons emitted in the second step but also the first step electrons which indicates that both processes are coherently coupled. This effect is shown for the two-photon ionization of the rare gases neon, argon and krypton measured at FLASH in a spherical chamber with up to 19 independently working elec-tron time-of-flight detectors. The angular distribution of 1s-shell double photon ionization of

helium should exhibit no β4 term because the neutral atom as well as the singly charged ion

have an isotropic electron distribution. This is confirmed by measurements at 79 eV and 91 eV.

Interestingly, a strong β4 distribution is measured in both ionization steps at a particular photon

energy of 61 eV. This unexpected result is interpreted as an autoionization process induced by a helium resonance into a doubly excited intermediate state. Since a large quantity of data is col-lected at FEL experiments, a CUDA-based evaluation and visualization software was developed and is discussed in detail.

The analogy of photoemission processes in homonuclear diatomic molecules to the Young type double slit experiment has been topic of a large amount of photoionization investigations. One

example is the oscillation in the photoionization cross section of N2 which Cohen and Fano

inter-preted as an interference effect. This two center interference is caused by electron non-locality in homonuclear molecules. Here, the oscillation in the cross section of molecular hydrogen is analyzed over a wide photon energy range (29 - 1200 eV) distinguishing between molecules that are oriented randomly in space and those with a molecular axis fixed in space (oriented sample). Furthermore, a transition effect from random to oriented target properties is detected if the de Broglie wavelength of the photoelectron is small enough to resolve the internuclear distance.

The ion fragment angular distribution given by the βm parameter is analyzed and found as a

reason for this transition. At high photon energies the βm value converges to −1 resulting in a

preferred molecular orientation for the ionization process. Several variants of ion time-of-flight detectors with the option of a coincidence measurement mode together with an electron time-of-flight spectrometer were used. The experiments were performed at BESSY II and PETRA

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1 Introduction

The description of atomic properties led classical physics to their limits. In 1802, Thomas Young performed the popular double slit experiment and proved the wave nature of light that way [35, 36]. However, the photoelectric effect, which was discovered in 1887 by Heinrich Hertz [59] and explained by Albert Einstein in 1905, impressively shows the particle properties of photons.

’Wir werden anzunehmen haben, daß bei der Ionisierung eines Gases durch ultra-violettes Licht je ein absorbiertes Lichtenergiequant zur Ionisierung je eines Gas-moleküles verwendet wird.’

Albert Einstein: ’Über einen die Erzeugung und Verwandlung des Lichtes betreffenden heuristischen Gesichtspunkt’ in [39].

Einstein interpreted Max Planck’s work [93] of 1900 about the thermal black body radiation and the emergent physical constant, the Planck’s constant. In 1911, Ernest Rutherford showed that

the size of the nucleus is in the order of 10−14_{m which is 4 orders of magnitude smaller than the}

total atom size of about 1 Å [100]. This discovery refuted Joseph John Thomson’s theory of a uniform mass distribution in the atom [116]. Niels Bohr created a model of atoms (1913) which in contrast to the classical model made the description of the emission lines in the hydrogen atom possible [12, 13]. In 1922, Arthur Holly Compton demonstrated in scattering experiments of X-rays with matter that the X-ray wavelength changes dependent on the scattering-angle. He explained this effect as a collision process in which the energy and momentum conservation for the two collision partners, electron and photon, remain valid [24]. In 1923, Louis de Broglie adapted the wave-particle dualism (known from the photons) to matter, such as electrons, and thus made significant progress possible in atomic physics [87]. Three years later, Erwin Schrödinger presented his famous equation [108] which was a huge step in quantum physics. In 1927, Max Born and Julius Robert Oppenheimer published the from then on known as Born-Oppenheimer approximation [15] separating the electron and nucleus motion in Schrödinger’s equation. In the same year, Werner Heisenberg introduced the uncertainty principle [56]. One year later, Paul Dirac discovered his important equation [37], which combined Einstein’s special relativity with quantum mechanics and explained the electron spin for the first time.

These were very important milestones discovered in a short period of time enabling the under-standing of the interaction of light with matter which has been subject to scientific research for a long time. A part of this interaction is the photoionization describing processes in which

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the incoming photon kicks out an electron, the so-called photoelectron, of the shell of the atom which remains as a positively charged ion.

The advent of synchrotron radiation sources facilities gave scientists the possibility to analyze photoionization processes with a broadband high photon energy light source. With the use of tunable X-ray monochromators atomic and molecular photoionization processes could be investigated in dependence of the photon energy.

One aspect of photoionization is the photoelectron angular distribution, i.e. the emission prob-ability of the electron as a function of the angle between the photon polarization vector and electron emission direction. With synchrotron radiation experiments for analysis of the angu-lar distribution and determination of the associated anisotropy parameter β [129] have been performed for a variety of atomic and molecular targets in the linear regime of one-photon ionization. The β parameter characterizes the single photon ionization as in rare gases [6]. Free electron lasers (FEL) [96], such as FLASH [117] in Hamburg, improved the light brilliance by more than 9 orders of magnitude compared to synchrotron radiation sources. This new light source allows to study non-linear multi-photon processes such as various sequential and simultaneous ionization processes involving two or more ionizing photons interacting with two or more electrons independently within one single FEL shot because of the high intensity of the FEL radiation. This high intensity is the main difference compared to photoelectron spectroscopy experiments with synchrotron radiation which have been done for a long time now, and from which we learned already about some simultaneous and sequential ionization processes initiated by one photon only.

One part of this thesis is devoted to the electron angular distribution of sequential two-photon double ionization processes in noble gases (helium, neon, argon, krypton) that occur only at high photon densities with sufficiently high probability. The second photon provides an additional angular momentum resulting in a more complicated angular distribution. This is described

by an additional term including a higher order anisotropy parameter β4. The setup used to

collect angle-resolved photoelectron signal data which are analyzed in this work consists of a spherical chamber with up to 19 simultaneously and independently working electron time-of-flight spectrometers and was deployed at FLASH on the most intense beamlines BL2 and BL3 [16, 17, 18, 20]. As the resulting data sets are large (in our case 45 terabytes), it is imperative to develop specific data analysis methods which are adapted to the specific setup circumstances. Here, an evaluation software based on the Compute Unified Device Architecture (CUDA) which uses GPUs for calculations has been developed. This allows the parallel analysis of many electron time-of-flight spectra.

The more complicated photoionization of homonuclear diatomic molecules is the second topic of this thesis. Interference effects occur as in the double slit experiments which has always fascinated scientists. According to the Physics World (2002) [30] ’the most beautiful experi-ment’ was conducted by Claus Jönsson in 1961 [67] showing interference effects of electrons in

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a double slit. In 1966, Howard D. Cohen and Ugo Fano [23] interpreted ’shoulders’ in the

pho-toabsorption cross section of N2 and O2 of Samson et al. [102] as interference phenomena by the

indistinguishability of the two nuclei in a homonuclear diatomic molecule forming a molecular double slit.

Rolles et al. [98] (2005) have shown that electron non-locality can be partially removed by

symmetry breaking using isotopic substitution in N2 (14,15N2), whereas Zimmermann et al.

[133] (2008) determined intramolecular scattering processes in core level ionization and loss

of coherence in CO in contrast to N2. The two center interference by electron non-locality in

homonuclear molecules can be turned off by using heteronuclear molecules similar to the famous Wheeler’s delayed choice experiment [65, 84, 115].

Recently, more detailed studies in this field appeared concerning cross section oscillation due

to interference: Canton et al. [21] (2011) analyzed valence shell ionization of H2, N2 and CO

vibrationally resolved and directly observed the oscillations [8]. Valence shell oscillations were

studied by Ilchen et al. [62] (2014) over a full period in N2 and O2 which was the previously

missing direct evidence of the oscillations described in the original publication by Cohen and Fano [23]. In addition, oscillations could be determined in the anisotropy parameter β as well which was predicted by Toffoli and Decleva in 2006 [120].

Hydrogen is both, the simplest atom as well as in its diatomic form the simplest molecule which was topic of many theoretical and experimental studies in the last decades [2, 5, 8, 14, 21, 22, 33, 43, 45, 52, 71, 74, 86, 88, 95, 103, 105, 106, 111, 113, 124].

In this dissertation, the popular Cohen-Fano oscillations [23] in H2 are analyzed over a wide

photon energy range (28 - 1200 eV) covering almost two oscillation periods. A lesser known fact is the characteristic behavior of these oscillations depending on whether the electrons are emitted from a randomly in space distributed or an oriented target. This should give rise to a 90° phase shift of both oscillations which is observed at low photon energies. The orientation

of the molecule to the light polarization axis leads to the anisotropy parameter βm which is

evaluated in the present study. A transition effect from random to oriented target properties is observed and explained by the ion fragment angular distribution. It takes place when the de Broglie wavelength of the photoelectrons resolves the internuclear distance of the two atomic emitter sites. The experimental setup consists of an ion time-of-flight spectrometer including the option of a coincidence mode with an electron time-of-flight spectrometer. The experiments were performed at BESSY II (U125/2, UE56/1-PGM-b, UE56/2 PGM1) and PETRA III (P04). The results are compared to work of Kossmann et al. (1989) [74], Chung et al. (1993) [22], Samson and Haddad (1994) [103], Stolterfoth et al. (2001) [113] and Fojon et al. (2004) [45]. Chapter 2 begins with the theoretical description of photoionization processes of rare gases. Then the hydrogen molecule is discussed with a focus on the oscillations in the cross section and on the molecular angular distributions. In the following chapter 3, the experimental setups are presented and the characteristics of the detectors are discussed. The analysis of the

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mea-surement data is explained in chapter 4 with particular focus on the evaluation of the electron and ion time-of-flight spectra and the handling of the flood of data received in our FLASH experiments. Chapter 5 presents the anisotropy parameters of the first and second ionization steps in sequential two-photon processes of noble gases as a function of photon energy and den-sity. Furthermore, a resonance effect is observed in helium. Chapter 6 deals with the results of the Cohen-Fano oscillations of the hydrogen molecule and the observed transition effect. After conclusions and an outlook on possible future studies in chapter 7, additional information is presented in the appendix.

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2 Photoionization of noble gases and molecular hydrogen

This chapter deals with aspects of the theoretical description of photoionization relevant for this

work1. After introducing the relevant quantum numbers in atomic quantum mechanics, some

example processes are presented. The atomic cross section is discussed and the photoelectron angular distribution as a function of the electron emission and the photon polarization vector is derived for single and sequential double ionization. Since the hydrogen molecule is the primary topic of this thesis’ investigations, the molecular photoionization is discussed afterwards.

2.1 Photoionization process

In atomic photoionization an electron is ejected by absorbing a photon. This photoelectron was bond and leaves the atom with the kinetic energy

Ekin = Ephoton− Ebin (1)

where Ephoton = hν is the photon energy at the frequency ν with the Planck’s constant2 h and

the electron binding energy Ebin.

The principal quantum number n of an atomic orbital defines the electron shell and has quantum numbers 1, 2, 3,... or in X-Ray notation K-, L-, M-,... shell, where the K-shell is the closest to the nucleus [107]. Equation (1) is the famous formula of Einstein’s photoelectric effect [39] which directly describes a photon as a particle.

Apart from conservation of energy, the angular momentum of the photon is transferred. In

dipole approximation the angular momentum of the photon ~lphotonis added to the orbital angular

momentum of the initial electron ~linitial and results in a photoelectron momentum ~lphotoelectron.

Since the angular momentum quantum number of the photon

lphoton = 1, (2)

the photoelectron angular momentum number is given by

lphotoelectron = linitial± 1. (3)

1_{This can be done in various mechanisms. A detailed description concerning valence shell ionization, inner}

shell ionization, correlation satellites, resonant excitations, Auger decay, autoionization or simultaneous double ionization can be found in [6, 53, 58, 63, 70, 73, 76, 78, 98, 99, 122, 125, 134] and an overview in section 2.2.

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The azimuthal quantum numbers l of atomic orbitals are named after their shape: • l = 0: s-orbital (sharp)

• l = 1: p-orbital (principal) • l = 2: d-orbital (diffuse) • l = 3: f-orbital (fundamental)

• for higher quantum numbers the letters are followed in alphabetical order: s, p, d, f, g, h, ...

The absolute value of the angular momentum is the given by ~l = p l (l + 1)~, (l = 0, 1, 2, ...n − 1) . (4)

Further quantum numbers are the magnetic quantum number ml (ml = 0, ±1, ±2, ..., ±l) and

the spin projection quantum number ms (ms = ±1₂) which characterize the orbital angular

momentum projection along a specified axis and the spin of the electron. More details and visualizations can be found in [36, 44, 87, 89, 107, 119].

A special case is the photoionization of an s-electron where the photoelectron is always a

p-electron (lphotoelectron = 0 + 1) as in helium 1s-ionization (n = 1, l = 0). The photoelectron

angular distribution as a function of the angle between the electron emission and the photon polarization vector will be presented in section 2.3.2. In molecular photoionization, the photon energy can also be used for vibrational or rotational excitation and the molecular orbitals and angular distributions become more complicated as discussed in section 2.5.

2.2 Photoionization examples

This section presents some selected photoionization processes in term scheme occurring during photoionization with synchrotron radiation within the energies from some eVs up to several hundred eV in order to give a quick overview of several mechanisms to ionize an atom or

molecule. Besides the direct photoionization processes (see figure 13_{), which describes valence}

or inner shell electrons emitted directly by absorbing a photon, there are a lot of multi-electron ionization processes. In the so-called ’shake’ processes, the photon energy is used for ionizing the atom and exciting another electron. If this other electron is excited to a higher state, the process is called ’shake-up’ (left panel of figure 2) and ’satellite’ because the kinetic energy difference of the direct ionization electron and the satellite electron is fixed for different photon energies (and the same excitations). If the other electron also leaves the atom, the process

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is called ’shake-off’ (right panel of figure 2). In an Auger process (named after Pierre Victor Auger [3]), another photoelectron, called Auger electron, is created by the filling of an inner shell vacancy which is unoccupied by an occurring ionization or excitation process before. This can be done in several mechanism (see figure 3 top). Furthermore, a resonant Auger decay or autoionization [68] is possible if the transition leads to a bound state above the first ionization energy distinguishing between the participator (if the excited electron is directly involved in the following decay) and the spectator decay as shown at the bottom in figure 3. A further occurring process is the fluorescence which is the spontaneous emission of a photon from an excited and relaxing atom or molecule, but this phenomena is not analyzed in this work. An introduction and closer inspection of fluorescence processes is given in [70].

ionization threshold unoccupied valence shell occupied valence shell core level (K-shell) direct ionization processes

valence shell ionization inner shell ionization

Figure 1: Valence shell and inner shell direct ionization process: The entire photon energy is transformed to the outgoing photoelectron.

ionization threshold unoccupied valence shell occupied valence shell core level (K-shell) 'shake' processes

shake-up (satellite) shake-off

Figure 2: Shake-up and shake-off ionization process: The photon energy is used to excite another electron.

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ionization threshold unoccupied valence shell occupied valence shell core level (K-shell) Auger processes

Auger decay

ionization threshold unoccupied valence shell occupied valence shell core level (K-shell) resonant Auger decay

spectator decay

Auger satellite satellite Auger decay

participator decay

Figure 3: Top: Auger ionization processes: An unoccupied inner shell vacancy is filled by an electron of higher potential energy resulting in several possible ionization and excitation processes. Bottom: Resonant Auger spectator and participator decay.

2.3 Photoionization cross section

This section gives a theoretical description of the photoionization process, starting with single photon ionization of atoms in dipole approximation without relativistic effects and describing the differential cross section, i.e. the photoelectron angular distribution.

2.3.1 Legendre polynomials

The Legendre polynomials (named after French mathematician Adrien-Marie Legendre) have many properties that make them an important tool in theoretical physics [11]. Here, Legendre

polynomials are used for the expansion of the cross section4 _{in the description of the}

photoion-ization process. They are produced by

Pn(x) = 1 2n_n! d dxn x2− 1n , (5)

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defined on the vector space of polynomial functions and are the solutions of the Legendre differential equation 1 − x2 d 2 dx2Pn(x) − 2x d dxPn(x) + n (n + 1) Pn(x) = 0, ∀n ∈ N0. (6)

Figure 4 shows the first seven polynomials in the interval [-1,1] which is the interesting region of

the Legendre polynomials. Pn has n roots in this interval. The polynomials form an orthogonal

system 1 ˆ −1 Pi(x) Pj(x) dx = 0, ∀i 6= j. (7) -1.0 -0.5 0.0 0.5 1.0 -1.0 -0.5 0.0 0.5 1.0 x P Hx L P6HxL P5HxL P4HxL P3HxL P2HxL P1HxL P0HxL

Figure 4: The first 7 Legendre polynomials in the interval [-1,1]. Hereafter, especially the P0,

P2 and P4 polynomials will be used for cross section expansion.

2.3.2 Partial and differential cross section

In photoionization, one or more electrons change its state by absorption of a photon. The quan-tum mechanical transition probabilities of the electrons from one state to another is determined

by Fermi’s golden rule [42]. Let H be a known and solved Hamiltonian, H1 the disturbance

operator where the states |E1i and |E2i defines the initial and final state to which the system

changes through the disturbance. One derives (approximately [90]) the transition probability per time unit

Γ12 =

2π

~ |hE2|H1| E1i|

2

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Generally, the partial cross section σ is a quantity to determine the probability at which a reaction between an incident particle that impinges on a target takes place. In gas phase experiments, the cross section of the atom or molecule corresponds to the surface on which the incident photon impinges punctiform. Based on the total area, the probability of a reaction can be calculated. The photoionization cross section is defined analogously with the final state in the continuum, i.e. the electron from the atom or molecule is released [6]

σ_1,2P hotoionisation = 4π 2_αa2 0 3 hν X 1,2,i |hE2|ri| E1i| 2 . (9)

In equation (9) only the dipole component of the light and non-relativistic conditions were

considered. α is the fine structure constant and a0 the Bohr radius. The spatial emission of

photoelectrons is a function of the emission angle in general. Here, the main interest is set on the angular distribution perpendicular to the linear polarized light propagation direction in the dipole plane. This relationship is obtained by differentiation of the cross section. It is customary to expand the differential cross section in terms of Legendre polynomials (partial wave expansion) dσ1,2 dΩ = σ1,2 4π ∞ X i=0 aiPi(cosθ) . (10)

θ is the angle between the polarization vector of the incoming photon and the electron emission

direction. For the coefficients ai of Pi Legendre polynomials follows from the consideration in

non-relativistic single ionization by one photon (l = 1) that ai = 0 for all i > 2. The emission

must be symmetrically along the axis of light polarization because there is no distinction of the

two spatial directions. This means that a1 must be equal to 0. Equation (10) yields to

dσ1,2

dΩ =

σ1,2

4π [1 + a2P2(cosθ)] . (11)

This was first described by C. N. Yang in 1948 [129].

For a known cross section value σ1,2, which generally is a function of the photon energy, the

angular distribution is merely depending of the coefficient a2 which is known as anisotropy

parameter β5_{. Because the differential cross section is positive definite (as meaningful value}

which can be interpreted), β is limited to the interval [-1,2]. Figure 5 shows the angular distribution for various β parameters. For a given l-state, the electron anisotropy parameter β is described by the Cooper Zare formula [27, 28]

β = l (l − 1) σ 2 l−1+ (l + 1) (l + 2) σl+12 − 6l (l + 1) σl+1σl−1cos (δl+1− δl−1) (2l + 1)lσ2 l−1+ (l + 1) σ2l+1 (12)

5_{For consistency with the following section, it should be noted here that by ’β parameter’ the ’β}₂_parameter’

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with the phase shift δl of the l-th partial wave and the partial cross section σl. A simple case is

the ionization of an s-orbital (l = 0) as mentioned, for which the Cooper-Zare-formula leads to

β = 2. Β2= 2.0 Β2= 1.8 Β2= 1.6 Β2= 1.4 Β2= 1.2 Β2= 1.0 Β2= 0.8 Β2= 0.6 Β2= 0.4 Β2= 0.2 Β2= 0.0 Β2=-0.2 Β2=-0.4 Β2=-0.6 Β2=-0.8 Β2=-1.0 0° 30° 330° 60° 300° 270° 90° 120° 240° 150° 210° 180° β β=2=2 ββ=1=1 β β=0=0 ββ=-1=-1

Figure 5: Top: Angular distribution for different values of the anisotropy parameter β in the allowed range [-1,2] in a polar plot. Bottom: 3D presentation for the values -1, 0, 1 and 2.

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2.4 Sequential two-photon double ionization in noble gases

The process of sequential double ionization is explained and the more complex angular distri-butions are introduced.

2.4.1 Two-step ionization

The sequential double ionization of an atom by two photons of a single light pulse of a FEL is performed in to steps: In the first step, an electron is released from the atom A by an incident

photon. Subsequently, an additional electron from the singly ionized atom A+ _{is also emitted}

by an absorbed photon and so produces a doubly ionized atom A2+ _{in the second step. This}

process is illustrated in figure 6. The detailed description of these two processes is performed in the following.

1

st

_photon

2

nd

_photon

1

stst

_step

₂

ndnd

_step

within

the same

FEL pulse

A

+

_A

+

_A

2+

neutral atom

single ionized

atom

double ionized

atom

1

1 photon per electron

photon per electron

FEL

Figure 6: Scheme of sequential double ionization of an atom by two photons of a single light pulse of a FEL.

2.4.2 Angular distributions of higher orders

The dipole approximation of the expansion in equation (10) of the cross section for single

pho-toionization containing only one angular distribution coefficient β2 is not sufficient to describe

sequential two-photon double ionization processes of rare gases. Higher order terms of the ex-pansion have to be included accounting for the additional angular momentum ∆l = 1 which is

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transferred from the second photon [48, 49]. This term for sequential two-photon ionization has a scaling factor given by the alignment of the ionic core left by the first ionization step. The alignment is caused by occupancy of the magnetic quantum number (non-symmetric population of states with different magnetic quantum number) [4, 48, 51, 54, 68]. If the electron of the first

step leaves an isotropic core, this alignment is zero and the corresponding P4 term should have

no effect on the photoelectron angular distribution. In order to illustrate the effect of higher

orders, figure 7 shows the influence of β4 parameter for a vanishing β2 term. The total cross

section is then described by the sum of both terms

dσ1,2

dΩ =

σ1,2

4π [1 + β2P2(cosθ) + β4P4(cosθ)] . (13)

Again, it should be noted that due to the demand of physical meaningful values of the differential cross section, which has to be greater than or equal to zero, only certain combinations of the

values for β2 and β4 are possible. The angular distributions derived from our measurements

(see chapter 5) for sequential double ionization of rare gases in the first and second steps are

characterized by positive β2 and negative β4 values in many cases. This combination is shown

in figure 8 for a constant β2 = 1.25 and varying β4.

Β4= 2.0 Β4= 1.8 Β4= 1.6 Β4= 1.4 Β4= 1.2 Β4= 1.0 Β4= 0.8 Β4= 0.6 Β4= 0.4 Β4= 0.2 Β4= 0.0 Β4=-0.2 Β4=-0.4 Β4=-0.6 Β4=-0.8 Β4=-1.0

0°

30°

330°

60°

300°

270°

90°

120°

240°

150°

210°

180°

Β

2 =0

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Β4=-1.0 Β4=-0.9 Β4=-0.8 Β4=-0.7 Β4=-0.6 Β4=-0.5 Β4=-0.4 Β4=-0.3 Β4=-0.2 Β4=-0.1 Β4= 0.0

0°

30°

330°

60°

300°

270°

90°

120°

240°

150°

210°

180°

Β

2 =1.25

Figure 8: Angular distribution for a constant β2 = 1.25 and various β4 values from 0 to -1.

A negative β4 decreases significantly the intensity for angles around 0◦ and 180◦, respectively.

These are typical angular distribution analyzed in chapter 5.

2.5 Theoretical description of the hydrogen molecule

The focus of this work is set on the oscillation of the relative cross section of molecular hydrogen which is shown schematically in figure 9.

The so-called Cohen-Fano oscillations of the molecular cross section are based on the analogy of a homonuclear diatomic molecule to the double slit experiment [23]. The indistinguishability of the two nuclei leads to interference effects in the molecular cross section

σmolecule ∝ (1 + sin[kR]/kR) . (14)

Where R is the distance between the two atoms and k is the wave number of the outgoing

electron. While in the original publication by Cohen and Fano [23] cross sections of N2 and O2

were used as a basis for interpretations, there were a number of photoionization experiments and theoretical investigations dealing with the hydrogen molecule’s cross section and its oscillations, as in [2, 14, 22, 21, 45, 88, 103, 113].

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æ

æ æ

0

2

4

6

8

10

-5

10

-4

0.001

0.01

0.1

1

10 k @R

_H2-1

_D

Cross

section

@arb

.

u.

D

molecular cross section atomic cross section

Relative cross section oscillation

2Σ

H

ΣH

2 P1=ÈΦ1È2 P2=ÈΦ2È2 P12=ÈΦ1+Φ2È2

R

H2

Figure 9: Schematic representation of the molecular cross section oscillation: The molecular

oscillation σH2 (magenta) caused by the interference of indistinguishable H-atoms, which is

analogous to the double slit experiment, takes place along the atomic cross section 2σH (cyan).

2.5.1 Hydrogen atom and molecule

The hydrogen atom represents a two-body problem and can be solved analytically. This is a simple and special feature in atomic physics (single electron system). The Hamiltonian of the hydrogen atom [44, 89, 90] is given by

H = − ~ 2 2me 4e− ~ 2 2mp 4p− e2 4πε0r , (15)

where me is the electron mass, mp is the mass of the core (proton), and r the distance between

the electron and nucleus. The Hamiltonian is thus composed of the kinetic energy of the nucleus and the electron as well as of the potential energy of the Coulomb interaction. The solution of the time-independent Schrödinger equation is a wave function in dependence of 6 coordinates. Using center of gravity and relative coordinates leads to a separation of Schrödinger’s equation: A center of mass motion equation whose solution is a plane wave (the movement of the whole atom) and an equation of motion for the relative motion obtained by relabeling a single particle equation with reduced mass in a spherically symmetric potential

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H = −~

2

2µ4 + Epot. (16)

Where µ is the reduced mass and Epot the potential energy of the electron in the Coulomb field

of the nucleus. The Hamiltonian is solved by transformation into spherical coordinates and a separation ansatz with respect to the radial component and the angular components presented in [107].

The atomic quantum numbers introduced in section 2.1 cannot be used in molecules since they

are not conserved quantities6_.

The H+

2 molecular ion has only one electron, but the Hamiltonian7 does not commute with the

square of the angular momentum operator l2 _[77]

H = p 2 A 2mA + p 2 B 2mB + p 2 e 2me − e 2 4πε0 1 rA + 1 rB − 1 R . (17)

When the z-axis coincides with the molecular axis, the Hamiltonian of the H+

2 commute with

lz. Analogous to atoms, the molecular states are denoted σ, π, δ, ...8 as a function of the value

of ml (=0, ±1, ±2, ...). This leads to a new molecular quantum number

λ = |ml| . (18)

The electronic states of inversion symmetric systems such as homonuclear diatomic molecules are eigenstates of the parity operator P , which commutes with the Hamiltonian, the so-called delocalized gerade and ungerade eigenstates [57, 60]. These states are coherent superpositions of both atomic sites of the molecule. The eigenfunctions of P are

Ψg = f (r) + f (−r) (19)

Ψu = f (r) − f (−r) (20)

where f is an arbitrary function from R3 _{in C. Analogously to atomic states, the main quantum}

number n is introduced collecting states of same symmetry [77], (see section 2.1) so that a state

such as 1σg is written (for n = 1, λ = 0, gerade).

6_{Not being a conserved quantity is also called a ’not good quantum number’.}

7_{with p}_n _{:= −i~∇}_n_.

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As a simple example one obtains a gerade and ungerade molecular state by combining the 1s

wave functions of two electrons as shown in figure 109

Ψ1σg = N [Ψ1s(ra) + Ψ1s(rb)] , (21)

Ψ1σu = N [Ψ1s(ra) − Ψ1s(rb)] . (22)

ra and rb are the position of the two ionic cores of atom A and B and N is the scaling factor

given by the overlap of these two states as presented in section 2.5.3. A further classification of a λ-state is given by the unified, i. e. R → 0, atomic quantum numbers as in figure 11 (for

instance 1sσg, 2pσu, etc.).

gerade case: ungerade case:

atom A atom B atom B atom A

Ψ

_1σ g

=

N [Ψ

1s

(

r

a

)+ Ψ

1s

(

r

b

)]

Ψ

1 σu

=

N [Ψ

1s

(

r

a

)−Ψ

1s

(

r

b

)]

∣Ψ

_1σ g

∣

2

∣Ψ

_1σ u

∣

2

Figure 10: Density distribution of two combined 1s wave functions for the gerade (left) and ungerade (right) case. The absolute squares are presented on top, while the wave functions are shown on bottom.

In a molecule with more than one electron, the total angular orbital momentum ~L is derived by the sum of the single angular momenta with

~ L = p L (L + 1)~. (23)

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Again, this leads to a new quantum number Λ, which characterize the angular momentum along the molecular axis

Lz = ±Λ~ (24)

with

Λ =X

i

λi (25)

and the notation Λ = 0 : Σ, Λ = 1 : Π, Λ = 2 : ∆ and so on. With the same proceed, one derives the total spin S of the molecule [46].

2.5.2 Potential energy curves

In a molecular ionization process, vibrational and rotational excitations has to be taken into

account. The hydrogen ion H+ _{creation processes of interest in this work are}

H2+ hν → H++ H, (26)

H2+ hν → H++ Hexcited, (27)

H2+ hν → H++ H+. (28)

Figure 11 shows a representation of the potential curves of the hydrogen molecule from Aoto et al. [2] and Walter et al. [124]. According to their potential energy, they start in succession with increasing photon energy. The compilation of individual curves to the convergence limit results

from the fact that the Q1 states [105] converge to the excited state 2pσu, the Q2 states [106] to

2pπu, the Q3 states to 2sσg and the Q4 states to 3pσu [43, 52]. 1sσg is the only H2+-state with a

local minimum, i. e. presenting a stable bound state. Since the velocity of the electrons is much faster compared to the nuclei caused by the greater mass of the proton, the Born-Oppenheimer approximation can be used which separates both motions [15]. In calculations, this means to derive the molecular state energies for a fixed internuclear distance R. The potential energy curves are determined by treating R as a parameter. The semiclassical Franck-Condon-principle [25, 26, 47, 86] is based on this very same thoughts; the electronic transitions are assumed to be instantaneously compared to the reaction of the protons, i.e. the electronic excitation has to be a vertical transition. The probabilities (Franck-Condon-factors) are proportional to the overlap integral of initial and final state. While close to the potential minimum a high similarity to a harmonic oscillator is observed, a modified potential (Morse potential) is used to describe increased vibrational excitations (more details in [98]).

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Figure 11: Left: Simple reduced nuclear potentials of H2, H+2 and H ++

2 with the equilibrium

positions of H2 (1.44 a.u.) and H+2 (2.00 a.u.). The figure is taken from [124]. Right: H2 potential

curves showing the allowed potential energy levels as a function of internuclear distance. The figure is taken from [2].

2.5.3 Molecular cross section

The wave properties of electrons and the indistinguishability of the nuclei in a hydrogen molecule leads to interference effects in the photoionization [23, 124]. Since this is a many-electron problem, approximations for the initial and final states of the electrons have to be made. For linearly polarized light, the cross section is given by

σ = 4πα

ω |Mf i(ω) |

2_. ₍₂₉₎

Where α is the fine structure constant, ω is the angular frequency of the photon and Mf i is the

transition matrix element10

Mf i(ω) = k1/2Ψf

eiKr∇r

Ψ_i . (30)

Here, k is the wave vector of the electron, K the wave vector of the photon and whose polarization vector. The simplest approximation for the final state is a plane wave which fails for calculating absolute values of the cross section. But for the description of interference effects

certainly has its place. This is presented as in Nagy et al. [88]11 _{in the following. This way is}

10_{of the transition from the initial Ψ}_i _{to the final Ψ}_f _state

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chosen, because of its simplicity and presentiveness. It is distinguished between velocity and length (index v or L) form of the dipole operator in the description of the matrix transition element

M_{f i}v = k1/2hΨf |∇r| Ψii (31)

and

M_{f i}L = −k1/2ω hΨf|r| Ψii . (32)

The 1s orbitals Ψ0 with the nuclear charge of the hydrogen ion ρ is given by

Ψ0(r) =

r

ρ3

π e

−ρr_. ₍₃₃₎

Then the initial (see section 2.5.1) and final states of the two nuclei at ra and rb are described

as Ψi = 1 p2 (1 + S)[Ψ0(ra) + Ψ0(rb)] , (34) Ψf(k, r) = 1 (2π)3/2e ikr_. ₍₃₅₎

Here S represents the overlap integral of the two 1s orbitals. For a fixed orientation the following cross section is obtained by using the velocity form [23, 88]

σv(D) =

128ρ5α

ωπ (1 + S)

(ek)2k cos2(kD/2)

(ρ2_{+ k}2₎4 , (36)

and in length form

σL(D) = 128αωρ5_k π (1 + S) sin(kD/2)D 2 (ρ2_{+ k}2₎2 + 4 cos(kD/2)k (ρ2_{+ k}2₎3 2 . (37)

Integration over all possible orientations of the molecular axis vector D to the polarization with

cos(θ) = k yields to σv = 64ρ5_α ω (1 + S) cos2_θ k3 (ρ2_{+ k}2₎4 1 + sin(kD) kD (38)

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and σL = 1024ωρ5_α (1 + S) (ρ2_{+ k}2₎6k 3_cos2_θ 1 +sin(kD) kD + 16ωρ 5_αk (1 + S) (ρ2_{+ k}2₎4 D2 3 − D k cos 2_θ sin(kD) + 1 k2(1 − 3 cos θ) cos(kD) −sin(kD) kD + 256ωρ 5_αk (1 + S) (ρ2_{+ k}2₎5 cos 2_θ sin(kD) kD − cos(kD) . (39)

The result of the integrated velocity form of Nagy et al. [88] depicts the results of Cohen and Fano [23] and provides an angular distribution as in atoms. Equations (36) and (38) show a 90° phase shift between oriented target and the random sample [113, 114]. The length form differs for both, the oriented and randomly distributed target, from the velocity form and the resulting angular distributions are more complicated. At low photon energies, the electron wave vector

is close to 0, which means sin(kD) ≈ 0 and cos(kD) ≈ 1, resulting in the same behavior of σv

and σL.

Figure 12: Angular distributions for H2 of σv (red) and σL(blue) after integration over molecular

axis orientation at photon energies of 25 eV, 125 eV, 250 eV and 500 eV. Whereas the velocity

form produces a constant (β2 = 2)-distribution, the result of the length form changes its angular

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In order to illustrate this difference, figure 12 presents the angular distributions integrated over all possible orientations of the molecule for different photon energies. Equations (36) and (37) describe the cross section as a function of the orientation of the molecule and the electric field vector of the linearly polarized photon. Figure 13 shows the resulting photoelectron angular distributions for parallelism between the molecular axis and the plane of polarization and an orientation of the two at 45° to each other for different photon energies. As an alternative description ’molecular frame photoelectron angular distribution’ (MFPAD) and ’electron frame photoion angular distribution’ (EFPAD) [81, 82] as a function of three angles can be used. This is shortly presented in appendix 8.1.

Figure 13: Angular distribution in velocity form with the molecular axis parallel (magenta) and in 45° (blue) to the polarization axis at the photon energies of 25 eV, 125 eV, 250 eV and 500 eV. This figure is also based on [88].

2.5.4 Relative cross section oscillation

Further studies with two-center wave functions as an assumption for the final state of Borbély and Nagy [14] have shown that the results are very sensitive to the choice of the final state.

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1680 terms12_{, for example, Fojon et al. [45] have attained an improved agreement with the}

measured data of Samson and Haddad [103]. This was achieved for the absolute cross section as well as in the interference effects, which is presented in figure 14.

As an interesting result for this thesis, Nagy et al. [88] have shown that by using the whole provided data by Samson and Haddad [103], the cross section ratio is approximately at a constant value in the photon energy region ≈100 - 300 eV (see figure 15).

Figure 14: Calculated cross sections of [14, 45] are shown and compared to experimental results of [103]. This figure is taken from [14]. The more complicated functions of Fojon et al. [45] describe the data better than the simple solution of [14], especially in the region k→0.

12_{Basis of products of B-splines and spherical harmonics; 280 B-spline functions per angular momentum with}

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Figure 15: The cross section oscillation comparison of Fojon et al. [45] (left) with experimental data [103] taken from [22] just covers low photon energies (18 - 124 eV). By using all data of Samson and Haddad [103] (up to 300 eV), Nagy et al. [88] (right) showed that in this region (100 - 300 eV) the cross section ratio stays almost at a constant value. The figures are taken from [45, 88].

2.5.5 βm parameter

The β-formula (11) presented in section 2.3.2 (figure 5) can also be used to describe the photoion angular distribution of the hydrogen molecule [33, 123] which gives the angular distribution of the molecular axis to the light polarization vector [54] when the molecule is photoionized. This

parameter is called βm then, where m labels molecular fragment, and is given in axial recoil

conditions [33, 74, 131] by βm = 2 (D2 Σ− D2Π) D2 Σ+ 2D2Π (40)

with the two transition amplitudes DΣ and DΠ from the ground state [33, 74]. This leads to

βm = 2for pure Σ → Σ transitions and βm = −1for pure Σ → Π transitions. The βm parameter

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3 Experiment

A quick overview of the used photon sources is given. The experimental setups used in our mea-surements are shown. Details of the ion and electron time-of-flight spectrometers are presented and the properties of the detectors are discussed.

3.1 Photon sources

Since the scientific objective is to study photoionization processes as a function of photon energy, it is unquestionably necessary to have a tunable light source over a large photon energy range. In this work, two different photon radiation sources, synchrotron and free electron laser radiation, were used for photoionization of atoms and molecules. The experiments were performed at

BESSY II (Berliner Elektronen-Speicherring Gesellschaft für Synchrotronstrahlung)13_{, PETRA}

III (Positron-Elektron-Tandem-Ring-Anlage) P04 beamline14 _{and FLASH}

(Freie-Elektronen-Laser in Hamburg)15_{. The principle of operation of these systems are presented in the following.}

In synchrotron radiation facilities, electrons or positrons are accelerated on a circular horizon-tal path (240 m circumference at BESSY II, 2304 m at PETRA III). Synchrotron radiation is generated through accelerated and relativistic charged particles, such as electrons. This effect is used in undulators that are utilized in third generation synchrotron radiation sources for creating constructive interference of the generated beams [72, 91]. These insertion devices are several meters long and consist of periodically alternating magnetic pairs as shown schematically on the left in figure 18. The electrons are accelerated to a sinusoidal route and emit photons

independently. At high kinetic energies, that is at velocities v with v2

/c2 . 1, the preferential

photon emission direction of the electrons (dipole transformed into the laboratory system) is their average direction of motion [110]. The undulator parameter K is introduced

K = eBλu

2πmec

with c the speed of light, e the elementary charge, me the electron mass, the B magnetic field

strength and λu undulator period. 1/K represents the amount of the constructive interference

occurring in the undulator. For an undulator K is smaller than 1 and with N pairs of magnets

the photon intensity so produced is proportional to N2_{. For K > 1 the device is called Wiggler}

and the electrons are accelerated strongly so that no constructive interference occurs. The resulting photon intensity is proportional to N.

13_{http://www.helmholtz-berlin.de/zentrum/historie/bessy/ and}

http://www.helmholtz-berlin.de/zentrum/grossgeraete/elektronenspeicherring/index_en.html

14_{http://petra3.desy.de/index_ger.html and}

http://photon-science.desy.de/facilities/petra_iii/beamlines/p04_xuv_beamline/index_eng.html

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While several of these undulators are installed in storage rings of third generation synchrotron light sources, the length of an undulator can be expanded up to more than a few 10 m (or even 100 m) at a linear accelerator. This condition is used in a free electron laser (FEL) [1, 32, 40, 61, 96] such as FLASH [117].

Starting in a superconducting linear accelerator, electrons are accelerated (up to 1.25 GeV) and arranged in packages. These so-called electron bunches, flying close to speed of light, are than compressed and lead into the 27 m long linear undulator (27.3 mm period length) with a

12 mm gap in which the self-amplified spontaneous emission (SASE-principle)16 _{occurs. As in}

synchrotron radiation sources the electrons start to emit photons spontaneously. These photons, moving on a linear line in the center of the undulator at speed of light, then start to interact with the electron bunches, which are on their slalom course. This leads to an acceleration (electrons out of phase) or deceleration (in phase) of the electrons. The so produced electron microbunches are synchronized and radiate coherently. The microbunching-effect with exactly one photon wavelength as longitudinal position difference of a microbunch to its neighbor is essential for the SASE-principle resulting in extremely high intense light pulses with a very short duration.

At our experiments 30-50 bunches formed a so-called ’train’ with a train repetition rate of 5 Hz and 10 Hz later on and a bunch repetition rate of 1 MHz (see figure 17). Compared to

synchrotron radiation the intensity is ≈ 106 _{times higher while the pulse duration is about 10}3

shorter. The FLASH experiments [18, 19, 20] were performed in 2006, 2007 and 2010. The used photon energies were from 38 eV to 91 eV for the sequential double ionization of rare gases. However, the possible wavelength range in 2012 was 4.2 - 45 nm (27.5 -295.2 eV). A comparison

of some beam properties of FLASH17_{and the PETRA III P04 XUV beamline [121] is presented}

in the table 1. It should be added that the photon energy range of P04 are ’at the moment’ status. The beamline is designed for 250 - 3000 eV and all parameters are given for a ring current of 100 mA. Additionally, at our P04 beamtimes only circularly polarized light was available. However, in the very future also linear polarization will be provided.

Our experiments to investigate the hydrogen molecule’s photoionization processes were per-formed at beamlines

• BESSY II: U125/2: 28 - 190 eV (2005), [95] • BESSY II: UE56/1-PGM-b: 100 - 650 eV (2010) • BESSY II: UE56/2 PGM1: 200 - 600 eV (2010-2013) • PETRA III: P04: 265 - 1200 eV (2013), details in [121].

16_{http://photon-science.desy.de/facilities/flash/machine/how_it_works/}

sase_self_amplified_spontaneous_emission/index_eng.html

17_{Great FLASH overview and introduction can be found in:}

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Beam properties PETRA III P04 beamline FLASH

undulator length 4.9 m 27 m

photon energy 0.56-4.6 nm 4.2-45 nm

(265-2200eV) (27.5-295.2eV)

photon flux >1012_ph/s ₁₀13_-1016_ph/s

single pulse energy 0.6-6 pJ 10-500 µJ

photons per pulse 2 · 106 1011-1013

pulse duration 90-100 ps 50-200 fs

focal spot size 10 µm 30 µm

brilliance

ph/s/mm2_/mrad2_/0.1 1020 1029-1031

Table 1: Beam properties comparison of a synchrotron and FEL radiation source (PETRA III P04 vs. FLASH).

For our experiments, we applied for beamtimes in the reduced bunch operation mode periods at BESSY II and PETRA III for which the time intervals between subsequent bunches are larger

than in the normal operation mode (BESSY II single bunch mode18_{: 4t = 800 ns, PETRA III}

40-bunches mode: 4t = 192 ns). This is especially suited for ToF spectroscopy because of the enhanced time-of-flight range without superimposition of spectra of the following bunch.

3.2 Experimental setups

The two experimental setups presented in the following are connected to the beamlines’ end-stations of the described light sources via a differential pumping section because of the pressure

difference between the beamline (ultra high vacuum19_{) and the experimental chamber (up to}

10−5mbar).

3.2.1 Spherical chamber

For the photoelectron angular distribution analysis, a spherical chamber with several electron

time-of-flight spectrometers20_{, short eTOF, is used, which was designed by AG Becker (Jens}

Viefhaus, Markus Braune, Burkhard Langer, Henrik Haak, Georg Prümper, Uwe Becker) and produced in 2000.

18_{http://www.helmholtz-berlin.de/forschung/grossgeraete/betrieb-beschleuniger/betriebsmodi_en.html}

19_{that means ∼ 10}−9_{mbar to 10}−10_mbar.

20_{up to 19 independently working eTOFs in the angles: 0°, 23.4°, 34.5°, 45.6°, 56.7°, 67.8°, 90°, 101.1°, 112.2°,}

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Trigger

PA

Gas Monitor

He, Ne, Ar, Kr

dt = 125/250/500 ps up to 50 bunches D ig iti ze r Computer

set of electron time of flight spectrometers

FLASH

Figure 16: Spherical vacuum chamber: The incoming FLASH light ionizes the noble gas cen-trally introduced by a gas needle in an arrangement with several independently working electron time-of-flight spectrometers perpendicular to the beam. Traces of amplified mirco-channel plate detector signals are recorded with a digitizer system. The modified figure is taken from [19]. This setup is presented in figure 16 and a sketch of the eTOF design is given in appendix 8.3 (figure 67). The eTOFs are arranged perpendicular to the beam in the dipole plane of the FEL radiation and the assignment of angles to the detectors are according to the horizontal polarization of the FEL (0° and 180° in horizontal direction, see inset of figure 16). In order to align all spectrometers to the center of the spherical chamber, which is also the focal spot of the radiation, the entrace apertures of the flight tubes are mounted in a internal ring holder. The nominal drift distance from the center to the detector is 309 mm, composed by 28 mm from the interaction point to the detector opening (opening angle 5°-6°on disk) and 281 mm of flight tube (see figure 67), which leads to time-of-flights of 100 ns up to 300 ns to a stack of three MCPs (micro-channel plates). The noble gas (helium, neon, argon or krypton) is centrally introduced by a gas needle.

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Figure 17: 12 traces of consecutive FLASH pulse trains consisting of 30 bunches each, with a time gap of 1µs between bunches. This series demonstrates the intensity fluctuations due to the statistical nature of the SASE process.

The MCP detector charge pulses created by the photoelectrons are amplified and recorded by a digitizer consisting of a multi-channel card system (Acqiris-system with time resolution of 125 ps, 250 ps or 500 ps). The flight tube is divided into 3 retarding sections, which can be used to decelerate the electrons for improving the time resolution. The retarding effects and photoelectron time-of-flight to kinetic energy conversion is discussed in section 3.3. By comparing the detector intensities one can derive the electron angular distributions, which is shown in the inset in figure 16. The blue lines represents the angles of the eTOFs.

The FLASH, BESSY II and PETRA III operation crew provide a bunch marker to the users. This signal is correlated with the generated photon pulse and used as a start signal for time-of-flight analysis (in the digitizer or TDC (see the following section), respectively). In addition, the gas monitor device (GMD) [117, 118] delivers a signal proportional to the single FLASH pulse energy, which was recorded as well. More details about the GMD signal and its analysis can be found in chapter 4.

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3.2.2 Ion time-of-flight spectrometer and electron-ion-coincidence measurement The experimental setup for the investigations of coherence properties of the hydrogen molecule consisted of an ion time-of-flight detector with the option of a coincidence mode with an electron time-of-flight spectrometer, which is shown in figure 18. The purpose of this setup is the analysis

of the resulting H+_{- and H}+

2-fragment ions (in addition to the He+- and He++-ions when using

a gas mixture) and to detect and determine the orientation of the molecule to the polarization vector of the photons when ionization takes place. This setup, called ARFMADS (Angular Resolved Fixed Molecule Angular Distribution Spectrometer) [7], and variants of it was designed by F. Heiser, O. Geßner [53], Georg Prümper and Jens Viefhaus and used by Oliver Gessner [53], Oliver Kugeler [76], Daniel Rolles [97, 98], Sanja Korica [73] and Axel Reinköster [95], for example. It is based on the Wiley-McLaren-setup [126] and mainly used as an ion time-of-flight spectrometer in this work (without the position sensitivity, this was only used in the data of 2005 [95]).

The photoelectron emerged in the interaction center, which flies in the direction of the eTOF, puts back the route of the drift tube in the time-of-flight spectrometer where it meets the MCPs. The generated charge pulse is amplified and discriminated by a CFD (Constant Frac-tion Discriminator). This particular discriminator is necessary since the pulses of the MCPs significantly differ in amplitude, but whose form remains approximately constant. In a electron-ion-coincidence mode, the detected electron triggers the pusher and extractor electrodes (see figure 18) of the ion detector by the next bunch. As a result, the generated ions are accelerated in the direction of the ion MCP detector. The ion drift tube length is about 27 cm (distance from interaction center to MCPs).

Instead of using the electron signals as a trigger, it is also possible to apply a static pusher and extractor voltage or to trigger by each 10th bunch marker, for example. In any mode, the

TDC21 _{(time-to-digital converter with an ion time-of-flight resolution of 120 ps) uses the bunch}

marker as the start signal and the CFD as a stop signal. These signals are converted and saved

by the software elecion22 _{in the form of event files. In these files an event is considered as the}

ordered tuple

{number of ions, number of electrons, ion time-of-flight, electron time-of-flight, ion position23_},

while all detected events are written one after the other. Coincidence maps such as electron-ion-coincidences (as in the right on top panel in figure 18), electron and ion time-of-flight spectra can be created from this data. At PETRA III P04 facilities a quite similar setup with a shorter ion drift tube and a magnetic eTOF bottle of Sascha Deinert et al. was used which is described in detail in [34].

21_{of the company GPTA with more details on: www.gpta.de.}

22_{of Kornel Wieliczek}

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BESSY II, U125/2:

28eV - 190eV

BESSY II, UE56/1-PGM-b:

100eV – 650eV

BESSY II, UE56/2-PGM1:

200eV - 600eV TDC Computer electron spectrometer electron spectrometer ion detector ion detector coincidence-map pusher bunch marker constant field H e- -+ +

time of flight spectra

amplifier, CFD MCPs MCPs MCPs amplifier, CFD dr ift t ub e dr ift t ub e ion tof ele ctr o n to f PETRA III, P04 265eV - 1200eV extractor

Figure 18: The ion spectroscopy is performed in several variants. In a electron-ion-coincidence mode the pusher and extractor electrodes of the ion spectrometer are triggered by a detected electron in the eTOF, thus accelerating the ions into the ion detector. However, this triggering can also be performed by the bunch marker or a constant voltage is applied.

3.3 Detector properties

3.3.1 Time-of-flight to kinetic energy conversion

In this section, it is shown how the electron TEV (Time-of-flight to Electron Volts) conversion is obtained from the time-of-flight spectra. For free space in non-relativistic conditions, the conversion is described by a quadratic relation between the time-of-flight of the electrons and their kinetic energy

Ekin = 1 2mev 2 = 1 2me s t 2 = 1 2mes 2 1 t 2 . (41)

Here, me is the electron mass and s the traveled distance corresponding to the time-of-flight t.

By applying retarding potentials to the drift tube, the electrons are decelerated and the arrival time difference between electrons of different kinetic energy is increased. This way the kinetic energy resolution is enhanced. However, the relation (equation (41)) becomes more complicated. In all cases, a calibration gas is used, for example neon as shown in figure 19, to record a series

of spectra for different photon energies Ephoton. At first, the prompt signal from these spectra

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fluorescence photons on the MCPs. The distance from the center of interaction to the MCP is known and the photons move at the speed of light. Based on the prompt channel number the

time zero t0 can be characterized. Since the binding energy Ebin of the transition(s) [6], their

corresponding peak(s) in the calibration gas spectrum and the photon energy are known, the photoelectron kinetic energy is given by equation (1) (see section 2.1). With this method one obtains a series of data points that assign channel numbers to kinetic energies.

As described above, a square approach for the conversion is enough at no or low retardation

Ekin(t) =

c2

(t − t0)2

. (42)

c2 can be treated as a fit parameter determined from the resulting data points. At higher

retarding voltages, c2 can not be chosen so that the fit curve is a satisfactory description of the

measured data. Since the precise potential curve inside the spectrometers is unknown, one does not attempt to obtain a conversion formula by solving the corresponding differential equation. In each case, equation (42) for the field-free space is interpreted as a second order polynomial

of 1

t−t0 in the measured energy range. It follows the general ansatz

Ekin(t) = n X i=0 ci 1 (t − t0) i . (43)

This represents an (inverse) expansion at t0 of the order of n. The time-of-flight electron

channel t here is always greater than t0. Already the fifth order produces good results for

all used retarding voltages (-400 V to 100 V). Figure 20 shows this for a retarding of -200 eV at the neon 2s and 2p line positions. The energies were chosen so that the resulting time-of-flight channels cover the region of interest. The TEV function only has to fit to the data points within this interval, which is the case for each conversion. The conversion function can be observed ’directly’ in a density plot of TOF channel vs. photon energy as in figure 19. Subtracting the binding energy from the photon energy moves all data points to one curve (see

figure 20). The asymptotic behavior towards 200 eV in figure 2024 _{is due to the fact that the}

retarding voltage sets the lower limit of the electron kinetic energy necessary to reach the MCP detector. Because the TEV conversion represents an important property of the detectors, figure 21 presents the TEV for several detectors at a retarding voltage of -100 V. All eTOFs show a very similar behavior. The small deviations must be explained mainly by not exact the identical fields inside the flight tubes for all eTOFs, adjustment discrepancies and statistical effects in the spectra.

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Figure 19: Top: 37 neoncalibration timeofflight spectra for different photon energies (148 -270 eV) with -100 V retarding voltage. Increasing photon energy leads to a higher kinetic energy and a lower time-of-flight. The signal at ≈ 3080 channels is the prompt peak. Bottom: Photon energy vs. electron time-of-flight density plot: The time to energy conversion function can be

derived by the 2p (lowest time-of-flight, Ebin = 21.57 eV) and 2s (Ebin = 48.48 eV) line positions

(39)

Figure 20: Time to energy conversion fit function of the Ne 2s and Ne 2p line positions for -200 V retarding voltage. æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æææ æ æ æ æ æææ æ ææ ææ_æ æ æææææææææææææææ æ æ æ æ æ æ æ æ ææ æ _æ æ æ ææ æ æ æ æ æ æ æ æ à à à à à à à à à à à à à à à à à à à à à àà à à à à ààà à àà àà_à à ààààààààààààààà à à à à à à à à àà à _à à à à à à à à à ààà à ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ì ììì ì ì ì ì ììì ì ì ì ìììììì_ì ì ì ìììììììììì ì ì ì ì ì ì ì ì ì ì ì ì ì ì _ì_ì _ì ì ì_ì ì ìì ì ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò ò òò_ò ò ò ò ò òòò ò ò ò òòòòò_ò òòòòòòòòòòòòò ò ò ò ò ò ò ò ò ò ò ò ò ò ò _ò òòòòòòòò ò ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ô ôô_ô ô ô ô ô ôôô ô ô ô ôôôôô_ô ôôôôôôôôôôôôô ô ô ô ô ô ô ô ô ô ô ô ô ô ô _ô ôôôôôôôôô ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç ç çç_ç ç ç ç ç ççç ç ç ç ççççç_ç ççççççççççççç ç ç ç ç ç ç ç ç ç ç ç ç ç ç _ç_ç _ç ç ç ç ç ç ç ç á á á á á á á á á á á á á á á á á á á á á á_á á á á á ááá á á á ááááá_á ááááááááááááá á á á á á á á á á á á á á á _á _á á á áá á á á á í í í í í í í í í í í í í í í í í í í í íí_í í íí í ííí í í í ííííí_í ííííííííííííí í í í í í í í í í í í í í í íííí_íí íííí ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó ó óóó ó ó ó ó óóó ó ó ó óóóóóó_ó ó óóóóóóóóóóó ó ó ó ó ó ó ó ó ó ó ó ó ó ó óó ó _ó_óóó ó _ó ó õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õ õõ õ õ õ õ õ õõ õ õ õ õõõõõõ_õ õõõõõõõõõõõõ õ õ õ õ õ õ õ õ õ õ õ õ õ õõõõ õ õ õ õ_{õõ õ} æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æææ æ æ æ æ æææ æ æ æ ææ_æ æ æææææææææææææææ æ æ æ æ æ æ æ æ ææ æ _æ æ ææ _{æ æ}_æ æ æ æ æææ à à à à à à à à à à à à à à à à à à à à à àà à à à à à àà à à à àà_à à ààààààààààààààà à à à à à à à à à à à _à à à àà à à à àà à à à 4000 5000 6000 7000 8000 9000 10 000 100 150 200 250 TOF channel Kinetic energy @eV D à 191.1° æ 336.6° õ _168.9° ó 292.2° í 134.4° á 123.3° ç 281.1° ô 101.1° ò 67.8° ì 45.6° à 34.5° æ 0°

Figure 21: Kinetic energy as a function of the electron time-of-flight for several electron spec-trometers with -100 V retarding voltage.

Coherence effects of diatomic homonuclear molecules and sequential two-photon processes of noble gases in the photoionization