# Publications by

Robert Alex Müller

## 2019

**Modification of multipole transitions by twisted light**

**100**, 043416 (2019)

**Abstract:** A theoretical analysis is presented for the excitation of single many-electron atoms and ions by twisted (or vortex) light. Special emphasis is put on excitations that can proceed via several electric and magnetic multipole channels. We argue that the relative strength of these multipoles is very sensitive to the topological charge and kinematic parameters of the incident light and can be strongly modified with respect to the plane-wave case. Most remarkably, the modification of multipole transitions by twisted radiation can be described by means of a geometrical factor. This factor is independent of the shell structure of a particular target atom and just reflects the properties of the light beam as well as the position of an atom with respect to the vortex axis. An analytical expression for the geometrical factor is derived for Bessel photons and for a realistic experimental situation in which the position of an atom is not well determined. To illustrate the use of the geometrical factor for the analysis of (future) measurements, detailed calculations are presented for the presented for the 3s 3p 3P1 -> 3s 3p 1P1 excitation in neutral Mg.

**Excitation of the ²²⁹Th nucleus via a two-photon electronic transition**

**99**, 042517 (2019)

**Abstract:** We investigate the process of nuclear excitation via a two-photon electron transition (NETP) for the case of the doubly charged thorium. The theory of the NETP process was originally devised for heavy-helium-like ions. In this work, we study this process in the nuclear clock isotope 229Th in the 2+ charge state. For this purpose we employ a combination of configuration interaction and many-body perturbation theory to calculate the probability of NETP in resonance approximation. The experimental scenario we propose for the excitation of the low-lying isomeric state in 229Th is a circular process starting with a two-step pumping stage followed by NETP. The ideal intermediate steps in this process depends on the supposed energy ℏωN of the nuclear isomeric state. For each of these energies, the best initial state for NETP is calculated. Special focus is put on the most recent experimental results for ℏωN.

**Investigation of Atomic Nuclei via Electronic Processes**

Technische Universität Carolo-Wilhelmina zu Braunschweig, Fakultät für Elektrotechnik, Informationstechnik, Physik (2019)

**Abstract:** In atomic physics, nuclei are often described as a point-like charges with an inﬁnite mass that binds the electrons. With more and more precise experimental techniques, however, this approximation is no longer suﬃcient and it is necessary to develop a better theoretical understanding of the ways atomic nuclei interact with the electron shell. We do observe for example small shifts in the lines of spectra of diﬀerent isotopes of the same atomic species. In this thesis, we present calculations for these isotope shifts and use them to derive the diﬀerence between the nuclear charge radii of two thorium isotopes, ²³²Th and ²²⁹Th as well as ²²⁹Th and the isomeric state ²²⁹mTh. These results are of particular interest for the development of a future

nuclear clock and coherent high-energy light sources. Moreover, we discuss precise isotope shift calculations for singly charged barium and compare them with a recent experiment. We motivate the relevance of such studies for the search for physics beyond the Standard Model.

Spectral lines, however, do not only shift but also split due to the non-point-like nature of atomic nuclei. From the spectroscopy of this hyperﬁne splitting, it is possible to extract the multipole moments of the nuclear electromagnetic ﬁeld. As a part of this thesis, we present the ﬁrst value of the nuclear magnetic dipole moment of the ²²⁹mTh nuclear isomer that does not rely on previous calculations or measurements.

Having extracted several important properties of the ²²⁹Th nucleus and the isomer ²²⁹mTh using atomic theory we invert our view in the second part of this thesis. Namely, we want to use processes in the electron shell to populate the ²²⁹mTh isomeric state. Preparatory to our calculations for the actual excitation of the isomer, we discuss the atomic structure of thorium. Of particular experimental interest is the level structure of singly charged thorium. In a recent study, we show the results of atomic structure calculations that help to interpret measured thorium spectra and can be used to estimate the probability of a nuclear excitation via the electron shell in this system. A deeper and more accurate discussion is performed for the comparably simple triply charged thorium ion. This study helps to test the various approximations necessary to discuss systems with a more complicated electronic structure. Bringing everything together the ﬁnal publication presented in this thesis proposes an experimental setup to excite the ²²⁹Th nucleus in a controlled way depending on the yet to be found energy of the nuclear isomeric state. This method is currently applied in an experiment at the German National Metrology Institute.

## 2018

**Elastic scattering of twisted electrons by diatomic molecules**

**98**, 042701 (2018)

**Abstract:** The elastic scattering of twisted electrons by diatomic molecules is studied within the framework of the nonrelativistic first Born approximation. In this process, the coherent interaction of incident electrons with two molecular centers may cause interference patterns in the angular distributions of outgoing particles. We investigate how this Young-type interference is influenced by the complex internal structure of twisted beams. In particular, we show that the corkscrewlike phase front and the inhomogeneous intensity profile of the incident beam can strongly modify the angular distribution of electrons, scattered off a single well-localized molecule. For the collision with a macroscopic target, composed of randomly distributed but aligned molecules, the angular-differential cross section may reveal valuable information about the transverse and longitudinal momenta of twisted states. To illustrate the difference between the scattering of twisted and plane-wave beams for both single-molecule and macroscopic-target scenarios, detailed calculations have been performed for a H2 target.

**Hyperfine interaction with the ²²⁹Th nucleus and its low-lying isomeric state**

**98**, 020503 (2018)

**Abstract:** The thorium nucleus with a mass number A=229 has attracted much interest because its extremely low-lying first excited isomeric state at about 8 eV opens the possibility for the development of a nuclear clock. Both the energy of this state as well as the nuclear magnetic dipole and electric quadrupole moment of the 229mTh isomer are subjects of intense research. The latter can be determined by investigating the hyperfine structure of thorium atoms or ions. Due to its electronic structure and the long lifetime of the nuclear isomeric state, Th2+ is especially suitable for such kinds of studies. In this Rapid Communication, we present a combined experimental and theoretical investigation of the hyperfine structure of the 229Th^(2+) ion in the nuclear ground state, where a good agreement between theory and experiment is found. For the nuclear excited state we use our calculations in combination with recent measurements to obtain the nuclear dipole moment of the isomeric state μ_iso=−0.35 μN, which is in contradiction to the theoretically predicted value of μ_iso=−0.076 μN.

## 2017

**Theoretical analysis of the electron bridge process in 229Th3+**

**408**, 84 (2017)

**Abstract:** We investigate the deexcitation of the 229Th nucleus via the excitation of an electron. Detailed calculations are performed for the enhancement of the nuclear decay width due to the so called electron bridge (EB) compared to the direct photoemission from the nucleus. The results are obtained for triply ionized thorium by using a B-spline pseudo basis approach to solve the Dirac equation for a local xα potential. This approach allows for an approximation of the full electron propagator including the positive and negative continuum. We show that the contribution of continua slightly increases the enhancement compared to a propagator calculated by a direct summation over bound states. Moreover we put special emphasis on the interference between the direct and exchange Feynman diagrams that can have a strong influence on the enhancement.

## 2016

**Two-color above-threshold ionization of atoms and ions in XUV Bessel beams and intense laser light**

**94**, 053420 (2016)

**Abstract:** The two-color above-threshold ionization (ATI) of atoms and ions is investigated for a vortex Bessel beam in the presence of a strong near-infrared (NIR) light field. While the photoionization is caused by the photons from the weak but extreme ultraviolet (XUV) vortex Bessel beam, the energy and angular distribution of the photoelectrons and their sideband structure are affected by the plane-wave NIR field. We here explore the energy spectra and angular emission of the photoelectrons in such two-color fields as a function of the size and location of the target atoms with regard to the beam axis. In addition, analog to the circular dichroism in typical two-color ATI experiments with circularly polarized light, we define and discuss seven different dichroism signals for such vortex Bessel beams that arise from the various combinations of the orbital and spin angular momenta of the two light fields. For localized targets, it is found that these dichroism signals strongly depend on the size and position of the atoms relative to the beam. For macroscopically extended targets, in contrast, three of these dichroism signals tend to zero, while the other four just coincide with the standard circular dichroism, similar as for Bessel beams with a small opening angle. Detailed computations of the dichroism are performed and discussed for the 4s valence-shell photoionization of Ca+ ions.

**Photoionization of neutral atoms by X waves carrying orbital angular momentum**

**94**, 041402 (2016)

**Abstract:** In contrast to plane waves, twisted or vortex beams have a complex spatial structure. Both their intensity and energy flow vary within the wave front. Beyond that, polychromatic vortex beams, such as X waves, have a spatially dependent energy distribution. We propose a method to measure this (local) energy spectrum. The method is based on the measurement of the energy distribution of photoelectrons from alkali-metal atoms. On the basis of our fully relativistic calculations, we argue that even ensembles of atoms can be used to probe the local energy spectrum of short twisted pulses.

**Sum rules for the polarization correlations in photoionization and bremsstrahlung**

**93**, 053421 (2016)

**Abstract:** The polarization correlations in doubly differential cross sections are investigated for photoionization and ordinary bremsstrahlung. These correlations describe the polarization transfer between incident light and ejected photoelectrons as well as between an incoming electron beam and bremsstrahlung light, respectively. They are characterized by a set of seven real parameters Cij. We show that the squares of these parameters are connected by simple “sum rules.” These sum rules can be applied for both one-electron systems and also for atoms, if the latter are described within the independent particle approximation. In particular, they are exact in their simplest form (i) for the photoionization of K-, LI,II-, and MI,II-atomic shells, as well as (ii) for bremsstrahlung in which the electron is scattered into s1/2 or p1/2 states, as in the tip (bremsstrahlung) region. Detailed calculations are performed to verify the derived identities and to discuss their possible applications for the analysis of modern photoionization and bremsstrahlung experiments. In particular, we argue that the sum rules may help to determine the entire set of (significant) polarization correlations in the case when not all Cij are available for experimental observation.

## 2015

**Effect of bound-state dressing in laser-assisted radiative recombination**

**92**, 053426 (2015)

**Abstract:** We present a theoretical study on the recombination of a free electron into the ground state of a hydrogenlike ion in the presence of an external laser field. Emphasis is placed on the effects caused by the laser dressing of the residual ionic bound state. To investigate how this dressing affects the total and angle-differential cross section of laser-assisted radiative recombination (LARR) we apply first-order perturbation theory and the separable Coulomb-Volkov continuum ansatz. Using this approach, detailed calculations are performed for low-Z hydrogenlike ions and laser intensities in the range from I_L=10^12 to 10^13W/cm2. It is seen that the total cross section as a function of the laser intensity is remarkably affected by the bound-state dressing. Moreover, the laser dressing becomes manifest as asymmetries in the angular distribution and the (energy) spectrum of the emitted recombination photons.

**Radiative recombination in the presence of an intense laser field**

Friedrich-Schiller-Universität Jena, Physikalisch-Astronomische Fakultät (2015)

**Abstract:** In this thesis we present a theoretical study on the radiative recombination of electrons into the ground state of hydrogen like ions in the presence of an intense external laser field. We employ for the description of this process Heisenberg’s S-matrix theory, where the final bound state of the electron is constructed using first order time dependent perturbation theory. Two different initial electron states are considered. First asymptotically plane-wave-like electrons with a separable Coulomb-Volkov continuum wave function and secondly twisted electrons with a well defined orbital angular momentum constructed from Volkov states. Using this approach we perform detailed calculations for the angle-differential and total cross section of laser assisted radiative recombination considering low-Z ions and laser intensities in the range from IL = 10^11 W/cm^2 to IL = 10^13 W/cm^2. Special emphasis is put on the effects arising due to the laser dressing of the residual bound state. It is seen that the bound state dressing remarkably affects the total cross section and manifests moreover as asymmetries in the angular and energy distribution of the emitted photons. For incident Coulomb-Volkov electrons we study moreover the polarization of the emitted recombination radiation. Here we find that the direction of polarization is rotated depending on the energy of the emitted recombination photons.

## 2014

**Relativistic theory for radiative forward electron emission in heavy ion-atom encounters**

**68**, 367 (2014)

**Abstract:** The forward electron emission with simultaneous photon production during the scattering of relativistic, highly stripped projectiles from light target atoms is calculated within the Dirac theory. The method of calculation is a simplification of the impulse approximation and is based on the relation of the cross section for radiative capture to continuum of loosely bound electrons to the frame-transformed electron bremsstrahlung cross section. It is demonstrated that such an approximation is well justified in a large region of energies and photon emission angles, with the exception of the extreme forward and backward emission and the soft-photon energy limit. The cusp spectrum and the corresponding angular distribution are compared to recent experimental data for the collision system 90.38 MeV/amu U88+ + N2.

**Polarization of atomic bremsstrahlung in coincidence studies**

**90**, 032707 (2014)

**Abstract:** We present a theoretical study of bremsstrahlung produced by high-energy electrons scattered by heavy atomic targets. Considering coincident observation of the emitted photons and the scattered electrons, we pay special attention to the polarization degree and direction of the outgoing light. To investigate these properties of atomic bremsstrahlung, we apply the density matrix approach and solutions of the Dirac equation. Detailed calculations are performed for initial electron energies ranging from 100 to 500 keV and different fixed electron scattering angles. The results of these calculations are compared with predictions obtained under the assumption that the scattered electrons remain unobserved. This comparison reveals that both the degree and the direction of linear polarization of bremsstrahlung are very sensitive to the direction of the scattered electron.

**Radiative-electron-capture-to-continuum cusp in U⁸⁸⁺+N₂ collisions and the high-energy endpoint of electron-nucleus bremsstrahlung**

**90**, 022707 (2014)

**Abstract:** The radiative electron capture of a target electron into the projectile continuum has been studied for the collision system U88+ + N2 → U88+ + [N+2]∗ + e− + γ at 90 MeV/u. Using a magnetic electron spectrometer, the energy distribution of cusp electrons emitted under an angle of 0∘ with respect to the projectile beam and with a velocity close to the projectile velocity has been measured in coincidence with the emitted photons under various observation angles. The experimental results provide a stringent test for the corresponding process in inverse kinematics, namely, the theory of electron-nucleus bremsstrahlung at the high-energy endpoint. For comparison this process is calculated using fully relativistic Dirac wave functions and using semirelativistic Sommerfeld-Maue wave functions.

## 2013

**Angular and Polarisation Properties of Bremsstrahlung Radiation in the Short-Wavelength Limit**

Friedrich-Schiller-Universität Jena, Physikalisch-Astronomische Fakultät (2013)

**Abstract:** In this work two methods for the description of atomic bremsstrahlung are discussed. The density matrix of the system after the scattering process is derived using a Rayleigh expansion of the photon interaction operator and a partial wave expansion of the free dirac electron. These derivations were done following Yerokhin and Surzhykov as well as Tseng and Pratt. From these results a new parametrisation of two observables of electron-atom bremsstrahlung is presented which expresses the angular distribution and the degree of linear polarisation in terms of spherical harmonics. That means once the coefficients are calculated the calculation of the bremsstrahlung properties is orders of magnitude faster than the calculations after Yerokhin and Surzhykov [6]. Also almost real-time calculations are possible when the tabulated coefficients are used. The coefficients yield a couple of symmetry relations and converge very fast against zero which reduces the needed expansion order remarkably. Also they behave very smooth when the other parameters are changed so we can get the coefficients for arbitrary parameter sets from an interpolation on a two dimensional grid. The number of coefficients needed increases with the photon energy but does not exceed 50 for energies up to several hundred keV while for energies less than 100keV for most applications a monadic number of coefficients is enough. Additionally the distance between the nodes on the grid can be increased for higher energies because the coefficients vary less for higher energies so less sets of coefficients are necessary to achieve the same accuracy.