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PhD Opportunities

Quantum chaos in multicharged positive ions

School of Mathematics and Physics | PHD
Reference Number
Application Deadline
11 February 2022
Start Date
1 October 2022


Project supervisors: Dr G. F. Gribakin and Dr C. P. Ballance Contacts:, State of the art and motivations Many-body quantum chaos is an area of research that spans the boundaries of atomic, nuclear and condensed matter physics [1]. Its starting point was a realisation by Wigner in the 1950's that the interaction between the nucleons in a heavy nucleus results in excited states of enormous complexity. This called for the development of a statistical theory that would describe the average properties of the spectrum and eigenstates: the level density, mean-squared amplitudes, etc. Notwithstanding the progress in computing power, the eigenstates of numerous many-particle systems still cannot be obtained by brute force methods. The project focusses on one such system { open-shell positive ions. Chaotic multiply excited states in positive ions lead to observable effects, such as strongly enhanced electron-ion recombination rates [2, 3, 4, 5] and secondary threshold laws in multiple ionization [6].

A convenient starting point for describing many-electron systems, e.g., atoms or ions, is the mean-field approximation. In this approximation every electron in the system moves in the static field of the nucleus and all other electrons. Quantum-mechanically, this motion is described by single-particle wavefunctions. The total wavefunction of the system is then given by an antisymmetrised product of the individual electron's wavefunctions. Accordingly, the atom or ion is characterised by a certain electronic configuration, i.e., a list of single-particle states, or orbitals, occupied by the electrons. Besides the orbitals occupied by electrons in the lowestenergy (ground) state of the system, one can also consider orbitals with higher energies, which are
normally empty. However, when energy is pumped into the system, electrons can be promoted into these orbitals, thereby creating an excited state of the atom or ion.

The above mean-field picture is only an approximation. In reality, when one electron changes its position, other electrons \notice" the change and respond to it, e.g., by changing their orbitals. This effect is known as electron correlations. It leads to a breakdown of the description based on fixed electron configurations. Different configurations become mixed by the residual electron interaction (i.e., that beyond the mean field). However, the configuration states can still be used as a basis for expanding the true eigenstates of the system. Their energies are found as eigenvalues of the Hamiltonian matrix in the space of configuration states.

The project focuses on atomic systems with extremely strong configuration mixing, in which all configurations with nearby energies are mixed together. Such mixing takes place in excited states of heavy nuclei, open-shell many-electron atoms, such as Ce, and multicharged many-electron ions, e.g., gold Au25+, tungsten W20+, etc. [2, 3, 4, 5, 8], and is termed many-body quantum chaos. Processes involving tungsten ions are of particular importance, because of this metal is a major component of ITER and a major contaminant of the thermonuclear reactor plasma, with potentially drastic consequences for the feasibility of fusion [7]. The complexity of these systems prevents one from doing a full configuration-interaction calculation, as the size of
the corresponding Hamiltonian matrix is too large. However, this complexity also allows one to
develop a statistical approach to the description of the system [2, 8, 9, 10].

Objectives & Methodology
The aim of the project is to develop further the ideas put forward in [2, 3, 4, 5, 8], focussing on the multicharged ions of tungsten, which is used as a wall material in nuclear fusion reactors.

The project will involve both analytical and numerical work, and will combine model calculations (e.g., random-matrix theory approaches) with realistic calculations for many-electron ions, based on the Dirac-Fock equation and configuration-interaction approaches. While full diagonalisation of the Hamiltonian matrix in this complex system is beyond reach, we can explore the mixing of individual important multiply excited configurations (along the lines of [3]). Particular attention should be given the so-called doorway configurations which provide a pathway for the initial excitation of the atomic system [9], such as the 4d-4f excitation in Xe ions [11].

A complimentary, and possibly more powerful, approach will be to construct the Hamiltonian matrix and investigate configuration mixing in the time domain by integrating the time-dependent Schrodinger equation. This procedure is computationally less expensive than the full diagonalisation, and can easily be used for Hamiltonian matrices of sizes up to 105. Such calculation can start with a simple single excited electron state, which would model electronion recombination or excitation by a photon. Its time evolution will the reveal the process of energy-sharing between this initial excitation and multiply excited many-electron states.
Such time-dependent approach will in some sense model the behaviour of atomic systems in short X-ray pulses, such as those created by the Linac Coherent Light Source (LCLS), the world's first hard hard X-ray free-electron laser (see, e.g., [12]).

This approach should give reliable quantitative information on the nature and extent of configuration mixing in many-electron multicharged ions. It should facilitate calculations of average values of observable quantities: recombination cross sections, radiative widths, etc. [4, 5]. From a broader perspective, the project will contribute to the theory of finite quantum systems of a few strongly-interacting particles [8], and understanding of many-body quantum chaos.

While the project will be based at Queen's, discussions, exchange of ideas and, possibly, joint work with the group of Prof. V. V. Flambaum and Dr. V. A. Dzuba at the School of Physics, UNSW (Sydney, Australia) is foreseen. This group is a world leader in applications of many-body theory in atomic physics and precise atomic calculations.

Required skills
The candidate is expected to have good working knowledge of and interest in Quantum Mechanics.
He/she will be expected to learn and master many-body-theory methods and their application to many-electron systems. Running, modifying and developing computer codes is an integral and substantial part of the project. In spite of the complexity of the codes, the underlying many-body theory should allow great transparency and good physical understanding of the results.

Further information
While focussed on the problems of quantum chaos and multicharged ions, the project will introduce the candidate to the general area of theoretical atomic physics, atomic structure calculations and computations methods. The experience of handling and writing computer codes, that will be acquired through the work on the project, will be useful in a wide range of future careers.

For further information, please contact Dr Gleb Gribakin.

[1] T. Guhr, A. Muller-Groeling, and H. A. Weidenmuller, Random-matrix theories in quantum physics: common concepts, Physics Reports 299, 189 (1998).
[2] V. V. Flambaum, A. A. Gribakina, G. F. Gribakin, and C. Harabati, Electron recombination with multicharged ions via chaotic many-electron states, Phys. Rev. A 66, 012713 (2002).
[3] G. F. Gribakin and S. Sahoo, Mixing of dielectronic and multiply-excited states in electron-ion
recombination: a study of Au24+, J. Phys. B 36, 3349 (2003).
[4] V. A. Dzuba, V. V. Flambaum, G. F. Gribakin, and C. Harabati, Chaos-induced enhancement of resonant multielectron recombination in highly charged ions: Statistical theory, Phys. Rev. A 86, 022714 (2012).
[5] V. A. Dzuba, V. V. Flambaum, G. F. Gribakin, C. Harabati, and M. G. Kozlov, Electron recombination,
photoionization, and scattering via many-electron compound resonances, Phys. Rev. A 88, 062713 (2013).
[6] G. F. Gribakin, S. Sahoo and V. N. Ostrovsky, Secondary threshold laws for multiple ionization of atoms, Phys. Rev. A 70, 062717 (2004).
[7] T. Putterich, R. Neu, R. Dux, A. D. Whiteford, M. G. O'Mullane and the ASDEX Upgrade Team, Modelling of measured tungsten spectra from ASDEX Upgrade and predictions for ITER, Plasma Phys. Control. Fusion 50, 085016 (2008).
[8] V. V. Flambaum and G. F. Gribakin, Statistical theory of Finite Fermi systems with chaotic excited eigenstates, Philosophical Magazine B 80, 2143-73 (2000).
[9] V. V. Flambaum, M. G. Kozlov, and G. F. Gribakin, Coherent and stochastic contributions of compound resonances in atomic processes: Electron recombination, photoionization, and scattering, Phys. Rev. A 91, 052704 (2015).
[10] J. C. Berengut, C. Harabati, V. A. Dzuba, V. V. Flambaum, and G. F. Gribakin, Level resolved
quantum statistical theory of electron capture into many-electron compound resonances in highly charged ions, Phys. Rev. A 92, 062717 (2015).
[11] A. Aguilar, J. D. Gillaspy, G. F. Gribakin, R. A. Phaneuf, M. F. Gharaibeh, M. G. Kozlov, J. D. Bozek, and A. L. D. Kilcoyne, Absolute photoionization cross sections for Xe4+, Xe5+, and Xe6+ near 13.5 nm: Experiment and theory, Phys. Rev. A 73, 032717 (2006).
[12] B. Rudek, S.-K. Son, D. Rolles et al., Ultra-efficient ionization of heavy atoms by intense X-ray free-electron laser pulses, Nature Photonics 6, 858 (2012).

Project Summary

Dr Gleb Gribakin

Mode of Study

Full-time: 3 years

Apply now Register your interest

Physics overview

The scientific research within the School of Mathematics and Physics was highly rated in the 2014 REF peer-review exercise, with 70% of research being judged as internationally excellent or world-leading. Physics at Queen's is currently joint 6th in the UK for Research Intensity and has been voted 4th in the UK for teaching satisfaction.

Physics research activity in the School is focused into five specific Research Centres; all members of academic staff belong to one of these Research Centres, listed below.

Astrophysics (PhD/MPhil)
Find out more below, or email Professor Mihalis Mathioudakis (

Atomistic Simulation (PhD/MPhil)
Find out more below, or email Dr Myrta Gruening (

Nanostructured Media (PhD/MPhil)
Find out more below, or email Dr Amit Kumar (

Plasma Physics (PhD/MPhil)
Find out more below, or email Professor Marco Borghesi (

Theoretical Atomic, Molecular and Optical Physics (PhD/MPhil)
Find out more below, or email Dr Alessandro Ferraro (

Registration is on a full-time or part-time basis, under the direction of a supervisory team appointed by the University. You will be expected to submit your thesis at the end of three years of full-time registration for PhD, or two years for MPhil (or part-time equivalent).

Physics Highlights
Career Development
  • Queen's graduates from Physics have secured employment through a number of companies such as Allstate, AquaQ Analytics, Citigroup, Deloitte, First Derivatives, PwC, Randox, Seagate, Teach First and UCAS. In addition, Belfast has been ranked as the world’s most business friendly small-medium sized city (Financial Times’ fDi Intelligence, 2018)
World Class Facilities
  • Since 2014, the School has invested over £12 million in new world-class student and staff facilities. Maths and Physics students have their own teaching centre that opened in 2016, housing brand experimental physics laboratories, two large computer rooms plus a student interaction area with a new lecture theatre and study rooms. In addition to this, Northern Ireland has the lowest student cost of living in the UK (Which? University, 2018) and is over £5000 per year cheaper for students to live in Northern Ireland compared to London
Internationally Renowned Experts
  • Queen's is joint 6th in the UK for Research Intensity for Physics and Astronomy (Complete University Guide 2021). The School has a continually growing international community of both undergraduate and postgraduate students and staff. Our research is conducted and recognised as excellent across the world. Staff are involved in cutting-edge research projects that span a multitude of fields.
Key Facts

  • Students will have access to our facilities, resources and our dedicated staff. The School of Maths & Physics is one of the largest Schools in the University. Staff are involved in cutting-edge research that spans a multitude of fields.

Course content

Research Information

Research Themes
Astrophysics (PhD/MPhil)

You’ll be involved in the search for distant supernovae and where they came from; study the asteroid and comet population in the Solar system; look for planets orbiting other stars in our Galaxy; study flares and other dynamic processes in the atmosphere of the Sun. You’ll have the opportunity to spend extensive periods at world-leading research centres such as the European Southern Observatory and NASA Goddard Space Flight Center.

At Queen’s we lead major European consortia and are supported by a multi-million pounds portfolio of research grants from a range of sources, including the UK Science and Technology Facilities Council, the Royal Society, and European Union.

Research Themes
Atomistic Simulation (PhD/MPhil)

Atomistic Simulation is the development and use of theoretical and computational methods to study structural, dynamical, and optical properties of molecules, liquids, solids and plasmas at the nanoscale. Computational experiments are used to interpret existing experimental data and to predict phenomena yet unobserved.

You’ll study problems at the interfaces between condensed matter physics, materials science, chemistry, biology, and engineering. You’ll interact with laboratory-based colleagues at Queen's and internationally, addressing fundamental and/or practical questions, and you will develop and program novel simulation methodologies to model for instance, electronic excitations, optical properties of materials, and the interaction between electric currents, heat and light.

Themes that are presently studied in the ASC include: non-linear optics in 2D materials, plasmonics, laser and ion-matter interactions, conduction in nanowires, aqueous Interfaces, nucleation, and crystallisation. Tools include time-dependent density-functional theory, many-body perturbation theory, classical molecular dynamics, Monte Carlo simulations and machine learning.

Research Themes
Centre for Nanostructured Media (PhD/MPhil)

Human history is defined by the materials we use to underpin our technology: stone, bronze, iron, silicon. As a PhD student in the Centre for Nanostructured Media, you will be playing a part in the development of materials systems which will, in some way, define our technology for the future. How can this not be exciting ? You will seek to reveal the physics of material behaviour at the boundary of current global knowledge and, at the same time, become proficient in techniques for materials growth, patterning and characterisation that are highly valued in high-tech companies and commercial research institutions, as well as in academic research settings. Our laboratories are extremely well-equipped for international-level research and our links to other research teams throughout the world in both academia and industry are strong and you should expect to travel, should you wish to, as part of your PhD experience.

Research Themes
Plasma Physics (PhD/MPhil)

Your research will involve identifying, and responding to, major contemporary issues within ionised matter physics, with major activities in laser- and electrically-produced plasmas, ultra-fast atomic and molecular physics and the interaction of ionising radiation and plasmas with matter, including biological systems. This research will employ local, national and international facilities, including some of the most powerful laser systems worldwide. You will also benefit from transferring your research findings into the industrial and medical sectors.

Research Themes
Theoretical Atomic, Molecular and Optical Physics (PhD/MPhil)

You’ll contribute to a body of work with recent major developments including strong field laser interactions with atoms and molecules, quantum information processing, quantum optics, and quantum thermodynamics, antimatter interactions with atoms and molecules, electron scattering by very complex targets such as the iron peak elements, and by Rydberg atoms, quantum many-body physics, ultra-cold atomic systems, and simulation of their features, and foundations of quantum mechanics.

Postgraduate research programmes within CNM provide experience and training in state-of-the art academic research: many of our research strands are world-leading, as evidenced by performance in REF2014. In addition, most of our postgraduate researchers are exposed to functional materials and photonics in major multinational companies.

Prof Marty Gregg - School of Mathematics and Physics
Career Prospects

Alumni Success
Many of our PhD graduates have moved into academic and research roles in Higher Education while others have progressed into jobs such as Data Scientist, Software Engineer, Financial Software Developer, IT Graduate Associate, Technology Consultant, Research Physicist, Telescope Operator and R&D Engineer.

People teaching you

Dr Amit Kumar
Head of Research Centre - Centre for Nanostructured Media
School of Maths and Physics

Dr Connor Ballance
Head of Centre - Centre for Theoretical and Atomic Molecular Physics
School of Maths and Physics

Dr Myrta Gruening
Head of Research Centre - Atomistic Simulation Centre
School of Maths and Physics

Prof Marco Borghesi
Head of Research Centre - Centre for Plasma Physics
School of Maths and Physics

Prof Mihalis Mathioudakis
Head of Research Centre - Astrophysics Research Centre
School of Maths and Physics

Learning Outcomes

Course structure



Assessment processes for the Research Degree differ from taught degrees. Students will be expected to present drafts of their work at regular intervals to their supervisor who will provide written and oral feedback; a formal assessment process takes place annually.

This Annual Progress Review requires students to present their work in writing and orally to a panel of academics from within the School. Successful completion of this process will allow students to register for the next academic year.

The final assessment of the doctoral degree is both oral and written. Students will submit their thesis to an internal and external examining team who will review the written thesis before inviting the student to orally defend their work at a Viva Voce.


Supervisors will offer feedback on draft work at regular intervals throughout the period of registration on the degree.


Our world-class facilities support research and teaching across a diverse range of areas designed to fulfil specific activities. The School contains 4,700m2 of purpose-built laboratory space which includes the ANSIN materials research hub, the Ewald Microscopy Facility (EMF) and the Taranis laser facility. The Teaching Centre (opened in 2016) includes experimental physics laboratories, two large computer rooms and plenty of student study and interaction space. Our laboratories and equipment are looked after by a dedicated team of technicians and are used by our researchers, students and industry.

Entrance requirements

The minimum academic requirement for admission to a research programme is normally an Upper Second Class Honours degree from a UK or ROI HE provider, or an equivalent qualification acceptable to the University. Further information can be obtained by contacting the School of Mathematics and Physics.

International Students

For information on international qualification equivalents, please check the specific information for your country.

English Language Requirements

Evidence of an IELTS* score of 6.0, with not less than 5.5 in any component, or an equivalent qualification acceptable to the University is required. *Taken within the last two years

International students wishing to apply to Queen's University Belfast (and for whom English is not their first language), must be able to demonstrate their proficiency in English in order to benefit fully from their course of study or research. Non-EEA nationals must also satisfy UK Visas and Immigration (UKVI) immigration requirements for English language for visa purposes.

For more information on English Language requirements for EEA and non-EEA nationals see:

If you need to improve your English language skills before you enter this degree programme, INTO Queen's University Belfast offers a range of English language courses. These intensive and flexible courses are designed to improve your English ability for admission to this degree.

Tuition Fees

Northern Ireland (NI) 1 £4,596
Republic of Ireland (ROI) 2 £4,596
England, Scotland or Wales (GB) 1 £4,596
EU Other 3 £23,850
International £23,850

1 EU citizens in the EU Settlement Scheme, with settled or pre-settled status, are expected to be charged the NI or GB tuition fee based on where they are ordinarily resident, however this is provisional and subject to the publication of the Northern Ireland Assembly Student Fees Regulations. Students who are ROI nationals resident in GB are expected to be charged the GB fee, however this is provisional and subject to the publication of the Northern Ireland Assembly student fees Regulations.

2 It is expected that EU students who are ROI nationals resident in ROI will be eligible for NI tuition fees. The tuition fee set out above is provisional and subject to the publication of the Northern Ireland Assembly student fees Regulations.

3 EU Other students (excludes Republic of Ireland nationals living in GB, NI or ROI) are charged tuition fees in line with international fees.

All tuition fees quoted are for the academic year 2021-22, and relate to a single year of study unless stated otherwise. Tuition fees will be subject to an annual inflationary increase, unless explicitly stated otherwise.

More information on postgraduate tuition fees.

Physics costs

Depending on the area of research chosen there may be extra costs which are not covered by tuition fees.

Additional course costs

All Students

Depending on the programme of study, there may also be other extra costs which are not covered by tuition fees, which students will need to consider when planning their studies . Students can borrow books and access online learning resources from any Queen's library. If students wish to purchase recommended texts, rather than borrow them from the University Library, prices per text can range from £30 to £100. Students should also budget between £30 to £100 per year for photocopying, memory sticks and printing charges. Students may wish to consider purchasing an electronic device; costs will vary depending on the specification of the model chosen. There are also additional charges for graduation ceremonies, and library fines. In undertaking a research project students may incur costs associated with transport and/or materials, and there will also be additional costs for printing and binding the thesis. There may also be individually tailored research project expenses and students should consult directly with the School for further information.

Bench fees

Some research programmes incur an additional annual charge on top of the tuition fees, often referred to as a bench fee. Bench fees are charged when a programme (or a specific project) incurs extra costs such as those involved with specialist laboratory or field work. If you are required to pay bench fees they will be detailed on your offer letter. If you have any questions about Bench Fees these should be raised with your School at the application stage. Please note that, if you are being funded you will need to ensure your sponsor is aware of and has agreed to fund these additional costs before accepting your place.

How do I fund my study?

1.PhD Opportunities

Find PhD opportunities and funded studentships by subject area.

2.Funded Doctoral Training Programmes

We offer numerous opportunities for funded doctoral study in a world-class research environment. Our centres and partnerships, aim to seek out and nurture outstanding postgraduate research students, and provide targeted training and skills development.

3.PhD loans

The Government offers doctoral loans of up to £26,445 for PhDs and equivalent postgraduate research programmes for English- or Welsh-resident UK and EU students.

4.International Scholarships

Information on Postgraduate Research scholarships for international students.

Funding and Scholarships

The Funding & Scholarship Finder helps prospective and current students find funding to help cover costs towards a whole range of study related expenses.

How to Apply

Apply using our online Postgraduate Applications Portal and follow the step-by-step instructions on how to apply.

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