Diffraction of the XFEL femtosecond pulse in a crystal BELARUSIAN STATE UNIVERSITY A.Benediktovich, I.Feranchuk, A.Leonov, D.Ksenzov, U.Pietsch The Actual Problems of Microworld Physics Gomel, July 22-August 2

Contents 1.Motivation 2.Basic assumptions and approximations 3.Master equations for electron density dynamics 4.Numerical results 5.Discussion & outlook

Contents 1.Motivation 2.Basic assumptions and approximations 3.Master equations for electron density dynamics 4.Numerical results 5.Discussion & outlook

Irradiation of a crystal with XFEL pulse 1 (23) X-ray Free Electron Laser (XFEL) – the perfect tool to study ultrafast dynamics with Ångström resolution Recording of the complete diffraction pattern by a single shot is possible Interaction of the XFEL fs-pulses with a crystal CAN NOT be described in the framework of the linear response theory because of the time evolution of the electron density is comparable with the time formation of the diffracted wave! The time scale we consider We consider the processes that take place during the passing of the XFEL pulse through the crystal ( t < 100 fs )

Contents 1.Motivation 2.Basic assumptions and approximations 3.Master equations for electron density dynamics 4.Numerical results 5.Discussion & outlook

Energy spectrum of a crystal Ground state Excited states IONIZATION ENERGY IN CRYSTAL IS SMALLER THAN THE PHOTON ENERGY 2 (23)

Features of ionization dynamics Characteristic photoelectron energy: The mean free path of electrons: Percentage of remaining free electrons: THE ROLE OF FREE ELECTRONS SHOULD BE ESSENTIAL 3 (23)

General dynamics scheme 4 (23)

Contents 1.Motivation 2.Basic assumptions and approximations 3.Master equations for electron density dynamics 4.Numerical results 5.Discussion & outlook

5 (23) Model to be considered Total system Free electrons EM field Beyond the present consideration Bound electrons

Atomic state population dynamics General form of rate equations: Constituting processes: - photoionization - Auger decay - electron-impact ionization - three-body recombination 6 (23) probability of transition from λ to μ configuration in unit time:

Free electron gas dynamics The Boltzmann kinetic equation: Set of simplifications: - lateral homogeneity: - anisotropy parameter: => - diffusion term: Reduced Boltzmann equation: - net force: 7 (23)

System of master equations 8 (23) coupled to ONE HAS TO SOLVE THE SYSTEM OF TWO INTEGRO-DIFFERENTIAL EQUATIONS SIMULTANEOUSLY

Effective charge model Hydrogen-like wave functions: Energy of a configuration: ALL CROSS-SECTIONS AND RATES CAN BE CALCULATED ANALYTICALLY 1 9 (23)

Contents 1.Motivation 2.Basic assumptions and approximations 3.Master equations for electron density dynamics 4.Numerical results 5.Discussion & outlook

Atomic population probabilities P λ (t) 10 (23)

Atomic population probabilities P λ (t) 11 (23)

Evolution of the atomic scattering factor 12(23)

Evolution of the atomic scattering factor 13 (23)

Evolution of the atomic scattering factor 14 (23)

15 (23) Pulse parameters: duration: 40 fs; photon energy: 8 keV; shape: Gaussian; fluence: 10 4 phs/Å 2. Evolution of the free electron density

Contribution of different channels 16(23)

Intensity of the XFEL pulse diffraction 17 (23)

Contents 1.Motivation 2.Basic assumptions and approximations 3.Master equations for electron density dynamics 4.Numerical results 5.Discussion & outlook

Discussion and outlook - the role of free electrons is essential - three-body recombination rate is extremely low Numerical algorithm and software are developed Simulation for the example of Si crystal is carried out Outlook - investigation of X-ray optics in the extremely intensive regime - similar calculations are required for description of interaction of relativistic dense electron bunches with the crystal (in order to use parametric X-ray radiation for compact XFEL sources 1 ) 19(23) - non-expensive based on basic principles - rates and cross-sections are fully analytical

Acknowledgements This work was supported by the BMBF under 05K10PSA 20 (23) Belarusian State University Prof. Dr. Ilya Feranchuk Dr. Andrei Benediktovitch Siegen University Prof. Dr. Ullrich Pietsch Dr. Dmitry Ksenzov Discussions Prof. Dr. Robin Santra and members of his group (CFEL, DESY) Dr. Ivan Vartaniants and members of his group (HASYLAB, DESY) Collaborators

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