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% Journal sections (see http://pkp.jinr.ru/index.php/PEPAN_LETTERS/about/editorialPolicies#focusAndScope)
%\issuearea{Physics of Elementary Particles and Atomic Nuclei. Theory}
% or in Russian
%\issuearea{ФИЗИКА ЭЛЕМЕНТАРНЫХ ЧАСТИЦ И АТОМНОГО ЯДРА. ТЕОРИЯ}

\title{The symmetry method as a way to visualize the spectrum of the ultrashort pulse interactions with polyatomic system}
\maketitle
\authors{A.\,Kharlamova$^{a, }$ \footnote{E-mail: kharlamova.anastasya2015@yandex.ru} , D.\,Makarov$^{a}$}
\setcounter{footnote}{0}
\from{$^{a}$\,Higher School of Natural Sciences and Technologies 
Northern (Arctic) Federal University named after M.V. Lomonosov, Arkhangelsk, Russia}



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\begin{abstract}
% Russian translation of the abstract
Представлен метод теоретического моделирования спектра взаимодействия ультракороткого лазерного импульса с различными многоатомными системами. В основе метода лежит поиск повторяющихся участков исследуемого объекта. Данный способ расчета дает возможность сократить время моделирования спектра взаимодействия импульса с веществом. Проведены расчеты на молекуле ДНК, РНК и кристалле алмаза. \
\vspace{0.2cm}

A method of theoretical modeling of the spectrum of interaction of an ultrashort laser pulse with various polyatomic systems is presented. The method is based on the search for duplicate sections of the object under study. This calculation method makes it possible to reduce the simulation time of the spectrum of interaction of a pulse with a substance. Calculations were performed on a DNA molecule, RNA and a diamond plate.
\end{abstract}
\vspace*{6pt}

\noindent
PACS: 44.25.$+$f; 44.90.$+$c

\label{sec:intro}
\section*{Introduction}

Currently, the European free electron laser is actively used to determine the structure of various materials [1,2]. The advantage of such a laser over other research methods is the ability to generate short beams that make it possible to study the structure without radiation exposure [3]. Nevetheless, the exposure time of such a pulse is commensurate with the characteristic atomic time, which makes it possible to study molecules without additional preparation, crystallization, etc. This was a revolution in several fields of study, mostly in structural biology [4]. However, the difficulty of obtaining and decoding diffraction patterns impede the application of a laser. The complexity of data processing and decoding, namely the phase problem, prevents the widespread use of ultrashort laser pulses to work with molecules [5]. It is necessary to search for new ways to decrypt the received data in order to create an effective three-dimensional visualization method.

\label{sec:preparation}
\section*{Theory}

In the previous work, the method proposed as the basis for decoding experimentally obtained diffraction patterns was described [6]. The method is based on a theoretical description of the interaction of an ultrashort pulse with the substance under study, using numerical modeling and using the Dirac–Hartree-Fock-Slater model. To simplify the decoding of molecular structures, it is proposed to conduct theoretical modeling of the interaction in order to obtain a reference diffraction pattern. In [7], the previous method was changed. It was proposed to simplify the calculation if there were symmetries or repeating sections in the structure under study. In this paper, the proposed method is tested in more detail on different structures, such as a DNA molecule (a larger site), an RNA molecule, and a diamond crystal. All these substances are of great interest in structural biology, medicine and physics. It is still difficult to conduct the research on them at the atomic level, which prompts the development of similar methods of their analysis. Using the symmetry method calculations could be carried out at the analytical level and processing time could be redused. The interaction spectra of an ultrashort pulse with molecules and a crystal will be obtained here, and the proposed theory will be tested by calculating the interaction spectra on these objects. The obtained interaction spectra will be compared with previous results.The calculation is based on the following dependence.


\begin{equation}
\small
\frac{d^2 \varepsilon}{d\Omega_\textbf{k}d\omega}=\frac{[\textbf{E}_\textbf{0}\textbf{n}]^2\left| \hslash(\omega) \right|^2}{(2\pi)^2c^3}\left[ \sum_{\mathrm{i=1}}^{A}N_{e,i}N_{A,i}\left( 1-\left| F_i\right|^2 \right)+\sum_{i,j=1}^{A}\delta_{i,j}N_{e,i} N_{e,j} F_i F_j \right]
\end{equation}


Where $\mathrm{\textbf{E}}_{0}$ is electromagnetic field strength USPs (ultrashort laser pulses),
\ ${\Omega}_{\textbf{k}}$ is unit of solid angle, $\varepsilon$ is scattering energy, n is the direction of the scattered pulse, с is the speed of light, $A$ is is the number of varieties of atom, $N_{e,i}$ is number of electrons in the atom of the i-th, $N_{A,i}$ is number of atoms of the i-th, $F_{i}$ is form factor of the atom of the i-th with the corresponding electron density, $\omega$ is pulse frequence. Similarly for atoms of a different sort, marked with the index j. The interference factor parameter $\delta$  is particularly interesting for calculating a system consisting of atoms of different sort. It reacts to changes in the coordinates of atoms in the system under study. It is calculated as follows. 
\begin{equation}
\delta_{i,j}=\sum_{\alpha=1}^{s}\sum_{n_{\alpha}=0}^{N_{\alpha}}e^{-i\textbf{p}\textbf{R}_{n_{\alpha}}}\sum_{A_{i}\in R_{\alpha,1}}e^{-i\textbf{p}\textbf{R}_{A_{i}}}\sum_{\beta=1}^{s}\sum_{n_{\beta}=0}^{N_{\beta}}e^{-i\textbf{p}\textbf{R}_{n_{\beta}}}\sum_{A_{j}\in R_{\beta,1}}e^{-i\textbf{p}\textbf{R}_{A_{j}}}
\end{equation}
Where $\textbf{p}$ is recoil pulse, when an ultrashort pulse is dissipated by a bound electron, $\delta_{i,j}$ is interference factor, $\alpha$ and $\beta$ are some asymmetry in the system, $s$ is the number of symmetries in the system, $\textbf{R}_{n_{\alpha}}$ and $\textbf{R}_{n_{\beta}}$ are radius is a vector that specifying the position of the symmetry $\alpha$ or $\beta$,  $\textbf{R}_{A_{i}}$ is radius is a vector specifying the position of i-th atoms in symmetry, $\mathrm{N}_{\alpha}^{}$ is the number of movements with a given symmetry $\alpha$, $\mathrm{N}_{\beta}^{}$ is the number of movements with a given symmetry $\beta$.

The symmetry method is illustrated in Fig.1. 
\begin{figure}[H]
\begin{center}
\includegraphics[width=90mm]{fig5.png}
%\includegraphics[width=127mm]{fig5.png}
\vspace{-3mm}
\caption{The scheme of calculation by the symmetry method. }
\end{center}
\labelf{fig01}
\vspace{-5mm}
\end{figure}
%============================= Fig. 1 ================================

Adenine repeats twice, cytosine once. It is necessary to calculate the interaction with atoms within the same symmetry, and then multiply the result. The calculations from the previous work shows that the increasing symmetry in the model change the spectra of the interaction of the pulse with matter. In this calculation, it is checked whether the proposed theory can find differences between different objects. 

\section*{Modeling}

The spectra of the interaction of an ultrashort pulse with molecules and a diamond are obtained in the article, and the proposed theory is verified by calculating the spectra of interaction with these objects. Two models of a polyatomic system, a DNA and RNA molecule, as well as diamond film models, as a simplified version, when the system consists of one kind of atoms, were chosen as objects of research.
The result of the simulation is a spectrum constructed in a spherical coordinate system. The main axis of DNA runs along the molecule, orthogonally to its bases. The pulse falls on the system at an angle of 90 degrees to the main axis system parameter $\theta$ -- is the angle between the helix axis and the scattering direction $\phi$ -- is the angle between the helix axis and the projection on the plane perpendicular to the helix axis, the calculations were performed in spherical coordinates. The results in all figures are normalized to the maximum value of the scattering spectrum (see Fig. 2 ).

\newpage
%============================= Fig. 2 ================================
\begin{figure}[H]
\begin{center}
\includegraphics[width=90mm]{fig7.png}
%\includegraphics[width=127mm]{fig7.png}
\vspace{-5mm}
\caption{Visualization of the theoretical model of the spectrum of interaction of a pulse with DNA, RNA and a diamond film.}
\end{center}
\labelf{fig02}
\vspace{-5mm}
\end{figure}
%============================= Fig. 2 ================================

On each model, areas of symmetry are highlighted, in DNA and RNA such areas are nitrogenous bases, in a diamond plate, a number of carbon is isolated as a repeating section. Characteristic stripes are visible on all contour graphs, this is due to the interference of repeating sections.

\section*{Conclusion}

As a result of the simulation, the spectra of the pulse interaction with three models were obtained: DNA, RNA and diamond film. The spectra are presented in the form of contour graphs, which makes it possible to analyze the resulting diffraction patterns. The differences in the shape of the spectra characteristic of each type of model are visible. The developed theory reacts to changes in the atomic composition of the molecule and to the number of established symmetries. The formula of the interference factor used here is simpler in comparison with previous works [8]. It allows not to calculate the scattering on all atoms of the investigated sample, and speeds up the calculation process.  

\section*{Acknowledgments}
The study was supported by a grant from the Russian Science Foundation, No. 23-12-20014; State assignment of the Russian Federation, No. FSRU-2024-0005.


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% Bibliography automatically generated via BibTeX (see template_bibtex.tex)

\begin{thebibliography}{1}
\def\selectlanguageifdefined#1{
\expandafter\ifx\csname date#1\endcsname\relax
\else\selectlanguage{#1}\fi}
\providecommand*{\href}[2]{{\small #2}}
\providecommand*{\url}[1]{{\small #1}}
\providecommand*{\BibUrl}[1]{\url{#1}}
\providecommand{\BibAnnote}[1]{}
\providecommand*{\BibEmph}[1]{\emph{#1}}
\ProvideTextCommandDefault{\cyrdash}{\hbox to.8em{--\hss--}}
\providecommand*{\BibDash}{\ifdim\lastskip>0pt\unskip\nobreak\hskip.2em\fi
\cyrdash\hskip.2em\ignorespaces}

\bibitem{Starodub:2017kwy}
\selectlanguageifdefined{english}
\BibEmph{Starodub D., Aquila A., Bajt S. et al. } {Single-particle structure determination by correlations of snapshot X-ray diffraction patterns}~//\href{http://doi.org/10.1038/ncomms2288}{ Nat Commun }
\BibDash
\newblock V.~3. \BibDash
\newblock P.~1276. \BibDash
\newblock 2012.

\bibitem{Neutze:2000kwy}
\selectlanguageifdefined{english}
\BibEmph{Neutze R., Wouts R.,  Weckert E., Hajdu J. } {Potential for biomolecular imaging with femtosecond X-ray pulses}~//\href{https://doi:10.1038/35021099}{ Nature }
\BibDash
\newblock V.~406. \BibDash
\newblock P.~752–-757 . \BibDash
\newblock 2000.

\bibitem{Howells:1984kwy}
\selectlanguageifdefined{english}
\BibEmph{Howells M. R. et al. } {An assessment of the resolution limitation due to radiation-damage in X-ray diffraction microscopy}~//\href{http://doi:10.1016/j.elspec.2008.10.008}{ J. Electron Spectrosc. }
\BibDash
\newblock V.~170.\BibDash
\newblock P.~4–-12 . \BibDash
\newblock 2009.

\bibitem{Gisriel:2019kwy}
\selectlanguageifdefined{english}
\BibEmph{Gisriel C., Coe J., Letrun R. et al. } {Membrane protein megahertz crystallography at the European XFEL }~//\href{https://doi.org/10.1038/s41467-019-12955-3}{ Nat Commun }
\BibDash
\newblock V.~10. \BibDash
\newblock P.~5021 . \BibDash
\newblock 2019.

\bibitem{Pandey:2020kwy}
\selectlanguageifdefined{english}
\BibEmph{Pandey S., Bean R., Sato T. et al.} {Time-resolved serial femtosecond crystallography at the European XFEL}~//\href{http://doi.org/10.1038/s41592-019-0628-z}{ Nat Methods }
\BibDash
\newblock V.~17. \BibDash
\newblock P.~73–-78 . \BibDash
\newblock 2020.

\bibitem{Bielecki:2019kwy}
\selectlanguageifdefined{english}
\BibEmph{Bielecki J., Mancuso A., et al} {Perspectives on single particle imaging with x rays at the advent of high repetition rate x-ray free electron laser sources}~//\href{http://doi:10.1063/4.0000024}{ Struct Dyn.}
\BibDash
\newblock V.~7.  \BibDash
\newblock 2019.


\bibitem{Makarov:2021kwy}
\selectlanguageifdefined{english}
\BibEmph{Makarov D.N., Kharlamova A.A.} {Calculation of the Spectrum of Interaction Model with Macromolecules by the Method of Finding Symmetries }~//\href{https://doi.org/10.1134/S1547477124701437}{ Phys. Part. Nucl. Lett. }
\BibDash
\newblock V.~21. \BibDash
\newblock P.~815--818. \BibDash
\newblock 2024.

\bibitem{Makarov:2021kwy}
\selectlanguageifdefined{english}
\BibEmph{Makarov D.N., Makarova K.A., Kharlamova A.A.} {Specificity of scattering of ultrashort laser pulses by molecules with polyatomic structure }~//\href{https://doi: 10.1038/s41598-022-09134-8}{ Scientific Reports. }
\BibDash
\newblock V.~12. \BibDash
\newblock P.~4976. \BibDash
\newblock 2022.

\end{thebibliography}


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