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DelBlock(SO2.aux,54,1)
  -- \@writefile{toc}{\contentsline {subsection}{\numberline {B}Theoretical Calculations - localized/ delocalized model}{3}{}}


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  ++ \@writefile{toc}{\contentsline {subsection}{\numberline {B}Theoretical Calculations - Symmetry broken and localized picture}{3}{}}


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  -- \citation{Gejo_2003}
  -- \citation{Sze_1987}


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  ++ \citation{Gejo_2003}


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  ++ \citation{Sze_1987}


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  ++ \citation{Gejo_2003}
  ++ \citation{Gejo_2003}


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  -- \@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Coincidence measurements}{5}{}}


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  -- \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Electron ion ion coincidence (PE(PI)$^2$CO) maps of the two-body dissociation channel O$^+$+SO$^+$. Each panel displays an ion pair and as insets the projection onto the time axes are shown. Also indicated in each panel is the slope of the coincidence plot. The measurement is performed with the spectrometer parallel to the electric field polarization vector of the incident synchrotron radiation.}}{7}{}}
  -- \newlabel{PEPIPICO_Op_SOp}{{6}{7}{}{}{}}
  -- \newlabel{PEPIPICO}{{IV\tmspace  +\thinmuskip {.1667em}A\tmspace  +\thinmuskip {.1667em}3}{7}{}{}{}}
  -- \@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Electron-ion-ion Coincidences}{7}{}}
  -- \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Electron ion ion coincidence (PE(PI)$^2$CO) maps for the three-body dissociation channel O$^+$+S$^+$. In each panel insets of the projection onto the time axes are shown as well as the slope of the coincidence plot. Also the slope of each coincidence plot is indicated. The measurement is performed with the spectrometer parallel to the electric field polarization vector of the incident synchrotron radiation.}}{7}{}}
  -- \newlabel{Sp_peaks}{{7}{7}{}{}{}}


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  -- \citation{Eland_PEPIPICO_1987}
  -- \citation{Eland_1991}
  -- \citation{Simon_1993}


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  ++ \@writefile{lof}{\contentsline {figure}{\numberline {6}{\ignorespaces Electron ion ion coincidence (PE(PI)$^2$CO) maps of the two-body dissociation channel O$^+$+SO$^+$. Each panel displays an ion pair and as insets the projection onto the time axes are shown. Also indicated in each panel is the slope of the coincidence plot. The measurement is performed with the spectrometer parallel to the electric field polarization vector of the incident synchrotron radiation.}}{7}{}}
  ++ \newlabel{PEPIPICO_Op_SOp}{{6}{7}{}{}{}}
  ++ \newlabel{PEPIPICO}{{IV\tmspace  +\thinmuskip {.1667em}A\tmspace  +\thinmuskip {.1667em}3}{7}{}{}{}}
  ++ \@writefile{toc}{\contentsline {subsubsection}{\numberline {3}Electron-ion-ion Coincidences}{7}{}}
  ++ \newlabel{seq_diss}{{7}{7}{}{}{}}
  ++ \citation{Wiesner_2006}


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  -- \newlabel{seq_diss}{{7}{8}{}{}{}}


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  ++ \@writefile{lof}{\contentsline {figure}{\numberline {7}{\ignorespaces Electron ion ion coincidence (PE(PI)$^2$CO) maps for the three-body dissociation channel O$^+$+S$^+$. In each panel insets of the projection onto the time axes are shown as well as the slope of the coincidence plot. Also the slope of each coincidence plot is indicated. The measurement is performed with the spectrometer parallel to the electric field polarization vector of the incident synchrotron radiation.}}{8}{}}
  ++ \newlabel{Sp_peaks}{{7}{8}{}{}{}}


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  -- \@writefile{toc}{\contentsline {subsubsection}{\numberline {1}The Configuration Interaction Calculations}{8}{}}


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  -- \@writefile{toc}{\contentsline {subsubsection}{\numberline {2}Ion-ion Coincidences}{8}{}}
  -- \@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces "PIPICO" data of the SO$^+$+O$^+$ fragments, measured: (a) below resonances at 500\nobreakspace  {}eV (b) at feature\nobreakspace  {}A (c) at feature\nobreakspace  {}B (d) at feature\nobreakspace  {}C (e) at feature\nobreakspace  {}D and (f) above resonances at 542\nobreakspace  {}eV. In (c)-(e) the line fitting with the experimental points is the sum of the two contributing profiles.}}{9}{}}
  -- \newlabel{PIPICO}{{8}{9}{}{}{}}


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  ++ \@writefile{lof}{\contentsline {figure}{\numberline {8}{\ignorespaces "PIPICO" data of the SO$^+$+O$^+$ fragments, measured: (a) below resonances at 500\nobreakspace  {}eV (b) at feature\nobreakspace  {}A (c) at feature\nobreakspace  {}B (d) at feature\nobreakspace  {}C (e) at feature\nobreakspace  {}D and (f) above resonances at 542\nobreakspace  {}eV. In (c)-(e) the line fitting with the experimental points is the sum of the two contributing profiles.}}{10}{}}
  ++ \newlabel{PIPICO}{{8}{10}{}{}{}}
  ++ \@writefile{toc}{\contentsline {section}{\numberline {V}Summary}{10}{}}
  ++ \@writefile{toc}{\contentsline {section}{\numberline {}Acknowledgments}{10}{}}
  ++ \@writefile{toc}{\contentsline {section}{\numberline {}References}{10}{}}


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  -- \@writefile{toc}{\contentsline {section}{\numberline {}References}{10}{}}


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  -- \subsection{Theoretical Calculations - localized/ delocalized model}


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  ++ \subsection{Theoretical Calculations - Symmetry broken and localized picture}


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  -- In the case of localized alignment , the reader should refer to \cite{Wiesner_2006}. We extend here the formalism for the case of non delocalized alignment, where the spectrometer axis is parallel to the polarization vector of the linearly polarized radiation. The intensity as a function of the amount of momentum along the spectrometer axis, $p_0$, for the different transition symmetries is given by \cite{Lindgren_thesis}:    


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  ++ In the case of symmetry breaking in alignment due to valence localization, the reader should refer to \cite{Wiesner_2006}. We extend here the formalism for the case of symmetry adapted alignment, where the spectrometer axis is parallel to the polarization vector of the linearly polarized radiation. The intensity as a function of the amount of momentum along the spectrometer axis, $p_0$, for the different transition symmetries is given by \cite{Lindgren_thesis}:    


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  -- In the next paragraphs we will look further into fragmentation mechanisms studying electron-ion/ions-coincidence measurements.  We have performed coincidence measurements at the energies indicated in Fig.~\ref{TIY}. In a first section information on the dissociation pathways and dynamics are investigated. These results lead us to study the alignment of the molecules upon fragmentation, as has been done previously \cite{Lindgren_2005, Burmeister_2006}, and compare it to the ozone case \cite{Wiesner_2006}.


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  ++ In the next paragraphs we will look further into fragmentation mechanisms studying electron-ion/ions-coincidence measurements.  We have performed coincidence measurements at the energies indicated in Fig.~\ref{TIY}. In a first section dissociation pathways, fragment ion angular anisotropy and dynamics are investigated. These results lead us to study the core hole  localization effect on the angular distribution of the molecules upon fragmentation, as has been done previously \cite{Lindgren_2005, Burmeister_2006}, and compare it to the ozone case \cite{Wiesner_2006}.


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  -- In the peak~A spectrum, Fig.~\ref{combined_hor_ver}~(d)-(f), the overall trend is that the measurements with the spectrometer parallel to the polarization of the synchrotron radiation have a narrower peak distribution than in the perpendicular measurement. Thus there is less momentum distributed parallel to the polarization direction of the radiation than perpendicular. This suggests an alignment with the molecular plane perpendicular to the polarization direction. This is consistent with an alignment along the molecular $x$-axis and thus an alignment with B$_1$ symmetry given by eqn~(\ref{B_1}), as predicted by the CI-calculations.


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  ++ In the peak~A spectrum, Fig.~\ref{combined_hor_ver}~(d)-(f), the overall trend is that the measurements with the spectrometer parallel to the polarization of the synchrotron radiation have a narrower peak distribution than in the perpendicular measurement. Thus there is less momentum distributed parallel to the polarization direction of the radiation than perpendicular. This suggests an alignment with the molecular plane perpendicular to the polarization direction. This is consistent with an alignment along the molecular $x$-axis and thus an alignment with B$_1$ symmetry given by eqn~(\ref{B_1}), in agreement with previous assignment \cite{Gejo_2003}.


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  -- For the transitions at peak~B, C and D alignment should be along the molecular $z$ or $y$ axes for polarization aligned in the molecular plane. From this we expect more (or at least the same) momentum distributed parallel to the direction of polarization compared to perpendicular. The measurements in  Fig.~\ref{combined_hor_ver}~(g)-(o) confirm this.  The S$^+$ fragment, however, shows another pattern which certainly arises from another dissociation mechanism which will be discussed in more detail in connection with other data.


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  ++ The measurements in  Fig.~\ref{combined_hor_ver}~(g)-(o), i.e. the transitions at peak~B, C and D, shows that more (or at least the same) momentum is distributed parallel to the direction of polarization compared to perpendicular. This is still in agreement with \cite{Gejo_2003} where alignment of the neutral excited states is expected to be along the molecular $z$ or $y$ axes and it is first hints that fast dissociation mechanism occurs.  The S$^+$ fragment, however, shows specific pattern which certainly arises from another dissociation mechanism which will be discussed in more detail in following.


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  ++ The elliptical shape of the contour plots provides information ondissociation. As discussed by Eland \cite{Eland_PEPIPICO_1987} and Simon \cite{Simon_1991, Simon_1993} multi-step dissociation where the time interval between dissociation steps is long enough for the molecule to rotate exhibit a rectangular shape in PE(PI)$^2$CO plots. All of the plots in Figs~\ref{PEPIPICO_Op_SOp} and \ref{Sp_peaks} have an overall elliptical shape, indicating little rotation before dissociation. Thus the induced alignment is preserved as it was suggested by the mass spectra.


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  -- The contour maps in Fig.~\ref{PEPIPICO_Op_SOp}~(c)-(e) exhibit two maxima along the diagonal. This is characteristic of a dissociation with the main fragment momenta imparted along the spectrometer axis. The molecule is thus in these cases aligned so that most of the momentum from the bond break is directed parallel to the light polarization vector, commensurate with A$_1$ or B$_2$ symmetry. This is consistent with the mass spectra and the CI-calculations. The distribution in plot Fig.~\ref{PEPIPICO_Op_SOp}~(b) exhibits a single peak characteristic of dissociations where most of the momentum is directed perpendicular to the measurement plane indicating B$_1$ symmetry. This is again consistent with the mass spectra and the CI-calculations. In Fig.~\ref{Sp_peaks} the "parallel" alignment of the S-O bond break is clearly seen for peaks~B, C and D again in agreement with the mass spectra.


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  ++ The contour maps in Fig.~\ref{PEPIPICO_Op_SOp}~(c)-(e) exhibit two maxima along the diagonal. This is characteristic of a dissociation with the main fragment momenta imparted along the spectrometer axis. The molecule is thus in these cases aligned so that most of the momentum from the bond break is directed parallel to the light polarization vector, commensurate with A$_1$ or B$_2$ symmetry. This is consistent with the mass spectra. The distribution in plot Fig.~\ref{PEPIPICO_Op_SOp}~(b) exhibits a single peak characteristic of dissociations where most of the momentum is directed perpendicular to the measurement plane indicating B$_1$ symmetry. This is again consistent with the mass spectra. In Fig.~\ref{Sp_peaks} the "parallel" alignment of the S-O bond break is clearly seen for peaks~B, C and D again in agreement with the mass spectra.


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  -- The elliptical shape of the contour plots provides information on multi step dissociation. As discussed by Eland \cite{Eland_PEPIPICO_1987} and Simon \cite{Simon_1991, Simon_1993} multi-step dissociation where the time interval between dissociation steps is long enough for the molecule to rotate exhibit a rectangular shape in PE(PI)$^2$CO plots. All of the plots in Figs~\ref{PEPIPICO_Op_SOp} and \ref{Sp_peaks} have an overall elliptical shape, indicating little rotation before dissociation. Thus the induced alignment is preserved.
  -- 


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  -- The contour plots can illuminate specific dissociation mechanisms. For the two-body dissociation channel O$^+$+SO$^+$ a slope of -1 is is clearly seen in Fig.~\ref{PEPIPICO_Op_SOp}. The three-body dissociation channel O$^+$+S$^+$+O shown in Fig.~\ref{Sp_peaks} shows several slopes in the plots. Below resonance and at peaks~A and D a slope of -1.0 is seen whereas at peaks~B and C a slope of -0.7 is seen. The -0.7 slope fits perfectly with the secondary decay process:


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  ++ The PEPIPICO spectra can illuminate specific dissociation mechanisms. For the two-body dissociation channel O$^+$+SO$^+$ a slope of -1 is is clearly seen in Fig.~\ref{PEPIPICO_Op_SOp}. The three-body dissociation channel O$^+$+S$^+$+O shown in Fig.~\ref{Sp_peaks} shows several slopes in the plots. From the elliptical contour plot, one should expect a concerted dissociation where the two bonds breaks simultaneously. The use of an impulsive model can certainly enlighten the influence of the valence repulsion force (due to the removal of electrons), however  the particular evolution of the slopes from peak A to D lead us however to use a classical fast step-wise picture to understand the dynamics within the very short timeframe of the dissociation. Indeed, below resonance and at peaks~A and D a slope of -1.0 is seen whereas at peaks~B and C a slope of -0.7 is seen. The -0.7 slope fits perfectly with a fast step wise decay process, such as:


DelBlock(SO2.tex,220,1)
  -- where a slope of -0.7 ($-m_\text{S}/m_\text{SO}$) is expected \cite{Eland_PEPIPICO_1987, Eland_1991, Simon_1993}. In the process above $U_1$ and $U_2$ are the kinetic energy released in each step. We can also conclude that $U_1>>U_2$ from the exact agreement with the -0.7 slope \cite{Lindgren_thesis}. The $\sim-1.0$ slope indicates that the dissociation mechanism is either deferred charge separation, where first a neutral oxygen atom is ejected from the doubly charged molecule which then dissociates into S$^+$ and O$^+$, or a fast concerted dissociation where the two bonds break "simultaneously".


AddBlock(SO2.tex,220,1)
  ++ where a slope of -0.7 ($-m_\text{S}/m_\text{SO}$) is expected. In the process above $U_1$ and $U_2$ are the kinetic energy released in each step. We can also conclude that $U_1>>U_2$ from the exact agreement with the -0.7 slope \cite{Lindgren_thesis}. The $\sim-1.0$ slope seems to indicate that the dissociation mechanism is like a fast deferred charge separation, where first a neutral oxygen atom is ejected from the doubly charged molecule which then dissociates into S$^+$ and O$^+$. Thus, studying the sequence of measurements from peak~B to peak D we see that with increasing photon energy the neutral evaporation seems to become faster. Even-though it is hard to get deeper information from these measurements, the validity of such step-wise picture would point out that the dissociation timeframe is long enough let the molecule slightly vibrate.


DelBlock(SO2.tex,222,4)
  -- Studying the sequence of measurements from peak~B to peak D we see that with increasing photon energy the dissociation dynamics changes, the neutral evaporation becomes faster. It is hard to tell from these measurements if the neutral evaporation is faster than the charge separation (deferred charge separation) or if it occurs at the same time (fast concerted dissociation).
  -- 
  -- 
  -- 


AddBlock(SO2.tex,224,2)
  ++ The 2 body dissociation is certainly on of the most sensitive decay channel of a single SO bond breaking. In the light of our results showing that the dissociation is fast enough to preserve the alignment of the core hole excited states, we will discuss in more details the influence of the core hole localization and valence localization/delocalization as it has been done for ozoneÊ\cite{Wiesner_2006}.
  ++ 


AddBlock(SO2.tex,293,1)
  ++ {\bf Comparison to O3???}
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Core localization and sigma* delocalization in the O-1s core-excited sulfur dioxide

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