^{1,2,3,4,a)}, Ji-Cai Liu (刘纪彩)

^{1,5}, Mau Hsiung Chen (陳茂雄)

^{6}, James P. Cryan

^{2,7}, Li Fang (方力)

^{8}, James M. Glownia

^{2,9}, Matthias Hoener

^{8}, Ryan N. Coffee

^{2,10}and Nora Berrah

^{8}

### Abstract

We devise a theoretical description for the response of nitrogen molecules (N_{2}) to ultrashort and intense x rays from the free electron laser Linac Coherent Light Source (LCLS). We set out from a rate-equation description for the x-rayabsorption by a nitrogen atom. The equations are formulated using all one-x-ray-photon absorption cross sections and the Auger and radiative decay widths of multiply-ionized nitrogen atoms. Cross sections are obtained with a one-electron theory and decay widths are determined from *ab initio* computations using the Dirac-Hartree-Slater (DHS) method. We also calculate all binding and transition energies of nitrogen atoms in all charge states with the DHS method as the difference of two self-consistent field (SCF) calculations (ΔSCF method). To describe the interaction with N_{2}, a detailed investigation of intense x-ray-induced ionization and molecular fragmentation are carried out. As a figure of merit, we calculate ion yields and the average charge state measured in recent experiments at the LCLS. We use a series of phenomenological models of increasing sophistication to unravel the mechanisms of the interaction of x rays with N_{2}: a single atom, a symmetric-sharing model, and a fragmentation-matrix model are developed. The role of the formation and decay of single and double core holes, the metastable states of , and molecular fragmentation are explained.

C.B. would like to thank Nikolai V. Kryzhevoi for fruitful discussions and a critical reading of the paper. We are grateful to Oleg Kornilov and Oliver Gessner for discussions on nitrogen molecules in intense x rays. C.B. was supported by the National Science Foundation (NSF) under Grant Nos. PHY-0701372 and PHY-0449235 and by a Marie Curie International Reintegration Grant within the 7th European Community Framework Program (call identifier: FP7-PEOPLE-2010-RG, proposal No. 266551). J.-C.L. thanks for support by the Fundamental Research Funds for the Central Universities. Additional funding was provided by the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy (DOE), for C.B. under Contract No. DE-AC02-06CH11357 and for L.F., M.H., and N.B. under Contract No. DE-FG02-92ER14299. M.H.C.’s work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344. R.N.C. is supported through the LCLS at the SLAC National Accelerator Laboratory by the U.S. Department of Energy, Office of Basic Energy Sciences. J.P.C. and J.M.G. are supported through both the LCLS and the PULSE Institute for Ultrafast Energy Science at the SLAC National Accelerator Laboratory by the U.S. Department of Energy, Office of Basic Energy Sciences. Portions of this research were carried out at the Linac Coherent Light Source (LCLS) at SLAC National Accelerator Laboratory. LCLS is an Office of Science User Facility operated for the U.S. Department of Energy Office of Science by Stanford University.

I. INTRODUCTION

II. ENERGIES AND DECAY WIDTHS OF MULTIPLY-IONIZED NITROGEN ATOMS

III. ION YIELDS OF A NITROGEN ATOM

IV. ION YIELDS OF A NITROGEN MOLECULE

A. Symmetric-sharing model

B. Fragmentation-matrix model

V. EXPLANATION OF THE OBSERVED PHENOMENA IN TERMS OF ELEMENTAL MOLECULAR PROCESSES

VI. CONCLUSION

### Key Topics

- Photons
- 32.0
- X-ray absorption spectra
- 22.0
- Ionization
- 21.0
- X-ray optics
- 15.0
- X-ray absorption near edge structure
- 13.0

## Figures

Classification of key types of core holes in N_{2} in the localized core hole picture:^{12,13} (a) a singly-occupied spatial core orbital on one of the atoms of N_{2}: a single core hole (SCH); (b) a fully-depleted spatial core orbital on one N atom: a double core hole (DCH) on a single site (ssDCH); and (c) a singly-occupied spatial core orbital on both N atoms: a DCH on two sites (tsDCH).

Classification of key types of core holes in N_{2} in the localized core hole picture:^{12,13} (a) a singly-occupied spatial core orbital on one of the atoms of N_{2}: a single core hole (SCH); (b) a fully-depleted spatial core orbital on one N atom: a double core hole (DCH) on a single site (ssDCH); and (c) a singly-occupied spatial core orbital on both N atoms: a DCH on two sites (tsDCH).

An N atom after irradiation by x rays of constant spatial beam profile with a photon fluence (photon flux integrated over all times) of 8 × 10^{11} μm^{−2} and a Gaussian temporal shape (7) with FWHM duration of 2 fs at varying photon energy ω_{X}. (a) Probabilities *P* _{ j }(ω_{X}) from Eq. (4) to find an N atom in specific charge states: neutral (black), singly (red), doubly (green), triply (blue), quadruply (magenta), quintuply (cyan), sextuply (purple), and septuply (orange). (b) Average charge state from Eq. (6).

An N atom after irradiation by x rays of constant spatial beam profile with a photon fluence (photon flux integrated over all times) of 8 × 10^{11} μm^{−2} and a Gaussian temporal shape (7) with FWHM duration of 2 fs at varying photon energy ω_{X}. (a) Probabilities *P* _{ j }(ω_{X}) from Eq. (4) to find an N atom in specific charge states: neutral (black), singly (red), doubly (green), triply (blue), quadruply (magenta), quintuply (cyan), sextuply (purple), and septuply (orange). (b) Average charge state from Eq. (6).

Ion yields of (a) an N atom and (b) an N_{2} molecule irradiated by x rays of a FWHM duration of 280 fs and a photon energy of 1100 eV. In (a), we show theoretical results from the single atom model of Sec. III for the x-ray flux (9). We assume that only 31% of the nominal pulse energy of 0.26 mJ are actually available in the LCLS experiment (Table I). In (b), we show experimental data for an N_{2} molecule. The comparison of atomic *Y* _{ j } with molecular ion yields for nitrogen clearly demonstrates the impact of molecular processes during and after the interaction with the x rays.

Ion yields of (a) an N atom and (b) an N_{2} molecule irradiated by x rays of a FWHM duration of 280 fs and a photon energy of 1100 eV. In (a), we show theoretical results from the single atom model of Sec. III for the x-ray flux (9). We assume that only 31% of the nominal pulse energy of 0.26 mJ are actually available in the LCLS experiment (Table I). In (b), we show experimental data for an N_{2} molecule. The comparison of atomic *Y* _{ j } with molecular ion yields for nitrogen clearly demonstrates the impact of molecular processes during and after the interaction with the x rays.

Average charge state from N_{2} subject to LCLS x-ray pulses of varying FWHM duration. We show deduced from the experimental ion yields of N_{2} (Fig. 5) for the nominal pulse durations 4, 7, 80, and 280 fs, alongside our calculations; the points are connected by interpolation. Specifically, we plot the experimental as black circles, from a single N atom as red squares (Sec. III), from the symmetric-sharing model as green diamonds (Sec. IV A), and from the fragmentation-matrix model (Sec. IV B) as blue triangles. See Table I for further LCLS pulse parameters.

Average charge state from N_{2} subject to LCLS x-ray pulses of varying FWHM duration. We show deduced from the experimental ion yields of N_{2} (Fig. 5) for the nominal pulse durations 4, 7, 80, and 280 fs, alongside our calculations; the points are connected by interpolation. Specifically, we plot the experimental as black circles, from a single N atom as red squares (Sec. III), from the symmetric-sharing model as green diamonds (Sec. IV A), and from the fragmentation-matrix model (Sec. IV B) as blue triangles. See Table I for further LCLS pulse parameters.

Molecular ion yields of N_{2} subject to LCLS x-ray pulses of varying duration: (a) 4 fs, (b) 7 fs, (c) 80 fs, and (d) 280 fs. Experimental are given by the blue bars and theoretical from the fragmentation-matrix model (Sec. IV B) by the red bars. See Table I for further LCLS pulse parameters.

Molecular ion yields of N_{2} subject to LCLS x-ray pulses of varying duration: (a) 4 fs, (b) 7 fs, (c) 80 fs, and (d) 280 fs. Experimental are given by the blue bars and theoretical from the fragmentation-matrix model (Sec. IV B) by the red bars. See Table I for further LCLS pulse parameters.

The probabilities to find N_{2} during the x-ray pulse in its ground state is given by the black lines; the red lines are the probability for a SCH; the green and blue lines stand for the probability to find a tsDCH and a ssDCH, respectively; the PAPs are indicated by magenta lines. Probabilities are determined for (a) a Gaussian short pulse (7) of a FWHM duration of 4 fs and (b) a Gaussian long pulse of a FWHM duration of 280 fs. See Table I for further LCLS pulse parameters.

The probabilities to find N_{2} during the x-ray pulse in its ground state is given by the black lines; the red lines are the probability for a SCH; the green and blue lines stand for the probability to find a tsDCH and a ssDCH, respectively; the PAPs are indicated by magenta lines. Probabilities are determined for (a) a Gaussian short pulse (7) of a FWHM duration of 4 fs and (b) a Gaussian long pulse of a FWHM duration of 280 fs. See Table I for further LCLS pulse parameters.

## Tables

Fragmentation constants for the most probable channels for the four pulse durations. Here, *f* _{{0, 2}} = 0.74 in all cases [Eq. X]. The LCLS FWHM pulse duration is τ_{X} [Eq. (7)], the LCLS photon energy is 1100 eV, and the actually available pulse energy in the LCLS experiment is *E* _{P}. A nominal pulse energy of 0.15 mJ is specified for 4 fs pulses and 0.26 mJ for the remaining three pulse durations.

Fragmentation constants for the most probable channels for the four pulse durations. Here, *f* _{{0, 2}} = 0.74 in all cases [Eq. X]. The LCLS FWHM pulse duration is τ_{X} [Eq. (7)], the LCLS photon energy is 1100 eV, and the actually available pulse energy in the LCLS experiment is *E* _{P}. A nominal pulse energy of 0.15 mJ is specified for 4 fs pulses and 0.26 mJ for the remaining three pulse durations.

Ratios to find SCH decay, the first tsDCH decay, the first ssDCH decay, and a PAPA process after averaging over a varying numbers of SASE pulses and from a Gaussian pulse (7) with a FWHM duration of τ_{X}. The ratios are probabilities from volume averaging (8) which are renormalized to the sum of the probabilities for (the first) decay of SCHs, tsDCHs, and ssDCHs. See Table I for further LCLS pulse parameters.

Ratios to find SCH decay, the first tsDCH decay, the first ssDCH decay, and a PAPA process after averaging over a varying numbers of SASE pulses and from a Gaussian pulse (7) with a FWHM duration of τ_{X}. The ratios are probabilities from volume averaging (8) which are renormalized to the sum of the probabilities for (the first) decay of SCHs, tsDCHs, and ssDCHs. See Table I for further LCLS pulse parameters.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content