Effect of two-particle correlations on x-ray coherent diffractive imaging studies performed with continuum models

Coherent diffraction imaging (CDI) of single molecules at atomic resolution is a major goal for the x-ray free-electron lasers (XFELs). However, during an imaging pulse, the fast laser-induced ionization may strongly affect the recorded diffraction pattern of the irradiated sample. The radiation tolerance of the imaged molecule should then be investigated a priori with a dedicated simulation tool. The continuum approach is a powerful tool for modeling the evolution of irradiated large systems consisting of more than a few hundred thousand atoms. However, this method follows the evolution of average single-particle densities, and the experimentally recorded intensities reflect the spatial two-particle correlations. The information on these correlations is then inherently not accessible within the continuum approach. In this paper we analyze this limitation of continuum models and discuss the applicability of continuum models for imaging studies. We derive a formula to calculate scattered intensities (including both elastic and inelastic scattering) from the estimates obtained with a single-particle continuum model under conditions typical for CDI studies with XFELs. We demonstrate through numerical simulations that it describes the scattered signal with good accuracy. Two-particle correlation effects manifest themselves only in the region of low momentum transfers, together with the effects of the finite size of the sample. We also show that inelastic scattering on bound electrons can have a significant impact on the measured intensities: it contributes to the background that reduces the contrast of the recorded image. This effect is even more pronounced at larger momentum transfers. Therefore, whereas inelastic scattering can be neglected for nanocrystals, where Bragg scattering dominates, and in experiments imaging single objects at low resolution, it should be taken into account when planning atomic resolution imaging of nonperiodic samples. Finally, we show the effect of the electronic damage on the recorded total signal. Progressing damage does not change the positions of intensity peaks that correspond to the (fixed) positions of imaged ions. It only changes the contrast between intensity minima and maxima, which reduces the image contrast. Our results have implications for imaging-oriented studies of radiation damage performed with continuum models, as they define the limits of applicability of these models for CDI simulations.

Figure 1

Average ion-charge-state populations within the imaged system as a function of time.

Figure 2

Time integrated intensity from diffractive scattering only by bound electrons as a function of the momentum transfer, $q$, along a random axis in reciprocal space. We show (i) the intensity constructed from average single particle densities ${\mathcal{I}}_b^C$, (ii) the scattered intensity ${\langle {\mathcal{I}}_{bb}\rangle }_R$, and (iii) the estimated uncorrelated intensity ${\langle {\mathcal{I}}_{bb}^{\mathrm{uncorr}}\rangle }_R$. The curves (ii) and (iii) almost overlap.

Figure 3

Time integrated intensity from diffractive scattering by trapped electrons as a function of the momentum transfer $q$. The electron temperature was set to 20 eV, and an external harmonic potential was applied. We show (i) the intensity constructed from average single-particle densities ${\mathcal{I}}_t^C$, (ii) the scattered intensity ${\langle {\mathcal{I}}_{tt}\rangle }_R$, (iii) the estimated uncorrelated intensity ${\langle {\mathcal{I}}_{tt}^{\mathrm{uncorr}}\rangle }_R$, (iv) the estimated correlated intensity ${\langle {\mathcal{I}}_{tt}^{\mathrm{corr}}\rangle }_R$, and (v) the intensity obtained from the theory of homogeneous, infinite-size, weakly coupled plasmas ${\mathcal{I}}_t^{\mathrm{DH}}$.

Figure 4

Time integrated intensity from diffractive scattering by trapped electrons as a function of the momentum transfer $q$. The electron temperature was set to 100 eV, and an external harmonic potential was applied. We show (i) the intensity constructed from average single-particle densities ${\mathcal{I}}_t^C$, (ii) the scattered intensity ${\langle {\mathcal{I}}_{tt}\rangle }_R$, (iii) the estimated uncorrelated intensity ${\langle {\mathcal{I}}_{tt}^{\mathrm{uncorr}}\rangle }_R$, (iv) the estimated correlated intensity ${\langle {\mathcal{I}}_{tt}^{\mathrm{corr}}\rangle }_R$, and (v) the intensity obtained from the theory of homogeneous, infinite-size, weakly coupled plasmas ${\mathcal{I}}_t^{\mathrm{DH}}$.

Figure 5

Average density of an electron system containing 200 electrons (solid lines) at temperatures 20 eV and 100 eV, with an external harmonic potential applied. Dashed lines show the radius and density values used for evaluating Eqs. (19) and (30).

Figure 6

Time integrated intensity from diffractive scattering by trapped electrons as a function of the momentum transfer $q$. The electrons were moving in the field of positive granular ions. The electron temperature was set to 20 eV. We show (i) the intensity constructed from average single-particle densities ${\mathcal{I}}_t^C$, (ii) the scattered intensity ${\langle {\mathcal{I}}_{tt}\rangle }_R$, (iii) the estimated uncorrelated intensity ${\langle {\mathcal{I}}_{tt}^{\mathrm{uncorr}}\rangle }_R$, (iv) the estimated correlated intensity ${\langle {\mathcal{I}}_{tt}^{\mathrm{corr}}\rangle }_R$, and (v) the intensity obtained from the Debye-Hückel theory ${\mathcal{I}}_t^{\mathrm{DH}}$.

Figure 7

Time integrated intensity from diffractive scattering by bound and trapped electrons as a function of the momentum transfer $q$ along a random axis in reciprocal space. We show (i) the intensity constructed from average single-particle densities ${\mathcal{I}}^C$, (ii) the full scattered intensity ${\langle \mathcal{I}\rangle }_R$, (iii) the estimated uncorrelated intensity ${\langle {\mathcal{I}}^{\mathrm{uncorr}}\rangle }_R$, and (iv) the estimated correlated intensity ${\langle {\mathcal{I}}_{tt}^{\mathrm{corr}}\rangle }_R$. The curves for intensities (ii), (iii), and (iv) almost overlap.

Figure 8

The effect of inelastic scattering and damage on the scattered intensity. We show: (a) the signal from the ‘ideal’ (undamaged) sample without inelastic scattering ${\mathcal{I}}_{\mathrm{ideal}}^{\mathrm{elast}}$, and with inelastic scattering ${\mathcal{I}}_{\mathrm{ideal}}^{}$, (b) the signal from the ‘ideal’ (undamaged) sample ${\mathcal{I}}_{\mathrm{ideal}}^{}$, and the signal from the damaged sample ${\langle \mathcal{I}\rangle }_R$, both including the inelastic scattering component.