## 3 Single-Particle Imaging with X-ray Lasers
### Light Source and Instruments  
There are seven X-ray instruments at the LCLS. The Coherent X-ray Imaging (CXI) instrument and the Atomic, Molecular & Optical Science (AMO) instrument are the instruments suitable for SPI. The CXI instrument is located in the Far Experimental Hall of LCLS. The hutch is located 440 m far away from the source. The instrument makes use of hard X-ray pulses to perform coherent X-ray imaging and serial femtosecond crystallography experiments. The primary operating photon energy range is 5-11 keV, with some capability for higher harmonics up to 25 keV. Three sample chambers exist with foci of \(\sim\)5 \(\upmu\)m, 1 \(\upmu\)m, and 0.1 \(\upmu\)m being possible. The AMO instrument is located in the Near Experimental Hall of LCLS and the photon energy range is 280-2000 eV. The focusing optics for the AMO instrument are bendable Kirkpatrick-Baez (KB) mirrors and the focused beam size is estimated to be 1.5-5 \(\upmu\)m. In both the CXI and AMO cases, a high-power laser synchronized with the X-ray laser  
Figure 4: Schematic layout of a single-particle imaging measurement using femtosecond XFEL pulses. Nanoparticles and viruses are injected to the XFEL focus by a particle injector or a fixed target system using thin membranes such as Si\({}_{3}\)N\({}_{4}\) and graphene. High dynamic range, high quantum efficiency and large panel detectors are employed to record patterns. In general, five main steps are required for a single-particle imaging experiment. After sample selection and preparation, they are delivered to the XFEL focus and the patterns are collected by the detector. This produces large data sets requiring advanced automatic processing. Hit finding, background subtraction and detector calibrations have to be applied to reduce the data. By using threshold analysis or diffusion maps, single-shot patterns are selected. After orientation determination and phase retrieval, a real-space three-dimensional image can in principle be achieved. is available. Figure 5 shows a simple layout of AMO and CXI instruments. More detailed information can be found from reference [144; 145; 146] and from the LCLS facility website [147; 148]. As mentioned in the setup and experiment procedure section, photon parameters should be determined before the experiment. The selection of the photon energy is very important to single particle imaging. As shown by momentum transfer \(q=4\pi\sin\theta/\lambda\) (\(2\theta\) is the scattering angle), to obtain higher resolution, higher photon energy is needed. On the contrary, for higher photon energy, the wavelength is shorter and the coherent scattering cross-section is weaker. In order to estimate the scattering intensity per shot, a sphere model with uniform electron density can be used [149],  
\[I(q)=I_{0}r_{e}^{2}\Delta\Omega|\Delta\rho|^{2}(4\pi\frac{\sin(Rq)-Rq\cos(Rq)} {q^{3}})^{2} \tag{1}\]  
where \(I_{0}\) is the incident intensity, \(r_{e}\) is the classical electron radius, \(\Delta\rho\) is the difference of the complex electron density of the sphere to the medium and \(R\) is the sphere radius. If we take the complex refractive index into the above equation, we can get the maximum ring intensity [150],  
\[I_{max}(q)=I_{0}r_{e}^{2}\Delta\Omega|\Delta\rho|^{2}16\pi^{2}R^{2}q^{-4} \tag{2}\]  
The intensity for high \(q\) will decrease rapidly. For the same sample size and scattering angle, shorter wavelength will contribute to higher resolution but lower intensity. Therefore, the optimized photon energy should be chosen depends on proposed resolution and sample types. Figure 5: Overview of Atomic, Molecular & Optical Science (AMO) and Coherent X-ray Imaging (CXI) instrument layout. Distances are indicated in meters from the interaction region (IR) for AMO, 1 \(\upmu\)m sample chamber (SC) and in parentheses from the 100 nm SC for CXI. The X-ray beam enters the hutch and passes through the diagnostics (D) and slits (S) and is then focused by KB mirrors. (**a**) An optical laser in-coupling (L-IN) is located 0.4m upstream of the IR.