## Measurements of two reproducible samples at random orientation can be considered as two measurements of the same sample with two different incident beam directions indicated by the vectors **K**

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**Figure 2.** Measurements of two reproducible samples at random orientation can be considered as two measurements of the same sample with two different incident beam directions indicated by the vectors **K**_{i1} and **K**_{i2}. The angles , θ and ψ are Euler's rotation angles. Points A and B are the centres of the corresponding Ewald's spheres. Coordinates on the first and second detector are indicated as *x*, *y* and *x*', *y*', respectively.

**Abstract**

Single-particle diffraction imaging experiments at free-electron lasers (FELs) have a great potential for the structure determination of reproducible biological specimens that cannot be crystallized. One of the challenges in processing the data from such an experiment is to determine the correct orientation of each diffraction pattern from samples randomly injected in the FEL beam. We propose an algorithm (Yefanov *et al* 2010 *Photon Science—HASYLAB Annual Report*) that can solve this problem and can be applied to samples from tens of nanometres to microns in size, measured with sub-nanometre resolution in the presence of noise. This is achieved by the simultaneous analysis of a large number of diffraction patterns corresponding to different orientations of the particles. The algorithm's efficiency is demonstrated for two biological samples, an artificial protein structure without any symmetry and a virus with icosahedral symmetry. Both structures are a few tens of nanometres in size and consist of more than 100 000 non-hydrogen atoms. More than 10 000 diffraction patterns with Poisson noise were simulated and analysed for each structure. Our simulations indicate the possibility of achieving resolution of about 3.3 Å at 3 Å wavelength and incoming flux of 10^{12} photons per pulse focused to 100**×**100 nm^{2}.