## 1 Introduction  
An X-ray free electron laser (XFEL) produces X-rays of unprecedented brilliance of about 10 billion times what was previously possible. As such, it has given rise to the speculation that it may be possible to determine the structures of uncrystallized individual biomolecules [1]. Although the ultimate aim is to determine the structures of biomolecules, it would be helpful to demonstrate the feasibility of the approach to simpler objects initially. In this vein, there has been some work already on reconstructing prolate spheroids [2] of metallic particles. What such experiments demonstrate is the feasibility of reconstructing the structure of particles of random unknown orientations. The aim of the present paper is to show that reconstruction of the structure of such particles is possible even with two particles in independent random orientations contributing to a single diffraction pattern. If it can be demonstrated for two randomly oriented nanoparticles, and we think that since the angular correlations seem to be identical, the number of identical particles may be increased without limit. In this respect, the method has some similarities with one proposed in the 1970s by [PERSON] [3] to generalize the methods then current for small angle X-ray scattering (SAXS). In SAXS, an ensemble of many particles is suspended in solution in random orientations. A major problem with SAXS is that one attempts to recover the structure of a particle from a set of experimental data consisting of a single line plot \(I(q)\). From the spherical harmonic expansion of a crude model of the particle, its shape is normally obtained by SAXS techniques. [PERSON]'s innovation was to point out that if an experiment could be performed that measures the diffraction patterns on a time scale shorter than the rotational diffusion time of the molecules, extra 3D information on the molecular structure could be obtained from the correlations in the angular variations of the measured intensities. There has also been work on determining the structure of Au nanoparticles to atomic resolution [4] by this technique. A major advantage of this method, if feasible, is the ability to determine the detailed structure of an ensemble of randomly oriented molecules. One would thereby be able to use the intensity enhancement caused by working with an ensemble of molecules rather than a single one, and yet not have to crystallize the molecules so that they are all in exactly the same orientation. [PERSON] et al. [5] have shown that even though there is nothing to be gained by having more particles per diffraction pattern since the noise goes up in the same proportion as the signal, one of the other results of their own analysis is that the noise can be reduced compared to the signal by averaging over many diffraction patterns. It should be pointed out that the need to avoid crystallization was the original aim of XFEL studies of biomolecules. If the experiment we propose is possible, it may be possible to avoid crystallization in structural studies and still avoid the very low scattered intensities that are inevitable in single particle studies even with an X-ray free electron laser (XFEL). An experiment has recently been reported in which the structure of the mimivirus has been determined experimentally by a variant of the single particle methods described earlier [6]. At least for an icosahedral particle, as the mimivirus largely is, it has been previously shown by us [7] that it is possible to determine the structure of the particle by an analogous method to that described here from simulated diffraction patterns. In an experiment to recover time-resolved structural variations, starting from a knowledge of a nearby structure, using many of the same quantities, namely, the pair correlations, have also been shown to be capable of recovering time-resolved changes in a structure from the knowledge of a closely related structure of a single molecule in realistic simulations, including shot noise [8].