Small-angle Scattering: More than nice curves. [Guest post by M. Gallagher-Jones]

[ed: Marcus Gallagher-Jones just finished his Ph.D. project on VUV and X-ray lasers for imaging of biological macromolecules]

I was happy to receive an invitation from Brian to write a post for his blog. I can honestly say that I owe a good deal of my knowledge of SAXS from helpful discussions with Brian and from reading LaN. Over the years we’ve shared a country, a workplace, and one more important thing: a scattering geometry. So without further ado I’m delighted to introduce to you the technique which has occupied my time for much of the past four years, Coherent Diffractive Imaging (CDI).

In a typical SAXS experiment the scattering patterns are continuous and diffuse because the photons are scattered from many different coherence volumes in the sample (the photons scattered from different regions add incoherently). The resultant scattering is therefore not solely dependent on the sample morphology. However, if the photons enveloping the sample are all coherent (i.e. are in phase),  the differential constructive and destructive interference of the scattered light produces a pattern of intermittent speckles.

These speckles intrinsically contain more information than a standard pattern as they have a radial dependency; the entire 2D pattern will be dependent on the relative positions of all point scatterers within the sample. For a well isolated object – one for which there are therefore also no interparticle interference effects — this has some very important consequences.

Theoretically, the far-field diffraction pattern of a well isolated object should be directly related to the objects’ 3D electron density via an inverse Fourier transform (IFT). Performing such an IFT is unfortunately not possible from standard diffraction experiments, as not all necessary information can be collected: scattered waves have both an intensity, which can be measured experimentally, and a phase, which cannot. With the phases missing the calculations necessary to compute a real space image from diffraction patterns will contain as many unknown variables as known variables and so solving them becomes a practical impossibility (the “phase problem” of diffraction).

However, in 1952 David Sayre presented a way to overcome this problem [1]. If the scattered intensity could be sampled twice finer than the Nyquist frequency (i.e. twice the inverse of the objects size) then the inverse Fourier transform would now have twice as many known variables as unknowns. In this way it should be possible to overcome the phase problem through iterative methods.

To achieve this “oversampled” condition, the object of interest must be surrounded by a region of zero electron electron density at least twice its size [2]. Phases can then be retrieved using iterative phase retrieval algorithms, originally developed for use in astronomy and electron microscopy [3]. The iterative phase retrieval progress typically follows the following steps:

1. An initial set of random phases will be combined with the measured intensities.
2. An inverse Fourier transform is performed to produce a real space image.
3. A set of constraints are applied to the reconstructed image, typically confining non-negative electron density to a specific ‘support’ region and setting density outside of this area to zero.
4. A Fourier transform is taken of the updated real space image to produce a new diffraction pattern.
5. The phases of the calculated pattern are combined with the measured intensities and the process is repeated.

So what are these reconstructed images and what can they tell us?

The reconstructed images represent a 2D projection of the total electron density of the object under investigation. They can therefore give some understanding of a materials complex internal density variations. Also, as they are projections of the total 3D density, it should be possible to retrieve a full 3D image of the object through combination with tomographic techniques [4].

All of this is not particularly new, and has already been done in electron microscopy for a long time. There are many advantages of using X-ray coherent diffractive imaging, however. Firstly, the higher penetrating power of X-rays allows objects several microns thick to be imaged, without the need for sectioning. Secondly, as the technique is diffraction based, the iterative algorithm takes the place of the objective lens, thereby theoretically only limiting the resolution by the wavelength of the incident radiation. In practice the true resolution is still far from this as CDI is still in its infancy (having been practically demonstrated only in 1999) [5]. There is still plenty of time for development.

As a biologist, I’m interested in CDI’s ability to image whole unstained samples in a near native environment. Interesting events in biology typically occur over a broad range of length scales and so being able to obtain quantitative electron-density information from cellular down to molecular scales is an attractive prospect. That being said, the technique lacks the inherent statistical rigour of SAXS and so, given the shared experimental geometry of the two techniques, I’m also interested in ways in which the two methods can be combined (something I spent a good portion of my PhD exploring [6]).

Eventually it would be nice to see this combination of scattering techniques being used to help fill in some of the current knowledge gaps in structural biology left unanswered by protein crystallography. Given that both SAXS and CDI can be employed on non-crystalline matter they do not have the strict sample limitations of diffraction. How and when such a convergence will happen is something I am not quite sure of yet, but I am certainly heavily invested in trying to make it possible. Hopefully, this post has sparked your interest in this alternative technique for looking at the nano–world with X-rays.

References

1. Sayre, D. Some implications of a theorem due to Shannon. Acta Crystallographica 5, 843–843 (1952).
2. Miao, J., Sayre, D. & Chapman, H. N. Phase retrieval from the magnitude of the Fourier transforms of nonperiodic objects. J. Opt. Soc. Am. B 15, 1662–1669 (1998).
3. Fienup, J. R. Phase retrieval algorithms : a comparison. Applied Optics 21, 2758–2769 (1982).
4. Jiang, H. et al. Quantitative 3D imaging of whole, unstained cells by using X-ray diffraction microscopy. PNAS 107, 11234–11239 (2010).
5. Miao, J., Charalambous, P. & Kirz, J. Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens. Nature 400, 342–344 (1999).
6. Gallagher-Jones, M. et al. Macromolecular structures probed by combining single-shot free-electron laser diffraction with synchrotron coherent X-ray imaging. Nature communications 5, 3798–3798 (2014).