intensity and scattering cross section in elastic and inelastic scattering pdf

Intensity And Scattering Cross Section In Elastic And Inelastic Scattering Pdf

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Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound , is forced to deviate from a straight trajectory by localized non-uniformities including particles and radiation in the medium through which they pass. In conventional use, this also includes deviation of reflected radiation from the angle predicted by the law of reflection. Reflections of radiation that undergo scattering are often called diffuse reflections and unscattered reflections are called specular mirror-like reflections.

Multi-slice simulations of electron diffraction by three-dimensional protein crystals have indicated that structure solution would be severely impeded by dynamical diffraction, especially when crystals are more than a few unit cells thick. In practice, however, dynamical diffraction turned out to be less of a problem than anticipated on the basis of these simulations. Here it is shown that two scattering phenomena, which are usually omitted from multi-slice simulations, reduce the dynamical effect: solvent scattering reduces the phase differences within the exit beam and inelastic scattering followed by elastic scattering results in diffusion of dynamical scattering out of Bragg peaks.

Transmission Electron Microscopy pp Cite as. The electron is a low-mass, negatively charged particle. As such, it can easily be deflected by passing close to other electrons or the positive nucleus of an atom. These Coulomb electrostatic interactions cause electron scattering, which is the process that makes TEM feasible.

Inelastic scattering and solvent scattering reduce dynamical diffraction in biological crystals

Multi-slice simulations of electron diffraction by three-dimensional protein crystals have indicated that structure solution would be severely impeded by dynamical diffraction, especially when crystals are more than a few unit cells thick. In practice, however, dynamical diffraction turned out to be less of a problem than anticipated on the basis of these simulations.

Here it is shown that two scattering phenomena, which are usually omitted from multi-slice simulations, reduce the dynamical effect: solvent scattering reduces the phase differences within the exit beam and inelastic scattering followed by elastic scattering results in diffusion of dynamical scattering out of Bragg peaks.

Thus, these independent phenomena provide potential reasons for the apparent discrepancy between theory and practice in protein electron crystallography. The Nobel Prize in Chemistry for Henderson, Frank and Dubochet who were key contributors to the development of cryo-electron microscopy cryo-EM for acquiring three-dimensional atomic, structural information of biological complexes, confirmed the enormous impact of methods that employ electron scattering for structure determination.

Because of the limited total electron dose that a biological sample can tolerate, the phase contrast of such a sample is weak and image quality is degraded by electron-optical distortions and potential drifts during the exposure, the signal-to-noise ratio SNR of cryo-EM data is poor at high resolution. Measuring electron scattering data from protein crystals in diffraction mode, rather than in imaging microscopy mode, circumvents or reduces several of the phenomena that compromise the signal-to-noise ratio in cryo-EM.

Thus, measuring in diffraction mode can result in a reduction by orders of magnitude of the electron dose required for achieving high resolution when imaging three-dimensional protein crystals. Analysis of published data for lysozyme nanocrystals of similar size and identical space group in either imaging or in diffraction mode under optimal conditions Nederlof et al.

The top-left panel in Fig. Subsequent images of the same crystal still diffracted, but high-resolution spots were already noticeably fading data not shown.

The resolution of the Bragg spots was isotropic, indicating that there was no noticeable beam - or sample drift. The plot in the lower-left panel in Fig. Significant data up to about 3. The small peaks at higher resolution are most likely due to noise fluctuations.

The top right panel in Fig. The plot in the lower-right panel in Fig. Significant diffraction data to 2. Imaging lysozyme nanocrystal in imaging and diffraction mode.

The lower-left panel indicates the signal-to-noise ratio of this imaging data. A peak search routine identified potential Bragg spots above background van Genderen et al. The average peak height of these spots was plotted as a function of resolution with a solid line. The average diffuse background was plotted with a dotted line.

The top right panel shows a diffraction pattern of a lysozyme nanocrystal. The crystal had a very similar size to the crystal of the left panel and had the same space group. The lower-right panel indicates the plot which was calculated in a similar way to the corresponding lower-left plot of the imaging data. For protein crystals, but likely also for non-crystalline biological samples, there are therefore clear benefits of collecting data in diffraction mode.

Multislice calculations indicated that dynamical diffraction would invalidate retrieving the phase information — that is lost in diffraction data, but that is essential for structure interpretation — by conventional crystallographic methods, which are based on kinematic diffraction theory and therefore assumes single elastic scattering. A prime hallmark of dynamical diffraction of a weak phase object is that Friedel pairs can have very different intensities, unlike in kinematic, single scattering diffraction data.

Nevertheless, protein structures have been solved by conventional phasing methods using electron diffraction data of relatively thick crystals e. Hattne et al. We therefore concluded that the theoretical analysis must have been incomplete. According to a classical particle description, upon elastic scattering, the electron can change its direction, but loses virtually no energy, whereas upon inelastic scattering it hardly changes direction, but loses energy and coherency.

The probability of scattering is determined by the atomic scattering cross section. Alternatively, treating electron as a wave, the scattering process can be described by the complex-valued scattering amplitudes. The intensity distribution of a scattered wave in the far-field gives the distribution of probabilities of detecting an electron at a selected location on the detector. Here we extend this analysis to include its effect on the quality of the elastic signal. Like kinematically scattered electrons, dynamically scattered electrons are diffracted in directions determined by the wavelength of the diffracted radiation, the reciprocal lattice parameters and the crystal orientation.

Since these values do not change upon multiple elastic diffraction, both kinematic and dynamic diffraction are focused into Bragg spots. However, the situation may change if at least one of the multiple scattering events was inelastic. Then the electron lost some energy, resulting in a longer wavelength and loss of coherency.

Please note that ignoring inelastic scattering in multislice calculations is not equivalent to filtering out inelastically scattered electrons using an energy filter. For instance, if we ignore inelastic scattering in our calculations, all electrons emerge from a thick sample being scattered dynamically, resulting in highly dynamical diffraction data. In reality, however, virtually all electrons will have scattered inelastically at least once, and would be removed by the energy filter, resulting in no diffraction data whatsoever.

Because it is disordered, it will not contribute significantly to higher resolution crystal diffraction, but instead give rise to a diffuse, mostly radially symmetrical diffraction signal. The signal has therefore frequently been ignored for studies at high resolution, since its signal can be considered to be part of the background when integrating Bragg spots.

Here, we make the point that the diffraction signal of the solvent scattering potential cannot be ignored in multislice calculations. An underlying assumption of the statement that in kinematic, single scatter electron diffraction, the Friedel pairs must have similar intensities, is that the sample is a weak phase object.

However, by ignoring the phase shift caused by the disordered solvent, this principle is violated. For instance, when the interprotein cavities align with the electron beam, the assumption that they do not contain solvent leads to a zero phase shift at these locations, and as a result to a strong phase shift difference between the locations occupied by the protein molecule and cavities. However, in reality the cavities are filled with solvent, and therefore induce a phase shift comparable to the phase shift experienced by the beam passing through protein.

So, ignoring the solvent potential incorrectly converts a protein crystal from a weak phase object into a strong phase object.

This exaggerates the differences within Friedel pairs. Below we provide simulations which include two important phenomena that are not usually included in calculations: inelastic scattering and bulk solvent diffraction. In this section we treat electron scattering as a particle phenomenon.

In elastic scattering, the electron can change direction, whilst its wavelength remains constant within measuring accuracy assuming the particle interpretation. Thus, after an electron has scattered elastically, any subsequent inelastic scattering events will hardly change its scattering angle.

For crystal diffraction, this means that electrons that only scattered inelastically end up in the direct beam. Electrons that scattered elastically and subsequently scattered one or more times inelastically, still end up in the Bragg peak, as they hardly changed direction.

So, for these electrons, inelastic scattering results in a small peak broadening. However, this does not mean that inelastic scattering does not affect diffracted amplitudes. Inelastic scattering is not a coherent event. Concomitant with the energy loss upon inelastic electron scattering, the electron therefore also loses coherency. It can be described by a wavefront with a strongly pronounced amplitude in the direction of the incident electron originating from the location of the inelastic event.

The consequence of this loss of coherency, is that subsequent elastic scattering is no longer coherent with elastically scattered electrons.

So elastic scattering that occurs after an inelastic event is no longer confined to Bragg spots, as its wavefunction no longer interferes laterally with the sample, but only along the direction of the electron. Therefore, its probability distribution has become radially symmetric about an axis defined by the direction of the last elastic scattering event.

For the subsequent discussion we distinguish several combinations of elastic and inelastic scattering events illustrated in Fig. Several types of multiple scattering that each have a different effect on the measured diffracted intensities. Elastic events are indicated by a change in direction, inelastic events are indicated by ovals.

Each combination has an associated probability, determined by the elastic and inelastic scattering cross sections and the thickness of the sample.

Deviations from translational symmetry or phonons cause incoherencies within the sample, resulting in diffusion of the diffraction data out of Bragg spots. Our discussion focuses instead on incoherencies of the beam, that are induced by the sample which could be fully or only partially ordered. For a disordered sample, the probabilities of these various types of multiple scattering can be calculated by assuming the scattering to be a Poisson process, that is exclusively determined by atomic scattering cross sections.

Such a Poisson process requires scattering to have time interval invariance: the probability of an electron being scattered should be independent of whether or when it has already been scattered before. This assumption is clearly not valid when atoms line up along a crystal lattice in tight columns parallel to the electron beam, such is the case for crystals with small unit cells that are aligned with the electron beam. In that case, if an electron encounters an atom in the column, it will certainly be encountering another atom further down, since high-energy electron scattering amplitudes have a narrow angular distribution.

This results in a process that has been referred to as electron channelling and enhances the dynamic effects. In imaging, it results in a thickness dependent modulation of observed image intensity at the projected positions of aligned columns e. However, in crystals of biological molecules, the atoms do not line up.

This allows us to model the probability of scattering P s z after traversing a sample over a distance z in nm as an exponential decay:. We assumed an average specific density for a protein of 1. The calculation shows that a protein on average contains about atoms per nm 3 including hydrogens.

Since proteins are hydrated, the contribution of solvating water also needs to be considered. Water has a fractional atomic composition of H 0. Similar considerations lead to a fractional atomic composition for nucleic acid of H 0. These values indicate that biological samples tend to scatter less elastically than was presumed in an earlier analysis: only one in five scattering events is elastic, instead of one in three.

The main reason for this discrepancy is that biological samples contain significant amounts of hydrogen atoms, for which only one in twenty scattering events is elastic, and that the earlier analysis approximated a biological sample as only containing carbon.

Using the atomic elastic and inelastic scattering factors for the average biological atom, the probabilities of multiple elastic and inelastic scattering as a function of sample thickness can now be calculated from a system of differential equations. First, we define:. Multiple scattering in an amorphous material can be modelled analytically using a Poisson distribution e. However, this equation is independent of the order of the elastic and inelastic events, while this order is relevant for the spatial distribution of the observed diffraction.

So instead of using an analytic approach, we used the finite difference method that can take the order of the events into account. The probabilities of the various types of single and multiple scattering as illustrated in Fig. Such unscattered electrons end up in the direct beam. Previously unscattered electrons that scatter inelastically within the slice d z contribute to P inc z. Electrons that have only scattered inelastically, but that scatter elastically within the slice d z reduce P inc z.

Electrons that only scattered inelastically end up in the central beam. Such electrons end up in a Bragg spot and can be used for structure solution using established, kinematic crystallographic theory.

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Issue contents. Article statistics. Download PDF of article. Acta Cryst. Towards quantitative treatment of electron pair distribution function. Gorelik , R. Neder , M.

Scattering and Diffraction

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Scattering and Diffraction

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