Mosaicity and integrated reflection intensity

Drenth, J.

doi:10.1107/97809553602060000658

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 2.1, pp. 58-59 | 1 | 2 |

Section 2.1.6. Mosaicity and integrated reflection intensity

J. Drenth^a ^*

^a Laboratory of Biophysical Chemistry, University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
Correspondence e-mail: j.drenth@chem.rug.nl

2.1.6. Mosaicity and integrated reflection intensity

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Crystals hardly ever have a perfect arrangement of their molecules, and crystals of macromolecules are certainly not perfect. Their crystal lattices show defects, which can sometimes be observed with an atomic force microscope or by interferometry. A schematic but useful way of looking at non-perfect crystals is through mosaicity; the crystal consists of a large number of tiny blocks. Each block is regarded as a perfect crystal, but the blocks are slightly misaligned with respect to each other. Scattering from different blocks is incoherent. Mosaicity causes a spread in the diffracted beams; when combined with the divergence of the beam from the X-ray source, this is called the effective mosaic spread. For the same crystal, effective mosaicity is smaller in a synchrotron beam with its lower divergence than in the laboratory. Protein crystals usually show a mosaic spread of 0.25–0.5°. Mosaic spread increases due to distortion of the lattice; this can happen as a result of flash freezing or radiation damage, for instance.

In Section 2.1.4.5, it was stated that the amplitude of the wave scattered by a crystal is proportional to the structure-factor amplitude [|F|] and that its intensity is proportional to $[|F|^{2}]$ . Of course, other factors also determine the intensity of the scattered beam, such as the wavelength, the intensity of the incident beam, the volume of the crystal etc. The intensity integrated over the entire region of the diffraction spot hkl is $[I_{\rm int} (hkl) = {\lambda^{3} \over \omega V^{2}} \left({e^{2} \over mc^{2}}\right)^{2} V_{\rm cr} I_{o} L P T |F(hkl)|^{2}. \eqno(2.1.6.1)]$

In equation (2.1.6.1), we recognize $[I_{o} (e^{2}/mc^{2})^{2}]$ as part of the Thomson scattering for one electron, $[I_{\rm el} = I_{o} (e^{2}/mc^{2})^{2} \sin^{2} \varphi]$ [equations (2.1.4.1a ) and (2.1.4.1b)] per unit solid angle. $[V_{\rm cr}]$ is the volume of the crystal and V is the volume of the unit cell. It is clear that the scattered intensity is proportional to the volume of the crystal. The term $[1/V^{2}]$ can be explained as follows. In a mosaic block, all unit cells scatter in phase. For a given volume of the individual blocks, the number of unit cells in a mosaic block, as well as the scattering amplitude, is proportional to [1/V] . The scattered intensity is then proportional to $[1/V^{2}]$ . Because of the finite reflection width, scattering occurs not only for the reciprocal-lattice point when it is on the Ewald sphere, but also for a small volume around it. Since the sphere has radius $[1/\lambda]$ , the solid angle for scattering, and thus the intensity, is proportional to $[1/(1/\lambda)^{2} = \lambda^{2}]$ .

However, in equation (2.1.6.1), the scattered intensity is proportional to $[\lambda^{3}]$ . The extra λ dependence is related to the time t it takes for the reciprocal-lattice `point' to pass through the surface of the Ewald sphere. With an angular speed of rotation ω, a reciprocal-lattice point at a distance [1/d] from the origin of the reciprocal lattice moves with a linear speed $[v = (1/d) \omega]$ if the rotation axis is normal to the plane containing the incident and reflected beam. For the actual passage through the surface of the Ewald sphere, the component perpendicular to the surface is needed: $[v_{\perp} = (1/d)\omega \cos \theta = \omega \sin 2\theta/\lambda]$ . Therefore, the time t required to pass through the surface is proportional to $[(1/\omega) (\lambda/\sin 2\theta)]$ . This introduces the extra λ term in equation (2.1.6.1) as well as the ω dependence and a $[1/\sin 2\theta]$ term. The latter represents the Lorentz factor L. It is a geometric correction factor for the hkl reflections; here it is $[1/\sin 2\theta]$ , but it is different for other data-collection geometries.

The factor P in equation (2.1.6.1) is the polarization factor. For the polarized incident beam used in deriving equation (2.1.4.1a ), $[P = \sin^{2} \varphi]$ , where φ is the angle between the polarization direction of the beam and the scattering direction. It is easy to verify that $[\phi = 90^{\circ} - 2\theta]$ , where θ is the reflecting angle (Fig. 2.1.4.9). P depends on the degree of polarization of the incident beam. For a completely unpolarized beam, $[P = (1 + \cos^{2} 2\theta)/2]$ .

In equation (2.1.6.1), T is the transmission factor: [T = 1 - A] , where A is the absorption factor. When X-rays travel through matter, they suffer absorption. The overall absorption follows Beer's law: $[I = I_{o} \exp (- \mu d),]$ where $[I_{o}]$ is the intensity of the incident beam, d is the path length in the material and µ is the total linear absorption coefficient . µ can be obtained as the sum of the atomic mass absorption coefficients of the elements $[(\mu_{m})_{i}]$ : $[\mu = \rho \textstyle\sum\limits_{i} g_{i} (\mu_{m})_{i},]$ where ρ is the density of the absorbing material and $[g_{i}]$ is the mass fraction of element i.

Atomic mass absorption coefficients $[(\mu_{m})_{i}]$ for the elements are listed in Tables 4.2.4.3 (and 4.2.4.1 ) of IT C (2004) as a function of a large number of wavelengths. The absorption is wavelength-dependent and is generally much stronger for longer wavelengths. This is the result of several processes. For the X-ray wavelengths applied in crystallography, the processes are scattering and photoelectric absorption. Moreover, at the reflection position, the intensity may be reduced by extinction.

Scattering is the result of a collision between the X-ray photons and the electrons. One can distinguish two kinds of scattering: Compton scattering and Rayleigh scattering. In Compton scattering , the photons lose part of their energy in the collision process (inelastic scattering), resulting in scattered photons with a lower energy and a longer wavelength. Compton scattering contributes to the background in an X-ray diffraction experiment. In Rayleigh scattering , the photons are elastically scattered, do not lose energy, and leave the material with their wavelength unchanged. In a crystal, they interfere with each other and give rise to the Bragg reflections. Between the Bragg reflections, there is no loss of energy due to elastic scattering and the incident beam is hardly reduced. In the Bragg positions, if the reduction in intensity of the incident beam due to elastic scattering can still be neglected, the crystal is considered an ideal mosaic . For non-ideal mosaic crystals, the beam intensity is reduced by extinction:

(1) The blocks are too large, and multiple reflection occurs within a block. At each reflection process, the phase angle shifts $[\pi/2]$ (Section 2.1.4.3.2). After two reflections, the beam travels in the same direction as the incident beam but with a phase difference of π, and this reduces the intensity.
(2) The angular spread of the blocks is too small. The incident beam is partly reflected by blocks close to the surface and the resulting beam is the incident beam for the lower-lying blocks that are also in reflecting position.

Extinction is not a serious problem in protein X-ray crystallography.

Absorption curves as a function of the X-ray wavelength show anomalies at absorption edges. At such an edge, electrons are ejected from the atom or are elevated to a higher-energy bound state, the photons disappear completely and the X-ray beam is strongly absorbed. This is called photoelectric absorption. At an absorption edge, the frequency of the X-ray beam ν is equal to the frequency $[\nu_{K}, \nu_{L}]$ or $[\nu_{M}]$ corresponding to the energy of the K, L, or M state. According to equation (2.1.4.4), anomalous scattering is maximal at an absorption edge.

References

International Tables for Crystallography (2004). Vol. C. Mathematical, physical and chemical tables, edited by E. Prince. Dordrecht: Kluwer Academic Publishers.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 2.1, pp. 58-59