Fiber symmetry

It is a 2D slice of the fiber pattern. Only fiber symmetry makes that it is completely represented by a curve as a function of a transversal (cf. Bonart [16]) coordinate... 

Definition. Fiber symmetry is uniaxial or cylindrical symmetry. Revolving the sample about the fiber axis does not change the scattering pattern, but tilting the sample with respect to the fiber axis does. 

Fiber symmetry with the 53 axis in fiber direction the pattern shows rotational symmetry in the plane ( 1, 2), thus 7(s) = 7 [ js +, 3) = 12, 3) is a function of S3 and of the distance from this axis only. 

It is complete because of fiber symmetry. The 2D Fourier transform of this image is not related to the searched slice, but to a projection of the correlation function. In contrast, the sought-after slice in real space... 

WAXS and MAXS. Fiber symmetry means that, even in WAXS and MAXS, the scattering pattern is completely described by a slice in reciprocal space that contains the fiber axis. Nevertheless, for 20 > 9° the tangent plane approximation is no longer valid and the detector plane is mapped on a spherical surface in reciprocal space. 

Figure 2.7. WAXS, 2D-detector and fiber symmetry unwarping of the detector surface to map it on the (sq, s )-slice. Fiber direction is tilted by y/ = —30° with respect to the primary beam, i = 10 cm, A. = 0.154 nm. On the detector the apparent warped grid is a square grid (edge length 3 cm)... 

Fiber Symmetry Equator and Meridian. Figure 4.1 sketches a scattering experiment of a polymer sample under uniaxial load. Let us assume that the material... 

A particularly simple case is the study of a fiber in the slit-focus camera, if the fiber is stretched out along the slit direction [31,62,63]. In this case the transversal structure according to Bonart [16] (cf. Sect. 8.4.3) is directly measured, as is established by change of variables S —> s, (fiber parallel to the slit) s2,s —> S 2 (fiber symmetry assumed)... 

Some experiments are aiming at the study of structure evolution. In general, the studied material is isotropic or exhibits simple anisotropy (e.g., fiber symmetry). Most frequently the material is irradiated in normal-transmission geometry. A synchrotron beamline is necessary, because in situ recording during the materials processing is requested with a cycle time of seconds between successive snapshots (time-resolved measurements). 

Sample Orientation If possible orient the sample in such a way that the beam-stop holder does not cut through an important region (peak). If you expect that the sample exhibits fiber symmetry, check it rotating the sample about the assumed fiber axis and take some patterns. 

For MAXS and WAXS the problem is more involved. If the scattering pattern shows fiber symmetry, the considerations of Sect. 2.8.22 apply. 

Spinning a crystal during measurement of WAXS patterns is an old method that turns any scattering pattern into a fiber pattern. The rotational axis becomes the principal axis. Thereafter isotropization of the scattering data is simplified because the mathematical treatment can resort to fiber symmetry of the measured data. In the literature the method is addressed as the rotating-crystal method or oscillating-crystal method. 

To specify the components of the scattering vector by S 2 and s3 is only a suggestion. The specification meets the case that is of highest practical importance (anisotropy with fiber symmetry). 

In general, only a 2D scattering pattern will be available. In this case isotropization can only be performed if the pattern shows fiber symmetry and the fiber axis is contained in the scattering pattern. This symmetry axis must be known. Complete is the available information under these conditions only if SAXS data are evaluated. For WAXS data there are blind regions about the meridian (cf. Fig. 2.6 on p. 28), and missing information must be completed either by extrapolation or by extra experiments in which the sample is tilted with respect to the primary beam. 

Isotropization in the Case of Fiber Symmetry. If methods for the analysis of isotropic data shall be applied to scattering patterns with uniaxial orientation, the corresponding isotropic intensity must be computed. By carrying out this integration (the solid-angle average in reciprocal space) the information content of the fiber pattern is reduced. One should consider to apply an analysis of the longitudinal and the transversal structure (cf. Sect. 8.4.3). 

Tchoubar. For the application to anisotropic scattering patterns Stribeck [26] has extended this principle to a space of deliberate dimensionality. Available technology constricts its practical use to the scattering of materials with fiber symmetry, and the fiber-symmetrical CDF... 

CDFs are computed from scattering data which are anisotropic and complete in reciprocal space. Thus the minimum requirement is a 2D SAXS pattern of a material with fiber symmetry taken in normal transmission geometry (cf. p. 37, Fig. 4.1). Required pre-evaluation of the image is described in Chap. 7. 

Figure 8.27. Steps preceding the computation of a CDF with fiber symmetry from recorded raw data The image is projected on the fiber plane, the equivalent of the Laplacian in real space is applied, the background is determined by low-pass filtering. After background subtraction the interference function is received... 

Perfection of Structure in Nanostructured Materials. An aim of modern nanotechnology is the fabrication of materials with highly perfect structure on the nanometer scale. The distortion of such nanostructured materials can be studied by SAXS methods. Frequently the material is supplied as a very thin film with predominantly uniaxial correlation among the nanodomains. Under these constraints the nanodomains are frequently arranged in such a way that the normal to the film is a symmetry axis rotation of the film on the sample table does not change the scattering (fiber symmetry). 

Whenever we are considering orientation, we are dealing with anisotropic scattering data. Orientation is most frequently analyzed in 2D scattering data with fiber symmetry or in pole-figure data recorded by means of a texture goniometer. 

Moreover, some issues concerning scattering patterns with fiber symmetry are discussed... 

Interpretation of for. For materials that exhibit fiber symmetry (i.e., g(structural entities with fiber symmetry... 

As are the other multipole-expansion coefficients, the uniaxial orientation parameter is computed from Eq. (9.6). For materials with fiber symmetry the relation simplifies4 and... 

In practice, either a pole figure has been measured in a texture-goniometer setup, or a 2D SAXS pattern with fiber symmetry has been recorded. In the first case we take the measured intensity g (pole figure. In the second case we can choose a reflection that is smeared on spherical arcs and project in radial direction over the range of the reflection. From the measured or extracted intensities I (orientation parameter by numerical integration and normalization... 

F2 Double Fiber Symmetry - Simplified Integral Transform... 

Figure 9.5. 3D geometrical relations in the scattering pattern for the case of double fiber symmetry (F2). Dash-dotted are both the axis of the observed pattern, I, and one of its reflection circles. Drawn in solid line are both the axis of a tilted representative structural entity and a reflection circle of its fiber-symmetrical intensity, Iopt. Important for the simplification are the relations in the spherical triangle plotted in bold... 

Basic Ideas. Let the average structural entity be anisotropic with fiber symmetry. Let its shape... 

Figure 10.3. Shape of the maximum peak intensity, J((p ), extracted from radial sections of a moderately anisotropic (Xj = 1.41) SAXS pattern with fiber symmetry as a function of the sectioning angle

fiber axis. Dots experimental values. Solid line Theoretical shape according to RULAND [265]... 

Fiber symmetry makes that every function gj is the sum of four quadrant functions,... 

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