Debye—Scherrer cone

Scattering from a polycrystaliine sample takes place into Debye-Scherrer cones with the k direction as axis and semiangles 2d defined by sm6=rl2k. The total cross section associated with each cone is [from (3.15)]... 

In three dimensions, the circular intersection of the smeared reciprocal lattice with the Ewald sphere results in the diffracted X-rays of the reflection hkl forming coaxial cones, the so-called Debye-Scherrer cones (Figure 1.11). 

To obtain the maximum amount of information, a spherical shell detector would be desirable, though currently impractical. Often, a flat two-dimensional detector, either film, image plate, or CCD is placed perpendicular to the direct beam. In this case, the Debye-Scherrer cones appear as circles as shown in Figure 1.12a. 

Figure 1.11 Comparison between the scattered beams originating from a single crystal (top) and a powder (bottom). For the latter, some Debye Scherrer cones are drawn.

For quantitative phase analysis it is generally accepted that the peak intensities need to be measured to an accuracy of about 1 — 2% relative. The ability to achieve this is strongly influenced by the size of the crystallites in the sample reproducible diffraction intensities require a small crystallite size to ensure that there is uniform intensity around the Debye-Scherrer cone. 

The detectors between 12° and 30° are arranged on a Debye-Scherrer cone to improve the -resolution. To reduce the background all internal... 

FIGURE 27.2 Debye-Scherrer powder method. Cones of reflected and transmitted radiation are produced. In this example the pattern is recorded with photographic film. Alternatively,... 

FIGURE 2.4 (a) Cones produced by a powder diffraction experiment (b) experimental arrangement for a Debye-Scherrer photograph. 

There are two different ways to get local diffraction patterns "Debye-Scherrer" type (such patterns are obtained if ( y ) is found within (E). In the first case (figure 15 the orientation of the local symmetry axis ( A ) is very close to that of (I) with the consequence tRat the whole cone ( e ) is located within the cone (C) even if the local orientation is relatively good, i.e., angle 0 is small. In the second case, the local orientation is poorer, i.e., 0 is fairly large. 

Fig. 4-16 Intersection of cones of diffracted rays with Debye-Scherrer film.

In all these methods, the diffracted beams lie on the surfaces of cones whose axes lie along the incident beam or its extension each cone of rays is diffracted from a particular set of lattice planes. In the Debye-Scherrer and focusing methods, only a narrow strip of film is used and the recorded diffraction pattern consists of short lines formed by the intersections of the cones of radiation with the film. In the pinhole method, the whole cone intersects the film to form a circular diffraction ring. 

Fig. 6-2 Geometry of the Debye-Scherrer method, diffraction cone.

MAUD MAUD (Material Analysis Using Diffraction) a user friendly Java program for Rietveld Texture Analysis and more, L. Lutterotti, S. Matthies and H. R. Wenk, Proceeding of the Twelfth International Conference on Textures of Materials (ICOTOM 12), 1999, Vol. l,p. 1599 Debye Scherrer film integration and of full 2D cones of diffraction... 

When a specimen has no particular orientation (for example, a randomly oriented mass of small crystals, as in a powder), each reflection is spread out into a right circular cone of radiation whose axis is the x-ray beam. The intersections of these cones with the photographic films are the Debye-Scherrer powder lines. If, however, the crystals are not randomly distributed but lie in preferred orientations, the powder rings become nonuniform in density, indicating orientation. One of the outstanding studies of orientation in the field of catalysis was that of Beeck (1) who showed by electron diffraction that nickel films deposited under certain conditions showed unusual catalytic activity and that this activity was accompanied by a particular orientation of the nickel crystals. 

A Debye-Scherrer camera consists of a metal cylinder provided with a photographic film. The primary beam is perpendicular to its axis. The distance between two symmetrical lines, produced by the intersection of a cone with the cylinder, is 46R, 6 being the Bragg angle (in radians) and R the radius of the camera. The interval is derived from Bragg s law. The powder method gives us only the norms of the reciprocal vectors. The set of norms corresponds to the projection of the reciprocal lattice onto a straight line. 

A) Schematic of the Debye-Scherrer method, developed in 1916, for X-ray diffraction of powders (polycrystdlline samples). Each characteristic interplanar spacing in the crystal gives rise to a cone of diffracted X-rays, segments of which are captured on the film strip placed inside the camera. 

Figure 5 2. The Debye-Scherrer powder method. An X-ray R passes through a collimator and then meets a powder preparation P. The reflections caused by P lie on cones of reflection, which form crescents or arcs on a cylindrical film F.

Debye-Scherrer method A method used in X-RAY diffraction in which a crystal in powder form is exposed to a beam of monochromatic x-rays. Because the crystal is in powder form all possible orientations of the crystal are presented to the x-ray beam. This has the result that the diffracted x-rays form cones concentric with the original beam. The Debye-Scherrer method is particularly useful for determining the lattice type of a crystal and the dimensions of its unit cell. The method was first developed by Peter Debye and Paul Scherrer. 

However, not every crystalline substance can be obtained in the form of macroscopic crystals. This led to the Debye-Scherrer (16) method of analysis for powdered crystalline solids or polycrystalline specimens. The crystals are oriented at random so the spots become cones of diffracted beams that can be recorded either as circles on a flat photographic plate or as arcs on a strip of film encircling the specimen (see Figure 6.4) (17). The latter method permits the study of back reflections as well as forward reflections. 

XRD on battery materials can be classified as powder dififaction, a technique developed by Peter Debye and Paul Scherrer. In powder dififaction the material consists of microscopic crystals oriented at random in all directions. If one passes a monochromatic beam of X-rays through a fiat thin powder electrode, a fraction of the particles will be oriented to satisfy the Bragg relation for a given set of planes. Another group will be oriented so that the Bragg relationship is satisfied for another set of planes, and so on. In this method, cones of reflected and transmitted radiation are produced (Fig. 27.2). X-ray diffraction patterns can be recorded by intercepting a... 

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