Extinction distances

One factor which should be noted for palladium, which also applies to the observation of the transition metals Is that not all crystallites have the same efficiency for diffracting electrons, l.e., as the atomic number decreases, the extinction distance for the crystallite Increases (13). Thus one would anticipate Chat as the mean atomic number decreases, the crystallites will have Co be progressively larger to enable visual observation on a support such as alumina. 

The method extends previous work of W. Sinkler, L.D. Marks and allows for a comparison with the Is-state model that approximates well an Argand plot in the first extinction distance while providing a promising path to go from an exit wave to the crystal structure ". J.R Jinschek et al. demonstrated by simulations and experiments on gold samples that a channeling plot (Figure 8) reveals the discrete nature of... 

In reflection, the intensity of the X-ray wavefield inside the crystal falls off very rapidly away from the surface, due to transfer of energy to the diffracted beam. Absorption also becomes important at low incident angles to the surface. By choosing the radiation and the reflection (inclnding its symmetry), the penetration may be varied between about 0.05 and 10 micrometres. This is ideally matched to device stmctures. This is quantified by the extinction distance g, defined as the depth at which the incident intensity has decreased to 1/e of its value at the surface. This may be calculated from diffraction theory, and some examples, for GaAs with CuK radiation, are shown in Table 3.2. It is assumed that the wafer surface is (001), hence the 004 reflection is symmetric and the others asymmetric. 

Table 3.2 Extinction distances in GaAs with CuK radiation...

Reflection Bragg angle (degrees) Rocking curve width (arc Extinction distance (/im)... 

The penetration in the absence of absorption is governed by the extinction distance,. This is the depth at which the intensity drops by a factor 1/e in a perfect crystal, and also the depth at which the Bloch wave in the crystal changes phase by a factor 2. ... 

An important feature of the use of soft radiation is that both the extinction distance and the absorption distance are small, and therefore the X-rays penetrate only a veiy small distance into the crystal. We therefore examine only a small slice of the crystal close to the surface. The strain fields of dislocations deeper into the crystal do not contribute significantly to the image and, in a transmission experiment, overlapping of the images leads to an upper limit of about 10" cm on the dislocation density for individual defect imaging. With the Berg-Barrett technique this can be pushed to about 10 cm ... 

For the Eikonal theory to be valid, the distortions must be suffieiently small that the eoneept of a ray is retained. The criterion is that the radius of curvature of the reflecting planes does not exceed a critical value R. approximately equivalent to an angular rotation of the Bragg planes by half the reflecting curve width in an extinction distance. The critical radius of curvature is thus... 

The high degree of X-ray polarisation in the electron orbit plane provides means of controlling both the signal/noise ratio and the penetration of the X-rays into the specimen. Depending on whether the incidence plane is chosen vertically or horizontally, sigma or pi polarisation may be selected. The strain sensitivity and the extinction distance can thus be varied while the normal photoelectric absorption conditions remain identical. 

The kinematic approximation breaks down at a certain crystal thickness when the diffracted intensity approaches that of the incident beam. A usefiil criterion for kinematic approximation is r < fg/4, where fg is the extinction distance of the strongest reflection in the diffraction pattern. The extinction distance is orientation dependent. In case only one set of lattice planes is strongly diffracting (the two-beam condition), the extinction distance is given by = h lmeX Vg. 

Near s = 0, the separation between dark (or bright) fringes corresponds roughly to the depth periodicity of the extinction distance tg. 

A small inclusion of thickness At tg) and extinction distance tgi increases the effective thickness of the foil by an amount... 

The Dauphine twin boundaries in a-quartz, which have been studied in detail by McLaren and Phakey (1969), are particularly interesting because the nature of their diffraction contrast fringes depends critically on the operating reflection g. For certain reflections, these twin boundaries behave as a-boundaries, whereas for other reflections, the fringe patterns originate not from any phase difference a, or any misorientation (change of s), but from the fact that the extinction distances in the two parts are significantly different. 

Table 8.4. Structure factor amplitudes and phase differences, and extinction distances for L quartz and its Dauphine twin for 100-kV electrons...

The action of the walls is both to cool the flame gases and to terminate the propagation chains. The theories of extinction distance have suggested that the extinction distance to a first approximation is inversely proportional to the burning velocity, and also depends somewhat on burner geometry. The major importance of extinction distance to kinetics in flames is, however, indirect its magnitude dictates the design of burner to be used in flame studies. 

It is useful to examine the time scales in flames a little more closely. The exact scale is a function of the particular mixture being burnt, and of the burner used, but the best burners are those designed to give a cylindrical flame, laminar rather than brush-shaped, with many small cones of reaction. These Meker type burners are composed of a bundle of capillary (hypodermic) tubes. To obviate pressure drop these should have the greatest possible diameter consistent with the extinction distance of the flame mixture used. The burnt gas is hot, so its volume Fjg is greater than that of the unbumt gas, F . The tubes are spaced out... 

The spatial resolution of the method is related to the condition of minimum thickness for the HOLZ lines where the extinction distances are very large (about 100 nm). The beam widening due to difussion phenomena inside this distance can increase the analyzed area by several orders depending of the atomic number. The small probes give rise to lower current producing very fine CBED patterns [11],... 

Absorption can phenomenologically be accounted for by assuming the extinction distances to be complex [151]. Formally l//g is replaced by l//g-i-i/Tg, where ig is called the absorption length. Empirically, it is found that tg ranges from 5 tg to IStg. The substitution can be made, either in the Equation set (2.5) or directly in the solution of this set. 

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