Second-order diffraction

Figure 4 Interference pettern created when regularly spaced atoms scatter an incident plane wave. A spherical wave emanates from each atom diffracted beams form at the directions of constructive interference between these waves. The mirror reflection—the (00) beam—and the first- and second-order diffracted beams are shown.

The second-order diffracted beams by themselves would produce a set of interference fringes having half the spacing of those of the first order, and an intensity proportional to that of the diffracted beams... 

Bohanon et al. [86] studied heneicosanoic acid (which contains 21 carbon atoms) and Lin et al. [87] studied this material with particular reference to the effect of pH and the presence of divalent cations in the subphase. The former authors made use of in-plane diffraction (method 2 above) and obtained first order and second order diffraction peaks. They were able to show that, at high pressures ( r=35 mN m-1), at low pH (pH = 2) and at temperatures in the region of 0-5 °C, the material packs into a distorted hexagonal structure with the tilt towards the nearest neighbours. However, in the region 5-10°C the tilt is towards the next nearest neighbours. In the latter study [87] in-plane diffraction was studied as a function of pH and the presence of Ca2+ or Cu2+ in... 

Fig. 10 X-ray diffraction data for (a) MVLG3/DOPC-DNA complexes and (b) MVLBisGI/ DOPC-DNA complexes at

Two measurements [31,10] were conducted at a low-inductance vacuum spark plasma and a tokamak plasma respectively. In both cases only the w line was reported. The first study used a double Johann spectrograph and characteristic K lines were used for calibration [31]. The energy of the w was 5.20558(55) keV or a 105 ppm result The second study [10] used a tokamak plasma and claimed an uncertainty of 40 ppm. Close lying Lyman series lines were used for calibration so this was a relative measurement chain assuming one-electron QED. Shorter wavelength calibration lines and helium-like resonances were observed in second order diffraction suggesting the significant systematic shifts discussed above. The third study [11] was a relative measurement to the w line and, as such, can not be compared to absolute measurements. 

The existence of the intensity maxima within the cones of scattering have been our experimental cornerstone for measuring the well-defined separation between the clay plates. The second remarkable feature of a comparison between parts (a) and (b) of Figure 11.1 is that the diffraction pattern from the polymer-added sample is much sharper it exhibits a more pronounced first-order diffraction maximum and a strong second-order diffraction maximum, which is rare for a pure aqueous sample [5], This effect was also observed for all the PVME-added samples and means that... 

FIGURE 11.4 I(Q) (arbitrary units) vs. Q plots obtained for r = 0.01 and c = 0.1-M gels at T = 8°C. The crosses, squares and circles denote the scattering from the pure aqueous system and those with v = 0.01 (1%) PVME and v = 0.01 (1%) PEO, respectively. The insets are the scattering on a magnified (x5) intensity scale, showing the clear second-order diffraction maxima for the polymer-added samples. 

This view runs into difficulties that have only recently been completely resolved. The principal one is that the pseudopotential form factor happens to be very small for this particular diffraction. In Fig. 18-4 is sketched the pseudopotenlial form factor for silicon obtained from the Solid Stale Table the form factor that gives the [220] diffraction is indicated. Because it lies so close to the crossing, it is small and the diffraction is not expected to be strong. Heine and Jones (1969) noted, however, that a second-order diffraction can take an electron across the Jones Zone this could be a virtual diffraction by a lattice wave number of [1 ll]27t/fl followed by a virtual diffraction by [I lT]27c/a. (Virtual diffraction is an expression used to describe terms in perturbation theory it can be helpful but is not essential to the analysis here.) This second-order diffraction would involve the large matrix elements associated with the [11 l]27t/a lattice wave number indicated in Fig. 18-4, and Heine and Jones correctly indicated that these are the dominant matrix elements. 

X rays of wavelength 2.63 A were used to analyze a crystal. The angle of first-order diffraction (n — 1 in the Bragg equation) was 15.55°. What is the spacing between crystal planes, and what would be the angle for second-order diffraction (n = 2) ... 

An alternate way of considering crystal planes, indexed as described earlier, is to consider the number of times a set of parallel planes intersects each crystal axis within one unit cell. The set of hkl planes cuts the a axis at h positions, the b axis at k positions, and the c axis at I positions [Figure 2.12(b)]. Ralph Steadman suggests To determine hkl move along the whole length of the unit cell vector a. Count the number of spaces crossed, and this is h. By spaces we mean spaces between planes. In Figure 2.12 hkl = 426 = 213, because, in usual crystallographic practice, any common divisor is factored out. In X-ray diffraction, described in Chapter 3, (426) is used to denote the second-order diffracted beam from a (213) plane. 

X-ray dififractograms of as synthesized ZGctab (b) and ZGdtab (a) are shown in Fig.4. Both patterns reveal the presence of a layered compound showing first and second order diffraction. The repeat distance in ZGctab and ZGdtab are 3.1 rnn and 3.0 nm,... 

The crystal plane (2h, 2k, 21) has half the spacing of the Qi, k, 1) plane and thus the conditions for fulfilment of the Bragg equation (equation 4.2) are the same for first order diffraction from the 2h, 2k, 21) plane as for second order diffraction from the (h, k, 1) plane, and usually all observed diffractions are treated as first order. 

Figure 17.6 X-ray diffraction patterns for neat CuPc and Cso films as well as for three mixed films grown at 375 K. The arrows indicate the weak (200) peak in the sample with a mixing ratio of 1 3 and the second order diffraction peak of the neat CuPc sample.

Diffraction intensities are shown in Fig. 34 with the notation [00], [10], and [20] indicating the specular, first- and second-order diffracted beams, respectively. It is immediately apparent from the contrast between the three figures that the availability of the PES sink has a marked effect on all the beams. The specular beam does not go through a minimum at 350 meV because as the energy is raised, dissociation becomes more probable, and thus more of the flux is channeled into this dissociation channel instead of into the reflected specular channel. Similarly, the peak values of the [10] and [20] beams are significantly reduced. In a nonreactive system, it is... 

The higher order grating signal after the photoexcitation of all-trans-f -carotene (Fig. 21) was interpreted by a two-step absorption model [117], By simultaneous fitting of the first- and second-order diffraction intensity versus the excitation intensity, the quantum yield of the photoisomerization (iso) and the kinetic parameters were determined. The results showed distinct solvent dependence. In hexane, the effect of 0iso was very small and the upper limit was obtained as 7 x 10-4. The nonlinear absorption in this case was caused by the excited absorption. In chloroform, saturation was observed in the absorption and it was attributed to formation of a trap state, most likely a photoisomer. By the fitting, (f>iso was obtained as 0.026 and the kinetic parameter and cross section of subsequent photoisomer absorption were determined. 

Diffuse X-Ray pattern of TTF-TCNQ at 60°K.. Satellite reciprocal planes (1-d scattering) are clearly observed at the wave vectors 0.295 b" (2Kp) and 0.59 tr (4Kp). The comparable intensity of thes e two types of 1-d precursor at 60 K and their different temperature dependence (see figure 4) rules out that the 4Kp scattering might arise from a second order diffraction from... 

Any radiation not absorbed by the sample falls on the detector, where the intensity is converted to an electrical signal that is amplified and read on a meter. The measuring phototube for the visible region has maximum response at 400 nm, with only 5% of this response at 625/nm. Measurements above 625 nm are best made by substituting a red-sensitive phototube (RCA 6953) along with a red filter to remove second-order diffraction from the grating (it passes the desired red radiation but not undesired higher orders). 

For Ihe grating in I robleni 7-13, calculate the wavelengths of first- and second-order diffraction at reflective angles of (a) 25 and (b) 0 Assume Ihe incident... 

The second-order diffraction ( = 2) for a gold crystal Is at an angle of 22.20° for X rays of 154 pm. What is the spacing between these crystal planes ... 

It is worth considering that in [97] an explanation of the origin of the diffraction maxima along the meridian at 0.40 and 0.80 A is provided, consistent with the paracrystalline model proposed by Lyndenmeyer and Hosemann [48] for PAN. The halo at f 0.80 is not necessarily the second-order diffraction of the maximum at f 0.40 A since it is apparent from calculations of Fig. 12 that these two meridional maxima may originate from different contributions. The maximum at 0.40 A arises from the average periodicity of lateral - CN groups alone (Fig. 12C), whereas the maximum at C = 0.8oA- arises from the contribution of only the backbone carbon atoms (Fig. 12D). 

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