Reflexion

The Fresnel equations predict that reflexion changes the polarization of light, measurement of which fonns the basis of ellipsometry [128]. Although more sensitive than SAR, it is not possible to solve the equations linking the measured parameters with n and d. in closed fonn, and hence they cannot be solved unambiguously, although their product yielding v (equation C2.14.48) appears to be robust. 

If necessary, the fit can be improved by increasing the order of the polynomial part of Eq. (9-89), so that this approach provides a veiy flexible method of simulation of a cumulative-frequency distribution. The method can even be extended to J-shaped cui ves, which are characterized by a maximum frequency at x = 0 and decreasing frequency for increasing values of x, by considering the reflexion of the cui ve in the y axis to exist. The resulting single maximum cui ve can then be sampled correctly by Monte Carlo methods if the vertical scale is halved and only absolute values of x are considered. 

A diffraction pattern of a single MWCNT (Fig. 1) contains in general two types of reflexions (i) a row of sharp oo.l (/ = even) reflexions perpendicular to the direction of the tube axis, (ii) graphite-like reflexions of the type ho.o (and hh.o) which are situated in most cases on somewhat deformed hexagons inscribed in circles with radii gho.o (or hh.o)-... 

Fig. 1. Typical ED pattern of polychiral MWCNT. The pattern is the superposition of the diffraction patterns produced by several isochiral clusters of tubes with different chiral angles. Note the row of sharp oo.l reflexions and the streaked appearance of 10.0 and 11.0 type reflexions. The direction of beam incidence is approximately normal to the tube axis. The pattern exhibits 2mm planar symmetry [9].

Simulated SWCNT ED patterns will be presented below. Tbe most striking difference with tbe MWCNT ED patterns is tbe absence of tbe row of sharp oo.l reflexions. In tbe diffraction pattern of ropes there is still a row of sharp reflexions perpendicular to the rope axis but which now corresponds to the much larger interplanar distance caused by the lattice of the tubes in the rope. The ho.o type reflexions are moreover not only asymmetrically streaked but also considerably broadened as a consequence of the presence of tubes with different Hamada indices (Fig. 3). 

An image of an MWCNT obtained by using all available reflexions usually exhibits only prominently the oo.l lattice fringes (Fig. 4) with a 0.34 nm spacing, representing the "walls" where they are parallel to the electron beam. The two walls almost invariably exhibit the same number of fringes which is consistent with the coaxial cylinder model. 

Several levels of interpretation have been proposed in the literature [9,16-19]. The 00./ reflexions are attributed to diffraction by the sets of parallel c-planes tangent to the cylinders in the walls as seen edge on along the beam direction their positions are independent of the direction of incidence of the electron beam. 

Fig. 8. Diffraction space according to the "disordered stacking model" (a) achiral (zigzag) tube (b) chiral tube. The parallel circles represent the inner rims of diffuse coronae, generated by streaked reflexions. The oo.l nodes generate sharp circles. In (a) two symmetry related 10.0 type nodes generate one circle. In the chiral case (b) each node generates a separate corona [9].

The "split" reflexions of the type ho.o (and hh.o) can be associated with the graphene sheets in the tangent planes perpendicular to the beam direction along... 

The average intercylinder spacing, which depends somewhat on the diameter, can be derived from the oo.l reflexions in the diffraction pattern, using the ho.o (or hh.o) spacing of graphite for internal calibrations since the latter seems to be independent of curvature. 

The angular splitting of the ho.o (or hh.o) reflexions is a measure for the chiral angle T). However the observed splitting depends as the direction of incidence of the electron beam and must thus be corrected for tilt [20,25]. 

On the other hand, TED patterns can assign the fine structure. In general, the pattern includes two kinds of information. One is a series of strong reflexion spots with the indexes of (00/), 002, 004 and 006, and 101 from the side portions of MWCNTs as shown in Eig. 1(b). The indexes follow those of graphite. The TED pattern also includes the information from the top and bottom sheets in tube. The helieity would appear as a pair of arcs of 110 reflexions. In the case of nano-probed TED, several analyses in fine structures have been done for SWCNT to prove the dependence on the locations [11,12]. 

Figure 9 shows angular distribution of EELS of an MWCNT with a diameter of 100 nm [16j. The core-loss spectra obtained from the 000 and 002 reflexions much resemble those of an MWCNT and graphite (Figs. 6(b) and 7(c)). The n excitation peak is smaller than that of a excitation peak. In contrast, the... 

Right Fig. 9. EEL spectra of an MWCNT obtained from the locations at 000, intermediate and 002 reflexions in the reciprocal space (modified from ref. 16). 

Relative reflexion. Measurements of the map of the relative reflexion are done with a sensitivity of 2 10 (Fig. 27). 

Figure 27. The relative reflexion bench for VIRGO coatings.

For any symmetry operator T = T 0) (rewritten r when operating on the domain of basis functions x)) for instance, the rotation-reflexion about the z-axis, with matrix representation... 

Berkeley, George. Siris a chain of philosophical reflexions and inquiries concerning the virtues of tar water, and divers other subjects connected together and arising one from another. / By the Right Rev. Dr. George Berkeley, Lord Bishop of Cloyne, and author of The minute philosopher [1 line of biblical quotation and 1 line of Horace in Latin], The second edition, / improved and corrected by the author ed. [London] Dublin printed, London re- printed, for W. Innys, and C. Hitch,... and C. Davis..., 1754. [3], 4-174, [2] p. 

Published also in the same year under titles A chain of philosophical reflexions and, Philosophical reflexions and inquiries. A variant edition first word of title measures 56 mm. Other editions from same year have first word of title measuring 73 mm. 

Uneven distributions of residuals. The MaxEnt calculations in presence of an overall chi-square constraint suffer from highly non-uniform distributions of residuals, first reported and discussed by Jauch and Palmer [29, 30] the error accumulates on a few strong reflexions at low-resolution. The phenomenon is only partially cured by devising an ad hoc weighting scheme [20,31, 32]. Carvalho et al. have discussed this topic, and suggested that the recourse to as many constraints as degrees of freedom would cure the problem [33]. 

Errors in the low-density regions of the crystal were also found in a MaxEnt study on noise-free amplitudes for crystalline silicon by de Vries et al. [37]. Data were fitted exactly, by imposing an esd of 5 x 10 1 to the synthetic structure factor amplitudes. The authors demonstrated that artificial detail was created at the midpoint between the silicon atoms when all the electrons were redistributed with a uniform prior prejudice extension of the resolution from the experimental limit of 0.479 to 0.294 A could decrease the amount of spurious detail, but did not reproduce the value of the forbidden reflexion F(222), that had been left out of the data set fitted. 

Finally, recent work of Iversen et al. has carefully examined the bias associated to the accumulation of the error on low-order reflexions, and attempted a correction of the MaxEnt density [39]. The study, based on a number of noisy data sets generated with Monte Carlo simulations, has produced less non-uniform distribution of residuals, and has given quantitative estimate of the bias introduced by the uniform prior prejudice. For more details on this work, we refer the reader to the chapter by Iversen that appears in this same book. 

Let us consider a collection H = (fr, h2,. . . , hA/) of symmetry-unique reflexions. We denote by Fj[ the target phased structure factor amplitude for reflexion h/, and with F rag the contribution from the known substructure to the structure factor for the same reflexion. We are interested in a distribution of electrons q( ) that reproduces these phased amplitudes, in the sense that, for each structure factor in the set of observations H,... 

The sum over symmetry operations in formula (16) can be rewritten by considering the effect of multiplying vector h7 by the rotation matrices The collection of distinct reciprocal vectors h7Rg is called the orbit of reflexion h7 [27] r7 is the set of symmetry operations in G whose rotation matrices are needed to generate the orbit ofh/ r, denotes the number of elements in the same orbit [50]. 

Under the simplifying assumption that the reflexions are independent of each other, K, can be written as a product over reflexions for which experimental structure factor amplitudes are available. For each of the reflexions, the likelihood gain takes different functional forms, depending on the centric or acentric character, and on the assumptions made for the phase probability distribution used in integrating over the phase circle for a discussion of the crystallographic likelihood functions we refer the reader to the description recently appeared in [51]. 

Let us assume an experimentally derived distribution P(R) for the amplitude R = F of a reflexion, normalised so as to have /0°° P(R) dR = 1. The P(R) distribution will be typically Gaussian around the measured Rohs = F obs with associated variance a2. P(R) may take a more involved functional form if the Gaussian has a substantial tail in regions of negative Robs. 

Under general hypotheses, the optimisation of the Bayesian score under the constraints of MaxEnt will require numerical integration of (29), in that no analytical solution exists for the integral. A Taylor expansion of Ao(R) around the maximum of the P(R) function could be used to compute an analytical expression for A and its first and second order derivatives, provided the spread of the A distribution is significantly larger than the one of the P(R) function, as measured by a 2. Unfortunately, for accurate charge density studies this requirement is not always fulfilled for many reflexions the structure factor variance Z2 appearing in Ao is comparable to or even smaller than the experimental error variance o2, because the deformation effects and the associated uncertainty are at the level of the noise. 

BUSTER has been run against the L-alanine noisy data the structure factor phases and amplitudes for this acentric structure were no longer fitted exactly but only within the limits imposed by the noise. As in the calculations against noise-free data, a fragment of atomic core monopoles was used the non-uniform prior prejudice was obtained from a superposition of spherical valence monopoles. For each reflexion, the likelihood function was non-zero for a set of structure factor values around this procrystal structure factor the latter acted therefore as a soft target for the MaxEnt structure factor amplitude and phase. 

For a number of 1907 acentric reflexions up to 0.463 A resolution, the mean and rms phase angle differences between the noise-free structure factors for the full multipolar model density and the structure factors for the spherical-atom structure (in parentheses we give the figures for 509 acentric reflexions up to 0.700A resolution only) were (Acp) = 1.012(2.152)°, rms(A( >) = 2.986(5.432)° while... 

Brindley, G. S. and E. N. Willmer (1952). The reflexion of light from the macular and peripheral fundus oculi in man. Journal of Physiology 116 350-356. 

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