Scattering point

Relative activation enthalpies (Aif) in Table 2 were converted to o% kx k ) at 298 K, and were plotted against Hammett a constants. Here, we used enthalpies, because the size of the entropy and hence the free energy depend much on low frequencies, which are less reliable than higher frequencies, especially for compounds with weak interactions such as TS (8). The use of free energy (AG ) gave similar correlations with more scattered points. As for the Hammett o constant, we used dual-parameter o constants in the form of the Yukawa-Tsuno equation (LArSR equation) (9) as defined in eq 3. Here, the apparent a constant (aapp) has a variable resonance contribution parameter (r), which varies depending on the nature of the reaction examined for t-cumyl... 

Mitchell has scattered points that do not conform to plotted data. It was uncertain which plot should have been used to compare to therefore the general consensus of all three was taken in this comparison and only obvious differences noted. For the single curve method the 6.4 amount in A was different from the plot. In B the 0.33, 0.082, 0.92, and 6.3 points were different. In F the 0.023 point was different and the 1.18 point appears to be in error. There was an improvement in precision when the multiple curve method was used. In A all three points conform. 

Figure 3. Total experimental intensities and background lines for tetramethylsilane. The two curves refer to two different scattering point to registration plane distances. ...

Angle between the forward direction of the incident beam and a straight line connecting the scattering point and the detector. 

Atoms in crystals cannot be regarded as scattering points the diameter of the electron cloud of an atom is of the same order of size as the distance between the centres of adjacent atoms—in fact, to a first approximation, the atoms in many crystals may be regarded as spheres of definite radius in contact with each other the electron clouds... 

Tomasik36 58-63 has investigated the polarographic reduction of nitropyridines in DMF at 20°, and finds a good correlation of with a for the 2-X,5-N02, 5-X,2-N02, 3-X,5NOz, and 2-X,4-N02 systems, with p values of 0.35, 0.41, 0.46, and 0.42, respectively. The nitropyridine 7V-oxide system also afforded a correlation in this latter case (p = 0.3), but for the 2-substituted-3-nitropyridines there were only scattered points.64 A series of 2-X,5Y-phenylazopyridine reductions also produced an Exjl-o correlation under the same conditions. For these 2X-5Y systems the defining equations were Eqs. (7) and (8) for the nitrobenzene and nitropyridine series and Eqs. (9) and (10) for the azobenzene and phenylazopyridine series, respectively. 

Up to this point we did not make any specific assumptions about the real space lattice. It could contain more than one atom per lattice point and more than more than one type of atoms. In such a case the lattice would be described using a Bravais lattice plus a basis (see Section 8.2.2. To obtain the intensity of the diffracted wave for crystals with a basis, we simply have to sum up the contributions from all scattering points within the unit cell. The scattering probability for a crystal of N unit cells with an electron density ne(r) is proportional to ... 

Fig. 7 Hydrogen bond acidity and vertical pK. Vertical pA a values show a reasonable correlation with Abraham s hydrogen bond acidity parameter, while the equilibrium pKa values show much more scatter. ( ) Points used in the least squares fit, ( ) vertical pATa values for carboxylic acids not used in the least squares fit, and ([>) equilibrium pK.A values not used in the least squares fit.

It is impossible to distinguish the two models visually. The subplots C and D are the socalled pseudophase plots of the two sequences of plots A and B, respectively each j/j is plotted against its consequent yi+1. The random sequence (A) produces scattered points (C) showing that there is no correlation between successive points. In contrast, the points of the deterministic sequence (B) he in a well-formed line (D). 

Since tan is equal to the ratio of the loss modulus to the storage modulus, one might expect a correlation of the toughness with the storage modulus. No such correlation was found—the plot appeared as scattered points—because at 10°C, 11 Hz, the loss moduli of the samples were neither negligible nor equal. 

In the kinematical theory, we consider the diffraction of a plane wave (of wavelength X) incident upon a three-dimensional lattice array of identical scattering points, each of which consists of a group of atoms and acts as the center of a spherical scattered wave. Our problem is to find the combined effect of the scattered waves at a point outside the crystal, at a distance from the crystal that is large compared with its linear dimensions. In developing the theory, we make several important assumptions ... 

There is no attenuation of the incident wave in the crystal so that the incident wave has the same amplitude at each scattering point. This is equivalent to neglecting any interaction between the incident wave in the crystal and the scattered waves. 

Each scattered wave travels through the crystal without being rescattered by other scattering points. 

Since the theory makes no assumptions about the nature of the wave or about the detailed mechanism of the interaction of the wave with the scattering points, it is applicable to x-rays, electrons, and neutrons. 

We begin by considering the scattering of a plane wave of wavevector Kq (of magnitude 1/X) by two identical scattering points Pi and P2 at the... 

The significance of the vector g can be understood with the aid of Figure 3.2. OP represents the incident wavevector Kq and OP the scattered wavevector Kg. P is a scattering point. The direction PP is given by the vector... 

Now if a, b, c are the primitive translation vectors that define the unit cell of the three-dimensional array of scattering points, then we have, using Eq. (3.6), the following conditions for diffraction maxima ... 

Equation (3.25), which was derived for the intensity of the radiation scattered by a unit cell, contains the atomic scattering factor / . This factor depends principally on the nature of the radiation (and therefore the scattering mechanism), the nature of the scattering point, and the scattering angle. 

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