Totally coherent scatterer

For a totally coherent scatterer like CF4, the first term in expression 22 will be zero (bine = 0) and the number of contributions in the third term will depend on the symmetry of the molecule. The tetrahedral symmetry of CF4 implies that the sum on the Legendre polynomials is zero for = 1 and = 2. The sum is non-zero only for = 3. This means that in the Q range, which is the most important to extract diffusivities (Q < 1 A ), the third term in ex-... 

In Figure 1, the total coherent scattering intensities from a 3.9 mol% aqueous solution of TBA (solid curve) and pure water (dotted curve) are shown. A monotonically decreasing curve (thin solid curve) is the theoretical self-scattering intensity. The difference in the intensity between water and solution is clearly seen in the large first peak and the second shoulder peak, but there is no difference in the phase (peak position) in the two curves. Note that, in spite of the very low concentration of TBA, the decrease in the intensity mentioned above is drastic. 

The total coherent scattering cross section from one set of nuclei can now be expressed in terms of the coherent scattering length (see table 1), the position of the nuclei, R , in the unit cell, and the scattering vector, Q,... 

The most fundamental and direct method of determining the degree of crystallinity is X-ray diffraction. It is based on the principle that the total coherent scattering from N atoms is independent of the state of aggregation. This statement leads to the... 

Weight fraction of crystallinity cp is then given by the ratio of the integral of the diffraction intensity scattered by the crystalline fraction to the total coherent scattered intensity ... 

The spin state of a paramagnetic system with total spin S wiU lift its (25 + l)-fold degeneracy under the influence of ligand fields (zero-field interaction) and applied fields (Zeeman interaction). The magnetic hyperfine field sensed by the iron nuclei is different for the 25 + 1 spin states in magnitude and direction. Therefore, the absorption pattern of a particular iron nucleus for the incoming synchrotron radiation and consequently, the coherently scattered forward radiation depends on how the electronic states are occupied at a certain temperature. 

The incoherent scattering is given by the difference between the total scattering and the coherent scattering. For a monatomic crystal with one atom per unit cell,... 

We shall assume, in addition to single scattering, that the particles are many and their separations random, which implies incoherent scattering. That is, there is no systematic relation among the phases of the waves scattered by the individual particles thus, the total irradiance scattered by the collection is just the sum of the irradiances scattered by the individual particles. Even, however, in a collection of randomly separated particles, the scattering is coherent in the forward direction, a subject to which we shall return in Chapter 3. 

As well as the coherent scattering, which has the same wavelength as the incident waves, an atom scatters waves of reduced wave-length. The intensity of these waves is not included in the formulae (I) and (2). For X-rays falling on a hydrogen atom, the total intensity scattered inelastically, due both to Raman effect and Compton effect, is given by the formula... 

Bragg invoked the Law of Reflectivity (or Reflections) that states that the scattering incident angle and exiting angle must be equal, 5in = 5out under the condition of coherent scattering. The wavelets scattered by the atoms combine to produce constructive inference if the total path difference 2 AP... 

We know that the total intensity scattered by the bound electron is equal to the coherent intensity scattered by the free electron, therefore the incoherent intensity is given by the following relation ... 

The scattering factor reaches its maximum for 0 = 0 and decreases with 0, meaning that the contributions to the total scattering from coherent scattering and Compton scattering behave as shown in Figure 1.5. 

The first difference with incoherent scattering is that the intensity varies as S(Q), leaving aside the influence of the Debye-Waller factor. One, therefore, expects a non-monotonic variation of the total intensity as a fimction of Q. In the small Q domain, the line-shape of the coherent scattering fimction is still Lorentzian, with a HWHM... 

Matchpoints in a two-component system denoted 1 and 2 can be calculated from a graph of 2 6 — Ps 1 against Ps, where Sb is the sum of coherent scattering lengths, V is the total volume, and X is the volume fraction matchpoint [39] ... 

The solid states represent bounded atomic associations and the diffraction intensity is a result of coherently scattered wave interference. The neutron scattering can be coherent elastic or inelastic and incoherent. In coherent total scattering experiment both elastically and inelastically scattered neutrons are detected, thus performing energy integration over the whole range gives a value for The total differential cross-section is,... 

Although it is one of the oldest techniques, it is probably the least understood, which probably explains why it has not found more widespread application. The basis of the model is that since the fluorescence intensity If is inversely proportional to the total matrix absorption or and since the mass absorption coefficient is approximately proportional to Z, it follows that If is proportional to Z. Similarly, since the sum 1 of the coherent and incoherent scatter is also inversely proportional to ax and since coherent scatter is roughly proportional to Z, it follows that if the scattering is mainly coherent, Ig/If is approximately proportional to Z. This ratio is thus far less matrix dependent than is If alone. Where incoherent scatter is significant, the ratio of Ig/If is proportional to somewhere between the first and second power of Z. 

The total coherent (Rayleigh) scattering cross-section per atom ctr can be calculated from... 

---