Neutron coherence length

The more recent neutron reflectivity studies have established that flattened surface micelle or fragmented bilayer structure in more detail and with more certainty, using contrast variation in the surfactant and the solvent [24, 31]. However, the extent of the lateral dimension (in the plane of the surface) and the detailed structure in that direction is less certain. From those neutron reflectivity measurements [24, 31] and related SANS data on the adsorption of surfactants onto colloidal particles [5], it is known that the lateral dimension is small compared with the neutron coherence length, such that averaging in the plane is adequate to describe the data. The advent of the AFM technique and its application to surfactant adsorption [15] has provided data that suggest that there is more structure and ordering in the lateral direction than implied from other measurements. This will be discussed in more detail in a later section of the chapter. At the hydrophobic interface, although the thickness of the adsorbed layer is now consistent with a monolayer, the same uncertainties about lateral structure exist. 

Additionally, a neutron transmission (NT) experiment on H20 / D20 mixtures was recently presented, reporting results in agreement with conventional theory and claiming to present a proof of the absence of the QE effect under consideration [Blostein 2003 (a)]. However, also this experiment does not provide the appropriate, and well defined, sub-femtosecond scattering time [Karlsson 2004], Moreover, it was discussed by Karlsson and Mayers that the neutron coherence length in this NT experiment is much shorter than that in NCS (being about 2.5 A, cf. [Karlsson 2003 (b)]). Thus there are no reasons whatsoever to expect deviations from the conventional, individual particle cross sections [Karlsson 2004],... 

Neutron reflectivity measures the variation in concentration normal to the surface of the specimen. This concentration at any depth is averaged over the coherence length of the neutrons (on the order of 1 pm) parallel to the sur ce. Consequendy, no information can be obtained on concentration variadons parallel to the sample surface when measuring reflectivity under specular conditions. More imponantly, however, this mandates that the specimens be as smooth as possible to avoid smearing the concentration profiles. 

Although the formalism for X-ray and neutron diffraction is essentially the same, it is appropriate to treat them separately because of the nature of the basic interaction. For the case of neutron diffraction, neutrons are scattered isotropically by all the nuclei of the system. The degree to which this takes place is determined by the coherent neutron scattering length b, which varies from isotope to isotope (Table I). Because the scattering is isotropic, the results of a given experiment can readily be presented in terms of a total radial distribution function. 

Coherent neutron seattering lengths (b) are 4.84, 5.88, and 8.185 fm for Ce, Rh and Ge, respectively. Since for these three elements the neutron scattering ability accidentally increases with the decreasing atomic number, Ge is determined from neutron data with better accuracy than Ce. On the contrary, the x-ray scattering factors are proportional to the atomic number, and therefore, Ce is determined from x-ray diffraction with a better precision than Ge. 

Neutron coherent scattering lengths b and coherent cross sections a shown in the form of a Periodic Table of the elements in which the radius of the circle is proportional to b and the area is proportional to a. For a few elements e.g. H) the value of b is negative this is indicated by the use of open circles. Black squares indicate elements with large absorption cross sections due to the occurrence of nuclear absorption edges. 

The end points define the lateral distance in the sample plane across which a given spherical wave front becomes out of phase by n degrees. For a spherical wavelength of 5 A emanating from a source 5 m away, the length I is of the order of 100 pm. In practice, the lateral coherence length depends on the energy spread and the wave vector spread of the incident neutron wave front If the instrumental... 

The distinct intramolecular and intermolecular contributions of the differential cross-section (5) are related to the sum of all partial structure factors, which is essentially proportional to the differential scattering cross-section, weighted by the respective coherent neutron scattering lengths (6). 

C is a constant, and < > denotes an equilibrium average. For small-angle neutron scattering (SANS), the scattering factor is proportional to the coherent neutron scattering-length density 6, giving the result that... 

Electron microscopic observations of the Ortho-III phase have been reported far less than of the Ortho-II phase [7.8, 7.34, 7.35]. Clear evidence for the occurrence of this phase is given in the [001] diffraction of Fig. 7.4(d). Due to the short structural coherence length of this phase, the results could not be confirmed either by X-ray, or by neutron diffraction. The Ortho-III phase ideally appears at an oxygen deficiency 0 =, and compared to the Ortho-I phase, has one out of three CuiOi chains depleted of oxygen in an ordered succession. (See Fig. 7.2(c)). Lattice parameters are am 3ai, bm b and cm c. The tripling of the unit-cell is reflected in the diffraction pattern by the appearance of two superstructure spots at positions h+ kl and h + kl. Superstructure spots for the Ortho-III phase are always very broad, due to substantial disorder in the cell-tripling alternation, or to the limited size of the domains. 

A study on polyacetylene of the Naarmann-Theo-philou type [14] has been published by Djurado et al. [82], using both x-ray and neutron diffraction. This type of polyacetylene is characterized by a high value of the longitudinal coherence length, Z. n = 170 A, and a reduced rate of isomerization at 300 K compared to... 

In their neutron diffraction studies of p-type doping of PPP, Stamm and Hocker [179] have determined the setting angle (p of the chains in the pristine polymer to be 57° 3°. In this refinement, the authors started from the data by Kovacic et al. [169], but assumed a monoclinic P2i/a structure with /i=I00°. While a number of later studies by others confirm that the value of 4> is around 57°, Stamm et al. [180] find a much lower value of 45° in their electron diffraction work. The neutron study yields lateral coherence lengths L of 60 A for Kovacic PPP and 150 A for Yamamoto PPP. 

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