Disorder of the first kind

This discussion has shown that the diffraction pattern can reveal three types of disorder, as discussed fully by several authors (Wiener and White, 1991a Blaurock, 1982 Hosemann and Bagchi, 1962). Thermal disorder is generally referred to as disorder of the first kind and lattice disorder as disorder of the second kind. The disorder due to the mosaic nature of the sample is referred to as orientational disorder. Thermal disorder and small amounts of orientational disorder are not particularly troublesome in the diffraction experiment. Lattice disorder, on the other hand, can be extremely problematic because one can never achieve a fully resolved image of the stmcture since there are too few stmcture factors available to obtain a faithful model. Thermal disorder simply means that the position of the atoms are .smeared in some fashion, determined by the equation of state of the molecules. If the lattice is excellent so that all of the stmcture factors observable within the limits of the... 

Finally, a note on disorder of the membrane stacks and on attempts to correct for it in the analysis of diffraction data. Generally, two kinds of disorder are being discussed in crystal structure Disorder of the first kind refers to displacements of the structural elements (for example the one-dimensional unit cell of a membrane stack) from the ideal positions prescribed by the periodic lattice. The effect on the diffraction pattern is indistinguishable from that of thermal vibrations and may, therefore, be expressed as a Debye-Waller temperature factor so that the structure factor, expressed as a cosine series, includes a Gaussian terra, according to... 

Fig. 5a-c. Diffraction patterns of lamellar stacks with perfect stacking order (a), with disorder of the first kind (b), and disorder of the second kind (c). For details of the model parameters, see Ref. 4. (From Ref. 4, with permission)... 

Two kinds of crystalline disorder can be identified in XRD data. The disorder of the first kind refers to the imperfections... 

Distortions of the First Kind and Thermal Disorder. In crystallography the best-known example for a lattice distortion of the first kind is the reduction of peak intensity from random temperature movement of the atoms. In materials science a frozen-in thermal disorder of nanostructures25 is observed as well. The result of this kind of disorder is a multiplicative26 attenuation of the scattering intensity by the Debye-Waller factor... 

Fig. 10a and b. Small-angle diffraction pattern from an oriented, stacked sample of sarcoplasmic reticulum vesicles, obtained in 1 s with synchrotron radiation at Stanford Synchrotron Radiation Laboratory. The peaks correspond to the first seven orders of a lamellar diffraction pattern with a lattice periodicity of 200 A and exhibit the significant effects of lattice disorder of the second kind. The intensity distribution in (b) was obtained by integration of the two-dimensional pattern (a) between the limits indicated in (a). (From Ref. 85, with permission)... 

Crystalline polymers characterized by disordered conformations of the chains are, for instance, polytetrafluoroethylene [10-12,49-54], cis-l,4-poly (isoprene) [100-102] and trans-1,4-poly( 1,3-butadiene) [8,9,55-58]. In these cases, disorder does not destroy the crystallinity because of the similar shape of the various conformational units. The occmrence of cases of conformational isomorphism of the first kind demonstrates that a polymer chain can remain straight, as if it was constrained to run inside the walls of a tight cylinder, while its conformational freedom remains of the same order of magnitude as that in the melt. 

For distortions of the first kind, neglecting thermal disorder and in absence of lattice disorder displacement, the diffuse part of the X-ray scattering (fdiff) is given by Eq. 3 (see for instance [152]) ... 

The results of the physicochemical analysis indicated that the CulhTe2-CdTe line consisted of a solid-solution region (with the zinc-blende structure) separated from the original compound CulhTe2 the chalcopyrite structure) by a two-phase region in which one of the phases was ordered in the chalcopjrrite form. This order-disorder transition was a phase transition of the first kind. 

As an example, consider the study by Banerjee et al. (1984) on the effect of electron irradiation on the order-disorder transformation in (DIJ Ni4Mo. Electron micrographs and diffraction patterns were obtained during in situ electron irradiations at 50-1050 K in a HVEM. At temperatures below 200 K, the alloy completely disorders. At 200-450 K, only SRO was observed, and the transition between LRO and SRO, which occurs via the completely disordered state, is consistent with the concentration-wave description of the SRO structure and supports the concept of spinodal ordering. It is believed that an interstitial mechanism is responsible for maintaining the SRO. Above 450 K, LRO persisted for samples initially in this state and SRO was only preserved up to 550 K for samples initially in that state. Between 550 and 720 K, a mixed SRO-LRO state occurred, and at temperatures above 720 K a complete transition to SRO was obtained. It is believed that maintenance of LRO requires a vacancy mechanism. At temperatures below 800 K the SRO-LRO transition occurred in a continuous fashion, while above 800 K a nucleation and growth mechanism was operative. This behavior is characteristic of an ordering transition of the first kind below and above the coherent instability temperature. 

There exist two kinds of bounds. To illustrate the first kind, suppose there is diagonal disorder only, with the e. independent random variables each having an average value e, the same as for the perfect crystal, and with the V. as in the crystal. Let p(ej be the... 

The order-disorder transformation is not unique to two-layer fluids, which is readily concluded from the second maximum of n in the vicinity of s 3.55 where the fluid consists of three strata. However, it turns out that only the innermost, middle stratum undergoes the same kind of structural reorganization just explained for the two-layer fluid the two contact strata (i.e., the strata closest to the substrate) do not participate in the transformation. The intensity of the second maximum in n is therefore reduced by roughly 2/3 compared with the first one, as one would expect. 

To conduct experiments of this kind it is very convenient to make use of disorder adsorbent provided by a film of amorphous selenium. During deposition under vacuum conditions, the pressure being no higher than lO Torr, the amorphous modification of selenium is being formed [38]. There are two forms of amorphous selenium which differ in coordination numbers and radii of coordination spheres. The first form is... 

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