Truncated crystal

The absorption spectrum of Cp3Nd(NCCH3) in a single crystal form at liquid nitrogen temperatures showed a truncated crystal field splitting pattern could be derived. The parameters of an empirical Hamiltonian were fitted [41] with the energies of 41 levels with a root mean square deviation of 26 cm1. 

Layered chemical sediments may contain a variety of dissolution features produced by contact with undersaturated waters such as rounded or truncated crystal surfaces, rounded dissolution cavities, and vertically-oriented dissolution pipes (Fig. 3D and 3E). The frequency and nature of these syndepositional dissolution features may be used to distinguish perennial saline lake deposits from saline pan (ephemeral lake) sediments (Li et al., 1996 Schubel Lowenstein, 1997). In very shallow water (centimeters deep) to subaerially-exposed environments, layered halite crusts have little to no protection from incoming... 

Fig. 5.3 Electron micrographs of single crystals of polyethylene crystallised from dilute solution in xylene (a) diamond-shaped crystals and (b) truncated crystals. (Reprinted by permission of John Wiley Sons, Inc.)...

The structure of a real (clean) surface may differ from that of an ideally truncated crystal. Two types of deviation are possible relaxation and reconstruction. 

All static studies at pressures beyond 25 GPa are done with diamond-anvil cells conceived independently by Jamieson [32] and by Weir etal [33]. In these variants of Bridgman s design, the anvils are single-crystal gem-quality diamonds, the hardest known material, truncated with small flat faces (culets) usually less than 0.5 nun in diameter. Diamond anvils with 50 pm diameter or smaller culets can generate pressures to about 500 GPa, the highest static laboratory pressures equivalent to the pressure at the centre of the Earth. 

In periodic boimdary conditions, one possible way to avoid truncation of electrostatic interaction is to apply the so-called Particle Mesh Ewald (PME) method, which follows the Ewald summation method of calculating the electrostatic energy for a number of charges [27]. It was first devised by Ewald in 1921 to study the energetics of ionic crystals [28]. PME has been widely used for highly polar or charged systems. York and Darden applied the PME method already in 1994 to simulate a crystal of the bovine pancreatic trypsin inhibitor (BPTI) by molecular dynamics [29]. 

Crystallization conditions such as temperature, solvent, and concentration can influence crystal form. One such modification is the truncation of the points at either end of the long diagonal of the diamond-shaped crystals seen in Fig. 4.11b. Twinning and dendritic growth are other examples of such changes of habit. 

Often (adsorption, reconstruction) the periodicity at the surface is larger than expected for a bulk-truncated surface of the given crystal this leads to additional (superstructure) spots in the LEED pattern for which fractional indices are used. The lattice vectors bi and b2 of such superstructures can be expressed as multiples of the (1 X 1) lattice vectors ai and Zx. ... 

Fullerenes are described in detail in Chapter 2 and therefore only a brief outline of their structure is presented here to provide a comparison with the other forms of carbon. The C o molecule, Buckminsterfullerene, was discovered in the mass spectrum of laser-ablated graphite in 1985 [37] and crystals of C o were fust isolated from soot formed from graphite arc electrodes in 1990 [38]. Although these events are relatively recent, the C o molecule has become one of the most widely-recognised molecular structures in science and in 1996 the codiscoverers Curl, Kroto and Smalley were awarded the Nobel prize for chemistry. Part of the appeal of this molecule lies in its beautiful icosahedral symmetry - a truncated icosahedron, or a molecular soccer ball, Fig. 4A. 

Figure 14.11 Crystal structure of HPF6.6H2O showing the cavity formed by 24 H2O molecules disposed with their O atoms at the vertices of a truncated octahedron, The PFe octahedra occupy centre and comers of the cubic unit cell, i.c. one PFr, at the centre of each cavily. ...

One of the more important uses of OM is the study of crystallization growth rates. K. Cermak constructed an interference microscope with which measurements can be taken to 50° (Ref 31). This app allows for study of the decompn of the solution concentrated in close proximity to the growing crystal of material such as Amm nitrate or K chlorate. In connection with this technique, Stein and Powers (Ref 30) derived equations for growth rate data which allow for correct prediction of the effects of surface nucleation, surface truncation in thin films, and truncation by neighboring spherulites... 

Great progress has been made, however, in our later study using in situ SXS [Stamenkovic et ak, 2007a], where, by simultaneously fitting the intensity ratio between two different sets of crystal truncation rod (CTR) data that constrain the fit to the full CTR data [Robinson, 1986 Warren, 1990], it was possible to reveal the elemental concentration profile at the surface (Fig. 8.13c). Based on the in situ SXS results depicted in Fig. 8.13a, the topmost surface layer is confirmed to be 100 at%... 

Robinson IK. 1986. Crystal truncation rods and surface roughness. Phys Rev B 33 3830-3836. 

In the case of supported metalhc particles, the construction is modified by introducing the adhesion energy (Wulff-Kaishew construction) [Henry, 1998]. The equilibrium shape is a Wulff polyhedron, which is truncated at the interface by an amount Ahs, according to the relation Ahs/hj = /3/(t where /3 is the adhesion energy of the crystal on the substrate. 

The exact amount of error introduced cannot immediately be inferred from the strength of the amplitudes of the neglected Fourier coefficients, because errors will pile up in different points in the crystal depending on the structure factors phases as well to investigate the errors, a direct comparison can be made in real space between the MaxEnt map, and a map computed from exponentiation of a resolution-truncated perfect m -map, whose Fourier coefficients are known up to any order by analysing log(<7 (x) tm (x)). 

In crystals, the scattering densities are periodic and the Bragg amplitudes are the Fourier components of these periodic distributions. In principle, the scattering density p(r) is given by the inverse Fourier series of the experimental structure factors. Such a series implies an infinite sum on the Miller indices h, k, l. Actually, what is performed is a truncated sum, where the indices are limited to those reflections really measured, and where all the structure factors are noisy, as a result of the uncertainty of the measurement. Given these error bars and the limited set of measured reflections, there exist a very large number of maps compatible with the data. Among those, the truncated Fourier inversion procedure selects one of them the map whose Fourier coefficients are equal to zero for the unmeasured reflections and equal to the exact observed values otherwise. This is certainly an arbitrary choice. 

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