# Extreme Symmetry

We need to point out that, if the wavelengths of laser radiation are less than the size of typical structures on the optical element, the Fresnel model gives a satisfactory approximation for the diffraction of the wave on a flat optical element If we have to work with super-high resolution e-beam generators when the size of a typical structure on the element is less than the wavelengths, in principle, we need to use the Maxwell equations. Now, the calculation of direct problems of diffraction, using the Maxwell equations, are used only in cases when the element has special symmetry (for example circular symmetry). As a rule, the purpose of this calculation in this case is to define the boundary of the Fresnel model approximation. In common cases, the calculation of the diffraction using the Maxwell equation is an extremely complicated problem, even if we use a super computer. [c.265]

The primary reason for interest in extended Huckel today is because the method is general enough to use for all the elements in the periodic table. This is not an extremely accurate or sophisticated method however, it is still used for inorganic modeling due to the scarcity of full periodic table methods with reasonable CPU time requirements. Another current use is for computing band structures, which are extremely computation-intensive calculations. Because of this, extended Huckel is often the method of choice for band structure calculations. It is also a very convenient way to view orbital symmetry. It is known to be fairly poor at predicting molecular geometries. [c.33]

It is possible that a transition structure calculation will give two negative frequencies (a second-order saddle point) or more. This gives a little bit of information about the potential energy surface, but it is extremely unlikely that such a structure has any significant bearing on how the reaction occurs. This type of structure will often be found if the starting geometry had a higher symmetry than the transition structure should have. [c.156]

There is, in principle, no reason why linear combinations should not be made between AOs which have the correct symmetry but very different energies, such as the lx orbital on the oxygen atom and the lx orbital on the phosphorus atom. The result would be that the resonance integral /i (see Figure 7.12) would be extremely small so that the MOs would be virtually unchanged from the AOs and the linear combination would be ineffective. [c.233]

Cu(II) complexed to four Cl ions can adopt arrangements from tetragonal to square planar. The latter arrangement gives maximum Cu—Cl bonding the former minimizes the Cl—Cl repulsion. In practice, complexes occur near both extremes and with almost all intermediate arrangements. The geometry of a given complex is determined by the counterion. Strongly hydrogen bonding counterions draw off electron density from the Cl and favor square planar symmetry. With pressure one finds that both neady square planar and neady tetrahedral complexes distort toward an intermediate symmetry with clear-cut changes in the electronic absorption spectmm. These transformations occur over relatively short ranges of pressure 1.5-2.5 GPa (15—25 kbar). The principle illustrated here is that at high compression the economy of best geometric packing overcomes the weaker van der Waals and hydrogen bonding forces. [c.168]

The CH3 group connected to the rest of the molecule by the fourth C-atom bond can rotate around this bond. The potential y((p) the rotating group experiences is three-fold. In the extreme case when the barrier is absent, the group rotates freely, while in the opposite limit of an infinite barrier, it oscillates in one of three minima. How the energy levels change when moving on from free rotation to torsional vibration is shown in fig. 49. Tunneling partially eliminates the triple degeneracy of each vibrational level creating a singlet (A) and doublet (E b) in accordance with the irreducible representations of the C3 symmetry group, which is isomorphous to the permutation group. Figure 49 shows that the A and E levels alternate in the progression of torsional multiplets n = 0,1,2,. .., and the sign of the tunneling splitting is (— 1)". [c.114]

From environmental to packaging to catalysis issues, the need to understand how molecules interact chemically and bond to a surface is paramount. XPD is an extremely good candidate for investigating adsorbate-substrate interactions because chemical shifts in the core-level transitions can lead to the identification of a specific species, and the scattering of core-level photoelectrons can lead to the determination of the structure in which they exist. Consider the initial interaction of gaseous CO on room temperature Fe (001). At this temperature Fe (001) has a bcc lattice structure with a fourfold symmetric surface. At a coverage of less than a monolayer, it was known that the CO adsorbs to the Fe, residing in fourfold hollow sites with only the C making direct contact with the Fe. The orientation of the C-O bond remained a question. It was proposed that this early stage of CO coverage on Fe (001) produced an intermediary state for the dissociation of the C and O, since that the CO bond was tilted with respect to the surface normal, unlike the upright orientation that CO was found to possess on Ni. Although near-edge X-ray adsorption fine structure (NEXAFS) results measured a tilt in the CO bond, the results were not very quantitative regarding the exact angle of the tilt. XPD, on the other hand, gave the CO bond angle as 35 2° relative to the surface, as determined from the large forward scattering peak depicted in by the solid line Figure 2a along the (100) azimuth. Here the ordinate is plotted as the C Is intensity divided by the O Is intensity. Plotting this ratio effectively removes instrumental contributions to the diffraction pattern, the oxygen atoms have no atoms above them from which their photoelectrons can scatter and thus should be featureless. The azimuthal scan shown in Figure 2b was taken at a polar angle of 35° to enhance the C Is diffraction signal. From the fourfold symmetry and knowledge of the crystallographic orientation of the Fe, it is clear that the tilt direction lies in the <100> planes, as depicted in Figure 2c the absence of a diffraction peak in the [ITO] polar scan shown by the dashed line in Figure 2a helps to confirm this. [c.245]

In the previous section we discussed wall functions, which are used to reduce the number of cells. However, we must be aware that this is an approximation that, if the flow near the boundary is important, can be rather crude. In many internal flows—where all boundaries are either walls, symmetry planes, inlets, or outlets—the boundary layer may not be that important, as the flow field is often pressure determined. However, when we are predicting heat transfer, it is generally not a good idea to use wall functions, because the convective heat transfer at the walls may be inaccurately predicted. The reason is that convective heat transfer is extremely sensitive to the near-wall flow and temperature field. [c.1038]

Experiments to test these remarkable theoretical predictions have been extremely difficult to carry out, largely because the electronic properties are expected to depend strongly on the diameter and the chirality of the CNT. Apart from the problems associated with making electronic measurements on structures just a nanometer across, it is also crucial to gain information on the symmetry of the CNT. Recently, the Delft [5] and Harvard [6] groups used scanning tunnelling microscope (STM) probes at low temperature to observe the atomic geometry (the CNT diameter and helicity), and to measure the associated electronic structure (DOS). The singularities in the DOS are very clearly seen in STS studies at 4 K [5]. Furthermore, a non-zero DOS at the Fermi level is reported [6] in the metallic CNTs, as expected and a vanishingly small DOS is measured in the semiconducting CNTs. Both groups also confirm an inverse linear dependence of the band gap on the CNT diameter. These experiments [5,6] have provided the clearest confirmation to date that the electronic DOS have singularities typical of a ID system. [c.110]

PLgel heads are made in a hatch process, and the characteristics of each hatch of material are assessed and quality controlled as described previously. As PS/ DVB is highly cross-linked and extremely stable, it therefore has a good shelf life and repeated assessment of material performance is not really an issue. Although the material properties remain consistent, by its very nature the column packing process can be susceptible to a number of variables which will ultimately affect the performance of the finished column. Therefore, every column that is packed must be tested individually to ensure a consistent product. The column performance, i.e., how well the column has been packed, is assessed by measuring the number of theoretical plates and the symmetry for a test probe eluting at total permeation (8). It is important to remember that this type of measurement will itself be affected by a number of chromatographic factors flow rate, temperature, choice of test probe, and system dispersion. Therefore, it is vital to establish a protocol for this test that is rigidly adhered to in order to ensure consistent column performance. [c.355]

It should be noted that, whereas ferroelectrics are necessarily piezoelectrics, the converse need not apply. The necessary condition for a crystal to be piezoelectric is that it must lack a centre of inversion symmetry. Of the 32 point groups, 20 qualify for piezoelectricity on this criterion, but for ferroelectric behaviour a further criterion is required (the possession of a single non-equivalent direction) and only 10 space groups meet this additional requirement. An example of a crystal that is piezoelectric but not ferroelectric is quartz, and ind this is a particularly important example since the use of quartz for oscillator stabilization has permitted the development of extremely accurate clocks (I in 10 ) and has also made possible the whole of modern radio and television broadcasting including mobile radio communications with aircraft and ground vehicles. [c.58]

Molecular oxygen, O2, is unique among gaseous diatomic species with an even number of electrons in being paramagnetic. This property, first observed by M. Faraday in 1848, receives a satisfying explanation in terms of molecular orbital theory. The schematic energy-level diagram is shown in Fig. 14.1 this indicates that the 2 least-strongly bound electrons in O2 occupy degenerate orbitals of n symmetry and have parallel spins. This leads to a triplet ground state,. As there are 4 more electrons in bonding MOs than in antibonding MOs, O2 can be formally said to contain a double bond. If the 2 electrons, whilst remaining unpaired in separate orbitals, have opposite spin, then a singlet excited state of zero resultant spin results, Ag. A singlet state also results if the 2 electrons occupy a single n orbital with opposed spins, S+. These 2 singlet states lie 94.72 and 157.85 kJmol above the ground state and are extremely important in gas-phase oxidation reactions (p. 614). The excitation is [c.605]

One fomi of electron diffraction is similar to the precession method, except that the single crystal is a grain of a poly crystalline foil. Figure B1.8.6 shows an electron diffraction pattern produced when the beam is directed down a fivefold symmetry axis of a quasicrystal. Because of the very short wavelength tire cone angle is so small that it lies within the mosaic spread of the grain, and the resulting diffraction pattern, after magnification by the electron optics, closely resembles a precession pattern made with x-rays. In this teclmique the divergence of the electron beam is extremely small, and the diffraction spots correspond to lattice points in a plane of the reciprocal lattice that passes tiirough the origin. The diffraction pattern therefore has a centre of symmetry. In convergent beam electron diffraction (CBED) [28] (see figure Bl.8.7 the divergence of the electron beam is still only a few tenths of a degree, but the resultant smearing of the Ewald sphere allows it to mtersect layers of the reciprocal lattice adjacent to the one passing through the origin, so that a region of broadened diffraction spots is surrounded by one or more rings of additional spots corresponding to points in these adjacent planes. Because Friedel s law does not apply m those planes, the pattern more closely reflects the true symmetry of the crystal. [c.1380]

Figure 3. Phase tracing for circling outside the ci pair for the model in A and states in symmetry. The Berry phase (half the angle shown at the extremity of the figure) is here —2tt. |

To convert this strategy into an actual result, we must examine the mechanics of two-dimensional rotation. When the shear force is first applied, the molecule experiences an acceleration. Within a short time, however, the shear force and the force of viscous resistance to the particle movement equalize and no further net acceleration occurs. This means that the particle rotates with a constant average angular velocity. This final situation is called a stationary-state condition and is the subject of our attention. Our first problem, then, is to evaluate this average angular velocity. Even though the particle is in a stationary state, its velocity is not absolutely constant, but only constant on the average. To see that this is the case, we return to an inspection of Fig. 2.11b. Those polymer segments along the y axis bear the full brunt of the velocity gradient as an inducement to rotation. It can be shown (e.g., see Schultz s Polymer Materials Science) by an analysis of the torque balance under stationary-state conditions that for segments along the y axis (subscript y), cOy = dv/dy. At the same time those segments which lie along the x axis (subscript x), experience no difference in velocity relative to the center of mass of the molecule co = 0. The angular velocity of a segment depends on its location relative to the center of mass in the molecule. Because of the symmetry of the rotation, the average angular velocity of a segment is the mean of these two extremes [c.108]

Spray Dynamic Structure. Detailed measurements of spray dynamic parameters are necessary to understand the process of droplet dispersion. Improvements in phase Doppler particle analyzers (PDPA) (29) permit m situ measurements of droplet size, velocity, number density, and Hquid flux, as weU as detailed turbulence characteristics for very small regions within the spray. Such measurements allow designers to evaluate differences in atomizers, changes in droplet size distributions, radial and circumferential symmetry, size—velocity correlations, interactions between droplets and the surrounding gas, droplet time-dependent behavior, and droplet drag and trajectories. In addition, the information can be extremely valuable for verifying physical models and estabHshing general correlations for atomizer performance. [c.331]

At the other extreme from rational continuum mechanics we have the study of elastic and plastic behaviour of single crystals. Crystal elasticity is a specialised field of its own, going back to the mineralogists of the nineteenth century, and involving tensor mathematics and a detailed understanding of the effects of different crystal symmetries the aforementioned Cauchy had a hand in this too. Crystal elasticity is of considerable practical use, for instance in connection with the oscillating slivers of quartz used in electronic watches these slivers must be cut to precisely the right orientation to ensure that the relevant elastic modulus of the sliver is invariant with temperature over a limited temperature range. The plastic behaviour of metal crystals has been studied since the beginning of the present century, when Walter Rosenhain (see Chapter 3, Section 3.2.1) first saw slip lines on the surface of polished polycrystalline metal after bending and recognised that plasticity involved shear ( slip ) along particular lattice planes and vectors. Crystal plasticity was studied intensely from the early 1920s onwards, and understanding was codified in two important experimental texts (Schmid and Boas 1935, Elam 1935) crucial laws such as the critical shear stress law for the start of plastic deformation were established. In the 1930s a start was also made with the study of plastic deformation in polycrystalline metals in terms of slip in the constituent grains. This required a combination of continuum mechanics and the physics of single-crystal plasticity. This branch of mechanics has developed fruitfully as a joint venture between mechanical engineers, applied (applicable) mathematicians, metallurgists and solid-state physicists. The leading spirit in this venture was Geoffrey (G.I.) Taylor [c.48]

My ehosen examples inelude rapid solidifieation, where the extremity is in eooling rate nanostruetured materials, where the extremity is in respeet of extremely small grains surface seienee, where the extremity needed for the field to develop was ultrahigh vaeuum, and the development of vacuum quality is traced thin films of various kinds, where the extremity is in one minute dimension and quasierystals, where the extremity is in the form of symmetry. Various further examples could readily have been ehosen, but this chapter is to remain relatively short. [c.393]

Inspection of the coefficients and a familiarity with the way they translate into symmetry properties of orbitals can be used in an extremely powerful way to aid in understanding a number of aspects of the properties of conjugated unsaturated compoimds. Such considerations apply particularly well to flie class of reactions classified as concerted, which will be described in detail in Chapter 11. It can be seen in Table 1.15 fliat flie coefficients are all of like sign in the lowest-energy orbital, j, and that flie number of times that a sign change occurs in the wave function increases with the energy of the orbital. A change in sign of the coefficients of the AOs on adjacent atoms corresponds to an antibonding interaction between flie two atoms, and a node exists between them. Thus, has no nodes, 2 has one, 3 has two, and so on up to which has five nodes and no bonding interactions in its combination of AOs. A diagrammatic view of flie bonding and antibonding interactions among the AOs of 1,3,5-hexatriene is presented in Fig. 1.10. Notice that for the bonding orbitals i d 3, fliete are more bonding interactions [c.33]

The bulk liquid state is a challenge for physicists. It has the trivial translational symmetry of the ideal gas, but the particles are strongly interacting as neighbors are roughly as close to each other as they are in a crystal. For those complex fluids where one has a reliable pair potential between the macro-particles it is usually a standard exercise to simulate the liquid structure. (Remember that we dealt with this problem in our tutorial.) However, this simphfied picture is not always justified. Once one has to take into account the solvent degrees of freedom, the simulation becomes a challenge one is faced with the different length scales of micro- and macro-particles and rapidly encounters the hmitations of today s computers. Nevertheless, the pressure of charged spherical macro-ions was investigated by Stevens et al. [10]. A system with an extreme asymmetric charge distribution was considered by Lowen and D Amico [11]. Simulations of highly asymmetric electrolytes with charge asymmetry up to 20 1 were done by Lobaskin and Linse [12], who found that an effective repulsive potential was operating between the macro-ions. Concerning the primitive model, a mechanism for counterion-mediated attraction between the hke-charged spherical macro-ions was proposed [13] this originates from a depletion zone of counterions between nearly touching macro-ions. Also, the effective triplet interactions in charged colloidal suspensions represents a significant attractive correction to the pairwise contributions [14]. [c.754]

All these compounds have (distorted) tetrahedral molecules, those of formula O2SX2 having C2v symmetry and the others Cj. Dimensions are in Table 15.15 the remarkably short O-S and S-F distances in O2SF2 should be noted (cf. above). Indeed, the implied strength of bonding in this molecule is reflected by the fact that it can be made by reacting the normally extremely inert compound SFg (p. 687) with the fluoro-acceptor SO3 [c.695]

S4N4 is kinetically stable in air but is endothermic with respect to its elements (AWf 460 8 kJmol ) and may detonate when struck or when heated rapidly. This is due more to the stability of elementary sulfur and the great bond strength of N2 rather than to any inherent weakness in the S-N bonds. On careful heating S4N4 melts at 178.2°. The structure (Fig. 15.33a) is an 8-membered heterocycle in the extreme cradle configuration it has D2J symmetry and resembles that of AS4S4 (p. 579) but with the sites of the Group 15 and Group 16 elements interchanged. The S-N distance of 162 pm is rather short when compared with the sum of the covalent radii (178 pm) and this, coupled with the equality of all the S-N bond distances in the molecule, has been attributed to some electron delocalization in the heterocycle. The trans-annular S- -S distances (258 pm) are intermediate between bonding S-S (208 pm) and [c.723]

The above considerations illustrate the inherent contradiction in designing highly accurate force fields. To get high accuracy for a wide variety of molecules, and a range of properties, many and ftmctional complex tenns must be included in the force field expression. For each additional parameter introduced in an energy term, the potential number of new parameters to be derived grows as the number of atom types to a power between 1 and 4. The higher accuracy that is needed, the more finely the fundamental units must be separated, i.e. more atoms types must be used. In the extreme limit, each atom which is not symmetry related is a new atom type in each new molecules. In this limit each molecule will have its own set of parameters to be used just for this one molecule. To derive these parameters, the molecule must be subjected to many different experiments, or a large number of electronic structure ealeulations. This is the approach used in inverting spectroscopie data to produce a potential energy surface. From a force field point of view the resulting function is essentially worthless, it just reproduces known results. To be useful a force field should be able to predict unknown properties of molecules from known data on other molecules, i.e. a sophisticated form of inter- or extrapolation. If the force field beeomes very complicated, the amount of work required to derive the parameters may be larger than the work required for measuring the property of interest for a given moleeule. [c.31]

The methods described in sections 14.1-14.3 can only locate the nearest minimum, which is normally a local minimum, when starting from a given set of variables. In some cases the interest is in the lowest of all such minima, the global minimum, in other cases it is important to sample a large (preferably representative) set of local minima. Considering that the number of minima typically grows exponentially with the number of variables, the global optimization problem is an extremely difficult task for a multidimensional function. " It is often referred to as the multiple minima or combinatorial explosion problem in the literature. Consider for example a determination of the lowest energy conformation of linear alkanes, CH3(CH,) , CH3, by a force field method. There will in general be three possible energy minima for a rotation around each C-C bond. For butane there are thus three conformations, one anti and two gauche (which are symmetry equivalent). These minima may be generated by starting optimizations from three torsional angles separated by 120°. In the CH3(CH2) + CFl3 case there are n such rotatable bonds, for a possible 3 " different conformations. In order to find the global minimum we need to calculate the energy of them all. Assume for the moment that each optimization of a conformation takes one second of computer time. Table 14.4 gives the number of possible conformations, and the time required for optimizing them all. [c.339]

See pages that mention the term

**Extreme Symmetry**:

**[c.391] [c.414] [c.567] [c.474] [c.266] [c.337] [c.416] [c.1106] [c.165] [c.879] [c.922] [c.931] [c.1159] [c.418] [c.132]**

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** The coming of materials science
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