Torus form

In the parabolic model the equations for caustics are simply Q+ = Q, and Q- = <2-- The periodic orbits inside the well are not described by (4.46), but they run along the borders of the rectangle formed by caustics. It is these trajectories that correspond to topologically irreducible contours on a two-dimensional torus [Arnold 1978] and lead to the quantization condition (4.47). 

Key Words —Carbon, molecular dynamics, torus, helix, graphitic forms. 

Here, the topological nature of the tori will be discussed briefly. Figure 1 shows the five possible prototypes of toroidal forms that are considered to be related to fullerenes. These structures are classified by the ratios of the inner and outer diameters r, and r, and the height of the torus, h. (Note that is larger than / ,) As depicted in Fig. 1, if r, = r, and h r, and h = — r,) then the toroidal forms are of type... 

Fig. 1. Five possible simple prototypes of the toroidal forms of graphitie carbon. All cross-sections of the tube are square. Here r , r,, and h are the outer and inner radii and height of the torus, respectively.

The relaxed structures of the various (rotational) symmetric toroidal forms were obtained by steepest decent molecular-dynamics simulations[15]. For the elongated tori derived from torus C240, the seven-fold rotational symmetry is found to be the most stable. Either five-fold or six-fold rotational symmetry is the most stable for the toroidal forms derived from tori Cjyo and C540, respectively (see Fig. 5). 

Because the cohesive energy of the fullerene Cyo is —7.29 eV/atom and that of the graphite sheet is —7.44 eV/atom, the toroidal forms (except torus C192) are energetically stable (see Fig. 5). Finite temperature molecular-dynamics simulations show that all tori (except torus Cm2) are thermodynamically stable. 

Fig. 10. Ffelically coiled form 540 one pilch contains a torus...

The number of atoms forming the smallest = l knee of the (9 ,0)-(5n,5 ) series is 57 and it leads to the C520 torus having adjacent heptagons. The number of atoms of any knee of that series is given by ... 

Fig. 7. (a) Torus C520 formed by 10 (9,0)-(5,5) knees, (b) torus C900 formed by 10 (9,0)-(5,5) knees extended by adding one circumferential ring on both sides. 

As a final example we consider noncovalent molecular complex formation with the macrocyclic ligand a-cyclodextrin, a natural product consisting of six a-D-glucose units linked 1-4 to form a torus whose cavity is capable of including molecules the size of an aromatic ring. Table 4-3 gives some rate constants for this reaction, where L represents the cyclodextrin and S is the substrate ... 

The pneumatic tire has the geometry of a thin-wallcd toroidal shell. It consists of as many as fifty different materials, including natural rubber and a variety ot synthetic elastomers, plus carbon black of various types, tire cord, bead wire, and many chemical compounding ingredients, such as sulfur and zinc oxide. These constituent materials are combined in different proportions to form the key components of the composite tire structure. The compliant tread of a passenger car tire, for example, provides road grip the sidewall protects the internal cords from curb abrasion in turn, the cords, prestressed by inflation pressure, reinforce the rubber matrix and carry the majority of applied loads finally, the two circumferential bundles of bead wire anchor the pressnrized torus securely to the rim of the wheel. 

FIGURE 16.44 The structure of a molecular magnet. The nano-size molecular torus contains 84 manganese atoms and is approximately 4 nm in diameter. The manganese atoms are bonded to groups of carbon atoms in the form of acetate ions, water molecules, and chlorine atoms. In this molecule the manganese atoms act as terromagnets. 

The answer is "NO." In the combustible mixture, an electric spark produces a flame kernel. Initially, its shape is elliptical (like an American football), and then becomes a torus (like an American doughnut). Afterwards, it changes into almost spherical shape, and propagates spherically in the unbumed mixture. This process is formed by the existence of spark electrodes, which is necessary for spark discharge. Spark electrodes lead not only to heat loss from the flame kernel but also a change in the kernel shape. Both affect the minimum ignition energy. 

False color images of adjacent flame interactions in the FF configuration. A torus of fresh reactants is formed in this case. 

A torus is formed by rotating a circle, radius r2, about an axis. 

A torispherical shape, which is often used as the end closure of cylindrical vessels, is formed from part of a torus and part of a sphere, Figure 13.7. The shape is close to that of an ellipse but is easier and cheaper to fabricate. 

There are not therefore three lone pair nonbonding domains but a nonbonding domain containing six electrons having the form of a torus around the fluorine atom. This is the situation for all atoms, except hydrogen atoms, in any diatomic molecule, as we have seen for the fluorine atom in HF, and for any singly bonded ligand. 

---