Expanding to Infinity

Objects or patterns which are periodic in one, two, and three directions will have one-, two-, and three-dimensional space groups, respectively. The dimensionality of the object/pattern is merely a necessary but not a satisfactory condition for the dimensionality of their space groups. We shall first 

CsnOn 3- 2. Quaerendo invenietts. Canon, coatrarium stricte re ersum (Oley) Andnnte t 

The one-dimensional space groups are the simplest of the space groups. They have periodicity only in one direction. They may refer to onedimensional, two-dimensional, or three-dimensional objects, cf. G, G, and GJ of Table 2-2, respectively. The infinite carbon chains of the carbide molecules 

Hargittai, I. Hargittai, Symmetry through the Eyes of a Chemist, 3rd ed., 

Objects or patterns which are periodic in one, two, and three directions will have one-, two-, and three-dimensional space groups, respectively. The dimensionality of the object/pattem is merely a necessary but not a satisfactory condition for the dimensionality of their space groups. We shall first describe a planar pattern after Budden [3] in order to get the flavor of space-group symmetry. Also, some new symmetry elements will be introduced. Later in this chapter, the simplest one-dimensional and two-dimensional space groups will be presented. The next Chapter will be devoted to the three-dimensional space groups which characterize crystal structures. 

Canon a 2. Queerendo iovemcfis. CiitoiiMBMtittm etrlctc Mwrsumr (Oley) 

---

A symmetric pattern expanding to infinity always contains a basic unit, a motif, which is then repeated infinitely throughout the pattern. Figure 8-la presents a planar decoration. The pattern shown is only part of the whole as the latter expands, in principle, to infinity The pattern is obviously highly symmetrical. Figure 8-lb shows the system of mutually perpendicular symmetry planes by solid lines. 

In the scalar case we obtained these conditions by analyzing (for the sake of simplicity) the asymptotic behavior of spherical waves. At the same time, as it has been demonstrated above, the same result could be obtained by analyzing the conditions required to ensure that the corresponding Kirchhoff integral goes to zero over a large sphere expanding to infinity. 

Exercise. Show that the terms omitted when expanding S(E — sn) in (4.3) tend to zero when the size of the heat bath goes to infinity ( thermodynamic limit ). 

Figure 10 The cross section of the potential surface of the H+ HF(v) —> H2 + F reaction taken at the HF bond length of 1.3 A. The potential felt by an H atom at coordinates X and Y approaching the H of HF positioned at the origin of the coordinate system is plotted as a function of the location of the attacking H. The H-F bond points down along the Y axis. Contour lines are spaced 0.5 kcal mof from each other. The energy is referred to an HF molecule expanded to 1.3 A and an H atom at the infinity.

The barrier crossing dynamics is determined by the memory kernel K(c), and Russell and Sceats suggested that it be modeled such that it gave the same velocity autocorrelation function in the vicinity of the outer minimum Rq of K(R) as exjjected from a superposition of the damp>ed conformational normal modes of the polymer, j,J = 1, N — 1. The potential P(R) is expanded about Rq with a force constant ficoi, where coq is defined by an integral such as Eq. (7.20), but with the range of integration extended to infinity. The result is that... 

We assume here that the scaling and wavelet functions, 4> and (p respectively, are defined over some interval [0, K] and then expanded to reside on L"(R) by regarding it as a periodic function with period K. This corresponds to wrapping around" a function over the interval [0. K] and extending it to infinity, as shown in Fig. 5. The circles in Fig. 5 indicate the end-points at which wrap-around is applied. 

As mentioned above, for conservative scattering, where = 1, the leading eigenvalue turns to infinity, i.e., the characteristic exponent A turns to zero. On expanding the eigenfunction vector for small true absorption in powers of k, we find, by means of Eqs. (2), (3), (4), and (6), the following asymptotic formulae for (1 - a) 1 ... 

In general, if o > 0 the first derivative of the power law approaches zero, but if 0 < 0, the first derivative of the power law diverges to infinity. Moreover, it is not possible to expand a power law in a Taylor series with an arbitrary number of terms. An example for a power law is given in Sect. 3.4.3.1. 

The methane molecule, CH4, has the geometry shown in Figure 2.19. Imagine a hypothetical process in which the methane molecule is expanded, by simultaneously extending all four C- - H bonds to infinity. We then have the process CH,(g) ---------------------> C(g) + 4H(g)... 

Let us check first whether if i = 0, then everything is OK. Yes, it is. Indeed, the denominator equals 1, and we have t = t and x = x. Let us see what would happen if the velocity of light were equal to infinity. Then, the Lorentz transformation becomes identical to the Galilean one. In general, after expanding t and jc in a power series of v /c, we obtain... 

To carry out the matching procedure between the inner and outer expansions, the outer solutions are first written in terms of the inner variable, t, and then are expanded using a Taylor series with respect to t around the point t = 0. The final stage is to equate to the Taylor expanded outer solution to the inner expansions in the limit when t tends to infinity. The terms that are matched between the inner and outer expansions are called the common parts. 

For simplicity we consider a spherically symmetric envelope expanding at constant velocity v. Then we can write some simple relations for the number density of H2 molecules, n(r), at radial distance r, for the radial column density of H2, N(r), from r to infinity, and for the radial extinction at 100 nm, Auy(r), of the external ultraviolet radiation field due to absorption and scattering by the circumstellar dust grains, as... 

In connection with the x integration in Eq. (2.63) we expand also the arc sine in Eq. (2.64). However, we may not cut off this expansion after a finite number of terms in view of the required integration of P(r) over r from d to infinity. Using... 

The potential within the volume of the solute molecule is a smooth function and can be expanded in an appropriate set of expansion functions gifr). This is a generalization of the usual multipole expansion. The latter uses Cartesian monomials 1, x, y, z, xy, y, xz, yz, x, ..., or the corresponding solid spherical harmonics to expand the potential. However, for reasons explained in [17], the origin-centered multipole expansion is unsuitable for most systems. We experimented with several expansion sets. One important requirement is that the expansion functions should not diverge to infinity like the solid harmonics do. We finally settled on a sine function expansion of the potential. It is conveniently defined in an outer box that is larger than the extent of the molecular electronic density, to avoid problems with the periodic nature of the sine expansion. Other... 

Equations 16.7 and 16.8 constitute a set of differential difference equations. Since the value of n can range from zero to infinity, the differential difference equation can be expanded into an infinite set of ordinary differential equations. 

Both of the above equations reduce to the Einstein limit [eqn. (5.1)] at low concentrations ( —> 0) and to oo as > K K The crowding factor K can be treated either as an adjustable parameter for fitting experimental data or as a theoretical parameter equal to the reciprocal of the volume fraction at which diverges to infinity. For random close packing of monodisperse hard spheres, we have 0niax = 0.64 and K = 1.56. An alternative theoretical route to K is to expand... 

---