Equatorial plane

Since h denotes the distance from the apex to the equatorial plane, then aty - h, p , and Eq. 11-33 becomes... 

This very simple result is independent of the value of the contact angle because the configuration involved is only that between the equatorial plane and the apex. 

As the spins precess in the equatorial plane, they also undergo random relaxation processes that disturb their movement and prevent them from coming together fiilly realigned. The longer the time i between the pulses the more spins lose coherence and consequently the weaker the echo. The decay rate of the two-pulse echo amplitude is described by the phase memory time, which is the time span during which a spin can remember its position in the dephased pattern after the first MW pulse. Tyy is related to the homogeneous linewidth of the individual spin packets and is usually only a few microseconds, even at low temperatures. 

Crystals of uranyl perchlorate, U02(C10[13093-00-0] have been obtained with six and seven hydration water molecules. The uranyl ion is coordinated with five water molecules (4) in the equatorial plane with a U—O(aquo) distance of 245 nm (2.45 E). The perchlorate anion does not complex the uranyl center. The unit cells contain two [0104] and one or two molecules of hydration water held together by hydrogen bonding (164). 

As distinct from ihe ideal connection of Dunlap, we now describe the series of nanotubule knees (9 ,0)-(5m,5 ), with n an integer. We call this series the perfectly graphitizahle carbon nanotuhules because the difference of diameter between the two connected segments of each knee is constant for all knees of the series (Fig. 4). The two straight tubules connected to form the = 1 knee of that series are directly related to Cfio, the most perfect fullerene[15], as shown by the fact that the (9,0) tubule can be closed by 1/2 Qo cut at the equatorial plane perpendicular to its threefold rotation symmetry axis, while the (5,5) tubule can be closed by 1/2 Qo cut at the equatorial plane perpendicular to its fivefold rotation symmetry axis [Fig. 5(a)]. 

Fig. 8. Planar representation of the (9M,0)-(5n,5n) knees, having a 36° bend angle produced by a heptagon-pentagon pair on the equatorial plane. The arrows show the dotted line of bonds where the knee N or N is connected to the corresponding straight tubules (a) knee N for n= 1 (b) stretched knee N , . for h = 1 and c=38 (c) general knees N and jV f.

The polyhedra of these complexes are actually flattened because the two trans U-0 bonds of the 002 group are shorter than the bonds to the remaining groups which form an equatorial plane. 

A chiral titanium complex with 3-cinnamoyl-l,3-oxazolidin-2-one was isolated by Jagensen et al. from a mixture of TiCl 2(0-i-Pr)2 with (2R,31 )-2,3-0-isopropyli-dene-l,l,4,4-tetraphenyl-l,2,3,4-butanetetrol, which is an isopropylidene acetal analog of Narasaka s TADDOL [48]. The structure of this complex was determined by X-ray structure analysis. It has the isopropylidene diol and the cinnamoyloxazolidi-none in the equatorial plane, with the two chloride ligands in apical (trans) position as depicted in the structure A, It seems from this structure that a pseudo-axial phenyl group of the chiral ligand seems to block one face of the coordinated cinnamoyloxazolidinone. On the other hand, after an NMR study of the complex in solution, Di Mare et al, and Seebach et al, reported that the above trans di-chloro complex A is a major component in the solution but went on to propose another minor complex B, with the two chlorides cis to each other, as the most reactive intermediate in this chiral titanium-catalyzed reaction [41b, 49], It has not yet been clearly confirmed whether or not the trans and/or the cis complex are real reactive intermediates (Scheme 1.60). 

An X-ray structure of the complex formed between 3-cinnamoyl-l,3-oxazohdin-2-one and a chiral TADDOL-Ti(IV) complex (see Chapters 1 and 6 by Hayashi and Gothelf, respectively) has been characterized [16]. The structure of this complex has the chiral TADDOLate and cinnamoyloxazohdinone ligands coordinated to titanium in the equatorial plane and the two chloride ligands in the axial plane and is similar to A in Fig. 8.8. The chiral discrimination was proposed to be due to... 

Alongside this, IR spectra of M3NbOF6 type compounds display a strong absorption band at 920 cm"1 indicating the presence of a double niobium-oxygen bond, Nb=0 [57, 115, 155, 156]. Kaidalova [156] explained this contradiction as a phenomenon that is related to the hampered rotation of the NbOF63 polyhedron in the equatorial plane. 

According to crystal analysis performed by Stomberg [173], Na2NbOF5 is made up of sodium ions and isolated NbOF52 complex ions and is similar in structure to FeWC>6. NbOFs2" polyhedrons comprise slightly distorted octahedrons that are located in one of two equivalent positions. The niobium atom is shifted 0.234 A from the equatorial plane towards the oxygen atom. 

The tantalum atoms are shifted from the equatorial planes of the octahedrons by 0.15 A. Potassium ions are situated between the chains and each ion is surrounded by nine fluorine atoms. The distances KrF and K2-F vary in the ranges of 2.63-2.95 and 2.85-3.07 A, respectively [215]. 

In this paper we amplify Powell s discussion, which is in some respects misleading. For example, Powell made the following statement Unlike the familiar four-lobed cubic d orbital, the pyramidal d orbital has only rather inconspicuous lobes of opposite sign. Each orbital is not quite cylindrically symmetrical about its own axis of maximum probability. In fact, the pyramidal d orbital that he discusses in detail is far from cylindrically symmetrical about its own axis of maximum probability, and the other pyramidal d orbital is also far from cylindrically symmetrical. In the equatorial plane about the axis of maximum probability the functions of Powell s first set (which we shall call II) vary from —0.3706 in two opposite directions to —1.7247 in the orthogonal directions. Each of these functions has almost the same value (strength) in the latter directions as in the principal directions, for which its value is 2.0950. The functions of the other set (which we call I) vary in this plane from —0.7247 to —1.4696, their value in the principal direction being 2.1943. 

Many types of phosphorus-phosphorus bonds are known, but it is rare to find such bonds in hexacoordinated phosphorus compounds (with the exception of 57). Cavell reported in 1998 the reaction of PCI5 with phenylbis(o-(trimethyl-siloxy)phenyl)phosphane, yielding the corresponding bischelate 61 in decent yield (52%) [99]. The octahedral nature of the central phosphorus atom was unambiguously determined by X-ray structural analysis. Two short axial bonds (2.202 A) lie perpendicular to the pseudo-octahedral equatorial plane. 

With three sulfur atoms above and three below the equatorial plane of the molecule, 190 was expected to be a good ligand for metal ions. However, a dichloromethane solution of 190 did not extract any ions from an aqueous solutions of various metal salts (in total 23 different cations including AP+,Ba +,... 

Table 2 Restricted Hartree-Fock energies (Hartrees) for Ceo and C70 and their muon adducts. AE is the difference in energy between the carbon allotrope and its adduct. In all cases, except where indicated by f, only the six carbon atoms in the immediate vicinity of the muon have had there positions optimised, f means that a full geometry optimisation has been carried out. The type specifies the defect and for C70 is identified in Table 1. is the spin density at the muon in atomic units (and the hyperfine coupling constant in MHz). JMuon constrained to lie in equatorial plane. indicates geometry not fully optimized.

The regular part, which is symmetrical with respect to the axis of rotation and the equatorial plane of the earth. Also we assume that this mass is equal to the total mass M of the earth, and that the center of this mass and that of the earth coincide. 

We have derived Equation (2.164), which shows how the field varies with the reduced latitude p on the surface of the spheroid. The reduced latitude is the angle between the radius vector and the equatorial plane. Fig. 2.7c. Also, it is useful to study the function y — y q>), where tp is the geographical latitude. This angle is formed by the normal to the ellipsoid at the given point p and the equatorial plane. Fig. 2.7b. First, we find expressions for coordinates v, y of the meridian ellipse. Its equation is... 

Now we introduce two moments of inertia one of them, A, around an arbitrary axis in the equatorial plane and the other, C, around the rotation axis, and taking into account the axial symmetry ... 

---