Approximate symmetry

The approximate symmetry of the band is due to the fact that Bi — Bq, that is, the vibration-rotation interaction constant (Equation 5.25) is small. If we assume that B = Bq = B and neglect centrifugal distortion the wavenumbers of the i -branch transitions, v[i (J)], are given by... 

From Equations (6.27) and (6.28) it follows that the zero gap, v[i (0)] — v[P(l)], is AB and that the spacing is 2B between adjacent i -branch lines and also between adjacent P-branch lines, hence the approximate symmetry of the band. 

XRD analysis of (n-BuLi)e (291b) reveals that the cluster of six Li atoms has an approximate symmetry (trigonal antiprismatic), with the six w-Bu groups placed around the axis of rotation. Each Li atom is coordinated to four Li atoms, one a-C and one /3-C atom the conformation of the six n-Bu groups is such that the a-C atom is coordinated to the three Li atoms of a face of the distorted octahedron and the fi-C atom to one of them only ". ... 

Antibiotic Cation Number of coordinating oxygen atoms ) /approximate symmetry of coordination sphere Approximate shape and size of complex Ref. 

Tetrakis O, O -dialkylthiophosphates have been obtained for niobium starting from oxidation states +V or +IV (Table 40). These chelates display a distorted dodecahedral geometry in the solid (X-ray) as well as in solution (ESR).561 An unusual cation, [ Nb(Cp)2 2(PS4)]+, in which a PS4 moiety is doubly bridging the terminal NbCp2 units, has been obtained from [NbCl2(Cp)2] and P4Si0. The metal has a distorted tetrahedral environment of approximate symmetry, while symmetry at phosphorus is reduced to C. ... 

In connection to control in dynamics I would like to take here a general point of view in terms of symmetries (see Scheme 1) We would start with control of some symmetries in an initial state and follow their time dependence. This can be used as a test of fundamental symmetries, such as parity, P, time reversal symmetry, T, CP, and CPT, or else we can use the procedure to discover and analyze certain approximate symmetries of the molecular dynamics such as nuclear spin symmetry species [2], or certain structural vibrational, rotational symmetries [3]. 

In certain special cases the approximate symmetries in atoms are sufficiently well explained using the quasispin formalism. In particular, the quasispin technique can be utilized to describe fairly accurately configuration mixing for doubly excited states of the two-electron atom. In the quasispin basis the energy matrix of the electrostatic interaction operator of such configurations is nearly diagonal, and hence the quantum number of total quasispin Q is approximately good . 

I am now at the end of my series of flashes on the Solvay Conferences in Physics. I hope that, in spite of its shortness and incompleteness, it may help in stimulating two kinds of considerations. Those of the first kind regard the extraordinary develoment undergone during the last 70 years by our views on the physical world, many parts of which in present days appear to be dominated by a few general concepts, such as those of exact and approximate symmetry, and to be treatable by mathematical procedures such as the application of the renormalization group. The other kind of considerations concerns the role that the Solvay Conferences in Physics have played in the development of physics during the last 70 years, and the unique value they will maintain, even in the future, as sources of information for the historians of science. 

Symmetry of the ligand field around a transition metal atom. An approximate symmetry is indicated in parentheses. 

Another term for approximate symmetry is broken symmetry. The symmetry breaking factor is whatever factor is responsible for the deviation of e from zero. As an example, any crystal has broken translational symmetry. The exact symmetry limit is an infinite crystal, obviously unattainable in practice. 

The whole concept of symmetry and law becomes more palatable against the backdrop of approximate symmetries. Inviolate laws that militate against the scientific spirit, are then prevented by broken symmetries and developments in science amount to relaxing the primitive laws, so as to describe more general situations around the special cases dictated by exact symmetries, i.e. by maximizing the parameter e. Any law that reflects a symmetry must then be considered as a useful starting point rather than a final result and conclusions based on perceived symmetries of space and time must be revisited to identify the effects of broken symmetry on the laws of Nature. 

To better appreciate the effects of approximate symmetries, the consequences of exact symmetries are examined first. 

Note that interpretations of the time-reversal experiments are only valid in strictly euclidean space-time. This condition is rarely emphasized by authors who state that all laws of physics are time-reversible, except for the law of entropy. Fact is that entropy is the only macroscopic state function which is routinely observed to be irreversible. One common explanation is to hint that entropy is an emergent property of macro systems and hence undefined for microsystems. Even so, the mystery of the microscopic origin of entropy remains. A plausible explanation may be provided if the assumed euclidean geometry of space-time is recognized as an approximate symmetry as demanded by general relativity. 

Not too many theories have been formulated from this point of view and some of the more interesting cases are at the speculative stage of development. Even so, it is remarkable how some of the most enigmatic of natural phenomena have no convincing explanation apart from broken-symmetry theories. Included are the initiation or nucleation of phase transitions, superconductivity (T4.5.1), the arrow of time (entropy) and the cosmic imbalance between matter and antimatter. The beauty of the world, indeed seems to lie in approximate symmetries. 

Despite the vast literature that deals with the observations of the previous paragraphs not even a speculative account to explain the paradoxical asymmetries could be found. Experience points to the conclusion that at a large enough scale the asymmetries in time, material and molecular activity must balance out, at least to within an approximate symmetry of high degree. If PCT invariance holds for the total Hamiltonian H, the matrix element of H between any two states A) and B) is related to that between their CPT-conjugate states A) and B)... 

We have here kept the description of the radiative decay rates fairly general and did not discuss their exact relations to the vibronic states (in particular within the Franck-Condon approximation). This description is sufficient for our purpose here and more adapted to the generalization to the case of MEF. It is nevertheless worth mentioning that there is an approximate symmetry between the transitions for absorption Sg(0) —> S 0) ) and those... 

The structure of compound 133 (Fig. 12) was determined by X-ray diffraction. It has an approximate symmetry with planar nitrogen atoms. An N—N unit is shared by the two five-membered rings, which are not planar. The torsion angles Si8-Nl 1-Nl 1 -Si8 and Sil2-Nll-Nll -Sil2 are found to be -45° and -37°. (See Table XX.)... 

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