Mixed symmetry

The index ms indicates that j s transforms according to the mixed symmetry representation of the symmetric Group 54 [33]. 7 5 is an irreducible tensor component which describes a deviation from Kleinman symmetry [34]. It vanishs in the static limit and for third harmonic generation (wi = u>2 = W3). Up to sixth order in the frequency arguments it can be expanded as [33] ... 

We note that 02 4 vanishs if two of the four the frequency arguments Wq, wi, u>2, W3 become zero. For the case that three frequency arguments are equal (third harmonic generation) both mixed-symmetry effective frequencies w 2 vanish. The coefficients Amsi Bms, etc. 

The dispersion coefficients for the mixed-symmetry component 7 5 which describes the deviation from Kleinman symmetry are for methane more than an order of magnitude smaller than coefficients of the same order in the frequencies for 7. Their varations with basis sets and wavefunction models are, however, of comparable absolute size and give rise to very large relative changes for the mixed-symmetry dispersion coefficients. 

Note, that an account of the electron correlation has a great influence on the symmetry of holes. For example, at the ROHF level the holes on the 02 and 03 ions, instead of the 2po symmetry, have a mixed symmetry 0.22 2pa + 0.13 2pjr although on the 01 and 04 ions, the symmetry of the uncorrelated holes is a pure 2pa. Thus, as in the case of the spin location, the allowance of the electron correlation is crucial for determining the symmetry of holes. 

In the last two decades, important discoveries in the field of superconductivity [1] have reopened the question of what is the symmetry of the superconducting state. In this contribution, after a brief historical introduction, the classification of superconducting states will be reviewed. Some consequences of the symmetry of the state will be then discussed, especially in view of the recent interest in the symmetry of high-7) superconductors. Finally, a novel approach particularly useful for mixed symmetry states will be introduced. 

Mixed-Symmetry Interpretation of Some Low-Lying Bands in Deformed Nuclei and the Distribution of Collective Magnetic Multipole Strength... 

The effect of neutron-proton symmetry breaking on the distribution of M1 strength in the SU(3) limit of the Interacting Boson Model (IBA-2) is studied. A possible alternative choice for the Majorana force is investigated, with a structure that resembles more closely that which is calculated in microscopic theories. It is found that the specific choice for the Majorana interaction has important consequences for the magnetic strength distribution function. In addition it allows for an alternative interpretation of the second excited K7T=0+ band in rare earth nuclei, as a mixed-symmetry state. 

In this section some of the E2 decay properties of the low-lying mixed symmetry band are discussed. In the IBA-2 model, E2 transitions are... 

Table 1. Calculated B(E2 values for various transitions involving members of the mixed-symmetry K -O band for two different choices for the boson effective charges, ...

In this contribution we have investigated the effect, on calculated observables in the IBA-2 model, of an alternative choice for the Majorana force, as suggested by microscopic calculations [Dru85]. This choice has the peculiar feature of producing in the SU(3) limit a spectrum in which there appears a mixed-symmetry band at approximately the same energy... 

Figure 15a. The degree of mixing symmetry indices vs. contact number. The convergence point indicates complete mixing. The contact number indicates the total number of black-white transitions in a scan of the field.

Figure 12. A representation of the coupled chromophore model. The orbitals designated (a) represent the frontier four-otbitals of one monomer chlorophyll unit, those labeled (b), the orbitals of the second monomer unit. Upon mixing, symmetry determines combinations of the monomer orbitals, yielding those of the dimer, labeled with (d).

One mates the connection by counting the independent components of the irreducible tensor. For the completely symmetric r = [71,0,0] states (see the discussion above equation (15) for an explanation of this notation) this is given by equation (18) of Reference [16]. In three dimensions the F = [71,1,0] tableau, ie either the two box antisymmetric or the mixed symmetry tableaux, is associate (see Reference [5] p. 396 and Reference [6] p. 164) to the F = [71,0,0] tableau, i.e. a completely symmetric tableaux. Thus a second rank antisymmetric or a mixed symmetry tensor has the same dimension as the associate completely symmetric tensor. The three box completely antisymmetric F = [1,1,1] tableau is the associate tableau to the tableau with no boxes and is therefore a scalar. There aie more general formulas for the number of independent components of an irreducible tensor in D dimensions, however they are not required to achieve the above identification, see D.E. Littlewood, The Theory of Group Characters (Oxford University Press, Oxford, 1950), equation (11.8 6), p. 236. 

A pair of mixed symmetry behaves like the real and the imaginary part of... 

Out of the spins of the two first quarks, one can build spin 5 = 0 and 5 = 1 states, which are respectively odd and even under P 2, the exchange of these quarks. When supplemented by the third spin, they lead to one spin 5 = 3/2 multiplet, which is symmetric, and two spin 5 = 1/2 doublets, which are partners of mixed symmetry. These spin wave functions are... 

Isospin wave functions are built in exactly the same way, with f replaced by u and i replaced by d. States with three identical quarks such as those of the ft family (sss) have a simple structure either the spin wave function corresponds to spin 5 = 3/2 and the space wave function has to be symmetric, or the total spin is 5 = 1/2 and one should combine the corresponding spin wave functions with a pair of mixed-symmetry space wave functions, as in eq. (3.33), to form an overall spin-space wave function which is symmetric. The above combinations are also found in qqq baryons made of ordinary quarks (q = u or d), when isospin is 7 = 3/2. This is the A family. When isospin is 7 = 1 /2, i.e., for the nucleon family, new arrangements exist. First, isospin 7 = 1/2 and spin 5 = 1/2 can be combined to form a symmetric spin-isospin wave function. This is what occurs for the nucleon itself and some of its excitations. The spin-isospin wave function can also be of mixed symmetry and is associated with a mixed-symmetry spatial wave function. Finally, there is the possibility of an antisymmetric spin-isospin wave function which allows for the use of an antisymmetric spatial wave function such as p x Aexp[-a(p + A )j. 

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