Charge compensation and the luminescence of Cr3+ in KMgF3

Studies of the photoluminescence spectra of Cr³⁺ ions in KMgF₃ crystals co-doped with Cr³⁺ and Ni²⁺ ions are reported. Several crystal field sites are identified by the different R-line spectra due to the ² E⁴ A ₂ transition and broadband luminescences associated with the ⁴ T ₂⁴ A ₂ transitions. Cr³⁺ ions substituting without local charge compensation on the octahedral Mg²⁺ site give rise to a low temperature R line in photoluminescence at =702.3 nm with a radiative decaytime of 3 ms at T=14 K. At T=300 K this isotropic centre gives rise to an unpolarized broadband ⁴ T ₂⁴ A ₂ emission, which results from the thermal occupancy of an excited ⁴ T ₂ state just above the ² E level which, at lower temperature, gives rise to emission in the R-line. Other crystal field sites are due to some Cr³⁺ ions having Mg²⁺ or K⁺ vacancies in nearest-neighbour positions, these vacancies being required to maintain charge neutrality in doped fluoride perovskites. The Cr³⁺–K⁺ vacancy complex results in the centre having trigonal symmetry, and low temperature, photoluminescence via R ₁ and R ₂ lines at 716.8 nm and 716.0 nm, respectively. Finally, Cr³⁺ ions having a nearest neighbour Mg²⁺ vacancy have tetragonal symmetry, experiencing weak crystal fields. In consequence, the ⁴ T ₂ level lies below ² E and the photoluminescence spectrum at low temperature takes the form of a polarized broad ⁴ T ₂⁴ A ₂ band with peak at 760 nm and radiative decaytime of 54 s.     

Jupiter's Great Red Spot and zonal winds as a self-consistent,one-layer,quasigeostrophic flow

We present the point of view that both the vortices and the east-west zonal winds of Jupiter are confined to the planet's shallow weather layer and that their dynamics is completely described by the weakly dissipated, weakly forced quasigeostrophic (QG) equation. The weather layer is the region just below the tropopause and contains the visible clouds. The forcing mimics the overshoot of fluid from an underlying convection zone. The late-time solutions of the weakly forced and dissipated QG equations appear to be a small subset of the unforced and undissipated equations and are robust attractors. We illustrate QG vortex dynamics and attempt to explain the important features of Jupiter's Great Red Spot and other vortices: their shapes, locations with respect to the extrema of the east-west winds, stagnation points, numbers as a function of latitude, mergers, break-ups, cloud morphologies, internal distributions of vorticity, and signs of rotation with respect to both the planet's rotation and the shear of their surrounding east-west winds. Initial-value calculations in which the weather layer starts at rest produce oscillatory east-west winds. Like the Jovian winds, the winds are east-west asymmetric and have Karman vortex streets located only at the west-going jets. From numerical calculations we present an empirically derived energy criterion that determines whether QG vortices survive in oscillatory zonal flows with nonzero potential vorticity gradients. We show that a recent proof that claims that all QG vortices decay when embedded in oscillatory zonal flows is too restrictive in its assumptions. We show that the asymmetries in the cloud morphologies and numbers of cyclones and anticyclones can be accounted for by a QG model of the Jovian atmosphere, and we compare the QG model with competing models.     

Perfect crystals andq-deformed Fock spaces

In [S], [KMS] the semi-infinite wedge construction of level 1U _q (A _n ⁽¹⁾ ) Fock spaces and their decomposition into the tensor product of an irreducibleU _q (A _n ⁽¹⁾ )-module and a bosonic Fock space were given. Here a general scheme for the wedge construction ofq-deformed Fock spaces using the theory of perfect crystals is presented.LetU _q (g) be a quantum affine algebra. LetV be a finite-dimensionalU _q (g)-module with a perfect crystal base of levell. LetV _aff V [z,z ^–1] be the affinization ofV, with crystal base (L _aff,B _aff). The wedge spaceV _aff V _aff is defined as the quotient ofV _aff V _aff by the subspace generated by the action ofU _q (g) [z ^a z ^b +z ^b z ^a ]_a,b onv v (v an extremal vector). The wedge space ^r V _aff (r ) is defined similarly. Normally ordered wedges are defined by using the energy functionH :B _aff B _aff . Under certain assumptions, it is proved that normally ordered wedges form a base of ^r V _aff.Aq-deformed Fock space is defined as the inductive limit of ^r V _aff asr , taken along the semi-infinite wedge associated to a ground state sequence. It is proved that normally ordered wedges form a base of the Fock space and that the Fock space has the structure of an integrableU _q (g)-module. An action of the bosons, which commute with theU _q (g)-action, is given on the Fock space. It induces the decomposition of theq-deformed Fock space into the tensor product of an irreducibleU _q (g)-module and a bosonic Fock space.As examples, Fock spaces for typesA _2n ⁽²⁾ ,B _n ⁽¹⁾ ,A _2n ^–1/(2) ,D _n ⁽¹⁾ andD _n ^+1/(2) at level 1 andA ₁ ⁽¹⁾ at levelk are constructed. The commutation relations of the bosons in each of these cases are calculated, using two point functions of vertex operators.     

Generalized evaluation subgroups of product spaces relative to a factor

For any -complexes and , we show that . We use this fact to compute generalized evaluation subgroups of generalized tori relative to a sphere.