Particle-rotor model

Since so little was known about the single particle structure of deformed A=100 nuclei, our approach in [VDH85] was to use a textbook version of the particle-rotor model, as outlined by [BUN71], that had been systematically used to study deformed rare earths. 

The N = 83, 85 and 87 isotones above the neutron-shell closure at N = 82 are described essentially by the f7 shell-model state. Calculations withing the particle-rotor model of Larsson et al. [77], assuming small nuclear deformations of the core, account qualitatively for the measured nuclear moments (cf. Fig. 3). The variation in the spectroscopic quadrupole moments reflects the successive filling of the f j neutron shell. This is realized from the formula [78] ... 

The nuclear spins and moments of the strongly deformed rare-earth nuclides have been discussed in detail previously within die Nilsson model in connection with the ABMR experiments mentioned above [79]. The addition of the data fiom coUinear fast-beam laser spectroscopy [48, 50, 71], the new reference values on spectroscopic quadrupole moments from muonic and pionic hfs [1], and the refined calculations within the partice-rotor model, including a number of orbitals close to the Fermi surface, have however resulted in a more complete picture and a better understanding of tiie nuclear single-particle stmcture in this region. 

To predict the properties of materials from the forces on the atoms that comprise them, you need to know the energy ladders. Energy ladders can be derived from spectroscopy or quantum mechanics. Here we describe some of the quantum mechanics that can predict the properties of ideal gases and simple solids. This will be the foundation for chemical reaction equilibria and kinetics in Chapters 13 and 19. Our discussion of quantmn mechanics is limited. We just sketch the basic ideas with the particle-in-a-box model of translational freedom, the harmonic oscillator model for vibrations, and the rigid rotor model for rotations. 

A major success of statistical mechanics is the ability to predict the thermodynamic properties of gases and simple solids from quantum mechanical energy levels. Monatomic gases have translational freedom, which we have treated by using the particle-in-a-box model. Diatomic gases also have vibrational freedom, which we have treated by using the harmonic oscillator model, and rotational freedom, for which we used the rigid-rotor model. The atoms in simple solids can be treated by the Einstein model. More complex systems can require more sophisticated treatments of coupled vibrations or internal rotations or electronic excitations. But these simple models provide a microscopic interpretation of temperature and heat capacity in Chapter 12, and they predict chemical reaction equilibria in Chapter 13, and kinetics in Chapter 19. 

Thus far, exactly soluble model problems that represent one or more aspects of an atom or molecule s quantum-state structure have been introduced and solved. For example, electronic motion in polyenes was modeled by a particle-in-a-box. The harmonic oscillator and rigid rotor were introduced to model vibrational and rotational motion of a diatomic molecule. 

The Micromeretics model 1001 [38] works on the following principle. A de-agglomerated stream of particles is sucked from a dispersion device into the center of a rotor where it divides into two streams. Air of either... 

The spin coating technique has attracted interest, since it maintains many aspects of technical catalysts prepared by pore volume or incipient wetness impregnation, and simultaneously allows the interpretation and analysis in a similar way as the more well-defined model systems discussed above [30]. Here, a solution of the desired catalyst precursor is dropped onto a wafer covered with an oxide film, which is spun on a rotor to create a liquid layer of uniform thickness in order to mimic traditional wet impregnation preparation of catalysts. Control of the catalyst loading and particle size is to some degree achieved by varying the rotation speed, concentration, and vapor pressure of the solute. Still the method suffers, however, from many of the drawbacks associated with wet-impregnated model catalysts, which imparts detailed mechanistic studies. 

The construction of a Hamiltonian is normally an easy problem. The solution of the Schrodinger equation, on the contrary, represents a serious problem. It can be solved exactly for several model cases a particle in a box (one-, two- or three-dimensional), harmonic oscillator, rigid rotor, a particle passing through a potential barrier, hydrogen atom, etc. In most applications only an approximate solution of the Schrodinger equation is attainable. 

Level introduced in [01Ga25] in addition to NDS [930hl2] see there a comparison of experimental data with predictions of the Rigid Triaxial Rotor plus Particle model (RTRP) and with data for other A-odd Xe isotopes. 

In the LL model local orientations of close-packed nematic molecular clusters are represented by free unit rotors Uj (particles) attached to lattice points of a cubic lattice. The nearest neighbors Uj and uy interact... 

The excitations of the crystal that originate from the MF model may correspond with strongly anharmonic translational vibrations or librations of the molecules they may even correspond with hindered- or free-rotor states. They remain single particle excitations, however, which do not show any dispersion (i.e. wave vector dependence) in their frequencies. The simplest manner to obtain this dispersion is by the so-called Exciton Model or Tamm-Dancoff Approximation [76]. From the crystal ground state, which is a product of (known) MF states ... 

These tests were performed using a universal Zwick tester, model 1435, coimected to a computer with appropriate software. Particle size was measured in paraffin oil using Zetasizemano S90 (Malvern Instmments) analyzer. Zeta potential of filler dispersion in water was studied by the means of Zetasizer 2000 (Malvern Instruments) apparams. Rheological properties of filler suspensions in paraffin oil were determined by viscometer RM500 (Rheometric Scientific). Dibutylphtalate (DBP) absorption was measured by means of an Absorptometer C (Brabender). The modification process was carried out in Brabender Measuring Mixer N50 at following parameters temperature 125°C rotor speed 40 RPM duration 0,5 h. 

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