title: Theoretical study of the structures of 4-(2,3,5,6-tetrafluoropyridyl)diphenylphosphine oxide and tris(pentafluorophenyl)phosphine oxide: Why does the crystal structure of (tetrafluoropyridyl)diphenylphosphine oxide have two different P=O bond lengths?

authors: Lane, Joseph R.

subjects: Science & Technology

abstract: The crystal structure of 4-(2,3,5,6-tetrafluoropyridyl)diphenylphosphine oxide (1) contains two independent molecules in the asymmetric unit. Although the molecules are virtually identical in all other aspects, the P=O bond distances differ by ca. 0.02 Å. In contrast, although tris(pentafluorophenyl)phosphine oxide (2) has a similar crystal structure, the P=O bond distances of the two independent molecules are identical. To investigate the reason for the difference, a density functional theory study was undertaken. Both structures comprise chains of molecules. The attraction between molecules of 1, which comprises lone pair–π, weak hydrogen bonding and C–H···arene interactions, has energies of 70 and 71 kJ mol⁻¹. The attraction between molecules of 2 comprises two lone pair–π interactions, and has energies of 99 and 100 kJ mol−1. There is weak hydrogen bonding between molecules of adjacent chains involving the oxygen atom of 1. For one molecule, this interaction is with a symmetry independent molecule, whereas for the other, it also occurs with a symmetry related molecule. This provides a reason for the difference in P=O distance. This interaction is not possible for 2, and so there is no difference between the P=O distances of 2.

text: Theoretical study of the structures of 4-(2,3,5,6-tetrafluoropyridyl)diphenylphosphine oxide and tris(pentafluorophenyl)phosphine oxide: Why does the crystal structure of (tetrafluoropyridyl)diphenylphosphine oxide have two different P=O bond lengths? The crystal structure of 4-(2,3,5,6-tetrafluoropyridyl)diphenylphosphine oxide (1) contains two independent molecules in the asymmetric unit. Although the molecules are virtually identical in all other aspects, the P=O bond distances differ by ca. 0.02 Å. In contrast, although tris(pentafluorophenyl)phosphine oxide (2) has a similar crystal structure, the P=O bond distances of the two independent molecules are identical. To investigate the reason for the difference, a density functional theory study was undertaken. Both structures comprise chains of molecules. The attraction between molecules of 1, which comprises lone pair–π, weak hydrogen bonding and C–H···arene interactions, has energies of 70 and 71 kJ mol⁻¹. The attraction between molecules of 2 comprises two lone pair–π interactions, and has energies of 99 and 100 kJ mol−1. There is weak hydrogen bonding between molecules of adjacent chains involving the oxygen atom of 1. For one molecule, this interaction is with a symmetry independent molecule, whereas for the other, it also occurs with a symmetry related molecule. This provides a reason for the difference in P=O distance. This interaction is not possible for 2, and so there is no difference between the P=O distances of 2.

year: 2020

title: Nonequilibrium transitions in quantum optical systems

authors: Drummond, Peter David

abstract: The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems.
In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function.
In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. 
In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. 
In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. 
Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.

text: Nonequilibrium transitions in quantum optical systems The topic of this thesis is a theoretical study of nonequilibrium transitions and quantum statistical properties of nonlinear quantum optical systems driven by an external radiation source. In certain limiting cases, a comparison is made between these transitions and the phase transitions found in equilibrium physical systems.
In chapters one and two, the mathematical tools are introduced. In operator terms, the time development is described by a Markovian master equation in the interaction picture. This is equivalent to a corresponding time development equation or Fokker-Planck equation in a vector space of c-numbers. In order to deal with the type of Fokker-Planck equation that results, we introduce a quantum classical correspondence resulting in a distribution function over a complex phase-space, which is a generalisation of the Glauber-Sudarshan P-function.
In chapter three this is applied to a model of a coherently driven mode with nonlinear dispersion and absorption. We find in the limit of zero temperature, that the spectrum is symmetric relative to the input frequency, and an exact solution is obtained for the distribution function. For a detuned driving field and nonlinear dispersion, optical bistability can occur. 
In chapter four a model of sub/second harmonic generation is introduced. This has several non-equilibrium transitions, including dispersive optical bistability, and bistable behaviour with coherent phase-locked input to both modes. Exact solutions occur in the limit of zero temperature and adiabatic elimination of one mode. In both chapters three and four, steady-state photon anti-bunching occurs with an absorptive nonlinearity. 
In chapter five we include interactions between the radiation mode and a fluorescent atomic system. In this case different behaviour occurs depending on the relative decay rates of the individual atoms and of the radiation mode. In the case of a high-Q interferometer, the atomic variables can be adiabatically eliminated. Both dispersive and absorptive bistability can occur. We show by analytic and numerical calculations (in the case of inhomogeneous broadening) that dispersive operation has advantages in requiring a lower atomic density and input field to observe bistability. When the input field has Gaussian (rather than coherent) photon statistics, there is no bistability, but enhanced photon bunching occurs. 
Finally, there is a different type of behaviour when the field mode decays rapidly and can be adiabatically eliminated. Within the cooperation lifetime the system can be described by a J²-invariant Hamiltonian, giving the special case of only collective damping. The result is a new type of critical point transition in the thermodynamic limit, with the appearance of a family of solutions like Lotka-Volterra cycles for a coherent driving field above threshold.

year: 1979

title: Te mātauranga whakatere waka

authors: Tuaupiki, Jackie Whetumarama

abstract: Hei tēnei tuhinga e whai ake nei, ka tirohia te hangarau waka a ngā tūpuna o Te Moana nui a Kiwa i whakawhiti ai rātou ki ō rātou moutere tae atu ki te mātauranga whakatere waka me te arotahi o te titiro ki te whakatinanatanga o aua mātauranga. Ka tāmutu i konā, ka pūhia te titiro i runga i ngā huarere o Poronīhia e kite ai tātou i ngā āhuatanga o Tāwhirimātea i ōna moutere me ngā wero ki te whakatere waka i ērā huarere.

text: Te mātauranga whakatere waka Hei tēnei tuhinga e whai ake nei, ka tirohia te hangarau waka a ngā tūpuna o Te Moana nui a Kiwa i whakawhiti ai rātou ki ō rātou moutere tae atu ki te mātauranga whakatere waka me te arotahi o te titiro ki te whakatinanatanga o aua mātauranga. Ka tāmutu i konā, ka pūhia te titiro i runga i ngā huarere o Poronīhia e kite ai tātou i ngā āhuatanga o Tāwhirimātea i ōna moutere me ngā wero ki te whakatere waka i ērā huarere.

year: 2017