The emerging concept of `beam by design' in free-electron laser (FEL) accelerator physics aims for accurate manipulation of the electron beam to tailor spectral and temporal properties of the radiation for specific experimental purposes, such as X-ray pump/X-ray probe and multiple wavelength experiments. `Beam by design' requires fast, efficient, and detailed feedback on the spectral and temporal properties of the generated X-ray radiation. Here a simple and cost-efficient method to extract information on the longitudinal Wigner distribution function of emitted FEL pulses is proposed. The method requires only an ensemble of measured FEL spectra and is rather robust with respect to accelerator fluctuations. The method is applied to both the simulated SASE spectra with known radiation properties as well as to the SASE spectra measured at the European XFEL revealing underlying non-linear chirp of the generated radiation. In the Appendices an intuitive understanding of time–frequency representations of chirped SASE radiation is provided.
У цiй роботi показано зв’язок мiж параметрами другого вiрiального коефiцiєнта для системи неабелевих енiонiв та двопараметричними модифiкацiями дробових статистик Голдейна–Ву та Полiхронакоса. Розраховано параметри, для яких неабелевi енiони можуть описуватись даними типами статистик. Розглянуто границю, в якiй параметр неадитивностi/неповноти q прямує до одиницi.
We consider two examples of real physical systems approximately described using fractional nonadditive Polychronakos statistics. The values of two statistics parameters are linked to properties of modeled systems using virial expansion. For a two-dimensional Fermi gas with contact interactions, accuracy up to the third virial coefficient is achieved. An approach to model the second virial coefficient of non-Abelian soft-core anyons is analyzed in detail.
We compare two approaches to the generalization of the ordinary Fermi-statistics based on the nonadditive Tsallis q-exponential used in the Gibbs factor instead of the conventional exponential function. Both numerical and analytical calculations are made for the chemical potential, fugacity, energy, and the specific heat of the ideal gas obeying such generalized types of statistics. In the approach based on the Gibbs factor containing the chemical potential, high temperature behavior of the specific heat significantly deviates from the expected classical limit, while at low temperatures it resembles that of the ordinary ideal Fermi-gas. On the contrary, when the fugacity enters as a multiplier at the Gibbs factor, the high-temperature limit reproduces the classical ideal gas correctly. At low temperatures, however, some interesting results are observed, corresponding to non-zero specific heat at the absolute zero temperature or a finite (non-zero) minimal temperature. These results, though exotic from the first glance, might be applicable in effective modeling of physical phenomena in various domains.
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