On symmetric commutator subgroups, braids, links and homotopy groups

In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for braid and surface groups.Comment: 24 pages, 1 figur

Direct and secondary nuclear excitation with x-ray free-electron lasers

The direct and secondary nuclear excitation produced by an x-ray free electron laser when interacting with a solid-state nuclear target is investigated theoretically. When driven at the resonance energy, the x-ray free electron laser can produce direct photoexcitation. However, the dominant process in that interaction is the photoelectric effect producing a cold and very dense plasma in which also secondary processes such as nuclear excitation by electron capture may occur. We develop a realistic theoretical model to quantify the temporal dynamics of the plasma and the magnitude of the secondary excitation therein. Numerical results show that depending on the nuclear transition energy and the temperature and charge states reached in the plasma, secondary nuclear excitation by electron capture may dominate the direct photoexcitation by several orders of magnitude, as it is the case for the 4.8 keV transition from the isomeric state of $^{93}$Mo, or it can be negligible, as it is the case for the 14.4 keV M\"ossbauer transition in $^{57}\mathrm{Fe}$. These findings are most relevant for future nuclear quantum optics experiments at x-ray free electron laser facilities.Comment: 17 pages, 7 figures; minor corrections made; accepted by Physics of Plasma