In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a computation of $B_{
2n}$ as a function of B$_{2n-2}$ only. Furthermore, we show that a direct computation of the Riemann zeta-function and their derivatives at k $in mathbb Z$ is possible in terms of these polynomial representation. As an explicit example, our polynomial Bernoulli number representation is applied to fast approximate computations of $zeta$(3), $zeta$(5) and $zeta$(7).
The magnitude of the spin polarization at the Fermi level of ferromagnetic materials at room temperature is a key property for spintronics. Investigating the Heusler compound Co$_2$MnSi a value of 93$%$ for the spin polarization has been observed at
room temperature, where the high spin polarization is related to a stable surface resonance in the majority band extending deep into the bulk. In particular, we identified in our spectroscopical analysis that this surface resonance is embedded in the bulk continuum with a strong coupling to the majority bulk states. The resonance behaves very bulk-like, as it extends over the first six atomic layers of the corresponding (001)-surface. Our study includes experimental investigations, where the bulk electronic structure as well as surface-related features have been investigated using spin-resolved photoelectron spectroscopy (SR-UPS) and for a larger probing depth spin-integrated high energy x-ray photoemission spectroscopy (HAXPES). The results are interpreted in comparison with first-principles band structure and photoemission calculations which consider all relativistic, surface and high-energy effects properly.
A theoretical frame for pump-probe photoemission is presented. The approach is based on a general formulation using the Keldysh formalism for the lesser Greens function to describe the real-time evolution of the electronic degrees of freedom in the i
nitial state after a strong pump pulse that drives the system out of equilibrium. The final state is represented by a time-reversed low-energy electron diffraction state. Our one-step description is related to Pendrys original formulation of the photoemission process as close as possible. The formalism allows for a quantitative calculation of time-dependent photocurrent for simple metals where a picture of effectively independent electrons is assumed as reliable. The theory is worked out for valence- and core-electron excitations. It comprises the study of different relativistic effects as a function of the pump-probe delay.
