We prove an isomorphism between the finite domain from 1 up to the product of the first n primes and the new defined set of prime modular numbers. This definition provides some insights about relative prime numbers. We provide an inverse function fro
m the prime modular numbers into this finite domain. With this function we can calculate all numbers from 1 up to the product of the first n primes that are not divisible by the first n primes. This function provides a non sequential way for the calculation of prime numbers.
We present our technique to create a magneto-optical trap for dysprosium atoms using the narrow-line cooling transition at 626$,$nm to achieve suitable conditions for direct loading into an optical dipole trap. The magneto-optical trap is loaded from
an atomic beam via a Zeeman slower using the strongest atomic transition at 421$,$nm. With this combination of two cooling transitions we can trap up to $2.0cdot10^8$ atoms at temperatures down to 6$, mu$K. This cooling approach is simpler than present work with ultracold dysprosium and provides similar starting conditions for a transfer to an optical dipole trap.
We present measurements of the hyperfine coefficients and isotope shifts of the Dy I $683.731 $nm transition, using saturated absorption spectroscopy on an atomic beam. A King Plot is drawn resulting in an updated value for the specific mass shift $d
elta u_mathrm{684,sms}^mathrm{164-162}=-534 pm 17 MHz$. Using fluorescence spectroscopy we measure the excited state lifetime $tau_{684}=1.68(5) mu$s, yielding a linewidth of $gamma_mathrm{684} = 95 pm 3 kHz$. We give an upper limit to the branching ratio between the two decay channels from the excited state showing that this transition is useable for optical pumping into a dark state and demagnetization cooling.
