## 1 Introduction  
Superconducting radio-frequency (SRF) cavities consisting of Niobium are enabling components of many modern particle accelerators from spallation neutron sources to CW (continuous wave) free electrons X-ray lasers. As CW applications become more and more important and repetition rates are increasing to obtain higher beam currents, the power dissipation in the cavity walls becomes a major limiting factor not only from a maximum-performance point of view, but also with respect to cost minimization. In the attempt to push superconducting materials to elevated performance, the degrading impact of trapped magnetic vortices must be investigated, understood and, ideally, eliminated. Over the past few years, several studies demonstrated in samples, in cavities and also in module-like operation how beneficial a reduction in trapped magnetic flux is. In particular, it was found that the size of trapped flux is affected by several different parameters, such as the cooldown conditions or material properties like RRR (residual resistivity ratio). This laid the basis for further, systematic investigations [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. These methods have in common that the magnetic flux is observed indirectly--either by measuring its impact on the cavity performance or by measuring the impact of the incomplete Meissnertransition on the ambient magnetic field or by small angle neutron scattering. From these indirect measurements one then needs to deduce, what the actual flux distribution in the superconductor is [2; 12; 13; 14; 15; 16]. In this paper, we first describe a direct method to measure magnetic fields inside the cavity material and then present results of our investigations of the influence of AC external magnetic fields on flux trapping behavior. The Meissner phase is the energetically favorable state of the whole sample, because it is a state of complete expulsion of any ambient magnetic field from the bulk material. Therefore, any deviation from this equilibrium state must have been caused by trapping a number of quantized flux lines within the Nb material before they could have been driven out. The idea of superimposing external AC magnetic fields is that a movement of flux-lines over macroscopic distances could be supported by those fields. Flux-lines can move freely in a normal-conducting medium, while they are driven out of superconducting areas. In the mixed state of coexisting superconducting and normal conducting regions, the movement of the flux-lines becomes viscous. The viscosity of the flux-line movement is rapidly changing in the vicinity of the transition temperature. In order to support the expulsion, it is necessary to have the sample at a temperature near or slightly below the transition temperature, where the flux lines are still sufficiently mobile. Furthermore, the driving force for the flux-line movement would have to exceed the blocking force on a flux-line. The movement of the flux-line itself can be suppressed by two mechanisms: (1) flux-trapping, i.e., encirclement of the flux-line by superconducting regions, or (2) flux-pinning, i.e., circumvention of the flux-line movement at features that deviate from the pure Nb matrix, such as dislocations or non-Nb inclusions. The mobility of the flux lines is highest at T\(\sim\)Tc, an alternating magnetic field applied to the sample during cooling is supposed to enhance the mobility of the flux lines and their movement in the axial direction by adding a constant field. In the frame of a series of previous experiments, different sample treatments such as heat treatment, spatial temperature gradients and different cooling rates have been tested with respect to their capability to minimize flux trapping. It could be shown that a recrystallizing heat treatment considerably reduces trapped flux. At the same time, it was observed that single crystal Nb samples trap less flux than polycrystalline samples [2; 17; 18].