# Noninvasive Magnetic-Marking-Based Flow Metering with Optically Pumped Magnetometers
## 2 Materials and Methods  
### Experimental Setup  
The experimental setup of the flow-metering apparatus and the orientation of the magnetic fields involved in the procedure is shown in Figures 1 and 2. The water reservoir consisted of a plastic syringe of 200 mL capacity. The syringe was operated by a syringe pump (Syringe Pump LA-800, Landgraf, Langenhagen, Germany), which can log pump rates digitally. For pre-polarization of the water sample the syringe was placed inside a 1 T permanent magnet (RHR-1T-50 Halbach magnet, BFLUX TECHNOLOGY, Dublin, Ireland). This way the whole volume of the syringe could be polarized before being pushed into the system. Pre-polarization was used to increase the magnetic signal for the OPM detector to a measurable level [7]. An interaction section and magnetic shielding (MS-2 Magnetic shield, Twinleaf LLC, Plainsboro Township, NJ, USA) containing magnetometers (Zero Field OPM, FieldLine Inc., Boulder, CO, USA.) were located downstream. The circuit ended in a water storage tank. The interaction section consisted of a Helmholtz-coil (Ferronato(r)-BHC-2, Servicencia, S.L.U., Toledo, Spain), which was used as an RF coil, and racetrack coils. The racetrack coils served a dual function of spin guiding between Halbach and shield, and providing a constant ambient magnetic field necessary for RF pulsing. The presence of a magnetic field is necessary for RF pulsing because a manipulation of the water polarization is not possible otherwise. In the presence of a magnetic field, the nuclear spin energy level of the hydrogen atom is split in two equidistant energy levels. This is called the Zeeman effect. To induce a transition between the energy levels, which define the macroscopic polarization of the water, an RF pulse with the transition frequency of the hydrogen can be applied. This transition frequency is called Larmor precession \(\omega_{L}\) and is defined as \(\omega_{L}=\gamma B\). Here, \(\gamma\) is the gyromagnetic ratio of the proton, which is roughly 42.577 MHz/T, and \(B\) is the strength of the ambient magnetic field. In the case of this experiment, the field generated by the racetrack coils was about 200 uT, which translates to an RF frequency of roughly 8 kHz. The magnetic shield downstream contained zero field OPM for the detection of polarization changes induced by the RF coil. OPM are very sensitive sensors that can reach a sensitivity of 10 fT/\(\sqrt{\mathrm{Hz}}\)[13] if they are operated in an environment where magnetic noise and the absolute magnetic field are strongly suppressed. The distance \(\Delta L\) between the center of the interaction section and the OPM was fixed at 212(1) mm throughout the experiment. A plastic tube with an inner diameter of 5 mm was used as a water guide. For validation, the data taken were compared to the pump rates indicated by the syringe pump in the end. Therefore, the flow velocity detected by the apparatus was converted to volumetric flow. Figure 1: Schematic overview of the experimental setup with applied magnetic fields and the trend of the medium polarization. (**a**) Schematic of the experimental setup: The dashed green line encircles the interaction section with the coil system. The green square in the center marks the location of the RF coil. The two black cylinders represent the racetrack coils. The distance \(\Delta L\) between the center of the RF coil and the OPM is known and used to determine the flow velocity. The red double arrows indicate the direction of the pumping mechanism. (**b**) Orientation and field strengths of the magnetic fields as they are found in the apparatus. The magnetic field of the Halbach array to pre-polarize the water and the field of the racetrack coils point in z-direction. (**c**) Trend of the macroscopic water polarization throughout the water circuit in z direction. The RF coil operates in x direction to reorient the water polarization away from its pre-polarization direction. The OPM sensors measure the z component of the field to detect how much water polarization is reoriented by the RF coil.