# Noninvasive Magnetic-Marking-Based Flow Metering with Optically Pumped Magnetometers
## 3 Results
### Short RF Pulsing  
In the experiments described above, the sinusoidal AM modulation of the RF pulses with 0.1 Hz limited both the data rate (one data point every 10 s) and the accuracy of the determination of \(\Delta t=t_{D}-t_{E}\). Shorter RF pulses are expected to produce narrower \begin{table}
\begin{tabular}{c c c} \hline
**Set Pump Rate (mL/min)** & **Flow Velocity (cm/s)** & **Calculated Volumetric Flow (mL/min)** \\ \hline
150 & 13.0(3) & 153(4) \\
175 & 14.3(6) & 168(7) \\
200 & 17.4(5) & 205(6) \\
225 & 19.2(3) & 226(4) \\
250 & 20.9(8) & 246(9) \\
275 & 22.4(5) & 264(6) \\ \hline \end{tabular}
\end{table} 
Table 1: Flow velocities and volumetric flows detected by the flow-metering apparatus. For comparison the pump rates of the syringe pump are listed in the left column. The manufacturer of the syringe pump noted an error of \(<\)1% on the pump rate. The error specifications represent one standard deviation. Figure 5: Comparison of the pump rate indicated by the syringe pump and the detected volumetric flow measured by the flow metering apparatus (blue). Error bars indicate one standard deviation. The black line indicates the ideal behavior. The error in x-direction is set to 1% (pump specification: \(<\)1% inaccuracy of the pump rate). transients in the polarization signal, allowing one to determine \(\Delta t\) more accurately and to increase the pulse rate. In the following, an experiment was conducted as described in Section 2.2 but rectangular RF pulses with a duration of 250 ms and a repetition rate of 0.4 Hz were applied. The flow velocity was set to of 12.7 cm/s. For comparison, the amplitude modulated RF pulse approach has a pulse length of 5 s. The RF amplitude in the experiment was 236 mV. The experimental results are shown in Figure 6. The mean full width at half-maximum (FWHM) of the observed transients of the magnetic field was 0.38(4) s. The specified error was calculated as the standard deviation (1 sigma) of the scattering data. This width of the observed B field \"pulses\" was about 50% larger than the duration of the applied RF pulses (0.25 s). To get a first insight into the action of the RF pulses on the flowing water, a simple model was used: The change of macroscopic polarization for a pre-polarized water voxel was assumed to be proportional to the integral \(\int Bdt\) experienced by the water voxel. Let \(v\) be the velocity of the water inside the tube. Then the magnetic field experienced by a water voxel \(S\) starting to move at a point \(x_{0}\) inside the interaction section is given by  
\[S(x_{0})=\int_{0}^{Pulse\ Length}B(x(t))dt\ [Ts], \tag{7}\]  
where \(x(t)=x_{0}+vt\) is the position of the water voxel and \(B(x(t))\) is the magnetic field at this position. In a first approximation, \(B\) is calculated using the Biot-Savart approach for the magnetic field along the symmetric axis of the Helmholtz coils:  
\[B(x)=\frac{\mu_{0}NI}{2}\left(\frac{R^{2}}{\left(R^{2}+\left(x-\frac{d}{2} \right)^{2}\right)^{\frac{3}{2}}}+\frac{R^{2}}{\left(R^{2}+\left(x+\frac{d}{2 }\right)^{2}\right)^{\frac{3}{2}}}\right) \tag{8}\]  
\(\mu_{0}\) is the vacuum permeability, \(N\) the number of coil windings of each coil, \(I\) the electric current created by the RF voltage, \(R\) the coil radius and \(d\) the distance between  
Figure 6: The experimental results for the application of short RF pulsing on the pre-polarized water. The black dotted lines mark the timing of the individual RF pulses. The asterisk marks the peak, which is used to compare the experimental data in Figure 7 with a signal estimate. the coils. \(B(x)\) is shown in Figure 7 for a range of \(\pm\)10 cm from the coil center. Due to the geometric situation of the setup, the outer layer of the magnetic shield has approximately 6 cm from the center of the Helmholtz coil (see the shaded area in Figure 7). For this reason, the \(B\) field distribution according to (8) shall only be considered a simple approximation.