The use of free spin precession of gaseous samples at ultra low-fields ($\approx\,\mu$T) for high sensitive magnetometers was already proposed in the early work of Cohen-Tannoudji$^1$. Due to the very long spin coherence times of several hours$^{1-3}$ hyperpolarized rare gases are prominent candidates. For the SQUID detected magnetometry the spin excitation was predominantly achieved by hard switching of the $B_0$ field non-adiabatically introducing free spin precession$^4$. To get the full flexibility of spin manipulation likewise in standard pulsed NMR we introduced two complementary methods to determine the flip angle achieved via AC excitation. As a first application the determination of the longitudinal spin relaxation of hyperpolarized gases in glass cells at ultra-low magnetic fields is shown.

Within a shielded room two crossed Helmholtz coils of $\approx\,1.6\,$m diameter were arranged to generate a constant $B_0$ holding field ($\approx\,1\,\mu$T,$\,\parallel\,y$) and perpendicular to a weak alternating $B_1$ field $\,\parallel\,y$ (likewise Fig.$\,2$ in Ref.$\,4$). On top of the $B_0$ Helmholtz coil a Maxwell coil was wound to shorten $T_2^{\star}$ by applying a gradient of $\partial B_y/ \partial y \,\approx\,90\,$nT/m. A sealed glass cell filled with $^3$He:Xe:N$_2$ gas (715:143:72 mbar) and a small droplet of Rb, was polarized in a remote lab and transported within a battery powered solenoid into the shielded room and put in the center of the Helmholtz coils right underneath the cryostat housing the liquid helium cooled SQUIDs sensing the magnetic flux change in $z-$direction. Fluxgate measurements of $B_1$ allow to calculate the flip-angle $\alpha = \gamma \tau B_1^+$ for a given pulse duration $\tau$.

To measure the effective flip angle we used two complementary pulse schemes. Fig.$\,$1a shows the repetitive coherent spin excitation with $T\!R \ll T_2^{\star}$. This allows to measure the free spin precession while turning the magnetization vector stepwise around the Bloch sphere and thus to deduce the flip angle $\alpha$ per step. To study these measurements further we modeled the $B_1$ frequency dependence with a custom Bloch-McConnell simulator. The second method (Fig.$\,1$b) implements fully incoherent repetitive excitations with two different $T\!R-$times. With a fixed $T\!R$ the measured initial FID amplitudes decay mono-exponentially with an effective relaxation rate $\frac{1}{T_{eff}} = \frac{1}{T_1} + \frac{1}{T_\alpha}$ with $\frac{1}{T_{\alpha}} = - \ln\left[\cos(\alpha)\right] / T\!R$. Thus using two different $T\!R$ and deducing the effective decay time for each series one can calculate $\alpha$ as well as $T_1$ from such a measurement:

$$T_1=\frac{T_{eff1}\,T_{eff2}\cdot\left(T\!R_2-T\!R_1\right)}{T\!R_2\,T_{eff1}-T\!R_1\,T_{eff2}} \quad\text{and}\quad \alpha = \arccos \left( \exp \left( T\!R_1 \cdot \left( \frac{1}{T_1}-\frac{1}{T_{eff1}} \right)\right)\right)$$

For the coherent spin excitation a superposition of the concomitant $B_1$ field $z-$component and the $^3$He spin precession signal is detected during the AC pulses (indents and spikes, Fig.$\,$2a/b). In between two AC excitations the free spin precession signal is measured. The phase of the spin signal and the excitation signal is changing by $180°$ when the spin signal is reaching a minimum. This is also seen in the Bloch simulation as long as the excitation frequency is not too far from resonance (Fig.$\,$3). By analyzing the simulated data one can show that the flip angle could be deduced precisely by fitting the function $A_0 \sin(n\,\alpha)$ to the amplitudes of the free spin precession up to the first maximum (Fig.$\,$2c/d). As this measurement does take just one or two minutes and could be stopped when minimum spin precession signal is seen, i.e. most of the initial polarization is preserved, one can perform consecutive measurements afterwards as shown in Fig.$\,2\,$a/b.

From the error propagation of the formula for calculating $\alpha$ for the second method it is seen that two distinct $T\!R-$ times have to be chosen leading to an experiment time of several hours (Fig.$\,4$). However, this measurement already implies the $T_1$ determination, which would be an own task for the first method. In addition this method also allows to determine the achievable $T_2^\star$ time, which shows up to change over time. This latter effect is not jet fully understood but is assumed to be tight to the remaining longitudinal polarization and not perfect spherical shape of our glass cell. In Fig.$\,4$b three measurements from two different polarization runs using the identical sample are shown.

The repetitive coherent spin excitation allows a fast determination of the effective flip-angle with the ability to save most of the longitudinal polarization for consecutive experiments. As an all in one solution the non coherent repetition of AC excitations allows for deducing all the relevant parameters $T_1$, $T_2^{\star}$ and $\alpha$. All this is essential for a precise $T_1$ determination and lays the basis for further dedicated spin excitation.