Use of Nuclear Spin Noise Spectroscopy to Monitor Slow Magnetization Buildup at Millikelvin Temperatures

Abstract At ultralow temperatures, longitudinal nuclear magnetic relaxation times become exceedingly long and spectral lines are very broad. These facts pose particular challenges for the measurement of NMR spectra and spin relaxation phenomena. Nuclear spin noise spectroscopy is used to monitor proton spin polarization buildup to thermal equilibrium of a mixture of glycerol, water, and copper oxide nanoparticles at 17.5 mK in a static magnetic field of 2.5 T. Relaxation times determined in such a way are essentially free from perturbations caused by excitation radiofrequency pulses, radiation damping, and insufficient excitation bandwidth. The experimental spin‐lattice relaxation times determined on resonance by saturation recovery with spin noise detection are consistently longer than those determined by using pulse excitation. These longer values are in better accordance with the expected field dependence trend than those obtained by on‐resonance experiments with pulsed excitation.


Introduction
High nuclear spin polarization levels can be attainedb yl owering the temperaturet ot he millikelvin range. Thise ffecth as been exploited as an alternative means of generating nuclear hyperpolarization,e ven at room temperature by using the "brute-force" approach. [1,2] It is critical for this approach to achieve high polarization levels within reasonably short time spans. Nuclear spin polarization buildup is al ongitudinal relaxation process characterizedb yarelaxation time constant T 1 , which is "astronomically" slow (i.e. in the order of magnitude of 10 7 s) below 1K. [3] Nanoparticles have been shown to have ah uge effecti na ccelerating spin polarization buildupa tv ery low temperatures. [4,5] These preceding studies demonstrated that the presenceo fm etallic nanoparticles, for example, 30 nm platinum or copper, could reduce the T 1 relaxation times by severalo rders of magnitude at ultralow temperatures. This findingo pens up the possibilityo fu sing high magnetic field and low temperature to grow significant polarization in ar easonable timescale, and thereby, encourage the development of ab rute-forcep olarizer.T he development of improved protocols for brute-force polarization requires the slow buildup of large nuclears pin polarization at low temperatures to be monitored.T here are severalo bstacles that need to be overcome to achieve this goal. One problem arises from strong radiation damping (RD) in the NMR circuit caused by large nuclear magnetization coupled to the feedback field of the radiofrequency (rf) coil. [6] As ac onsequence,t he spectral line shapes may be heavily distorted, which interferesw ith quantitative interpretation. [7] In addition, uniform rf excitation over the entire spectralw idth is virtually impossible due to the large spectral line widths that are characteristicf or static low-temperature solid-state NMR spectra.For these two main reasons, the quantitativee valuationo ft he line shapesi np ulsed NMR spectra of supercooled samples is highly problematic (see the Supporting Information). However, high RD rates, l 0 R ,a lso enable the use of an alternative detection method:s pinn oise detection. [8][9][10][11][12][13][14] Observation of dynamic nuclearp olarization( DNP)-derived nuclear hyperpolarization through spin noise was achieved both after dissolution [15] and more recently in situ. [16] Because spin noise detection avoids the conversion of large longitudinal magnetization to transverse magnetization by rf pulses,t here is no need to measure off-resonance to avoid RD-related lineshape artifacts. As af urther benefit, spin noise data are intrinsically free from any perturbationsb yr f-pulse imperfections and interference.
Herein, we demonstrate spin noise detection as an alternative way of monitoring the very slow buildup of nuclear spin polarization of as ample in an ultralow temperature cryostat.
At ultralowt emperatures, longitudinal nuclear magnetic relaxation times become exceedingly long and spectral lines are very broad.T hese facts pose particular challenges for the measurement of NMR spectra and spin relaxation phenomena. Nuclear spin noise spectroscopy is used to monitor protons pin polarization buildup to thermale quilibrium of am ixture of glycerol, water,a nd copper oxide nanoparticles at 17.5 mK in as tatic magnetic field of 2.5 T. Relaxation times determined in such aw ay are essentially free from perturbations caused by excita-tion radiofrequency pulses, radiation damping, and insufficient excitation bandwidth. The experimental spin-lattice relaxation times determined on resonanceb ys aturation recovery with spin noise detection are consistently longert han those determined by using pulse excitation.T hese longer values are in better accordance with the expected field dependence trend than those obtained by on-resonance experiments with pulsed excitation.
The nuclears pin noise signal integral is related to the total longitudinal spin polarization by the current RD rate.

Theory
RD is am ajor source of complexityi nN MR spectroscopy experiments on highly polarized samples. The origin of RD lies in the nature of the rf-probe circuits used in NMR spectroscopy experiments. [6] Any transversem agnetization induces ac urrent in the receiver coil, which is normally detected as the free inductiond ecay (FID). The same current also causesamagnetic feedback field (the RD field), which acts on the spins in turn, tilting the longitudinal magnetization away from the magnetic field axis, and thus, generating transversem agnetization. The rf-field generated by this secondary transverse magnetization interferes with the original transverse signal in ac omplexw ay, depending on the phase shift of the feedback field. [17,18] Pronounced effects of RD are, for example, increased resonance line widths, line shapes that are, even for small flip angle spectra, no longerL orentzian and maye xhibit "wiggles" in the case of larger tipping angles. [7] Although these effects are alleviateds omewhat by fast transverse relaxation, they heavily interferewith quantitative interpretation.
For all NMR spectroscopy experiments with strong nuclear magnetization in resonant circuits,R Dn eeds to be considered. The RD rate, l 0 R ,i sameasure for its impact, which is defined by Equation (1): [6] l 0 R ¼ in which h is the filling factor, m 0 is the vacuum magnetic permeability, g is the gyromagnetic ratio, M 0 z is the thermale quilibrium magnetization, and Q is the quality factor of the rf circuit.
The high levels of magnetization, M z ,t hat are achievable by hyperpolarizationo rb rute-force methods, together with resonance circuits of high quality factors, Q,a nd large filling factors, h,c an cause huge RD rates. In some cases, these can counteractt he originalp urpose of signal enhancement by circuit optimization and spin polarization.
To avoid more severe effects of RD on line shapes, small flip angle pulse spectra can be used, [7] but even then the observed line shapes result from ac omplex interplay of T 1 , T* 2 ,a nd l 0 R , as well as RD. [7,19] RD can destroy the usually linear relationship between the observeds ignal amplitude and the amount of recoveredm agnetization. [20] As ac onsequence the buildup curves observedo nh igh polarization samples by pulsed NMR saturation recovery experiments may not deliver reliable information on the recovery of equilibrium magnetization, M 0 .
Spin noise spectra depend on the line width, l* 2 ¼ pT* 2 ÀÁ À1 , and on the equilibrium RD rate, l 0 R ,a nd the actual RD rate, l R t ðÞ . [8] The last one depends on the fraction of recovered magnetization, K(t) = M z (t)/M 0 ,a nd thus, l R t ðÞ¼l 0 R Kt ðÞ .N otably,b ecause we cannotc learly distinguish between homogeneous and inhomogeneous broadening herein, we use l 2 ¼ l* 2 and T 2 ¼ T* 2 herein and in the Supporting Information. Assum-ing perfect tuning (w LC = w 0 ), the spin noise powers pectrum is given by Equation (2): in which k B is the Boltzmann constant, T is the temperature of the coil and sample, R P is the equivalent parallel resistanceo f the circuit, and W U a is an additional noise source.T he conditions of the experiments described herein are adequately described by Equation (2), so there is no need to use the refined theoryr ecently introduced by Ferrand et al., [21] which covers more complex situations encountered under high-resolution conditions. The influence of the temperature ratio between the sample and coil, #,a sw ella so ft he tuning of the NMR receiver circuit have been described previously. [11,22,23] After subtraction of the thermalc ircuit noise baseline [obtained from Eq. (2) with M z = 0, and thus l 0 ,i ntegration yields the theoretical spin noise signali ntegral [Eq. (3)]. [11,16] lim D!1 in which D is used as an auxiliary variable for the integration limits of the symmetrical Lorentziand efined by Equation (2). This is the quantitative relationship between spin noise signal integral and fractiono fr ecovered magnetization, K(t), which is discussed in more detail in the Supporting Information.

Results and Discussion
Protons aturation recovery experimentsc onducted by using either conventional pulse excitationorthe spin noise detection methodi ntroduced by McCoy andE rnst, [8] shown in Figure 1a and b, respectively,w ere performed at 2.5 T, at the resonant frequency of the rf circuit on as ample of am ixture of glycerol, water,and CuO nanoparticles (see the Experimental Section). The magnetic field was adjusted such that ap ure dip signal in the 1 Hs pin noise spectra waso btained (see Figure 2). The "tuning" offset (here:o ffset of the Larmor frequency to the coil resonantfrequency,asadjusted by the magnetic field strength) causedadispersive contribution to the spin noise line shape. [8] Af ully negative absorptive line shape is not obtained at the Larmorf requency;t his phenomenonc annot be explained by McCoya nd Ernst's derivations [8] and has been described as the spin noise tuning optimum (SNTO). [21,22] Adjusting for an egative purely absorptive spin noise peak (green trace in Figure 2) is an important prerequisite for performing the spin noise experiments shown herein.
Three representative spectra acquired over the courseo ft he spin noise saturation recovery experiment at 2.5 Ta re displayedi nF igure 3. For comparison, additional experiments were performed by using the pulse method (Figure 1a (on-resonance), 2.0, and 3.0T (off-resonanceo ft he rf-circuit). These data allow us to compareb uildup rates determined by the standard saturation recovery method and to assess the suitability of the spin noise data for evaluation with respectt o T 1 field dependence. The saturation step was performed by using at rain of hard pulses;i ts successw as confirmedb yt he absence of as ignal in the pulse spectra.
The buildup curves obtained from all experiments have been plottedi nF igure 4, each normalized to its maximum and superimposed for comparison.
The spin noise buildup data at 2.5 T( on-resonance) measured by the spin noise acquisition scheme given in Figure 1b are mostly located between the data points obtaineda t2 .0 and 3.0 T( off-resonance), which are determined by pulsed NMR spectroscopy experiments by using the schemes hown in Figure 1a.T his conforms to the expected behaviorb ecause, generally, T 1 increases with the strength of the magnetic field. The spin noise buildup data (red crosses in Figure 4) are closer to the two off-resonanced ata sets (black and gray dots) than the pulse data points at 2.5 T( on-resonance;b lue dots in Figure4). To analyze the differences between the data series in more detail,t he experimental data were initially fitted to am onoexponential model.S imilart oprevious saturation-recovery pulse experiments under similarc onditions, [4] magnetization buildup was not monoexponential.
Because the mechanisms of relaxation in this complex system are still insufficiently explored,w ef ocuso nt he quality of the experimental data with respectt os ystematic interference. Monoexponential target functions were insufficientf or describing the experimentalb uildup data, so at wo-compo-   1 Hs pin noisep ower spectra recorded during buildup.The blue curve is the first acquired spectrum (recoveryt ime:10min), the red curve is the last one (recovery time:4 3h31 min), and one intermediate spectrum (recoveryt ime:1h30min) is shown in green. The dip at + 600 kHz is the growings pin noises ignal.T he Lorentzian-shaped baseline is due to Nyquist noise. [16,26] Further peaks are artifacts: ac entral spikea t1 05 MHz and an externalstray rf-interferencea tano ffset of À1200 kHz. b) An expanded region of the spectraindicated by the rectangle.  At ðÞ¼1 À ae Àt=T 1a À be Àt=T 1b ð4Þ in which a and b are the relative amplitude coefficients,a nd T 1a and T 1b are the relaxation time constants. The parameters obtained by fitting Equation (4) to the experimental data in Figure 4r esulted in the curves shown in Figure 4, which are also summarized in Ta ble 1.
For experiments with pulse excitation, the additional condition a + b = 1w as used in the fitting procedure. Because this assumption does not hold for spin noise based data, since even at full saturation an et spin noise signal should be observable, [8,11] a and b were treated asi ndependent variablesi n the spin noise detected case.
The relaxation times determined by using on-resonance spin noise detection are thus similar to those observed by pulse spectra off-resonance, whereas on-resonance strong RD causes systematic errors in pulse spectra. For both components, especially the more slowly recovering one, longerr elaxation times are found by using spin noise detection. We see this as experimental evidence of the fact that spin noise observation does not interfere as much with the buildup process as pulsed observation experiments. The longitudinal relaxation times are expected to increase with increasing field strengthsd ue to spin diffusion in the presence of the paramagnetic nanoparticles. [27] Therefore, the relaxationt imes at 2.5 Tare expected to be between those determined at 2.0 and 3.0 T. This is clearly obeyed for the spin noise data at 2.5 To ft he faster component, whereas the pulsed experiment clearly violates this expectation.
No consistentt rend can be derived for coefficients of the relaxation components. Ad etailed discussion of the aspects of quantitative interpretationo fs pin noise derived buildup curves, taking into account the intrinsically nonlinear behavior of spin noise integralsa nd potentials ystematic errors, can be found in the SupportingI nformation.
Notwithstanding the detailed analysis of the relaxation mechanisms in the presence of nanoparticles, we have compared the magnetic field dependence of the relaxation times to the simple power law dependence predicted spin diffusion modelsr eported in Refs. [5,27].A sd etailed in the Supporting Information, the on-resonance spin noise relaxation times de-termined from the spin noise data on resonancea gree with the expectedm onotonous field dependence much better than the values obtainedb yp ulsed excitation. The field dependences of both longitudinal relaxation components, T 1a and T 1b ,i n the sample investigated suggestr apid diffusionb ehavior, according to the definition of Blumberg. [27] 4. Conclusions During 1 HNMR spectroscopy experiments at millikelvin temperatures, high polarization caused strong RD, which interfered with the monitoring of polarization buildup by pulsed excitation spectra. We introduced the observation of nuclear spin noise as av iable alternative. With this approach, monitoring of the magnetization buildup was possible on-resonance of the rf circuit.B uildup rates determinedbymeans of spin noise detection on-resonance were in the same range as those determined off-resonance with pulses pectra.T he application of the spin noise technique gave predictable responses to RD effects, and thus, allowed us to avoid using off-resonance detection; this removed some experimental complexity and associated sourceso fu ncertainty.I na ddition, even extremelyb road line shapes could be detected reliably,s ince no limitation due to finite excitation bandwidth existed in spin noise detected NMR spectroscopy.A se laborated in the Supporting Information, potentials ystematic errors in relaxation times determined in this way were significantly below the statistically random errors usually encountered in pulsed relaxation experimentsu nder conditions such as those used herein.
Preparation for spin noise detection was relativelys traightforward to implement for slowly relaxing systemsa te xtremely low temperatures:I tr equired the spin noise signalt ob el ocated and the tuning of the coil or magnetic field strength to be adjusted for ap urely absorptive spin noise line shape.
In particular, for monitoring the buildup of magnetization in very slowly relaxing systems, spin noise detection could improve the data quality.T his method avoided anyi nterference between polarization buildup and rf irradiation and could be appliedc ontinuously;t hus making up for the inherently lower sensitivity.

Experimental Section
All experimental data reported herein and in the Supporting Information were obtained from the protons in as ample of 23 mLt otal volume, which was prepared from 20 parts 2 m Na[1-13 C]acetate in H 2 O/glycerol = 50:50 (v/v) and one part CuO nanoparticles (< 50 nm). [5] We performed the experiments at 17.5 mK on the millikelvin NMR spectrometer described in Refs. [4,24,25],w hich employed a 3 He- 4 He dilution refrigerator and as uperconducting magnet with av ariable field strength of up to 15 T; we applied between 2.0 and 3.0 T. The NMR probe was tuned to af ixed frequency of 104.5 MHz, which corresponded to an approximate Larmor frequency of 1 Hat2 .5 T.
The pulse sequences used herein are shown in Figure 1. The magnetic field was kept constant during the entire duration of the saturation recovery experiments. The saturation sequence always consisted of 500 pulses of 2.5 msa t1 8W .F or the conventional pulse experiments (Figure 1a)u sed to monitor the recovery,1msp ulses at an rf power of 22 Ww ere used. Due to the long relaxation times and frequent changes of the magnetic field, no exact pulse calibration was performed. Avoiding exact pulse angle calibration could be seen as an added advantage of spin noise based methods. Saturation recovery experiments with spin noise detection were performed analogously to the procedure introduced in Ref. [8] using the sequence in Figure 1b for the monitoring of the extremely slow magnetization buildup pertinent here. To prevent pick-up of rf signals through the transmitter cable and circuit acting as an antenna, the transmitter cable was manually disconnected from the probe after the saturation pulse train and two 50 W terminators were attached to avoid open connectors. Disconnecting the transmit channel was not ar equirement for the experiment as such, but was ap recaution required in the particular setup and environment. By using optimal spectrometer hardware and in an rf-quiet environment (in particular,d evoid of radio stations working in the respective frequency range), the connection could be left in place. The delay, t 0 ,r equired for this task and for starting the monitoring experiment was short (< 1min) relative to the relaxation times involved and the same for pulse and spin noise experiments, and thus, was negligible in the calculation of the time coordinates of the buildup curves. During the whole spin noise detected experiment, noise blocks of 1536 data points each were acquired with as pectral width of about 2.86 MHz, resulting in ad uration of 0.26 ms per noise block. Each block was Fourier transformed and the power spectrum was calculated. At otal of 16 kp ower spectra (corresponding to ap eriod of 20 min of recovery time each) were then combined. To eliminate constant noise contributions from the spectrometer and environment, we subtracted ab ackground noise power spectrum obtained by noise acquisition in an identical manner,e xcept for the NMR Larmor frequency being set off-resonance by changing the magnetic field to 3.00 T. (Notably,s ince that frequency was also off-resonance with respect to the rf-coil's resonance frequency,n os pin noise signal was obtained.) Af ifth-order polynomial baseline correction was applied before final integration of the peaks. The resulting negative [due to the intrinsic properties of spin noise, see Eq. (2)] integral values were multiplied by À1t oe nable direct comparison with conventionally determined buildup curves. For the spin noise detected spectra, the time coordinates in the buildup curves were calculated as the average of the start and end times of the noise block acquisition.