Journal of Magnetic Resonance 255 (2015) 114–121
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Dynamic nuclear polarization in the hyperfine-field-dominant region Seong-Joo Lee ⇑, Jeong Hyun Shim, Kiwoong Kim, Kwon Kyu Yu, Seong-min Hwang Center for Biosignals, Korea Research Institute of Standards and Science (KRISS), 267 Gajeong-ro, Yuseong-gu, Daejeon 305-340, Republic of Korea
a r t i c l e
i n f o
Article history: Received 19 December 2014 Revised 27 March 2015 Available online 24 April 2015 Keywords: Dynamic nuclear polarization Ultra-low field NMR SQUID detection Hyperfine-field-dominant region
a b s t r a c t Dynamic nuclear polarization (DNP) allows measuring enhanced nuclear magnetic resonance (NMR) signals. Though the efficiency of DNP has been known to increase at low fields, the usefulness of DNP has not been throughly investigated yet. Here, using a superconducting quantum interference device-based NMR system, we performed a series of DNP experiments with a nitroxide radical and measured DNP spectra at several magnetic fields down to sub-microtesla. In the DNP spectra, the large overlap of two peaks having opposite signs results in net enhancement factors, which are significantly lower than theoretical expectations [30] and nearly invariant with respect to magnetic fields below the Earth’s field. The numerical analysis based on the radical’s Hamiltonian provides qualitative explanations of such features. The net enhancement factor reached 325 at maximum experimentally, but our analysis reveals that the local enhancement factor at the center of the rf coil is 575, which is unaffected by detection schemes. We conclude that DNP in the hyperfine-field-dominant region yields sufficiently enhanced NMR signals at magnetic fields above 1 lT. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Signal detection in low-field nuclear magnetic resonance (NMR) requires extraordinary sensors, such as electromagnetic resonators [1], superconducting quantum interference devices (SQUIDs) [2], and atomic magnetometers [3]. Among them, SQUIDs exhibit unique performance manifested as low 1=f noise and nearly flat frequency response over tens of kilohertz [4]. Up to date, in most SQUID-based ultra-low field (ULF)-NMR/magnetic resonance imaging (MRI) studies [2,5–25] a so-called prepolarization field (Bp ) has been applied before acquisition. The NMR signal is, thus, proportional to Bp , which is electrically switched on and off. A stronger Bp is, of course, advantageous for obtaining a stronger signal; however, it also creates unwanted side effects, such as the induction of eddy currents on the walls of a magnetically shielded room (MSR) [15–17] and flux trapping in superconducting pickup coils [24]. Both effects generate a temporally varying field, which broadens the NMR linewidth and distorts the MR image. Dynamic nuclear polarization (DNP) based on the Overhauser double-resonance effect [26] has been considered as one of the alternatives to applying Bp , as it utilizes radio frequency (rf) instead of direct current (dc). DNP enhances nuclear spin polarization by polarization transfer from excited unpaired electron
⇑ Corresponding author. E-mail address: [email protected] (S.-J. Lee). http://dx.doi.org/10.1016/j.jmr.2015.04.004 1090-7807/Ó 2015 Elsevier Inc. All rights reserved.
spins via the process of cross-relaxation. Several DNP experiments have been reported [12,13,27–33] using field intensities ranging from the Earth’s field intensity to a few milliteslas. Some studies [28,30,33] reported enhancement factors of few thousands at the Earth’s field, whereas at high fields the enhancement factors were limited to 109 and 165 for 14 N and 15 N, respectively [30]. To the best of our knowledge, no DNP experiments have been performed for field intensities below the Earth’s field intensity. Nitroxide radicals, which are typically exploited in DNP, have hyperfine interactions between unpaired electron and 14 N (or 15 N) nuclear spins. Below the Earth’s magnetic field is, therefore, a so-called hyperfine-field-dominant region. In the present work, we report experimental measurements of DNP spectra in the hyperfine-field-dominant region. Investigations of magnetic field and rf power dependencies show that lowering magnetic fields below the Earth’s field does not make DNP with nitroxide radicals more efficient. In the DNP spectra, the large overlap of two peaks having opposite signs results in net enhancement factors, which are significantly lower than theoretical expectations [30] and nearly invariant with respect to magnetic fields. Our numerical analysis qualitatively agrees with those results. The local enhancement factors at the center of the rf coil was estimated by considering the rf field inhomogeneity and spatial profile of detection sensitivity. The experimental results suggest that 1 lT is an approximate lower limit for reliable single-shot acquisitions via DNP with nitroxide radicals in our SQUID-based system.
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2. Methods We used commercially available 4-Hydroxy-2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPOL; Sigma–aldrich, Cat. No. 176141), containing an unpaired electron. Fig. 1(a) shows the chemical structure of this material. 2 mM of TEMPOL solution were prepared with deionized water. The main interaction in TEMPOL is the interaction between the 14 N (K = 1) and the unpaired electron in the free radical. Therefore, the effective Hamiltonian is:
H ¼ cS B S cK B K þ AS K;
ð1Þ
where A is the hyperfine field constant (47.81 MHz for TEMPOL [27]) and cS and cK are the gyromagnetic ratios of the unpaired electron spin S and the 14 N nuclear spin K, respectively. Eigenvalues of this Hamiltonian are obtained from Breit-Rabi equation [34]. Because the hyperfine field becomes dominant below the Earth’s field intensity, the eigenstates are expressed not using the general jmS ; mK i basis, but using the jF; mF i basis. Here, F is the total spin quantum number (F ¼ S þ K) and mS ; mK , and mF (¼ mS þ mK ) are the magnetic moments of spins S; K, and F, respectively. Considering the 14 N nucleus and the unpaired electron, the energy states are defined as j1i ¼ jF; mF i ¼ j3=2; 3=2i; j2i ’ j3=2; 1=2i; j3i ’ j3=2; 1=2i; j4i ¼ j3=2; 3=2i; j5i ’ j1=2; 1=2i, and j6i ’ j1=2; 1=2i [29,30]. The energy levels for each energy state are plotted as a function of Bp in Fig. 1(b). The theoretical curves in this study were calculated by using the equations described in Ref. [30]. The transition T ij is defined as the difference between i-th and j-th energy levels. Fig. 1(c) shows the experimental setup for the ULF-DNP experiment. To avoid the effects of external magnetic noise, all equipment was mounted inside the MSR which was specially designed for the ULF-NMR/MRI experiments [21,22]; inner-most aluminum shell is partitioned into small panels for reducing the eddy current
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effect, whereas outer-most aluminum shell forms a closed surface to prevent the SQUID system from the external rf environment. A dc SQUID was used as the NMR signal detector, which was connected with a 2nd order gradiometric pickup coil. The Bp and the measurement field (Bm ) were generated using the Helmholtz coil and the double Helmholtz coil [35], respectively. The Bp strengths were 0.71, 2.04, 4.91, and 47.68 lT, measured using a fluxgate magnetometer. The strength of Bm was about 4.94 lT, corresponding to the proton resonance frequency of 210.5 Hz. The directions of the Bm , the rf field (Brf ), and the Bp were perpendicular to each other. A surface-type coil [36–38] was used for the rf resonance circuit, which is labeled as ‘‘Tank circuit’’ in Fig. 1(c). More detailed description can be found in the section Appendix A. The pulse sequence for the ULF-DNP experiment is displayed in Fig. 1(d). Initially, Brf and Bp are applied simultaneously during the time t rf . After these fields are turned off, Bm is applied, following which the SQUID sensor measures the enhanced free-precession-decay (FPD) signal. If the external Brf is applied to induce the transitions, 1 H nuclear spin repolarization occurs owing to the interaction between the 1 H nuclear spin and spins S, K. Hence, an enhanced FPD signal can be obtained. The data were measured 4 times at each point and the Fourier transforms of all FPD time traces were averaged. The error bars, representing the standard deviation, are plotted along the experimental data. In this study, we define that the experimentally obtained enhancement factor is the DNP enhanced NMR signal (SDNP ) divided by the NMR signal (SNMR ), i.e., enhancement factor = SDNP =SNMR . The SNMR obtained without Brf was only measurable after at least 20 iterations at 47.68 lT. However, because the NMR intensity is proportional to Bp it is possible to predict the intensity of the SNMR below 4.91 lT by using the SNMR obtained at 47.68 lT (20 iterations). Therefore, the enhancement factors for field intensities below 4.91 lT were calculated by this procedure.
Fig. 1. (a) Chemical structure of TEMPOL containing an unpaired electron. (b) Energy levels plotted as a function of Bp for 14 N (K = 1). (c) Schematic diagram of the ULF-DNP experimental setup. The axes of the Bp ; Bm , and Brf coils are perpendicular to each other. The sample, contained in a bottle with 5 cm outer diameter and 7.8 cm length, was placed in the Brf coil of the tank circuit. (d) Illustration of the FPD pulse sequence. The following experimental parameters were used in this study: t rf ¼ 1 s, tmeasurement ¼ 2 s, trepetition ¼ 2 s.
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3. Results and discussion 3.1. Dependence of the DNP-NMR spectrum on Bp Fig. 2 shows the DNP spectra of 2 mM TEMPOL solution at 47.68, 4.91, 2.04, and 0.71 lT, obtained with the rf power of about 23 W. We successfully obtained the DNP spectrum at ultra-low intensity magnetic field (below 1 lT) and measured the two main peaks in all spectra. In addition, the peak in the higher frequency region was inverted in real spectra. In low magnetic field, eight p and two r transitions occur when the direction of Brf is perpendicular and parallel to that of Bp , respectively. However, because Bp and Brf were perpendicular to each other in our experimental setup, only p transitions could be measured. Among the eight p transitions, the related transitions (T 16p ; T 25p ; T 36p , and T 45p ) within the measured frequency range are plotted as a function of Bp in the upper inset in Fig. 3(a). Fig. 3(a) shows the dependence of the enhancement factors eij for individual T ij transitions on Bp , for field
intensities below the Earth’s field intensity. The spectrum is dominated by strong e16 and e45 , whereas e25 and e36 are negligible (lower inset in Fig. 3(a)). However, because the negative enhancement of T 16p and the positive enhancement of T 45p overlap for field intensities below the Earth’s field intensity, as described in Refs. [30,32], the actual spectrum does not exhibit strong enhancement. The net enhancement factor Enet , which is determined by the large overlap of the transitions, can be expressed as [30]:
X Enet Bp ; f rf ¼ eij Bp g Ei Ej f rf ; C ;
ð2Þ
ij
where f rf is the applied rf, g is the shape function, Ei and Ej are the eigenvalues from Eq. (1), and C is the linewidth. (The Planck constant h is assumed to be 1.) We chose a normalized Lorentzian function as the shape function:
ðC=2Þ2 g Ei Ej f rf ; C ¼ 2 Ei Ej f rf þ ðC=2Þ2
ð3Þ
Fig. 2. DNP enhanced NMR signal (SDNP ) plotted vs. the rf for the 2 mM TEMPOL solution at Bp of (a) 47.68, (b) 4.91, (c) 2.04, and (d) 0.71 lT, obtained with rf power of about 23 W. The closed and the open circles represent the magnitude and the real spectra, respectively. The real spectra were obtained with the phase correction.
Fig. 3. The theoretical curves (see Appendix B): (a) The theoretical enhancement factors (eij ) of T 16p ; T 25p ; T 36p , and T 45p plotted as a function of Bp . The lower inset shows the magnified view of the graph for the T 25p and T 36p transitions. The upper inset shows the transition frequency vs. Bp . (b) The theoretical enhancement factors (Enet ) determined by the overlap of the transitions plotted as a function of rf. The curves were obtained with the values shown in (a) and Eq. (3) (see Eq. (2)). For a convenient comparison with other figures, the theoretical spectra with absolute magnitude were plotted.
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Fig. 3(b) shows the theoretical Enet at various Bp . Since the derived equations and the shape function in Ref. [30] describe the individual transition, one can plot the theoretical Enet as shown in Fig. 3(b) though the strong overlap exists at magnetic fields below the Earth’s magnetic field. These curves were obtained by the averaged linewidth (C) of 7.45 MHz (see Appendix B.2). Although eij increased exponentially with decreasing Bp ; Enet was almost unchanged. This result will be compared with the experimental results in the following paragraph. The variation of Enet is plotted as a function of Bp in Fig. 4. The theoretical curves, solid line for the higher frequency-side peak (HP) and dashed line for the lower frequency-side peak (LP), are nearly constant, varying less than 5%. Although the experimental result seems not to follow the theoretical curves, the deviation is not significant, compared with the variation of Bp . The enhancement varied only 45% while Bp decreased approximately 70 times. Therefore, it is a plausible conclusion that, with respect to Bp ; Enet is nearly constant for the magnetic fields below the Earth’s field intensity. This explains why the NMR intensity in Fig. 2 decreased quite linearly with decreasing Bp ; when Enet is a constant, the NMR signal intensity is simply proportional to the initial thermal polarization. As explained above, the large overlap of the two transitions (T 16p and T 45p ) made Enet uncorrelated to Bp . 3.2. Dependence of the DNP-NMR spectrum on rf power The ratios of LP and HP for all spectra in Fig. 2 are different from those reported in theoretical curves in Fig. 3(b). This could be explained by the rf power dependence of LP and HP. In Fig. 5(a), the normalized magnitude spectra at 47.68 lT are plotted for different values of rf power. Each spectrum was normalized by its corresponding SDNP measured at 74.4 MHz. As the rf power increased, the relative intensity of LP decreased. This result suggests that the LP to HP ratio becomes similar to the theoretical one (Fig. 3) when the full saturation is created by large Brf . Fig. 5(b) shows Enet ðHPÞ and Enet ðLPÞ as a function of the rf power. Below 15.6 W, Enet ðLPÞ was larger, but, above 15.6 W, became smaller than Enet ðHPÞ. The transition probability of T 45p is lower than that of T 16p [30]. In addition, the difference between the energy levels of the T 16p transition is larger than those of the T 45p transition, which affects eij . Therefore, e45 is smaller than e16 at the same rf power. This mechanism can explain why Enet ðHPÞ is larger than Enet ðLPÞ above 15.6 W. However, the inverse behavior that Enet ðLPÞ is higher than Enet ðHPÞ below 15.6 W is unclear. We believe the properties, such as T1 and T2 of individual transitions should be separately
Fig. 5. (a) Normalized spectra vs. the rf power, for field intensity of 47.68 lT. Each spectrum was normalized by the NMR intensities measured at the rf of 74.4 MHz (marked by the arrow). (b) Enet vs. the rf power. Open squares and circles represent, respectively, the Enet values of HP and LP in (a). The inset shows the magnified view of the graph for the Enet values of the HP and LP below 1 W. The maximal NMR intensities at each peak were used for calculations. The solid lines represent the exponential fitting curves.
investigated for a better explanation of such power dependencies. However, HP and LP are produced as a result of large overlap of two transition peaks, of which widths are 7.45 MHz. A clear discrimination them, thus, may not be easily accessible and thereby will be a challenging work in future. 3.3. Net enhancement factor Enet with high rf power In order to observe the saturation of Enet , a higher rf power is required to be applied. Fig. 6 shows Enet at higher values of rf power. In our experiments, Enet started to converge toward the maximal value for power above 120 W, and increased up to about 325 for rf power of 164 W at 4.91 lT. 3.4. Local enhancement factor Eloc ð~ r; PÞ The local enhancement factor (Eloc ) at the center of the rf coil was larger than all values shown in this study. Two points must be considered for the calculation of Eloc . First, Brf is not sufficient to saturate the entire sample with approximate dimensions of
p 252 78 mm3. Although surface-type resonance circuits
Fig. 4. The net enhancement factors (Enet ) vs. Bp , for higher and lower frequencyside peaks, denoted as HP and LP (open squares and circles). The maximal SDNP values of HP and LP in Fig. 2 were used for calculations of Enet obtained by the experiments. Solid and dashed lines represent the theoretical curves, which are rescaled by matching the values at 47.68 lT to the experimental values. The theoretical curves show that the net enhancement factor is nearly independent of magnetic field lower than the Earth’s field.
require less power to saturate the electron spin resonance (ESR) transition than volume resonance circuits [38], the former have only a single-turn rf coil, which cannot cover our sample. Therefore, the NMR signal obtained without Brf was generated by a larger volume than that obtained with Brf . Second, in NMR signal detection, the sensitivity profile of the pickup coil exhibits hemisphere-like shapes [22]. We numerically calculated the two factors to obtain Eloc at the center of the single-turn rf coil. Fig. 7 shows the configuration of the sample, the pickup coil, and the rf coil. Over the volume of the sample, the sensitivity of
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where qð~ r Þ represents the density of the sample, and gð~ r Þ is the sensitivity profile for the pickup coil used in the experiment. The enhancement factor Eloc ð~ r; PÞ also exhibits spatial variation, which is associated with the rf field inhomogeneity generated by the rf coil. P is the amount of rf power injected into the tank circuit. According to Eqs. (4) and (5), the net enhancement factor Enet r Þ will should be affected by detection schemes; a variation in gð~ lead to a different Enet value. Instead, the local enhancement factor Eloc at the center (~ r c ) of the rf coil is expected to be independent of detection methods, presenting the efficiency of DNP neutrally, and can be used as a reference, validating the efficiency of DNP in the hyperfine-field-dominant region. The numerical procedure to estimate Eloc ð~ r c ; P Þ is described below.
Fig. 6. Enet vs. the rf power. The data were obtained with the 74.4 MHz Brf corresponding to the maximum peak in HP at 4.91 lT. In most cases, the error bars, denoting the standard deviation, are smaller than the circles size.
Fig. 7. Illustration of the experimental configuration including the pickup coil, the sample, and the rf coil. The pickup coil is a 2nd order gradiometer. The rf coil is a single-turn circle, and the sample is in the cylindrical bottle. All dimensions, such as the diameters of coils, the gap between the bottom of the pickup coil and the sample, are uniformly scaled down with respect to the experimental values. The sensitivity profile is shown by the surface plot with the color scale on the left. The intensity of Brf generated by the rf coil with 1 A current is displayed for the plane through the coil with the color scale on the right. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the NMR signal, given by the geometry of the pickup coil, cannot be uniform, as illustrated by the surface plots through the sample volume in Fig. 7. In addition, the Brf generated by the rf coil is not uniform either. The variation of the Brf intensity in the plane through the rf coil is shown in the density plot of Fig. 7. Therefore, the NMR and DNP enhanced NMR signals can be expressed as:
SNMR ¼ SDNP ¼
Z Z
qð~rÞgð~rÞd~r
ð4Þ
qð~rÞgð~rÞEloc ð~r; PÞd~r
ð5Þ
3.4.1. Pickup coil sensitivity profile gð~ rÞ During free precession, nuclear spins generate variable magnetic fields, i.e., magnetic flux, in the pickup areas. By generating the screening currents due to the variation of magnetic flux, the pickup coil transfers the magnetic flux to the SQUID sensor. As shown in Fig. 7, the sample approaches the pickup coil’s bottom, and the sample dimensions are comparable to the size of the pickup coil. In this case, local NMR signals cannot be detected with an equal efficiency. Thus, the sensitivity profile gð~ r Þ, which is illustrated by the contour plot in Fig. 7, should be calculated in order to determine Eloc ð~ r; P Þ in Eq. (5). The principle of reciprocity makes the calculation simple. One needs to know the receiving field at ~ r, which the pickup coil would produce when the same current corresponding to the magnetic fluxes flows through the pickup coil. The actual dimensions were used in the calculation. The sensitivity for the top area closest to the pickup coil is nearly fivefold higher than that for the bottom. The numerically obtained gð~ r Þ will be used below to extract r; P Þ. Eloc ð~
3.4.2. Estimation of Eloc at the center The spatial dependence of the DNP enhancement factor originates from the spatially varying Brf intensity Brf ð~ r; PÞ produced by r; P Þ can be easily calculated with the single-turn rf coil. The Brf ð~ analytic solutions [39]. The relation between the local Brf ð~ r; PÞ and the Eð~ r; PÞ can be deduced from the solution of Bloch equation [40,41]. The enhacemenet factor should be proportional to the degree of saturation, which can be expressed as 1 1=ð1 þ aPÞ. We introduce a parameter cð~ rÞ that links the applied current I r; PÞ ¼ cð~ rÞI. The final expression and the generated rf field as Brf ð~ for the enhancement factor is, thus, given by:
" Eloc ð~ r; PÞ ¼ E0 1
#
1 2 2
1 þ cð~ rÞ k P
;
ð6Þ
pffiffiffi in which k is defined as cT 1 = Z , and Z represents the impedance of the rf circuit. In order to estimate the unknown parameters, E0 and k in Eq. (6), we fitted the measured SDNP =SNMR data (HP in Fig. 5(b)) with Eqs. (4)–(6). The comparison is shown in Fig. 8. With the estimated E0 and k for HP, the enhancement at the center of r c ; P Þ, was obtained. Conclusively, we can notice that, the rf coil, Eloc ð~ at the center, Eloc is 1.77 times larger than the measured value. The histogram of the Brf intensities over the volume of the sample is shown in Fig. 9. The value at the center of the coil is indicated by an arrow. Notably, most areas are weakly excited compared with the center; however, those areas have weak sensitivities. This can explain why the estimated enhancement at the center is only 1.77 times larger than the measured values.
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Acknowledgment This work was supported by a World Class Laboratory (WCL) Grant from the Korea Research Institute of Standards and Science.
Appendix A. Experimental setup
Fig. 8. The power dependence of Enet (blue circle) is fitted (orange line) with Eqs. (4)–(6). With the estimated parameters from the fitting, Eloc at the center of the rf coil is plotted (green line with square), in order to obtain the actual values at the center. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
A dc SQUID (CE2Blue; Supracon AG, Germany) was used as the NMR signal detector, which was shielded by two different cans. The inner shield and the outer shield were a superconducting Nb cast can (99.9%) and a superconducting lead can (99.5%), respectively. A 2nd order gradiometric pickup coil connected with the dc SQUID was made of 125 lm Nb wire with 50 mm baseline and 65 mm diameter. The dc SQUID sensor and the pickup coil were additionally wrapped in a crumpled Mylar film to protect the SQUID system from the influence of the external radiofrequency (rf) field. The environmental noise in our system was pffiffiffiffiffiffi about 2.3 fT= Hz. Fig. A.1 shows a schematic diagram of the surface-type resonance circuit. The lparallel length is given by [37]:
lparallel ¼
0:663c 276 log ðb=aÞ pffiffiffi ; tan1 2pf rf 2pf rf L
ðA:1Þ
where c is the speed of light, f rf is the target rf, L is the inductance of the single-turn rf coil, is the dielectric constant of the insulator, and a and b are the inner and outer radii of the coaxial line, respectively. More detailed description of parameters of the surface-type resonance circuit is given in Table A.1. The tuning range of the rf resonance circuit was from 63 MHz to 84 MHz. Low power (ZHL-1A; Minicircuits, USA) and high power (1000A225; rf/microwave instrumentation, USA) amplifiers were used in the experiments with rf power below 1 W and above 9 W, respectively.
Fig. 9. The histogram of Brf intensities over the volume of the sample. The value at the center of the coil is indicated by the arrow.
4. Conclusion Fig. A.1. Schematic diagram of the surface-type resonance circuit.
In summary, we measured the enhanced 1 H NMR signals of water in the 2 mM TEMPOL solution in the hyperfine-field-dominant region. The obtained DNP spectra had two peaks corresponding to the overlap of two main transitions, T 16p and T 45p . The dependence of the DNP spectrum on Bp showed that the net enhancement factor Enet was nearly constant for field intensities below the Earth’s field intensity owing to the overlap of the two transitions. The dependence of the DNP spectrum on rf power showed that below 15.6 W Enet of LP was larger than that of the HP. On the other hand, for power above 15.6 W Enet of HP became larger than that of LP. Considering the numerical calculation of the local effect of our system, we obtained the local enhancement factor of HP exceeding 575 at 4.91 lT, which exhibits the DNP efficiency unaffected by detection methods. This value is, however, still lower than the theoretical prediction, that is nearly 1000. 1 lT is an approximate lower limit for reliable single-shot acquisitions via DNP with nitroxide radicals in our SQUID-based system.
Table A.1 Parameters of the surface-type resonance circuit. Target rf f rf
74.4 MHz
Single-turn rf coil (copper wire) Thickness Diameter Inductance L
1.3 mm 55 mm 404 nH
Coaxial line (RG58) Inner radius a Outer radius b Dielectric constant of the insulator
0.4675 mm 1.8 mm 2.27496
Parallel line lparallel (RG58 coaxial line) k=2 balun (RG58 coaxial line) Tuning capacitor (Voltronics Corp.; NMNT70–6) Matching capacitor (Voltronics Corp.; NMNT23–12ENL)
218.4 mm 1337 mm 2.5–70 pF 3–23 pF
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Appendix B. Calculation of theoretical enhancement factor (Fig. 3) B.1. Assumptions for calculation The enhancement factor is expressed as qfsðhSz i S0 Þ=I0 [30], where q is the coupling factor describing the nucleus-electron interaction, f is the leakage factor, s is the saturation factor,
Sz is the spin polarization after the electron spin resonance saturation, and S0 and I0 are the spin polarizations of the electron and the nucleus at the thermal equilibrium, respectively. During the enhancement factor calculations, we assumed that q; f ; s, and I0 are 0.5 (pure dipolar coupling), 1, 1 (complete saturation), and cN Bp =4kB T [40], where cN is the gyromagnetic ratio of 1 H, kB is the Boltzmann’s constant, and T is the temperature, respectively.
Fig. B.1. Lorentzian-fitted curves plotted vs. the rf frequency at (a) 47.68, (b) 4.91, (c) 2.04, and (d) 0.71 lT. Solid lines represent the fitting curves with Lorentzian broadening function. The open circles represent the real spectra obtained by the experiments, the same spectra shown in Fig. 2.
Fig. B.2. Theoretically obtained individual enhancement factor eij plotted vs. the rf at Bp of (a) 47.68, (b) 4.91, (c) 2.04, and (d) 0.71 lT. The (red) solid and the (blue) dotted lines represent e16 and e45 , respectively. e25 and e36 were plotted with the (black) solid lines. The sum of the eij of four transitions, the Enet , were plotted with dashed line. The spectra of jEnet j were the same spectra shown in Fig. 3(b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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B.2. Estimation of the linewidth C of the shape function In order to determine the linewidth (C) of the shape function described in Eq. (3), we fitted the experimental data with normalized Lorentzian broadening function (Fig. B.1). We chose the fitting function F fit as:
X F fit Bp ; f rf ; C ¼ eij Bp g Ei Ej f rf ; C sij ;
ðB:1Þ
ij
where sij is saturation ratios for individual T ij transitions. The curves were obtained with the assumptions of s25 ¼ s36 ¼ s45 . Since the rf power used in the experiments was not enough to create full saturation, it was reasonable assumptions that the sij for individual T ij were different to each other. In addition, we assumed the s25 and s36 as above, because the contributions of T 25p and T 36p to Enet were very weak. The linewidths obtained by the fit at Bp of 47.68, 4.91, 2.04, and 0.71 lT were 7.37, 7.46, 7.51, and 7.45 MHz, respectively. B.3. The theoretically obtained individual enhancement factor eij at various Bp To aid the understanding of the theoretical curve shown in Fig. 3(b), we plotted the theoretically obtained individual enhancement factor eij as a function of the rf at various Bp (Fig. B.2). Here, eij is defined as:
eij Bp ; f rf ¼ eij Bp g Ei Ej f rf ; C ;
ðB:2Þ
where eij and the shape function g were defined in the main text. Although the maximum peaks of e16 and e45 exponentially increased with decreasing Bp , those were also coming close to each other. Therefore, Enet , the sum of the eij of four transitions, became almost constant for magnetic fields below the Earth’s magnetic field. References [1] B. Blümich, J. Perlo, F. Casanova, Mobile single-sided NMR, Prog. Nucl. Magn. Reson. Spectrosc. 52 (2008) 197–269. [2] J. Clarke, M. Hatridge, M. Mößle, SQUID-detected magnetic resonance imaging in microtesla fields, Annu. Rev. Biomed. Eng. 9 (2007) 389–413. [3] D. Budker, M. Romalis, Optical magnetometry, Nat. Phys. 3 (2007) 227–234. [4] J. Clarke, A.I. Braginski, The SQUID Handbook: Fundamentals and Technology of SQUIDs and SQUID Systems, vol. I, Wiley, Weinheim, 2006. [5] R. McDermott, A.H. Trabesinger, M. Mück, E.L. Hahn, A. Pines, J. Clarke, Liquidstate NMR and scalar couplings in microtesla magnetic fields, Science 295 (2002) 2247–2249. [6] S.K. Lee, M. Mößle, W. Myers, N. Kelso, A.H. Trabesinger, A. Pines, J. Clarke, SQUID-detected MRI at 132 lT with T1-weighted contrast established at 10 lT–300 mT, Magn. Reson. Med. 53 (2005) 9–14. [7] V.S. Zotev, A.N. Matlashov, P.L. Volegov, I.M. Savukov, M.A. Espy, J.C. Mosher, J.J. Gomez, R.H. Kraus Jr., Microtesla MRI of the human brain combined with MEG, J. Magn. Reson. 194 (2008) 115–120. [8] R.H. Kraus Jr, P. Volegov, A. Matlachov, M. Espy, Toward direct neural current imaging by resonant mechanisms at ultra-low field, Neuroimage 39 (2008) 310–317. [9] M. Espy, M. Flynn, J. Gomez, C. Hanson, R. Kraus, P. Magnelind, K. Maskaly, A. Matlashov, S. Newman, T. Owens, M. Peters, H. Sandin, I. Savukov, L. Schultz, A. Urbaitis, P. Volegov, V. Zotev, Ultra-low-field MRI for the detection of liquid explosives, Supercond. Sci. Technol. 23 (2010) 034023. [10] M. Burghoff, H.H. Albrecht, S. Hartwig, I. Hilschenz, R. Körber, N. Höfner, H.J. Scheer, J. Voigt, L. Trahms, G. Curio, On the feasibility of neurocurrent imaging by low-field nuclear magnetic resonance, Appl. Phys. Lett. 96 (2010) 233701. [11] S.H. Liao, K.W. Huang, H.C. Yang, C.T. Yen, M.J. Chen, H.H. Chen, H.E. Horng, S.Y. Yang, Characterization of tumors using high-T c superconducting quantum interference device-detected nuclear magnetic resonance and imaging, Appl. Phys. Lett. 97 (2010) 263701. [12] V.S. Zotev, T. Owens, A.N. Matlashov, I.M. Savukov, J.J. Gomez, M.A. Espy, Microtesla MRI with dynamic nuclear polarization, J. Magn. Reson. 207 (2010) 78–88.
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Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Dynamic nuclear polarization in the hyperfine-field-dominant region Seong-Joo Lee ⇑, Jeong Hyun Shim, Kiwoong Kim, Kwon Kyu Yu, Seong-min Hwang Center for Biosignals, Korea Research Institute of Standards and Science (KRISS), 267 Gajeong-ro, Yuseong-gu, Daejeon 305-340, Republic of Korea
a r t i c l e
i n f o
Article history: Received 19 December 2014 Revised 27 March 2015 Available online 24 April 2015 Keywords: Dynamic nuclear polarization Ultra-low field NMR SQUID detection Hyperfine-field-dominant region
a b s t r a c t Dynamic nuclear polarization (DNP) allows measuring enhanced nuclear magnetic resonance (NMR) signals. Though the efficiency of DNP has been known to increase at low fields, the usefulness of DNP has not been throughly investigated yet. Here, using a superconducting quantum interference device-based NMR system, we performed a series of DNP experiments with a nitroxide radical and measured DNP spectra at several magnetic fields down to sub-microtesla. In the DNP spectra, the large overlap of two peaks having opposite signs results in net enhancement factors, which are significantly lower than theoretical expectations [30] and nearly invariant with respect to magnetic fields below the Earth’s field. The numerical analysis based on the radical’s Hamiltonian provides qualitative explanations of such features. The net enhancement factor reached 325 at maximum experimentally, but our analysis reveals that the local enhancement factor at the center of the rf coil is 575, which is unaffected by detection schemes. We conclude that DNP in the hyperfine-field-dominant region yields sufficiently enhanced NMR signals at magnetic fields above 1 lT. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Signal detection in low-field nuclear magnetic resonance (NMR) requires extraordinary sensors, such as electromagnetic resonators [1], superconducting quantum interference devices (SQUIDs) [2], and atomic magnetometers [3]. Among them, SQUIDs exhibit unique performance manifested as low 1=f noise and nearly flat frequency response over tens of kilohertz [4]. Up to date, in most SQUID-based ultra-low field (ULF)-NMR/magnetic resonance imaging (MRI) studies [2,5–25] a so-called prepolarization field (Bp ) has been applied before acquisition. The NMR signal is, thus, proportional to Bp , which is electrically switched on and off. A stronger Bp is, of course, advantageous for obtaining a stronger signal; however, it also creates unwanted side effects, such as the induction of eddy currents on the walls of a magnetically shielded room (MSR) [15–17] and flux trapping in superconducting pickup coils [24]. Both effects generate a temporally varying field, which broadens the NMR linewidth and distorts the MR image. Dynamic nuclear polarization (DNP) based on the Overhauser double-resonance effect [26] has been considered as one of the alternatives to applying Bp , as it utilizes radio frequency (rf) instead of direct current (dc). DNP enhances nuclear spin polarization by polarization transfer from excited unpaired electron
⇑ Corresponding author. E-mail address: [email protected] (S.-J. Lee). http://dx.doi.org/10.1016/j.jmr.2015.04.004 1090-7807/Ó 2015 Elsevier Inc. All rights reserved.
spins via the process of cross-relaxation. Several DNP experiments have been reported [12,13,27–33] using field intensities ranging from the Earth’s field intensity to a few milliteslas. Some studies [28,30,33] reported enhancement factors of few thousands at the Earth’s field, whereas at high fields the enhancement factors were limited to 109 and 165 for 14 N and 15 N, respectively [30]. To the best of our knowledge, no DNP experiments have been performed for field intensities below the Earth’s field intensity. Nitroxide radicals, which are typically exploited in DNP, have hyperfine interactions between unpaired electron and 14 N (or 15 N) nuclear spins. Below the Earth’s magnetic field is, therefore, a so-called hyperfine-field-dominant region. In the present work, we report experimental measurements of DNP spectra in the hyperfine-field-dominant region. Investigations of magnetic field and rf power dependencies show that lowering magnetic fields below the Earth’s field does not make DNP with nitroxide radicals more efficient. In the DNP spectra, the large overlap of two peaks having opposite signs results in net enhancement factors, which are significantly lower than theoretical expectations [30] and nearly invariant with respect to magnetic fields. Our numerical analysis qualitatively agrees with those results. The local enhancement factors at the center of the rf coil was estimated by considering the rf field inhomogeneity and spatial profile of detection sensitivity. The experimental results suggest that 1 lT is an approximate lower limit for reliable single-shot acquisitions via DNP with nitroxide radicals in our SQUID-based system.
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2. Methods We used commercially available 4-Hydroxy-2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPOL; Sigma–aldrich, Cat. No. 176141), containing an unpaired electron. Fig. 1(a) shows the chemical structure of this material. 2 mM of TEMPOL solution were prepared with deionized water. The main interaction in TEMPOL is the interaction between the 14 N (K = 1) and the unpaired electron in the free radical. Therefore, the effective Hamiltonian is:
H ¼ cS B S cK B K þ AS K;
ð1Þ
where A is the hyperfine field constant (47.81 MHz for TEMPOL [27]) and cS and cK are the gyromagnetic ratios of the unpaired electron spin S and the 14 N nuclear spin K, respectively. Eigenvalues of this Hamiltonian are obtained from Breit-Rabi equation [34]. Because the hyperfine field becomes dominant below the Earth’s field intensity, the eigenstates are expressed not using the general jmS ; mK i basis, but using the jF; mF i basis. Here, F is the total spin quantum number (F ¼ S þ K) and mS ; mK , and mF (¼ mS þ mK ) are the magnetic moments of spins S; K, and F, respectively. Considering the 14 N nucleus and the unpaired electron, the energy states are defined as j1i ¼ jF; mF i ¼ j3=2; 3=2i; j2i ’ j3=2; 1=2i; j3i ’ j3=2; 1=2i; j4i ¼ j3=2; 3=2i; j5i ’ j1=2; 1=2i, and j6i ’ j1=2; 1=2i [29,30]. The energy levels for each energy state are plotted as a function of Bp in Fig. 1(b). The theoretical curves in this study were calculated by using the equations described in Ref. [30]. The transition T ij is defined as the difference between i-th and j-th energy levels. Fig. 1(c) shows the experimental setup for the ULF-DNP experiment. To avoid the effects of external magnetic noise, all equipment was mounted inside the MSR which was specially designed for the ULF-NMR/MRI experiments [21,22]; inner-most aluminum shell is partitioned into small panels for reducing the eddy current
115
effect, whereas outer-most aluminum shell forms a closed surface to prevent the SQUID system from the external rf environment. A dc SQUID was used as the NMR signal detector, which was connected with a 2nd order gradiometric pickup coil. The Bp and the measurement field (Bm ) were generated using the Helmholtz coil and the double Helmholtz coil [35], respectively. The Bp strengths were 0.71, 2.04, 4.91, and 47.68 lT, measured using a fluxgate magnetometer. The strength of Bm was about 4.94 lT, corresponding to the proton resonance frequency of 210.5 Hz. The directions of the Bm , the rf field (Brf ), and the Bp were perpendicular to each other. A surface-type coil [36–38] was used for the rf resonance circuit, which is labeled as ‘‘Tank circuit’’ in Fig. 1(c). More detailed description can be found in the section Appendix A. The pulse sequence for the ULF-DNP experiment is displayed in Fig. 1(d). Initially, Brf and Bp are applied simultaneously during the time t rf . After these fields are turned off, Bm is applied, following which the SQUID sensor measures the enhanced free-precession-decay (FPD) signal. If the external Brf is applied to induce the transitions, 1 H nuclear spin repolarization occurs owing to the interaction between the 1 H nuclear spin and spins S, K. Hence, an enhanced FPD signal can be obtained. The data were measured 4 times at each point and the Fourier transforms of all FPD time traces were averaged. The error bars, representing the standard deviation, are plotted along the experimental data. In this study, we define that the experimentally obtained enhancement factor is the DNP enhanced NMR signal (SDNP ) divided by the NMR signal (SNMR ), i.e., enhancement factor = SDNP =SNMR . The SNMR obtained without Brf was only measurable after at least 20 iterations at 47.68 lT. However, because the NMR intensity is proportional to Bp it is possible to predict the intensity of the SNMR below 4.91 lT by using the SNMR obtained at 47.68 lT (20 iterations). Therefore, the enhancement factors for field intensities below 4.91 lT were calculated by this procedure.
Fig. 1. (a) Chemical structure of TEMPOL containing an unpaired electron. (b) Energy levels plotted as a function of Bp for 14 N (K = 1). (c) Schematic diagram of the ULF-DNP experimental setup. The axes of the Bp ; Bm , and Brf coils are perpendicular to each other. The sample, contained in a bottle with 5 cm outer diameter and 7.8 cm length, was placed in the Brf coil of the tank circuit. (d) Illustration of the FPD pulse sequence. The following experimental parameters were used in this study: t rf ¼ 1 s, tmeasurement ¼ 2 s, trepetition ¼ 2 s.
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3. Results and discussion 3.1. Dependence of the DNP-NMR spectrum on Bp Fig. 2 shows the DNP spectra of 2 mM TEMPOL solution at 47.68, 4.91, 2.04, and 0.71 lT, obtained with the rf power of about 23 W. We successfully obtained the DNP spectrum at ultra-low intensity magnetic field (below 1 lT) and measured the two main peaks in all spectra. In addition, the peak in the higher frequency region was inverted in real spectra. In low magnetic field, eight p and two r transitions occur when the direction of Brf is perpendicular and parallel to that of Bp , respectively. However, because Bp and Brf were perpendicular to each other in our experimental setup, only p transitions could be measured. Among the eight p transitions, the related transitions (T 16p ; T 25p ; T 36p , and T 45p ) within the measured frequency range are plotted as a function of Bp in the upper inset in Fig. 3(a). Fig. 3(a) shows the dependence of the enhancement factors eij for individual T ij transitions on Bp , for field
intensities below the Earth’s field intensity. The spectrum is dominated by strong e16 and e45 , whereas e25 and e36 are negligible (lower inset in Fig. 3(a)). However, because the negative enhancement of T 16p and the positive enhancement of T 45p overlap for field intensities below the Earth’s field intensity, as described in Refs. [30,32], the actual spectrum does not exhibit strong enhancement. The net enhancement factor Enet , which is determined by the large overlap of the transitions, can be expressed as [30]:
X Enet Bp ; f rf ¼ eij Bp g Ei Ej f rf ; C ;
ð2Þ
ij
where f rf is the applied rf, g is the shape function, Ei and Ej are the eigenvalues from Eq. (1), and C is the linewidth. (The Planck constant h is assumed to be 1.) We chose a normalized Lorentzian function as the shape function:
ðC=2Þ2 g Ei Ej f rf ; C ¼ 2 Ei Ej f rf þ ðC=2Þ2
ð3Þ
Fig. 2. DNP enhanced NMR signal (SDNP ) plotted vs. the rf for the 2 mM TEMPOL solution at Bp of (a) 47.68, (b) 4.91, (c) 2.04, and (d) 0.71 lT, obtained with rf power of about 23 W. The closed and the open circles represent the magnitude and the real spectra, respectively. The real spectra were obtained with the phase correction.
Fig. 3. The theoretical curves (see Appendix B): (a) The theoretical enhancement factors (eij ) of T 16p ; T 25p ; T 36p , and T 45p plotted as a function of Bp . The lower inset shows the magnified view of the graph for the T 25p and T 36p transitions. The upper inset shows the transition frequency vs. Bp . (b) The theoretical enhancement factors (Enet ) determined by the overlap of the transitions plotted as a function of rf. The curves were obtained with the values shown in (a) and Eq. (3) (see Eq. (2)). For a convenient comparison with other figures, the theoretical spectra with absolute magnitude were plotted.
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Fig. 3(b) shows the theoretical Enet at various Bp . Since the derived equations and the shape function in Ref. [30] describe the individual transition, one can plot the theoretical Enet as shown in Fig. 3(b) though the strong overlap exists at magnetic fields below the Earth’s magnetic field. These curves were obtained by the averaged linewidth (C) of 7.45 MHz (see Appendix B.2). Although eij increased exponentially with decreasing Bp ; Enet was almost unchanged. This result will be compared with the experimental results in the following paragraph. The variation of Enet is plotted as a function of Bp in Fig. 4. The theoretical curves, solid line for the higher frequency-side peak (HP) and dashed line for the lower frequency-side peak (LP), are nearly constant, varying less than 5%. Although the experimental result seems not to follow the theoretical curves, the deviation is not significant, compared with the variation of Bp . The enhancement varied only 45% while Bp decreased approximately 70 times. Therefore, it is a plausible conclusion that, with respect to Bp ; Enet is nearly constant for the magnetic fields below the Earth’s field intensity. This explains why the NMR intensity in Fig. 2 decreased quite linearly with decreasing Bp ; when Enet is a constant, the NMR signal intensity is simply proportional to the initial thermal polarization. As explained above, the large overlap of the two transitions (T 16p and T 45p ) made Enet uncorrelated to Bp . 3.2. Dependence of the DNP-NMR spectrum on rf power The ratios of LP and HP for all spectra in Fig. 2 are different from those reported in theoretical curves in Fig. 3(b). This could be explained by the rf power dependence of LP and HP. In Fig. 5(a), the normalized magnitude spectra at 47.68 lT are plotted for different values of rf power. Each spectrum was normalized by its corresponding SDNP measured at 74.4 MHz. As the rf power increased, the relative intensity of LP decreased. This result suggests that the LP to HP ratio becomes similar to the theoretical one (Fig. 3) when the full saturation is created by large Brf . Fig. 5(b) shows Enet ðHPÞ and Enet ðLPÞ as a function of the rf power. Below 15.6 W, Enet ðLPÞ was larger, but, above 15.6 W, became smaller than Enet ðHPÞ. The transition probability of T 45p is lower than that of T 16p [30]. In addition, the difference between the energy levels of the T 16p transition is larger than those of the T 45p transition, which affects eij . Therefore, e45 is smaller than e16 at the same rf power. This mechanism can explain why Enet ðHPÞ is larger than Enet ðLPÞ above 15.6 W. However, the inverse behavior that Enet ðLPÞ is higher than Enet ðHPÞ below 15.6 W is unclear. We believe the properties, such as T1 and T2 of individual transitions should be separately
Fig. 5. (a) Normalized spectra vs. the rf power, for field intensity of 47.68 lT. Each spectrum was normalized by the NMR intensities measured at the rf of 74.4 MHz (marked by the arrow). (b) Enet vs. the rf power. Open squares and circles represent, respectively, the Enet values of HP and LP in (a). The inset shows the magnified view of the graph for the Enet values of the HP and LP below 1 W. The maximal NMR intensities at each peak were used for calculations. The solid lines represent the exponential fitting curves.
investigated for a better explanation of such power dependencies. However, HP and LP are produced as a result of large overlap of two transition peaks, of which widths are 7.45 MHz. A clear discrimination them, thus, may not be easily accessible and thereby will be a challenging work in future. 3.3. Net enhancement factor Enet with high rf power In order to observe the saturation of Enet , a higher rf power is required to be applied. Fig. 6 shows Enet at higher values of rf power. In our experiments, Enet started to converge toward the maximal value for power above 120 W, and increased up to about 325 for rf power of 164 W at 4.91 lT. 3.4. Local enhancement factor Eloc ð~ r; PÞ The local enhancement factor (Eloc ) at the center of the rf coil was larger than all values shown in this study. Two points must be considered for the calculation of Eloc . First, Brf is not sufficient to saturate the entire sample with approximate dimensions of
p 252 78 mm3. Although surface-type resonance circuits
Fig. 4. The net enhancement factors (Enet ) vs. Bp , for higher and lower frequencyside peaks, denoted as HP and LP (open squares and circles). The maximal SDNP values of HP and LP in Fig. 2 were used for calculations of Enet obtained by the experiments. Solid and dashed lines represent the theoretical curves, which are rescaled by matching the values at 47.68 lT to the experimental values. The theoretical curves show that the net enhancement factor is nearly independent of magnetic field lower than the Earth’s field.
require less power to saturate the electron spin resonance (ESR) transition than volume resonance circuits [38], the former have only a single-turn rf coil, which cannot cover our sample. Therefore, the NMR signal obtained without Brf was generated by a larger volume than that obtained with Brf . Second, in NMR signal detection, the sensitivity profile of the pickup coil exhibits hemisphere-like shapes [22]. We numerically calculated the two factors to obtain Eloc at the center of the single-turn rf coil. Fig. 7 shows the configuration of the sample, the pickup coil, and the rf coil. Over the volume of the sample, the sensitivity of
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where qð~ r Þ represents the density of the sample, and gð~ r Þ is the sensitivity profile for the pickup coil used in the experiment. The enhancement factor Eloc ð~ r; PÞ also exhibits spatial variation, which is associated with the rf field inhomogeneity generated by the rf coil. P is the amount of rf power injected into the tank circuit. According to Eqs. (4) and (5), the net enhancement factor Enet r Þ will should be affected by detection schemes; a variation in gð~ lead to a different Enet value. Instead, the local enhancement factor Eloc at the center (~ r c ) of the rf coil is expected to be independent of detection methods, presenting the efficiency of DNP neutrally, and can be used as a reference, validating the efficiency of DNP in the hyperfine-field-dominant region. The numerical procedure to estimate Eloc ð~ r c ; P Þ is described below.
Fig. 6. Enet vs. the rf power. The data were obtained with the 74.4 MHz Brf corresponding to the maximum peak in HP at 4.91 lT. In most cases, the error bars, denoting the standard deviation, are smaller than the circles size.
Fig. 7. Illustration of the experimental configuration including the pickup coil, the sample, and the rf coil. The pickup coil is a 2nd order gradiometer. The rf coil is a single-turn circle, and the sample is in the cylindrical bottle. All dimensions, such as the diameters of coils, the gap between the bottom of the pickup coil and the sample, are uniformly scaled down with respect to the experimental values. The sensitivity profile is shown by the surface plot with the color scale on the left. The intensity of Brf generated by the rf coil with 1 A current is displayed for the plane through the coil with the color scale on the right. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
the NMR signal, given by the geometry of the pickup coil, cannot be uniform, as illustrated by the surface plots through the sample volume in Fig. 7. In addition, the Brf generated by the rf coil is not uniform either. The variation of the Brf intensity in the plane through the rf coil is shown in the density plot of Fig. 7. Therefore, the NMR and DNP enhanced NMR signals can be expressed as:
SNMR ¼ SDNP ¼
Z Z
qð~rÞgð~rÞd~r
ð4Þ
qð~rÞgð~rÞEloc ð~r; PÞd~r
ð5Þ
3.4.1. Pickup coil sensitivity profile gð~ rÞ During free precession, nuclear spins generate variable magnetic fields, i.e., magnetic flux, in the pickup areas. By generating the screening currents due to the variation of magnetic flux, the pickup coil transfers the magnetic flux to the SQUID sensor. As shown in Fig. 7, the sample approaches the pickup coil’s bottom, and the sample dimensions are comparable to the size of the pickup coil. In this case, local NMR signals cannot be detected with an equal efficiency. Thus, the sensitivity profile gð~ r Þ, which is illustrated by the contour plot in Fig. 7, should be calculated in order to determine Eloc ð~ r; P Þ in Eq. (5). The principle of reciprocity makes the calculation simple. One needs to know the receiving field at ~ r, which the pickup coil would produce when the same current corresponding to the magnetic fluxes flows through the pickup coil. The actual dimensions were used in the calculation. The sensitivity for the top area closest to the pickup coil is nearly fivefold higher than that for the bottom. The numerically obtained gð~ r Þ will be used below to extract r; P Þ. Eloc ð~
3.4.2. Estimation of Eloc at the center The spatial dependence of the DNP enhancement factor originates from the spatially varying Brf intensity Brf ð~ r; PÞ produced by r; P Þ can be easily calculated with the single-turn rf coil. The Brf ð~ analytic solutions [39]. The relation between the local Brf ð~ r; PÞ and the Eð~ r; PÞ can be deduced from the solution of Bloch equation [40,41]. The enhacemenet factor should be proportional to the degree of saturation, which can be expressed as 1 1=ð1 þ aPÞ. We introduce a parameter cð~ rÞ that links the applied current I r; PÞ ¼ cð~ rÞI. The final expression and the generated rf field as Brf ð~ for the enhancement factor is, thus, given by:
" Eloc ð~ r; PÞ ¼ E0 1
#
1 2 2
1 þ cð~ rÞ k P
;
ð6Þ
pffiffiffi in which k is defined as cT 1 = Z , and Z represents the impedance of the rf circuit. In order to estimate the unknown parameters, E0 and k in Eq. (6), we fitted the measured SDNP =SNMR data (HP in Fig. 5(b)) with Eqs. (4)–(6). The comparison is shown in Fig. 8. With the estimated E0 and k for HP, the enhancement at the center of r c ; P Þ, was obtained. Conclusively, we can notice that, the rf coil, Eloc ð~ at the center, Eloc is 1.77 times larger than the measured value. The histogram of the Brf intensities over the volume of the sample is shown in Fig. 9. The value at the center of the coil is indicated by an arrow. Notably, most areas are weakly excited compared with the center; however, those areas have weak sensitivities. This can explain why the estimated enhancement at the center is only 1.77 times larger than the measured values.
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Acknowledgment This work was supported by a World Class Laboratory (WCL) Grant from the Korea Research Institute of Standards and Science.
Appendix A. Experimental setup
Fig. 8. The power dependence of Enet (blue circle) is fitted (orange line) with Eqs. (4)–(6). With the estimated parameters from the fitting, Eloc at the center of the rf coil is plotted (green line with square), in order to obtain the actual values at the center. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
A dc SQUID (CE2Blue; Supracon AG, Germany) was used as the NMR signal detector, which was shielded by two different cans. The inner shield and the outer shield were a superconducting Nb cast can (99.9%) and a superconducting lead can (99.5%), respectively. A 2nd order gradiometric pickup coil connected with the dc SQUID was made of 125 lm Nb wire with 50 mm baseline and 65 mm diameter. The dc SQUID sensor and the pickup coil were additionally wrapped in a crumpled Mylar film to protect the SQUID system from the influence of the external radiofrequency (rf) field. The environmental noise in our system was pffiffiffiffiffiffi about 2.3 fT= Hz. Fig. A.1 shows a schematic diagram of the surface-type resonance circuit. The lparallel length is given by [37]:
lparallel ¼
0:663c 276 log ðb=aÞ pffiffiffi ; tan1 2pf rf 2pf rf L
ðA:1Þ
where c is the speed of light, f rf is the target rf, L is the inductance of the single-turn rf coil, is the dielectric constant of the insulator, and a and b are the inner and outer radii of the coaxial line, respectively. More detailed description of parameters of the surface-type resonance circuit is given in Table A.1. The tuning range of the rf resonance circuit was from 63 MHz to 84 MHz. Low power (ZHL-1A; Minicircuits, USA) and high power (1000A225; rf/microwave instrumentation, USA) amplifiers were used in the experiments with rf power below 1 W and above 9 W, respectively.
Fig. 9. The histogram of Brf intensities over the volume of the sample. The value at the center of the coil is indicated by the arrow.
4. Conclusion Fig. A.1. Schematic diagram of the surface-type resonance circuit.
In summary, we measured the enhanced 1 H NMR signals of water in the 2 mM TEMPOL solution in the hyperfine-field-dominant region. The obtained DNP spectra had two peaks corresponding to the overlap of two main transitions, T 16p and T 45p . The dependence of the DNP spectrum on Bp showed that the net enhancement factor Enet was nearly constant for field intensities below the Earth’s field intensity owing to the overlap of the two transitions. The dependence of the DNP spectrum on rf power showed that below 15.6 W Enet of LP was larger than that of the HP. On the other hand, for power above 15.6 W Enet of HP became larger than that of LP. Considering the numerical calculation of the local effect of our system, we obtained the local enhancement factor of HP exceeding 575 at 4.91 lT, which exhibits the DNP efficiency unaffected by detection methods. This value is, however, still lower than the theoretical prediction, that is nearly 1000. 1 lT is an approximate lower limit for reliable single-shot acquisitions via DNP with nitroxide radicals in our SQUID-based system.
Table A.1 Parameters of the surface-type resonance circuit. Target rf f rf
74.4 MHz
Single-turn rf coil (copper wire) Thickness Diameter Inductance L
1.3 mm 55 mm 404 nH
Coaxial line (RG58) Inner radius a Outer radius b Dielectric constant of the insulator
0.4675 mm 1.8 mm 2.27496
Parallel line lparallel (RG58 coaxial line) k=2 balun (RG58 coaxial line) Tuning capacitor (Voltronics Corp.; NMNT70–6) Matching capacitor (Voltronics Corp.; NMNT23–12ENL)
218.4 mm 1337 mm 2.5–70 pF 3–23 pF
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Appendix B. Calculation of theoretical enhancement factor (Fig. 3) B.1. Assumptions for calculation The enhancement factor is expressed as qfsðhSz i S0 Þ=I0 [30], where q is the coupling factor describing the nucleus-electron interaction, f is the leakage factor, s is the saturation factor,
Sz is the spin polarization after the electron spin resonance saturation, and S0 and I0 are the spin polarizations of the electron and the nucleus at the thermal equilibrium, respectively. During the enhancement factor calculations, we assumed that q; f ; s, and I0 are 0.5 (pure dipolar coupling), 1, 1 (complete saturation), and cN Bp =4kB T [40], where cN is the gyromagnetic ratio of 1 H, kB is the Boltzmann’s constant, and T is the temperature, respectively.
Fig. B.1. Lorentzian-fitted curves plotted vs. the rf frequency at (a) 47.68, (b) 4.91, (c) 2.04, and (d) 0.71 lT. Solid lines represent the fitting curves with Lorentzian broadening function. The open circles represent the real spectra obtained by the experiments, the same spectra shown in Fig. 2.
Fig. B.2. Theoretically obtained individual enhancement factor eij plotted vs. the rf at Bp of (a) 47.68, (b) 4.91, (c) 2.04, and (d) 0.71 lT. The (red) solid and the (blue) dotted lines represent e16 and e45 , respectively. e25 and e36 were plotted with the (black) solid lines. The sum of the eij of four transitions, the Enet , were plotted with dashed line. The spectra of jEnet j were the same spectra shown in Fig. 3(b). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
S.-J. Lee et al. / Journal of Magnetic Resonance 255 (2015) 114–121
B.2. Estimation of the linewidth C of the shape function In order to determine the linewidth (C) of the shape function described in Eq. (3), we fitted the experimental data with normalized Lorentzian broadening function (Fig. B.1). We chose the fitting function F fit as:
X F fit Bp ; f rf ; C ¼ eij Bp g Ei Ej f rf ; C sij ;
ðB:1Þ
ij
where sij is saturation ratios for individual T ij transitions. The curves were obtained with the assumptions of s25 ¼ s36 ¼ s45 . Since the rf power used in the experiments was not enough to create full saturation, it was reasonable assumptions that the sij for individual T ij were different to each other. In addition, we assumed the s25 and s36 as above, because the contributions of T 25p and T 36p to Enet were very weak. The linewidths obtained by the fit at Bp of 47.68, 4.91, 2.04, and 0.71 lT were 7.37, 7.46, 7.51, and 7.45 MHz, respectively. B.3. The theoretically obtained individual enhancement factor eij at various Bp To aid the understanding of the theoretical curve shown in Fig. 3(b), we plotted the theoretically obtained individual enhancement factor eij as a function of the rf at various Bp (Fig. B.2). Here, eij is defined as:
eij Bp ; f rf ¼ eij Bp g Ei Ej f rf ; C ;
ðB:2Þ
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