Solid State Nuclear Magnetic Resonance 59-60 (2014) 45–47
Contents lists available at ScienceDirect
Solid State Nuclear Magnetic Resonance journal homepage: www.elsevier.com/locate/ssnmr
Simultaneous measurement of very small magnetic fields and spin-lattice relaxation B. Kresse a,n, A.F. Privalov a, A. Herrmann b, M. Hofmann b, E.A. Rössler b, F. Fujara a a b
Institut für Festkörperphysik, TU Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany Experimentalphysik II, Universität Bayreuth, 95440 Bayreuth, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 6 September 2013 Received in revised form 7 March 2014 Available online 17 March 2014
A field cycling (FC) NMR experiment is presented which allows for the simultaneous determination of very small magnetic fields down to about 3 μT and the concomitant measurement of nuclear spin-lattice relaxation times in these fields. The technique will enable broadband spin-lattice relaxation dispersion experiments down to about 100 Hz 1H Larmor frequency. Limitations of its applicability are discussed. & 2014 Elsevier Inc. All rights reserved.
Keywords: FC NMR Small Larmor frequencies Relaxometry
1. Introduction In a recent paper [1] we described technical improvements of our home built electronical FC relaxometer [2]. With the help of a specially designed 3-dimensional resistive coil system including three coils for producing a magnetic field in the longitudinal and two coils for that in the transverse directions we attained magnetic fields of arbitrary size ðμT…TÞ corresponding to 1H NMR-frequencies from some 10 Hz to about 40 MHz. The lowest magnetic fields reached by this setup are controlled and stabilized using an active field compensation scheme making use of a fluxgate magnetic field sensor. Test experiments have shown that arbitrarily oriented magnetic fields down to less than 1 μT can be established, kept stable and measured by observing the 1H NMR Larmor frequency in these fields. Thereby, the smallest 1H Larmor frequency measured and reported in [1] was about 12 Hz – corresponding to a magnetic field strength of about 0:3 μT. The essential idea of that experiment consists in a field cycle such that after having polarized the sample in a high magnetic field Bpol (typically of the order of 1 T) the field is switched to the low desired magnetic field, denoted as evolution field Bev. If the field switch is performed quickly enough and Bev is chosen not parallel to Bpol then the nuclear spin magnetization M may not point into the Bev-direction. As a consequence, the nuclear spins will start to precess about Bev (Fig. 1). After a certain evolution time τ, the field is switched up adiabatically to some detection field, pointing into the z-direction. The amplitude S of the FID signal, recorded after a
n
Corresponding author. Fax: þ 49 6151 16 2833. E-mail address: [email protected] (B. Kresse).
http://dx.doi.org/10.1016/j.ssnmr.2014.03.001 0926-2040/& 2014 Elsevier Inc. All rights reserved.
901 pulse, is proportional to the longitudinal component of the magnetization at the end of the evolution time. If the components Bev, ? and Bev,z are stable enough, the precession of M can be traced by repeating the field cycle for different τ. The oscillation frequency of S denotes the Larmor frequency and the oscillation amplitude yields the direction of Bev relative to Bpol. In one of those experiments (Fig. 6 of [1]) the misalignment of Bev with respect to Bpol was small enough such that the damped FID decayed toward a high plateau value. In the “conclusions and outlook” of [1] it was already anticipated that this plateau value will decay with the spin-lattice relaxation time (in this small field Bev). Thereby, the experiments might contain the potential of providing information about the absolute field value and the spinlattice relaxation time in this field, measured in one and the same experiment. It is the purpose of the present paper to substantiate this hypothesis [3].
2. Experiments For our experiments we used a liquid toluene-h8 sample, which has a melting point of T m ¼ 178 K. This choice was motivated since toluene is a van der Waals-liquid, from which we expect dispersionless spin-lattice relaxation. And we performed all experiments at 183 K, only slightly above Tm, in order to reduce T1. (At room temperature T1 is an order of magnitude larger). The sample was polarized at a field of 0:7 T ð c ¼ 30 MHz 1H Larmor frequency) and the detection field was 1 T ð c ¼ 42 MHz). After a π =2pulse of 1 μs length the amplitude of a FID is recorded. The signal S, which is the initial amplitude of the sum of 4 accumulations, has
46
B. Kresse et al. / Solid State Nuclear Magnetic Resonance 59-60 (2014) 45–47
Fig. 1. Adapted from [1]. Left: Behavior of the magnetization M in a small evolution field Bev which is tilted with respect to the polarization field Bpol ¼ Bz. Right: Measured Larmor precession of the longitudinal component of M as probed by the FID amplitude. The shown example displays a Larmor period of 1.7 ms which corresponds to Bev 14 μT. From the ratio of the amplitude and the baseline level of S the components of Bev can be estimated as Bev; ? 10:5 μT and Bev;z 9:5 μT. Fig. 3. 1H magnetization decay curve of a toluene sample at 183 K in a 291 tilted Bev.
Fig. 2. 1H magnetization decay curve of a toluene sample at 183 K in a 231 tilted Bev.
been measured as the absolute value of the complex FID in order to reduce noise due to slight instabilities of the detection field. This method is applicable as long as we do not expect negative signals. The baseline was corrected by subtraction of the noise level. Switching down from the polarization field to the evolution field takes 2.7 ms, afterwards the fluxgate sensor takes another 2 ms to recover. The field is then stabilized within 0.5 ms. Switching up and stabilizing the detection field takes 10 ms. Figs. 2–5 show decay curves of the longitudinal magnetization in very low evolution fields which are tilted away from the polarization field direction. For short evolution times, the curves show a damped oscillation due to the Larmor precession about the tilted Bev yielding directly the field Bev. As already mentioned in [1], inhomogeneities of Bev lead to a decay of the oscillation amplitude to a plateau with the time constant T n2ev . This plateau then decays further monoexponentially with T1 at longer evolution times. Note that due to the given field inhomogeneity the oscillation damping and the relaxation are well separated, T 1 ⪢T n2ev . During the oscillation the values of the longitudinal magnetization have been recorded on a linear evolution time scale, afterwards the recording was switched to a logarithmic time scale. The following function was used for fitting the data: SðτÞ ¼ ½Sn0 cos ðωev τ þ φÞ expð τ=T n2ev Þ þS00 expð τ=T 1 Þ þc
ð1Þ
Sn0 is the amplitude of the z-component of the Larmor precession and S00 is the value of the plateau of the oscillation decay. The oscillation frequency ωev ¼ γ Bev yields directly the field strength.
Fig. 4. 1H magnetization decay curve of a toluene sample at 183 K in a 391 tilted Bev.
Fig. 5. 1H magnetization decay curve of a toluene sample at 183 K in a 441 tilted Bev.
The fit parameter c is introduced to account for the possibility that due to a magnetization build up during switching up to the detection field S(τ) does not necessarily decay toward a plateau value which corresponds to the population equilibrium at the very low evolution fields.
B. Kresse et al. / Solid State Nuclear Magnetic Resonance 59-60 (2014) 45–47
47
For samples with a short T1 at low fields/frequencies, like polymers, this method is limited to about those Larmor frequencies for which one period is shorter than T1. To exemplify this, Fig. 6 shows a magnetization decay curve for a polybutadiene sample with a molecular weight of 441,000 g/mol at 393 K. In this case T1 ¼(5.0 70.2) ms at an evolution field of νev ¼(33479) Hz which has a period of 3 ms. The fit still describes the data reasonably well, but this is about the limit. It is not possible to go to significantly lower evolution fields, where the Larmor period increases while T1 decreases and extract T1 and νev from the magnetization curve. The whole relaxation dispersion of this polybutadiene sample is published in [4] together with other measurements of different molecular masses and temperatures.
3. Conclusions Fig. 6. 1H magnetization decay curve of a polybutadiene sample with a molecular mass of 441,000 g/mol at 393 K in a 231 tilted Bev.
The curves in Figs. 2–4 were recorded with one and the same amplitude of the transversal component of the evolution field Bev; ? ¼ ð3:7 7 0:3Þ μT, but with different longitudinal components Bev,z. The components Bev,? , Bev,z and thereby the total evolution field strengths Bev ¼ ðB2ev; ? þ B2ev;z Þ0:5 ¼ 1γ ωev ¼ 2p γ νev and the spin-lattice relaxation times T1 were obtained by fitting the experimental data with Eq. (1). The fit results, represented by the solid curves in the figures, are given in the insets. The fitted T1 values at these low fields do well coincide with T1-times at higher frequencies (extreme motional narrowing). Bev; ? ¼ Bev ðS00 =Sn0 þ 1Þ 0:5 is also evaluated from the fit. The relative mean error of Bev,? is much higher than the one of 0 Bev,z because it also depends on the ratio of the plateau value S0 and n the oscillation amplitude S0. One notes from the figures that at a given Bev, ? the oscillation amplitude of the NMR signal increases as Bev,z decreases. Therefore, for avoiding negative signals for even smaller Bev,z we should also reduce Bev, ? . An example is given in Fig. 5 with Bev; ? ¼ ð2:4 7 0:8Þ μT and other fit results again given in the inset. The experiments show the asset of this method. The evolution field strength can be determined with high accuracy and a mean error of only about a few Hz. Also the uncertainty of the spinlattice relaxation time is only a few percent if Bev is not tilted more than about 301, see Figs. 2 and 3.
Measuring the spin-lattice relaxation times at very low magnetic fields by FC-NMR requires a careful calibration of the evolution field. Our FC relaxometer not only provides the possibility to compensate for the earth field and other unwanted field components but also to switch down to an evolution field arbitrarily tilted with respect to the polarization field. Using this feature we can measure T1 and the Bev with high accuracy in one and the same experiment by recording the Larmor precession and the spin-lattice relaxation decay in an evolution field tilted with respect to the polarization field direction.
Acknowledgments We thank the DFG for financial support (Grant no. FU 308/14). References [1] B. Kresse, A. Privalov, F. Fujara, NMR field-cycling at ultralow magnetic fields, Solid State Nucl. Magn. Reson. 40 (2011) 134–137. [2] O. Lips, A. Privalov, S. Dvinskikh, F. Fujara, Magnet design with high B0 homogeneity for fast-field-cycling NMR applications, J. Magn. Reson. 149 (2001) 22–28. [3] European Patent: Publication Number EP 2 577 341 A1. [4] A. Herrmann, B. Kresse, J. Gmeiner, A.F. Privalov, D. Kruk, F. Fujara, E.A. Rössler, Protracted crossover to reptation dynamics: a field cycling 1H NMR study including extremely low frequencies, Macromolecules 45 (2012) 1408–1416.
Contents lists available at ScienceDirect
Solid State Nuclear Magnetic Resonance journal homepage: www.elsevier.com/locate/ssnmr
Simultaneous measurement of very small magnetic fields and spin-lattice relaxation B. Kresse a,n, A.F. Privalov a, A. Herrmann b, M. Hofmann b, E.A. Rössler b, F. Fujara a a b
Institut für Festkörperphysik, TU Darmstadt, Hochschulstr. 6, 64289 Darmstadt, Germany Experimentalphysik II, Universität Bayreuth, 95440 Bayreuth, Germany
art ic l e i nf o
a b s t r a c t
Article history: Received 6 September 2013 Received in revised form 7 March 2014 Available online 17 March 2014
A field cycling (FC) NMR experiment is presented which allows for the simultaneous determination of very small magnetic fields down to about 3 μT and the concomitant measurement of nuclear spin-lattice relaxation times in these fields. The technique will enable broadband spin-lattice relaxation dispersion experiments down to about 100 Hz 1H Larmor frequency. Limitations of its applicability are discussed. & 2014 Elsevier Inc. All rights reserved.
Keywords: FC NMR Small Larmor frequencies Relaxometry
1. Introduction In a recent paper [1] we described technical improvements of our home built electronical FC relaxometer [2]. With the help of a specially designed 3-dimensional resistive coil system including three coils for producing a magnetic field in the longitudinal and two coils for that in the transverse directions we attained magnetic fields of arbitrary size ðμT…TÞ corresponding to 1H NMR-frequencies from some 10 Hz to about 40 MHz. The lowest magnetic fields reached by this setup are controlled and stabilized using an active field compensation scheme making use of a fluxgate magnetic field sensor. Test experiments have shown that arbitrarily oriented magnetic fields down to less than 1 μT can be established, kept stable and measured by observing the 1H NMR Larmor frequency in these fields. Thereby, the smallest 1H Larmor frequency measured and reported in [1] was about 12 Hz – corresponding to a magnetic field strength of about 0:3 μT. The essential idea of that experiment consists in a field cycle such that after having polarized the sample in a high magnetic field Bpol (typically of the order of 1 T) the field is switched to the low desired magnetic field, denoted as evolution field Bev. If the field switch is performed quickly enough and Bev is chosen not parallel to Bpol then the nuclear spin magnetization M may not point into the Bev-direction. As a consequence, the nuclear spins will start to precess about Bev (Fig. 1). After a certain evolution time τ, the field is switched up adiabatically to some detection field, pointing into the z-direction. The amplitude S of the FID signal, recorded after a
n
Corresponding author. Fax: þ 49 6151 16 2833. E-mail address: [email protected] (B. Kresse).
http://dx.doi.org/10.1016/j.ssnmr.2014.03.001 0926-2040/& 2014 Elsevier Inc. All rights reserved.
901 pulse, is proportional to the longitudinal component of the magnetization at the end of the evolution time. If the components Bev, ? and Bev,z are stable enough, the precession of M can be traced by repeating the field cycle for different τ. The oscillation frequency of S denotes the Larmor frequency and the oscillation amplitude yields the direction of Bev relative to Bpol. In one of those experiments (Fig. 6 of [1]) the misalignment of Bev with respect to Bpol was small enough such that the damped FID decayed toward a high plateau value. In the “conclusions and outlook” of [1] it was already anticipated that this plateau value will decay with the spin-lattice relaxation time (in this small field Bev). Thereby, the experiments might contain the potential of providing information about the absolute field value and the spinlattice relaxation time in this field, measured in one and the same experiment. It is the purpose of the present paper to substantiate this hypothesis [3].
2. Experiments For our experiments we used a liquid toluene-h8 sample, which has a melting point of T m ¼ 178 K. This choice was motivated since toluene is a van der Waals-liquid, from which we expect dispersionless spin-lattice relaxation. And we performed all experiments at 183 K, only slightly above Tm, in order to reduce T1. (At room temperature T1 is an order of magnitude larger). The sample was polarized at a field of 0:7 T ð c ¼ 30 MHz 1H Larmor frequency) and the detection field was 1 T ð c ¼ 42 MHz). After a π =2pulse of 1 μs length the amplitude of a FID is recorded. The signal S, which is the initial amplitude of the sum of 4 accumulations, has
46
B. Kresse et al. / Solid State Nuclear Magnetic Resonance 59-60 (2014) 45–47
Fig. 1. Adapted from [1]. Left: Behavior of the magnetization M in a small evolution field Bev which is tilted with respect to the polarization field Bpol ¼ Bz. Right: Measured Larmor precession of the longitudinal component of M as probed by the FID amplitude. The shown example displays a Larmor period of 1.7 ms which corresponds to Bev 14 μT. From the ratio of the amplitude and the baseline level of S the components of Bev can be estimated as Bev; ? 10:5 μT and Bev;z 9:5 μT. Fig. 3. 1H magnetization decay curve of a toluene sample at 183 K in a 291 tilted Bev.
Fig. 2. 1H magnetization decay curve of a toluene sample at 183 K in a 231 tilted Bev.
been measured as the absolute value of the complex FID in order to reduce noise due to slight instabilities of the detection field. This method is applicable as long as we do not expect negative signals. The baseline was corrected by subtraction of the noise level. Switching down from the polarization field to the evolution field takes 2.7 ms, afterwards the fluxgate sensor takes another 2 ms to recover. The field is then stabilized within 0.5 ms. Switching up and stabilizing the detection field takes 10 ms. Figs. 2–5 show decay curves of the longitudinal magnetization in very low evolution fields which are tilted away from the polarization field direction. For short evolution times, the curves show a damped oscillation due to the Larmor precession about the tilted Bev yielding directly the field Bev. As already mentioned in [1], inhomogeneities of Bev lead to a decay of the oscillation amplitude to a plateau with the time constant T n2ev . This plateau then decays further monoexponentially with T1 at longer evolution times. Note that due to the given field inhomogeneity the oscillation damping and the relaxation are well separated, T 1 ⪢T n2ev . During the oscillation the values of the longitudinal magnetization have been recorded on a linear evolution time scale, afterwards the recording was switched to a logarithmic time scale. The following function was used for fitting the data: SðτÞ ¼ ½Sn0 cos ðωev τ þ φÞ expð τ=T n2ev Þ þS00 expð τ=T 1 Þ þc
ð1Þ
Sn0 is the amplitude of the z-component of the Larmor precession and S00 is the value of the plateau of the oscillation decay. The oscillation frequency ωev ¼ γ Bev yields directly the field strength.
Fig. 4. 1H magnetization decay curve of a toluene sample at 183 K in a 391 tilted Bev.
Fig. 5. 1H magnetization decay curve of a toluene sample at 183 K in a 441 tilted Bev.
The fit parameter c is introduced to account for the possibility that due to a magnetization build up during switching up to the detection field S(τ) does not necessarily decay toward a plateau value which corresponds to the population equilibrium at the very low evolution fields.
B. Kresse et al. / Solid State Nuclear Magnetic Resonance 59-60 (2014) 45–47
47
For samples with a short T1 at low fields/frequencies, like polymers, this method is limited to about those Larmor frequencies for which one period is shorter than T1. To exemplify this, Fig. 6 shows a magnetization decay curve for a polybutadiene sample with a molecular weight of 441,000 g/mol at 393 K. In this case T1 ¼(5.0 70.2) ms at an evolution field of νev ¼(33479) Hz which has a period of 3 ms. The fit still describes the data reasonably well, but this is about the limit. It is not possible to go to significantly lower evolution fields, where the Larmor period increases while T1 decreases and extract T1 and νev from the magnetization curve. The whole relaxation dispersion of this polybutadiene sample is published in [4] together with other measurements of different molecular masses and temperatures.
3. Conclusions Fig. 6. 1H magnetization decay curve of a polybutadiene sample with a molecular mass of 441,000 g/mol at 393 K in a 231 tilted Bev.
The curves in Figs. 2–4 were recorded with one and the same amplitude of the transversal component of the evolution field Bev; ? ¼ ð3:7 7 0:3Þ μT, but with different longitudinal components Bev,z. The components Bev,? , Bev,z and thereby the total evolution field strengths Bev ¼ ðB2ev; ? þ B2ev;z Þ0:5 ¼ 1γ ωev ¼ 2p γ νev and the spin-lattice relaxation times T1 were obtained by fitting the experimental data with Eq. (1). The fit results, represented by the solid curves in the figures, are given in the insets. The fitted T1 values at these low fields do well coincide with T1-times at higher frequencies (extreme motional narrowing). Bev; ? ¼ Bev ðS00 =Sn0 þ 1Þ 0:5 is also evaluated from the fit. The relative mean error of Bev,? is much higher than the one of 0 Bev,z because it also depends on the ratio of the plateau value S0 and n the oscillation amplitude S0. One notes from the figures that at a given Bev, ? the oscillation amplitude of the NMR signal increases as Bev,z decreases. Therefore, for avoiding negative signals for even smaller Bev,z we should also reduce Bev, ? . An example is given in Fig. 5 with Bev; ? ¼ ð2:4 7 0:8Þ μT and other fit results again given in the inset. The experiments show the asset of this method. The evolution field strength can be determined with high accuracy and a mean error of only about a few Hz. Also the uncertainty of the spinlattice relaxation time is only a few percent if Bev is not tilted more than about 301, see Figs. 2 and 3.
Measuring the spin-lattice relaxation times at very low magnetic fields by FC-NMR requires a careful calibration of the evolution field. Our FC relaxometer not only provides the possibility to compensate for the earth field and other unwanted field components but also to switch down to an evolution field arbitrarily tilted with respect to the polarization field. Using this feature we can measure T1 and the Bev with high accuracy in one and the same experiment by recording the Larmor precession and the spin-lattice relaxation decay in an evolution field tilted with respect to the polarization field direction.
Acknowledgments We thank the DFG for financial support (Grant no. FU 308/14). References [1] B. Kresse, A. Privalov, F. Fujara, NMR field-cycling at ultralow magnetic fields, Solid State Nucl. Magn. Reson. 40 (2011) 134–137. [2] O. Lips, A. Privalov, S. Dvinskikh, F. Fujara, Magnet design with high B0 homogeneity for fast-field-cycling NMR applications, J. Magn. Reson. 149 (2001) 22–28. [3] European Patent: Publication Number EP 2 577 341 A1. [4] A. Herrmann, B. Kresse, J. Gmeiner, A.F. Privalov, D. Kruk, F. Fujara, E.A. Rössler, Protracted crossover to reptation dynamics: a field cycling 1H NMR study including extremely low frequencies, Macromolecules 45 (2012) 1408–1416.
