Solid state Field-Cycling NMR relaxometry: Instrumental improvements and new applications

Highlights

• Recent developments of Field-Cycling NMR instrumentation.

• Relaxation theory including residual dipolar and quadrupolar interactions.

• Experimental cases: ionic motions, polymer dynamics, structure determination, etc.

Abstract

The paper reviews recent progress in field cycling (FC) NMR instrumentation and its application to solid state physics. Special emphasis is put on our own work during the last 15 years on instrumentation, theory and applications. As far as instrumentation is concerned we report on our development of two types of electronical FC relaxometers, a mechanical FC relaxometer and a combination of FC and one-dimensional microimaging. Progress has been achieved with respect to several parameters such as the accessible field and temperature range as well as the incorporation of sample spinning. Since an appropriate analysis of FC data requires a careful consideration of relaxation theory, we include a theory section discussing the most relevant aspects of relaxation in solids which are related to residual dipolar and quadrupolar interactions. The most important limitations of relaxation theory are also discussed. With improved instrumentation and with the help of relaxation theory we get access to interesting new applications such as ionic motion in solid electrolytes, structure determination in molecular crystals, ultraslow polymer dynamics and rotational resonance phenomena.

Introduction

One of the earliest applications of Nuclear Magnetic Resonance (NMR), pursued in the forties, has been spin–lattice relaxation studies of molecular dynamics in condensed matter. Historically, the probably most important step toward the understanding of spin–lattice relaxation was taken by Bloembergen, Purcell and Pound (BPP) [1] who interpreted the phenomenological relaxation terms included in the Bloch equations [2] in terms of molecular motions. Already in the fifties both, relaxation experiments and relaxation theory have undergone a tremendous development. A marvelous account of early studies is Abragam’s “bible” [3] which has probably remained the most frequently cited NMR textbook until today. Apart from high resolution NMR in solution, which also quite early became a standard method in analytical chemistry, spin–lattice relaxometry was one of the most important and widespread NMR methods up to the middle of the sixties or even beyond; and a lot of research concentrated on studying molecular dynamics (correlation times, activation energies) has been performed. However, the amount of information of a typical spin–lattice relaxation experiment always remained quite limited as the only experimental parameter to be varied was the temperature.

Generally speaking, the spin–lattice relaxation rate $T_1^{-1}$ is determined by a spectral density function which is the Fourier transform of a time correlation function characterizing stochastic temporal fluctuations of the interactions causing the relaxation. The dominant relaxation mechanism for nuclei of spin quantum number I = 1/2 (typically ¹H and ¹⁹F) is provided by dipole–dipole interactions. In simple cases (only one spin species, isotropic motion) the relaxation rate measured at a Larmor frequency ω is given by the BPP formula [1]:

$$T_1^{-1}(\omega )=K[J(\omega )+4J(2\omega )]$$

with a spectral density function J(ω) characterizing the fluctuations of the dipolar interactions and a coupling constant K. Assuming an exponential correlation function (not specifying at this stage the mechanism of the fluctuations), C(t) ∝ exp (−t/τ_c), where τ_c is a characteristic time constant, referred to as a correlation time, the spectral density is Lorentzian [3]:

$$J(\omega )=\frac{{\tau }_c}{1+{(\omega {\tau }_c)}^2}$$

Thus, relaxation experiments at a single frequency and varying temperature, give access to a temperature dependence of the correlation time, τ_c. In this context, we like to refer to an early review paper [4] which, from an experimentalist’s point of view, nicely describes the basics of relaxometry, introduces spectral density functions for many relevant cases of molecular dynamics and discusses typical experimental examples.

As long as only fixed frequency NMR spectrometers were available, there was hardly any potential for further developments of NMR relaxometry. Therefore, essentially the whole information to be gained on the molecular process under study was contained in the T₁(T) behavior. At this point the most crucial limitation of a T₁ measurement becomes apparent: One is interested in the frequency dependent behavior of J(ω) but one only gets its value at the given spectrometer frequency ω. Therefore it has been obvious that already quite early researchers were looking for ways out of this limitation. An obvious procedure is to just perform the experiment at various NMR frequencies, i.e. using several NMR spectrometers or spectrometer settings. In fact, quite a number of studies have pursued this strategy and have received decent results. For two impressive early examples see Refs. [5], [6] which beautifully demonstrates the benefit gained from analyzing relaxation data obtained at many spectrometer frequencies. However, apart from the fact that nobody can afford to run too many NMR spectrometers there is also a physical reason which prohibits the extension of this procedure. The NMR signal-to-noise ratio, which scales quadratically (or at least with the power of 3/2) with the applied magnetic field, at low fields eventually becomes too small to allow for a reasonable detection of the NMR signal.

At this point FC relaxometry sets in serving as a suitable tool to study the field dependent and thereby the broad band Larmor frequency dependent spin–lattice relaxation rate, $T_1^{-1}(\omega )$. As the term “field cycling” tells, the experiment makes use of a time dependent, i.e. cycled (or periodically switched), magnetic field in the following way (see Fig. 1)

In a first step (polarization period), a well-defined equilibrium state is attained. This may, for instance, be realized by applying a “polarization” field, B_pol, as high as possible. After having reached the corresponding equilibrium Boltzmann polarization the magnetic field is rapidly switched to a variable “evolution” field, B_ev, differing from B_pol. After this field switch, the nuclear spin magnetization relaxes toward its new equilibrium value. In case of an exponential relaxation the nuclear magnetization will approach this new equilibrium with a rate $T_1^{-1}(\omega ={\omega }_{\mathit{ev}}=\gamma B_{\mathit{ev}})$. Experimentally, this polarization equilibration can be traced by a fast up-switch of the field to a suitable “detection” field, B_det, as a function of the hold time (evolution time) t_ev of B_ev. B_det is chosen such that at its corresponding Larmor frequency ω_det a suitable rf π/2 – pulse can be applied measuring a free induction decay (FID) amplitude, S, which is proportional to the instantaneous value of the nuclear polarization (magnetization) M for the set t_ev. Accumulating several scans is often necessary to obtain a sufficient signal-to-noise ratio. Then, t_ev is varied yielding the M(t_ev) dependence. A fit to M(t_ev) yields T₁ at the set B_ev, i.e. at the set ω = ω_ev. Obviously, thereby only one rf set-up is needed to measure the whole relaxation dispersion curve $T_1^{-1}(\omega )$. For obtaining a M(t_ev) curve with a reasonably high amplitude of the polarization field may be chosen such that the evolution field differs from it as much as possible. In practice, this implies that for low B_ev it is recommended to apply a strong B_pol and for a high B_ev one should use a small (or possibly zero) B_pol. Of course, only relaxation times T₁ not much shorter than the switching times are accessible. Such relaxation dispersion curves cover a frequency range of 4 or – in advanced relaxometers – even up to 6 orders of magnitude, typically for ¹H frequencies v = ω/(2π) of the order of several 10 MHz down to several 10 Hz.

Let us, as an appetizer, demonstrate some experimental features, the beauty and the rich information density of FC by presenting unpublished results of the ⁷Li spin–lattice relaxation rate dispersion measured in Li-metal [7]. First, for illustrating the experimental procedure just discussed in context with Fig. 1 and to get a feeling for some experimental peculiarities we will have a glance at some typical magnetization recovery curves. Fig. 2a shows, for a fixed sample temperature, a couple of typical magnetization decay curves measured at different evolution fields, i.e. at different ⁷Li Larmor frequencies, including fit curves according to a stretched exponential (Kohlrausch) law: ∝exp[−(t_ev/T₁)^β]. There are quite a few things which we immediately learn from these data:

• At all Larmor frequencies the Kohlrausch exponent β turns out to be unity, within experimental error. That is, the relaxation is always monoexponential with high precision.

• Spin–lattice relaxation times are quite dependent on the Larmor frequency, getting longer for higher Larmor frequencies.

• At low Larmor frequencies we measure a signal drop since, as explained in the context of Fig. 1, the experiment was performed in the mode using a high polarization field. At the other edge, at high Larmor frequencies, we measure a signal increase because a zero polarization field was chosen.

• The data, normalized in a way such that the maximum signal amplitude at the highest Larmor frequency is set to unity, show some interesting features with respect to their relative height. At the lowest frequency (1 kHz) a reduced initial amplitude (instead of 1) and a non-zero long time plateau value (instead of almost zero) is seen. These are – unwanted but harmless – effects of a finite switching time. For instance, during the downswitch to the evolution field the polarization already drops by about a factor of 2, and during upswitch to the detection field the polarization also goes up to some extent, implying an initial amplitude of only about 0.5 instead of 1. And the nonzero long time plateau value is due to a polarization buildup already during the upswitch to the detection field. At 5.774 MHz the decrease of the initial signal becomes less pronounced, and the long time plateau value is now about 0.5 as it should be because of the increased thermal polarization being kept at this field. For the same reason, at 6.906 MHz, now working in the high field mode using zero polarization field, we see again a final plateau value of about 0.5, the initial value being nonzero because of polarization buildup during upswitch to the detection field, as above. In this way we can, finally, also interpret the magnetization recovery curve at 12 MHz.

• When talking about long time plateau values of magnetization recovery curves one should note that there are, in addition to relaxation effects during switching periods, other possible sources of bias caused by instrumental imperfections. We will deal with them in more detail in Section 2.

Fig. 2b contains the resulting relaxation dispersion curves obtained at various temperatures. This large set of recent experimental data can readily be interpreted qualitatively by visual inspection:

As well-known from literature [8] and immediately seen from the figure, in metals there are two contributions to $T_1^{-1}$: $T_1^{-1}={\left(T_1^{-1}\right)}_{\mathit{diff}}+{\left(T_1^{-1}\right)}_{\mathit{el}}$. The first contribution, ${\left(T_1^{-1}\right)}_{\mathit{diff}}$ gives rise to a “Lorentzian like” frequency dependence of the relaxation rate decaying to a frequency independent background relaxation rate given by the second term, ${\left(T_1^{-1}\right)}_{\mathit{el}}$. The Lorentzian like contribution is essentially due to the diffusive motion of the metal ions. Its temperature dependence reflects that of the Li ion jump correlation time. The frequency independent background value becomes best visible at high frequencies and low temperatures where the motional part of the relaxation has already decayed. It obeys the famous Korringa relation [9] (T_l)_el · T = const and is due to inelastic scattering processes of conduction electrons in the vicinity of the Fermi surface. With increasing temperature the Fermi surface becomes more smeared out and scattering processes are more probable.

Two ways of field switching can be conceived, called mechanical and electronical FC. In case of mechanical FC the sample is moved mechanically in between different positions of an inhomogeneous magnetic field whereas electronical FC requires electronical switching of the current across a field producing coil. Typically, for mechanical FC one may use the (position dependent) stray field of a superconducting magnet which allows for very high B_pol and B_det. However, the switching time, which in this case is the time needed for the sample displacement, one can hardly reach values below 100 ms. Consequently, only T₁ values above a few tens of ms are accessible which limits the applicability range of this method. The advantages and disadvantages of electronical FC are just inverted. Here, superconducting high field magnets cannot be used since in this case fast electronical field switching is unfeasible. Therefore, the method can only make use of resistive magnets which typically are able to provide fields of not more than, say, 1 T. The strong advantage of a resistive magnet is, however, that it can be quickly switched if properly designed. Typically, a switching time of the order of 1 ms can be reached. Summing up we note that mechanical and electronical FC are quite complementary: Mechanical FC reaches high fields but cannot measure T₁ much below 100 ms, typically; electronical FC allows to measure short T₁ down to 1 ms (or even below) but is limited in the upper field. One may even combine mechanical and electronical FC by making use of the fact that often at higher fields the relaxation times tend to be much longer than at lower fields.

Our work to be reviewed here is based upon a lot of previous achievements in the field of FC NMR. There are two important reviews, written by Noack [10] and Kimmich and Anoardo [11]. Both reviews pointed at all important early attempts in FC relaxometry such that we allow ourselves not to double the citations of the early work but just refer to these involved and rather complete reviews. Noack, whose important contributions to the instrumental development of electronical FC cannot be overestimated and who therefore should be considered as the true pioneer in this field, discussed in length all basic concepts and many technological considerations. Kimmich and Anoardo, almost twenty years later, discussed the progress in the field and focused on the wide range of applications of the FC technique in soft matter physics. So the question arises: Why should there be another review paper on FC NMR relaxometry?

The answer to this question involves some science sociological aspects. The history of NMR shows that the progress in this field has mostly been driven by technological progress. For instance, high resolution NMR in liquids for chemical analytics is linked to the development of homogeneous high field magnets. Or, the great successes of NMR structure determination in proteins and other complex molecules have been triggered by the progress in digital electronics enabling involved pulse sequences and multidimensional techniques. Or, magnetic resonance imaging (MRI) could become a routine real time method only with the availability of fast computers. When we discuss NMR relaxometry in this frame, we have to state that despite the significant early development of relaxation theory (linked with names like Bloembergen, Purcell, Pound, Redfield, Solomon, Morgan, Abragam) and despite important early experimental contributions the instrumentation technology did not sufficiently keep on to allow for a development of relaxometry in a way comparable to that of the above mentioned fields. There is another aspect which links technological development with the existence of a “market”. Whereas high resolution NMR and MRI have become highly desired tools in the nonacademic world (chemical industry and medicine), FC relaxometry has not. The lower acceptance of FC relaxometry in industry is mainly due to the large theoretical effort which has to be invested if experimental relaxation dispersion curves are to be interpreted. As we will discuss in more detail later in this paper the information on dynamical processes hidden in experimental relaxation data requires a whole lot of theoretical effort to get revealed. We do not exaggerate if we state: “There are a lot of measured FC relaxometry data around but only very few neatly interpreted ones”. This calls for a close cooperation between instrumentalists, experimentalists and theoreticians.

In this sense, during our work we have tried (i) to identify scientific problems which had not been dealt with using the existing FC technology, (ii) to improve the instrumentation accordingly, (iii) to apply these achievements experimentally in close cooperation with relaxation theory. The present contribution, which will report on about 15 years of research work of ourselves, will reflect these three aspects. Of course, the reader should be aware that the article only presents a snapshot of current and ongoing work. By no doubt FC instrumentation contains a lot of potential for further instrumental development. Only occasionally, we will also take up some scientific highlights of other research groups.

The structure of the paper will account for the fact that our group is strongly instrumentation oriented. Therefore, a large section (Section 2) is devoted to our instrumental work. Starting with a short overview on instrumental desires (Section 2.1) we will proceed with some basic considerations (Section 2.2). Then, our “working horse”, the high power FC machine FC-I will be presented (Section 2.3). A few remarks on the sample environment for this machine, i.e. our high and low temperature probeheads as well as our sample spinning device, are found in Section 2.4. A major recent implementation into FC-I has been the possibility to go down to very low evolution fields in the μT range (Section 2.5). It turns out that in such small fields – but also otherwise – the problem of finite field stability becomes most crucial. We devote some considerations to our ways of handling such field fluctuations (Section 2.6). A second electronical FC machine (FC-II) developed in our group uses a joke and will be described in Section 2.7. Two Sections 2.8 Mechanical FC relaxometer, 2.9 Combination of mechanical FC relaxometry with one dimensional imaging are devoted to our work on mechanical FC relaxometers, the second one attempting to combine FC relaxometry with one dimensional microimaging. In this context we allow ourselves to finally point at an outstanding project of utmost relevance, namely the development of a combined FC/MRI spectrometer (Section 2.10) in the Aberdeen group.

Section 3 contains an outline of relaxation theory with focus on solid state specific aspects. The outline begins with an introduction to relaxation theory including two-exponential relaxation processes (Section 3.1) and effects of more complex dynamics, i.e. several motional processes (Section 3.2). Then, in Section 3.3, the influence of nuclei possessing quadrupole moments on the relaxation of neighboring nuclei of spin quantum number 1/2 is discussed; specifically, theories of polarization transfer and quadrupole relaxation enhancement are presented. Similarities and differences between these two effects are discussed in connection with residual dipolar and quadrupolar interactions typically present in solids. In Section 3.4 limitations of the perturbation theory of relaxation are specified and the principles of a relaxation theory based on the stochastic Liouville equations, valid beyond the perturbation range, are briefly described. Finally, spectral densities associated with different types of motion are discussed (Section 3.5).

Section 4 reviews our most important FC applications. We start with a report on a typical solid state application of FC where – in combination with some other NMR methods – electronical FC has been used to study the ion dynamics in a class of superionic conductors (Section 4.1). Here, we profit from many of the features of our FC-I machine: high power, fast switching, broad temperature range. Another important solid state phenomenon in the presence of quadrupolar nuclei, namely the appearance of polarization transfer resonances in the relaxation dispersion curves, is extensively discussed in Section 4.2. Here, it is adequate not only to review our more recent results but also to cite some work of other authors and some very old work of one of us using a quite different method (“β-NMR”). Section 4.3 will deal with the study of slow polymer motion. In this current project we have been able, thanks to the availability of very small evolution fields, to test involved theories of anomalous polymer dynamics (Doi-Edwards and de Gennes theory of reptation dynamics). An absolutely unusual application of FC on a spinning sample is presented in Section 4.4. We observe resonances when the Larmor frequency is a (rational) multiple of the sample rotation frequency. This project is just at its beginning but contains a high potential for structure determination. The last part (Section 4.5) will review an application for which the combination of mechanical FC and one dimensional imaging is perfectly suited, namely the study of heavy ion induced point defects in ionic crystals. As will be shown, the experiments result in spatially resolved – that is: dependent on the heavy ion penetration depth – relaxation dispersion curves, thereby yielding highly valuable information on the nature and the distribution of radiation defects.