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Abstract

Over centuries mineralogy has developed from a mainly descriptive scientific discipline to a quantitative experimental research field covering a wide area. A broad spectrum of different mineralogical disciplines, ranging in their topics from the macroscopic physical and chemical properties of igneous rocks up to the atomic structures and characteristics of minerals, crystals and materials is applied today. Mineralogy can be regarded as a field of both Earth sciences and materials science and is an extraordinarily manifold scientific discipline with numerous points of contact to geology, chemistry, physics and materials science, characterised by the large variety of methods used in basic and applied research.

A wide variety of analytical methods such as diffraction methods, microscopy, thermal analysis and spectroscopy is used in the different research fields of mineralogy, each method contributing small pieces to the large puzzle of unsolved scientific questions. Often several methods fail in the investigation of structural aspects as in the case of amorphous materials (e.g. glasses from magmatic melts) or microcrystalline materials and for specific structural questions, e.g. the location and dynamics of hydrogen atoms in minerals. Solid state nuclear magnetic resonance can be an ideal complementary method here.

All nuclei that possess a magnetic moment (I > 0) are able to provide detailed information about their local environments as local probe, e.g. about bond angles, neighbouring atoms (1st and 2nd coordination sphere), local symmetry, the coordination number, as well as being sensitive to dynamic processes (Chandrakumar & Subramanian, 1987; Ernst et al., 1987; Fyfe, 1983; Slichter, 1990).

Introduction

Over centuries mineralogy has developed from a mainly descriptive scientific discipline to a quantitative experimental research field covering a wide area. A broad spectrum of different mineralogical disciplines, ranging in their topics from the macroscopic physical and chemical properties of igneous rocks up to the atomic structures and characteristics of minerals, crystals and materials is applied today. Mineralogy can be regarded as a field of both Earth sciences and materials science and is an extraordinarily manifold scientific discipline with numerous points of contact to geology, chemistry, physics and materials science, characterised by the large variety of methods used in basic and applied research.

A wide variety of analytical methods such as diffraction methods, microscopy, thermal analysis and spectroscopy is used in the different research fields of mineralogy, each method contributing small pieces to the large puzzle of unsolved scientific questions. Often several methods fail in the investigation of structural aspects as in the case of amorphous materials (e.g. glasses from magmatic melts) or microcrystalline materials and for specific structural questions, e.g. the location and dynamics of hydrogen atoms in minerals. Solid state nuclear magnetic resonance can be an ideal complementary method here.

All nuclei that possess a magnetic moment (I > 0) are able to provide detailed information about their local environments as local probe, e.g. about bond angles, neighbouring atoms (1st and 2nd coordination sphere), local symmetry, the coordination number, as well as being sensitive to dynamic processes (Chandrakumar & Subramanian, 1987; Ernst et al., 1987; Fyfe, 1983; Slichter, 1990). This is due to the fact that, besides the outside static magnetic field, there are small internal local fields that contain the appropriate structural information, which influence the effective magnetic field at the nucleus. The two most important interactions in this context are the chemical shift interaction for all nuclei and the electric quadrupolar interaction for nuclei with a nuclear spin of I > 1/2. The chemical shift (CS) is caused by the external magnetic field, which induces motion of the electrons in the electron shell of the nucleus such that a secondary magnetic field, characteristic of the local environment, is set up. The electric quadrupolar interaction is caused by the phenomenon that the charge distribution in the nucleus is not spherical for nuclei with I > 1/2 giving rise to a quadrupolar moment. This quadrupolar moment can interact with a possible electric field gradient (EFG) in the electron shell of the nucleus by electric Coulomb forces which are again characteristic of the local environment (Chandrakumar & Subramanian, 1987; Ernst et al., 1987; Fyfe, 1983; Slichter, 1990). In conclusion, NMR spectroscopy is a powerful complementary method to X-ray structure analysis and to many other methods used in mineralogy, probing short-range structural effects rather than long-range order.

Since the discovery of nuclear magnetic resonance (NMR) by Purcell et al. (1946) and Bloch et al. (1946), the field has developed rapidly over the last fifty years. Pulse Fourier transform NMR has emerged as the method of choice, allowing maximum flexibility in the combination and application of multiple pulse sequences (Ernst et al., 1987). Recent developments in linear and digital radio frequency (RF) transmitter technology in combination with highly sophisticated digital operating units, data digitisation and processing facilitate highly precise RF pulses of any phase in very short times. The development of special probe heads for the magic angle spinning (MAS) method has completed this rapid progress (Chandrakumar & Subramanian, 1987; Ernst et al., 1987; Fyfe, 1983; Slichter, 1990).

Recent developments in Multi Quantum coherence experiments in combination with MAS (MQMAS) (Frydman & Harwood, 1995; Medek et al., 1995), heteronuclear correlation (HETCOR) (Fyfe et al., 1992) and the detection of connectivities and internuclear distances by means of the determination of the heteronuclear dipolar interaction by applying a rotational echo double resonance experiment (REDOR) (Gullion & Schaefer, 1989; Fyfe et al., 1997) have both enhanced resolution and also increased the quantity of new structural information.

The following chapter will give an introduction into basic solid state NMR theory, present routine methods and recent technical developments. In the second part three examples will show how solid state NMR can be an ideal complementary method in Rietveld powder diffraction refinements of crystalline rock-forming minerals.

The NMR spectroscopic study of 207Pb in pure lead phosphate shows that it is possible to determine the local environment of the Pb sites and get structural information even without knowing the crystal structure.

The question of dynamics is highlighted in the paragraph on sodium cation dynamics in the room- and high-temperature nitrate cancrinite structure. The comparison of theoretical 23Na quadrupolar interaction parameters from a simple point charge model and Rietveld methods with the NMR experimental data gives important information for the understanding of the high-temperature structure of nitrate cancrinite.

The introduction of solid state NMR in crystal chemistry and crystal structure determination in the structural study of cation order and disorder in Na8[AlSiO4]6(CO3)x(HCOO)2–2x(H2O)4x, 0.2 ≤ x ≤ 1 provides important information on the question of order and disorder of the carbonate and formate anions in the sodalite structure and helps in solving the different structures of the “as-synthesised”, the calcined sodalites with Rietveld powder refinement methods.

It should be noted that this chapter cannot give a complete overview about NMR applications in mineralogical research. There are numerous more extended treatments, some of which also have more accessible, phenomenological introductions to the method. Examples include chapters in several books of “Reviews in Mineralogy” of the Mineralogical Society of America (e.g. Hawthorne, 1988), in the AGU “handbook of physical constants” (Ahrens, 1995), in Engelhardt & Michel (1987), Duer (2001), or MacKenzie & Smith (2002). Many of these also have quite useful tabulations of NMR data for minerals, and cover a wide range of applications.

Common magnetic and electric interactions in nuclear magnetic resonance

Zeeman interaction

A lot of nuclei possess a nuclear spin (I > 0) and thus a magnetic moment. The magnetic moment forumla is related to the dimension-free nuclear spin by the magnetogyric ratio γ and Planck's constant h. In the quantum mechanical consideration it can be expressed by operators (Abragam, 1961; Chandrakumar & Subramanian, 1987; Ernst et al., 1987; Fyfe, 1983; Haeberlen, 1986; Slichter, 1990).  

formula
The eigenvalues can be determined by the application of the nuclear spin operator Î on the spin function uI,m. Two solutions are possible: Application of the square of the spin operator Î2 on the spin function gives the eigenvalue I, where I is the nuclear spin quantum number. Application of the z component of the spin operator Îz (relative to a static magnetic field) on the spin function gives the magnetic quantum number m as solution.  
formula
 
formula
The energy levels are degenerated if an external magnetic field forumla is absent, and split into (2I + 1) eigenstates under the influence of an external magnetic field. The corresponding Hamiltonian forumla for the Zeeman interaction can be described as follows.  
formula
The solution of the Schrödinger equation gives the energy eigenvalues Em for the Zeeman interaction.  
formula
Transitions between the energy levels are only allowed for Δm = ±1. This selection rule can be determined by solution of the transition moment,  
formula
where forumla is the perturbation field operator.  
formula
A transition between two energy levels is excited if an electromagnetic wave is irradiated whose energy exactly meets ΔE. The matching resonance frequency ω0 is the Larmor precession frequency of the concurrent nucleus.

Chemical shift interaction

Besides the static magnetic field the effective magnetic field at the nucleus is influenced by internal local magnetic and electric fields.

One of the most important interactions, the chemical shift (CS) is caused by the external field, which induces motion of the electrons such that a secondary magnetic field is set up. The secondary field produced by this current loop opposes the main field at the nucleus. Thus, the nucleus is either shielded (diamagnetic effect) or de-shielded (paramagnetic effect). The resulting field is lower or higher than the static magnetic field. The chemical shift Hamiltonian forumla is described as  

formula
where σ is the chemical shielding tensor related to the outer static field forumla.

To simplify the mathematical treatment of the chemical shift interaction, the point symmetry and the local environment of the nucleus will be considered in a Cartesian coordinate system. The origin is placed at the centre of the nucleus and the coordinate system is called the principal axes system (PAS). The chemical shielding tensor forumla in the PAS can be transformed by the Wigner rotation matrices forumla (α,β,γ)CS into the laboratory axes system (LS), which corresponds to the outer static magnetic field (z direction of the LS is equal to the z direction of the static magnetic field). α, β and γ are the Euler angles between the axes of both coordinate systems. The chemical shift Hamiltonian contains a sum of two terms describing the isotropic (σiso) and anisotropic part of the chemical shielding tensor.  

formula
The spatial coordinates of the chemical shielding tensor σ in the PAS are defined in such a way that the direction of the strongest absolute interaction is fixed as the z direction. σxx, σyy and σzz are the principal elements of the tensor.  
formula
Three parameters can be extracted from each CS lineshape in the NMR spectrum. The isotropic chemical shift σiso, which is defined by the trace (Tr) of the symmetric chemical shielding tensor, the chemical shift anisotropy σaniso, which characterises the difference between z axis and xy plane of the PAS, and finally the asymmetry parameter η, which describes the deviation from axial symmetry and can reach values between 0 and 1 (Spiess, 1978).  
formula
 
formula
 
formula
In solid state NMR usually powdered samples are measured where the local environments of the nuclei are randomly orientated in the sample and thus the principal axes systems of the chemical shielding tensors. Consequently, a typical frequency distribution results.

If we consider a spherical coordination of the observed nucleus, a Lorentzian signal appears in the spectrum as shown in Figure 1. With increasing chemical shift anisotropy a typical lineshape results where the covered range is increasing proportional to the chemical shift anisotropy. If asymmetry exists, an additional shoulder occurs in the lineshape (Fig. 1).

Fig. 1

Chemical shift anisotropy lineshapes under static conditions for σaniso = 0, σaniso > 0, and σaniso < 0. The asymmetry parameter η is zero and η = 0.4.

Fig. 1

Chemical shift anisotropy lineshapes under static conditions for σaniso = 0, σaniso > 0, and σaniso < 0. The asymmetry parameter η is zero and η = 0.4.

Dipolar interaction

The resulting magnetic field for the observed nucleus is also influenced by the magnetic moments of neighbouring nuclei. The dipolar Hamiltonian forumla for two interacting nuclei i and j depends on the magnetogyric ratios γi and γj of the nuclei and the internuclear distance rij as shown in Figure 2. Θ is the angle between the magnetic field direction forumla and the internuclear vector forumla.  

formula

Fig. 2

Dipolar interaction axes system with internuclear vector forumla and angle Θ between connecting axis and static magnetic field.

Fig. 2

Dipolar interaction axes system with internuclear vector forumla and angle Θ between connecting axis and static magnetic field.

The interaction tensor forumla is a symmetric tensor which contains the spatial dependence of the interaction. μ0 is the permeability of vacuum.

The resulting interaction in the laboratory frame in spherical coordinates for homonuclear dipolar interaction between nuclei of the same sort (i = j) is described by the following term:  

formula
The resulting interaction for heteronuclear dipolar interaction between nuclei of a different sort (ij) is described as follows:  
formula
An isolated pair of spins (two-spin system) in polycrystalline material provides a typical lineshape, the Pake doublet (Pake, 1948). It presents a frequency distribution of the dipolar interaction for all possible orientations of the two-spin system in a powder sample. In this special case the internuclear distance between both nuclei can be directly determined from the corresponding lineshape.

In the case of a multi-spin system in polycrystalline or amorphous solids the spectrum usually shows broad and featureless resonances. Those lineshapes can often be described by a Gaussian frequency distribution g(ω) with normalised area.  

formula
Δω is the full width at half height of the signal. ω0 is the frequency where the function g(ω) has its maximum. In most cases Equation 17 does not describe the lineshape properly. In these cases the lineshape is best characterised by the so-called second moment M2, which presents the mean quadratic linewidth (Abragam, 1961).  
formula
In the special case when a pure Gaussian curve, as in Equation 17, describes the signal, the second moment M2 can be determined directly from the full width at half height Δω. 
formula
The van Vleck equation (van Vleck, 1948) delivers a direct connection between the second moment and the atom arrangement in the solid. For a polycrystalline or amorphous material with observed nuclei i which see the same local environment in the average, the following equation can be formulated.  
formula
with the term K1 for homonuclear dipolar interaction with nuclei of the same sort:  
formula
and the term K2 for heteronuclear dipolar interaction with other nuclei:  
formula
rij and rik are the average internuclear distances between neighbouring nuclei and I is the nuclear spin quantum number.

Quadrupolar interaction

Nuclei with a nuclear spin of I > 1/2 possess an electric quadrupolar moment Q. The quadrupolar moment is a degree for the deviation of charge (e) distribution from ideal spherical symmetry. This quadrupolar moment can interact via Coulomb forces with a non-spherical charge distribution in the surrounding electron shell, the electric field gradient (EFG). The electric field gradient Vik can be described as the first derivative of the electric field forumla and the second derivative of the electrostatic potential V, where xi and xk are spatial coordinates.  

formula
The Hamiltonian forumla for the quadrupolar interaction can be formulated as  
formula
After insertion of the Wigner rotational matrixes and transformation to the laboratory frame the Hamiltonian can be described as follows, where Θ and Φ are the Euler angles.  
formula
The Hamiltonian is only valid in the limit of a first-order perturbation, which means that in comparison to the Zeeman interaction, the quadrupolar interaction is low. The central transition (1/2 → –1/2) of non-integral spins is not influenced in the case of first order perturbation. In addition the following definitions are valid. The principal elements of the EFG are defined as follows:  
formula
The quadrupolar coupling constant CQ, the quadrupolar frequency VQ and the asymmetry parameter η are defined as  
formula
 
formula
 
formula
If the Zeeman interaction is comparable to or smaller than the quadrupolar interaction, the first-order perturbation theory is not sufficient to describe the interaction; it must be described by second-order terms. Second-order quadrupolar interaction leads to a shift of the central transition and the satellite transitions (e.g. 1/2 → 3/2). The additional quadrupolar shift σQS is defined as follows:  
formula
where ω0 is the Larmor frequency of the nucleus. Second-order quadrupolar interaction leads to broad lineshapes for the central transition. The anisotropy cannot be averaged out by line-narrowing techniques like magic angle spinning (MAS). The angular distribution in the powder sample leads to a frequency distribution. Figure 3 shows typical lineshapes of quadrupolar interaction under static and MAS conditions.

Fig. 3

Quadrupolar lineshapes under static and MAS conditions with CQ > 0. The asymmetry parameter η is zero and η = 0.5.

Fig. 3

Quadrupolar lineshapes under static and MAS conditions with CQ > 0. The asymmetry parameter η is zero and η = 0.5.

Routine techniques in solid state nuclear magnetic resonance

A lot of routine techniques are applied in solid state NMR today to enhance resolution or to separate distinct interactions. The magic angle spinning (MAS) technique is one of the most commonly used methods in solid state NMR to obtain highly resolved solid state NMR spectra. Multiple pulse sequences like the Hahn echo are used under static or MAS conditions. Cross-polarisation is applied as a double resonance method to enhance the excitation intensity of nuclei with low natural abundance and low magnetogyric ratio. Spin-lattice and spin-spin relaxation times give information about the dynamics of the nuclei in the solid.

Magic angle spinning (MAS)

Most of the interaction Hamiltonians contain the angular term 3(cos2Θ – 1) after transformation of the interaction from the principal axes system to the laboratory frame. The magic angle spinning technique uses this fact by spinning a sample fast about an axis with an angle of Θ = 54°44' to the external static magnetic field (Andrew et al., 1959; Lowe, 1959).

For a better understanding of this technique the averaging mechanism of MAS is derived in the followings. In our previous considerations the Hamiltonians of the interactions were directly transformed from the principal axes system (PAS) into the laboratory axes system (LS) via Wigner rotational matrices. In the case of fast sample spinning in a cylindrical container about the cylinder axis, a third coordinate system emerges: the spinner system as shown in Figure 4. To consider the influence of the sample rotation as part of the Hamiltonian in the laboratory frame, the transformation from the PAS to the LS cannot be done via direct transformation but has to include a transformation step into the spinner system, where ωr is the rotation frequency and Θ the angle between the z axis of laboratory frame and the rotation axis.

Fig. 4

Coordinate system scheme and transformation operators for the MAS technique.

Fig. 4

Coordinate system scheme and transformation operators for the MAS technique.

Each interaction operator forumla is formulated as follows:  

formula
Cλ is a time-independent constant depending on the interaction, forumla contains the spin operator and forumla the spatial dependence, which is the product of the Wigner rotation matrices forumla (which transform the interaction from the PAS to the LS via the current Euler angles αλ, βλ and γλ) and the interaction tensor forumla in the PAS. Assuming an interaction of first-order perturbation (m = 0) and that only symmetric parts of forumla must be considered (l = 0, 2) the Hamiltonian under spinning conditions (see Fig. 4 for axes systems and Euler angles) is  
formula
After insertion of the Wigner rotation matrices and separation of the different terms the Hamiltonian can be displayed as a sum of three different terms (Maricq & Waugh, 1979):  
formula
The isotropic part forumla is present in the case of chemical shift interaction while it is absent for dipolar interaction and quadrupolar interaction of first order.  
formula
The second term forumla is a time-independent angular term, which becomes zero if the spinning axis and the z axis of the static magnetic field include an angle of Θ = 54°44'.  
formula
The last term is time-dependent and is time-averaged if the rotation frequency ωr >> 2πΔv, where Δv is the spectral frequency width of the static signal.  
formula
Cn and Sn are complicated, time-independent trigonometric terms (Maricq & Waugh, 1979). If the rotation frequency becomes lower than Δv, the time-dependence has to be considered. In this case, the time-dependent term is easier to handle if we observe the development of the Hamiltonian during one rotation period. This can be done by using the Average Hamiltonian theory of Maricq & Waugh (1979). The periodic time-dependent Hamiltonian forumla can then be substituted by a time-averaged Hamiltonian which will be approximated by a progression.  
formula
The progression converges if  
formula
If ωr < 2πΔv, then in the distance of ±r, n = 0, 1, 2 there are rotational echoes, the so-called spinning sidebands. The overall intensity of these sidebands corresponds to the lineshape of the static spectrum and thus can be used to determine e.g., the chemical shift anisotropy.

For quadrupolar shift interaction of second order, terms with m ≠ 0 also have to be considered in Equation 31. Under MAS conditions the quadrupolar frequency forumla (Freude & Haase, 1993; Samoson, 1985) is  

formula
where  
formula
In the case of second-order quadrupolar interaction, the interaction cannot be averaged out by MAS. Instead a powder lineshape g(φ,Θ,η) results (Eqn. 40), which depends on two orientation angles. MAS leads to line-narrowing and the resulting lineshape in the case of MAS are presented in Figure 3.

The MAS technique has developed rapidly from its beginnings in 1959. The first spinner reached frequencies of 1.7 kHz (Andrew et al., 1959; Lowe, 1959). Today different commercial MAS probe heads are available for NMR experiments. All designs distinguish themselves by separated double-bearing and drive air circuits. The produced air-cushion enables rotation frequencies of 7–8 kHz for 7 mm rotors, 12–18 kHz for 4 mm rotors, and 50 kHz for 2.5 mm rotors. The spinning speed is detected either optically by a glass fibre or acoustically via a copper wire. Rotors usually are built from zirconia and rotor caps are made of Kel-F, Delrin, boron nitride, zirconia and Vespel.

Pulse sequences

Most pulse sequences, similar to magic angle spinning, average or select certain interactions by manipulation of the spin operator components of the Hamiltonians. This opens the possibility to create a selective solid state NMR experiment in combination with MAS (Combined Rotation and Multi-pulse Sequence, CRAMPS) or as a static experiment. Multiple pulse sequences can be applied either as one-dimensional or two-, three or n-dimensional experiments. Pulse sequences that are used routinely are, e.g., WAHUHA-4 (Waugh et al., 1968) and MREV-8 (Mansfield et al., 1973) to remove the homonuclear dipolar interaction, i.e., for protons. Homonuclear correlation spectroscopy like COSY (correlated spectroscopy) or INADEQUATE (incredible natural abundance double quantum transfer experiment) use those pulse sequences to show neighbourhoods of nuclei like 29Si or 13C. Double resonance pulse sequences like cross-polarisation are applied to transfer magnetisation between different nuclei or to show connectivities by HETCOR. Multi-quantum excitation pulse sequences are used in two-dimensional experiments to separate interactions, e.g. in the MQMAS experiment (Frydman & Harwood, 1995; Medek et al., 1995), or to show connectivities, e.g. double quantum resonance for 1H.

One of the most common multi-pulse sequences used is the solid echo or Hahn echo (Hahn, 1950). Under static conditions the expected signal usually covers a broad frequency range and thus the free induction decay (FID) in the time domain is short. The receiver cannot directly acquire the FID after the 90° pulse because of pulse ring-down. A dead time of at least 8 μs arises depending on the frequency, probe design, etc., where no signal can be detected. The signal loss leads to a decreased signal-to-noise ratio, in addition to a phase error because the signal cannot be detected starting at t = 0. The phase of a signal is the corresponding angle of the sine-function describing the signal form at a specific time. A phase correction of second order has to be carried out, which distorts the baseline. A baseline correction is nearly impossible because in respect of broad signals it is hard to decide which part is signal or baseline.

The solid echo is presented in Figure 5. After a preparation pulse of 90°, the magnetisation decays until a time of τ1 is reached. Then a 180° pulse is applied which inverts the spin precessing motion and leads to a refocusing of the macroscopic magnetisation during τ2. At the time of τ1 = τ2, the FID has totally refocussed and decays again. An echo has built up. If the acquisition is started right on top of the echo, the full FID can be acquired without loss of signal by dead time and no second-order phase correction has to be applied because the acquisition is started at t = 0.

Fig. 5

Solid echo pulse scheme. The first pulse is a 90° pulse. A 180° pulse follows after a time τ1.

Fig. 5

Solid echo pulse scheme. The first pulse is a 90° pulse. A 180° pulse follows after a time τ1.

The solid echo can also be applied to determine the homonuclear dipolar interaction. As obvious from Table 1, all interactions except the homonuclear dipolar interaction are completely refocused after a period of τ1 +τ2 = 2τ1. The corresponding Hamiltonians show different signs of the spin operator Îz during the evolution period τ1 and τ2. Only the homonuclear dipolar interaction operator does not change the sign because it contains the spin operator Îz in quadratic dependence and thus will not be refocused in the spin echo maximum. The echo maximum intensity, as a function of τ1, gives the homonuclear dipolar interaction, which is proportional to the spin-spin relaxation time T2.

Table 1.

Interaction Hamiltonians under application of a solid state echo (see Figure 5) before and after the 180° pulse during τ1 and τ2.

InteractionDuring τ1during τ2

Homonuclear dipolarforumlaforumla
Heteronuclear dipolarforumla
Chemical shiftforumlaforumla
InteractionDuring τ1during τ2

Homonuclear dipolarforumlaforumla
Heteronuclear dipolarforumla
Chemical shiftforumlaforumla

Relaxation times

Internal molecular dynamic processes – such as translation or diffusion of atoms, ions, molecules or reorientational jumps of molecules or groups of molecules about certain axes – cause nuclear spin relaxation if the frequency spectrum of the dynamic processes contains the Larmor frequencies of the nuclei involved, and if dynamics and nuclear spins are coupled. The latter condition is fulfilled if the interacting magnetic moments of the nuclei move relatively to each other and/or to the outer static magnetic field. In this case magnetic dipole-dipole interactions are (in solid state NMR) time-dependent and thus spin-lattice relaxation occurs if the dynamic frequency distribution contains the appropriate transition frequencies. Estimation of relaxation times is a suitable procedure for the study of dynamics in addition to other research methods (heat capacity, neutron diffraction etc.).

The spin-lattice relaxation time T1 is directly correlated to the time the magnetisation needs to get back into thermodynamic equilibrium following a radio frequency pulse. After a 180° pulse, turning the complete z magnetisation into the –z direction, the magnetisation Mz along the z axis after a time τ shows the following correlation (Bloch, 1946):  

formula
T1 is determined in the so-called “inversion recovery experiment” by incrementing τ and estimating the rebuilt magnetisation by the application of a 90° pulse and detecting the magnetisation in the xy area. T1 is directly delivered by the slope of the logarithmic function. The described experiment is suitable for nuclei in solids with short spin-lattice relaxation times because the next experiment after detection of the magnetisation cannot be started before the system has reached thermodynamic equilibrium. Long spin-lattice relaxation times can be better determined by the so-called “saturation recovery experiment”. A pulse train of about fifty 90° pulses at the start of the experiment destroys the Mz magnetisation totally so that it is zero at τ = 0. The rebuilt magnetisation is estimated similarly as in the inversion-recovery experiment after a period τ by application of a 90° pulse. The spin-spin relaxation time T2 displays the direct homonuclear dipolar interaction. It can be determined by a 90xτ–180yτ–(acq) sequence (see Fig. 5). The spin-lattice relaxation time in the rotating frame T gives information about very slow dynamic processes, because the nuclei can relax only if the motional frequencies are in the range of the strength of the applied radio frequency pulse. The nutation frequency of a radio frequency pulse ω1 is usually in the range of 50–120 kHz.

A correlation time τc is used for the description of dynamic processes. τc is defined as the mean time for a dynamic translation step in the order of magnitude of a molecular dimension. For a reorientation process τc corresponds to the mean time, which is necessary for a considerable change of the angle. The stochastic theory yields a more precise definition of τc using the “correlation function” for observed dynamic processes. As far as thermally activated dynamic jumps over a potential barrier with activation energy EA are considered, one can expect statistically that a fraction of molecules possesses the necessary energy to pass the potential barrier. The process is described best by an Arrhenius equation where τc∞ is the correlation time at infinite high temperature.  

formula
Each of the existent dynamic processes in a solid can be described by a correlation time τc. Maximum relaxation takes place at that temperature where the Larmor frequency of the observed nucleus ω0τc−1. A “half classical” theory for the spin-lattice relaxation was developed by Bloemburgen, Purcell & Pound (1948).  
formula
CI is an extent for the dipolar interaction strength and contains the second moment M2 according to Equation 18. The function possesses its maximum at ω0τc = 0.625. With decreasing Larmor frequency ω0 the maximum shifts to lower temperatures. In the limit of high temperatures the equation can be reduced to  
formula
In the limit of low temperatures Equation 43 can be reduced to  
formula
The activation energy can be determined from either side (low or high temperature) by determination of the slope of the linear increase.

The T1 relaxation curve of tetramethyl ammonium iodide (CH3)4NI is shown in Figure 6. The proton relaxation rate of tetramethyl ammonium iodide can be described by two dynamic processes and the sum of two relaxation curves with different slopes and maxima. First, the (CH3)4N+ tetrahedra are able to rotate about their axes. Second, the one-dimensional rotation of methyl groups about their threefold axis is present. The resulting function shows two maxima for a Larmor frequency of ω0/2π = v0 = 15 kHz and different linear slopes for the low- and high-temperature regions. The methyl group rotation is responsible for the slope in the low-temperature region while the tetrahedron rotation dominates the high-temperature region. An activation energy EA = 23 ± 1 kJ/mol for the methyl groups and EA = 46 ± 3 kJ/mol for the whole tetrahedron can be estimated (Albert et al., 1972).

Fig. 6

T1 relaxation time curve in half logarithmic scaling as a function of the reciprocal temperature of tetramethyl ammonium iodide. The Larmor frequency is v0 = 15 MHz.

Fig. 6

T1 relaxation time curve in half logarithmic scaling as a function of the reciprocal temperature of tetramethyl ammonium iodide. The Larmor frequency is v0 = 15 MHz.

Cross-polarisation (CP)

A number of nuclei suffer in spectral excitation intensity from a low magnetogyric ratio and low natural abundance. The cross-polarisation (CP) experiment is a double resonance experiment which uses the strong magnetic polarisation of nuclei with high natural abundance and high magnetogyric ratio such as 1H or 19F to transfer magnetisation on nuclei with low spectral resolution such as 29Si and 13C via heteronuclear dipolar interaction (Pines et al., 1971; Pines et al., 1973). The increase of spectral sensitivity is proportional to γH/γX. An additional advantage is that the spin system relaxes over the proton spin reservoir which often shows short spin-lattice relaxation times. The pulse sequence is displayed in Figure 7. In the first step the proton magnetisation is turned by a 90° pulse into the xy area of the laboratory frame. Subsequently the 1H magnetisation is hold (spin-lock) in the xy plane of the coordinate system while the X nucleus magnetisation is turned into the xy plane. If both magnetisations have the same angular frequency ω1 = ωrf in the rotating coordinate frame, they can exchange energy, because the energy levels of both nuclei have the same energy distance under the influence of the radiofrequency field B1. This condition, the Hartmann-Hahn condition (Hartmann & Hahn, 1962) is fulfilled if  

formula

Fig. 7

Cross-polarisation pulse scheme. The X nucleus is the observed nucleus and 1H the decoupled nucleus.

Fig. 7

Cross-polarisation pulse scheme. The X nucleus is the observed nucleus and 1H the decoupled nucleus.

Thermodynamically this process can be considered as a heat exchange between the cold proton spin reservoir and the hot X nucleus spin reservoir. The signal intensity M(t) is decisively dependent on how long both reservoirs are in contact, i.e. the contact time t (Walther et al., 1990). M0 is the total intensity.  

formula
The exponential increase of magnetisation is a function of the cross polarisation time TIS and thus depends on the number of proton spins and the distance of these spins rIS to the X nuclei. On the other hand the proton spin reservoir relaxes exponentially during excitation of the radio frequency pulse to get thermodynamical equilibrium for the occupancy of the energy levels during the lower B1 field. The intensity curve decays with the spin-lattice relaxation time in the rotating frame T. The cross-polarisation time TIS contains the heteronuclear second moment as shown in Equation 48 and thus distance information. This information can be used, e.g., to determine if a proposed structural model for an unknown structure is valid. The theoretical and TIS can be calculated (Eqns. 18, 20 and 48) for the different crystallographic X nucleus sites and compared with the experimental ones (Walther et al., 1990; Fyfe et al., 1997).  
formula
The structural models can also be composed by Monte Carlo methods and compared to the experimental results. If only separated spin pairs with distinct distances are present, Equation 47 has the following form. The magnetisation M(t) oscillates with a cosine function with frequency b. In addition, spin diffusion processes between protons take place as expressed by the spin diffusion time TII (Walther et al., 1990).  
formula
 
formula
The constant b directly contains the distance rIS for the distinct spin pair.

Latest methodical developments in solid state nuclear magnetic resonance

A number of new important solid state NMR methods have been invented during recent years. The progress can be attributed to the large developments in the high-frequency technique, where digital receivers, transistor transmitters etc. made it possible to adjust the precession variability of pulse amplitude and phase, to allow fast shifting of pulse phases as well as to shorten delays between pulses to much shorter times. Moreover, the probe head equipment has developed intensively, e.g. the turbines can reach frequencies of 50 kHz. In this subsection the developments of REDOR, SATRAS, DOR, DAS and MQMAS will be introduced.

Rotational echo double resonance (REDOR)

A new interesting experiment to investigate the atomic arrangement of different nuclei in solids by heteronuclear dipolar interaction is the REDOR (Rotational Echo Double Resonance) experiment. In the REDOR experiment (Gullion & Schaefer, 1989) a rotation-synchronised echo pulse sequence is applied to the observed nucleus. The echo (S0) refocuses the chemical shift anisotropy and the heteronuclear dipolar interaction (see pulse sequence; Fig. 5). During the echo period a sequence of two 180° pulses per rotor period is applied to the other nucleus which perturbs the dipolar MAS refocusing process and consequently diminishes the echo intensity (SF). Quantitative distance information can be extracted directly for an isolated two-spin system. Figure 8 shows a theoretical example for a 13C–15N two-spin system with an internuclear distance of r = 2.54 Å. The signal loss, the (S0SF)/S0 ratio, shows a characteristic oscillation (Fig. 8) as a function of the number of rotor periods Nc (Gullion & Schaefer, 1989), the rotation period time tr = 2π/ωr, the placement of the first dephasing pulse in the first half rotation period P (t1 = 1/P*tr), as well as the dipolar coupling constant DHD (Pan et al., 1990). The dipolar frequency under magic angle spinning conditions ωD is given by  

formula

Fig. 8

Theoretical REDOR curve for a 13C-15N two-spin system with a dipolar coupling constant of DHD = 1/Tr and one dephasing pulse in each half of a rotor cycle (P = 2) (Guillon & Schaefer, 1989).

Fig. 8

Theoretical REDOR curve for a 13C-15N two-spin system with a dipolar coupling constant of DHD = 1/Tr and one dephasing pulse in each half of a rotor cycle (P = 2) (Guillon & Schaefer, 1989).

α is the azimuthal and β the polar angle defined by the internuclear vector in the spinner coordinate system with the z axis parallel to the rotor axis. The time evolution of ωD is expressed in terms of the average Hamiltonian. This leads to an evaluation of the average dipolar transition frequency forumla for one rotor cycle. The application of a single dephasing 180° pulse at a time t1 during a rotor cycle changes the average dipolar frequency to  

formula
The non-zero integral in Equation 52 measures the accumulation of the phase ΔΦ. The accumulated phase for two equally-spaced 180° dephasing pulses per rotor cycle is  
formula
Averaging over all directions of the internuclear vectors in the powder leads to the echo signal intensity SF.  
formula
However, the interpretation of more complex spin-systems with three or more nuclei coupling to each other is much more difficult (Fyfe et al., 1997). As an example, the proton, fluorine, and aluminium spin clusters in phlogopite represent a multi-spin system. Figure 9 shows examples of {19F} 27Al REDOR NMR and {1H} 27Al REDOR NMR experiments for Al-rich phlogopites for Nc = 26 rotor periods. The spectra (S0 signal) in the first row are the Fourier-transformed spectra of the echo amplitude without dephasing pulses in the 19F or 1H channels. The heteronuclear dipolar interaction is averaged out by MAS in this experiment. The upper left spectrum from fluorine- and aluminium-rich phlogopite shows three signals. The two signals in the tetrahedral region are due to tetrahedral aluminium from phlogopite (δ(27Al) = 67.6 ppm) and tetrahedral aluminium from an impurity phase (kalsilite, δ(27Al) = 57.9 ppm). A superposition of three resonances from the octahedral Al sites of phlogopite (δ(27Al) = 1.5 ppm), corundum (δ(27Al) = 11.8 ppm), and potassium aluminium hexafluoride (δ(27Al) = -3.5 ppm) is observed in the octahedral region. In the upper right spectrum of a hydroxyl- and Al-rich material the signal components from kalsilite and potassium aluminium hexafluoride are absent (Fechtelkord et al., 2003). The middle spectra (SF signal) have dephasing pulses in the 19F (left spectrum) and 1H channel (right spectrum) which re-introduce the heteronuclear dipolar interactions between fluorine and aluminium (left spectra) or protons and aluminium (right spectra). Signal components which result from Al sites with protons or fluorine atoms as neighbours are diminished. Al sites without protons or fluorine atoms as neighbours such as those in kalsilite and corundum should experience no change in signal intensity in comparison to the S0 experiment. The difference signal (S0SF) contains mainly those signal components which experience heteronuclear interactions to the dephased nucleus. An excellent example is the complete disappearance of the kalsilite signal in the lower left {19F} 27Al REDOR NMR spectrum. In addition, the intensity of the broader octahedral phlogopite signal has increased compared to that of the narrow corundum signal.

Fig. 9.

Representative {19F} 27Al (left) and {1H} 27Al REDOR spectra (right) of two Al-rich phlogopites with composition K(Mg3-iAlx)[Al1+iSi3-iO10](OH)γ,(F)2-γ,. × and y are as indicated in the figure.

Fig. 9.

Representative {19F} 27Al (left) and {1H} 27Al REDOR spectra (right) of two Al-rich phlogopites with composition K(Mg3-iAlx)[Al1+iSi3-iO10](OH)γ,(F)2-γ,. × and y are as indicated in the figure.

Bertmer & Eckert (1999) showed in a recent paper that for a multi-spin system the (S0SF)/S0 ratio in the limit of short evolution times is independent of the specific spin geometry involved and can be approximated by a parabola. The curvature of this parabola is defined by the second moment M2 (Slichter, 1990) of the observed nuclei. The slope of the REDOR dephasing curve can be simulated by a first-order approximation given in Equation 55 (Bertmer & Eckert, 1999) and remains valid up to about (S0SF)/S0 = 0.3:  

formula
The theoretical second moment M2 can be estimated from van Vleck calculations (van Vleck, 1948) for specific bond distances where the range is usually limited to the nearest neighbours (up to 4.6 Å, Equation 20). Figure 10 shows the {19F} 27Al and {1H} 27Al REDOR NMR curves for an Al-rich phlogopite composition. The intensities of the SF and S0 signal components have been derived from the spectra by integration of the octahedral and tetrahedral regions. Separation of the three octahedral signal compounds (corundum, phlogopite, and potassium aluminium hexafluoride) and the two tetrahedral signals (phlogopite and kalsilite) is not possible. However, corundum and kalsilite should not influence the REDOR curve; their signal contribution is automatically removed in the difference spectrum due to the lack of heteronuclear dipolar interactions.

Fig. 10.

{19F} 27Al REDOR curves (a) and {1H} 27Al REDOR curves (b) of an Al-rich phlogopite with composition K(Mg3–xAlx)[Al1+iSi3–xO10](OH)γ,(F)2–γ, with x = 0.8 and (a) y = 0,5, (b) y = 1,8 The integrated intensity ratios of the tetrahedral and octahedral regions of the spectra are plotted as a function of rotor cycles Nc.

Fig. 10.

{19F} 27Al REDOR curves (a) and {1H} 27Al REDOR curves (b) of an Al-rich phlogopite with composition K(Mg3–xAlx)[Al1+iSi3–xO10](OH)γ,(F)2–γ, with x = 0.8 and (a) y = 0,5, (b) y = 1,8 The integrated intensity ratios of the tetrahedral and octahedral regions of the spectra are plotted as a function of rotor cycles Nc.

Unfortunately, the range of experimental points in the short evolution time range is limited to a maximum of 5 to 6 points and the SF ratio in that region is very low. Thus, the reliability of a fit is bad and an extraction of M2 would be not possible with adequate tolerances. However, the slope of each REDOR curve gives a good hint about the strength of heteronuclear dipolar coupling between Al and F or Al and H. The dipolar coupling between the octahedral sites and OH/F should be stronger due to the smaller distance to OH and fluorine groups compared to IVAl sites. The trend can be observed for all the REDOR curves. The slope of the octahedral REDOR curves is always higher than for the tetrahedral curve. In addition {1H} ↔ 27Al dipolar interaction is stronger than the {19F} ↔ 27Al dipolar interaction due to the higher magnetogyric ratio of 1H which can be also observed in the graphs resulting in an increased slope for the {1H} 27Al REDOR curves compared to the {19F} 27Al REDOR curves.

Satellite transition spectroscopy (SATRAS)

For many structure-clarifying questions the isotropic chemical shift σiso is of fundamental importance. σiso is accessible for spin one-half nuclei under MAS conditions directly from the spectra, since only interactions of first order are present whose anisotropic parts are averaged out by MAS.

For quadrupolar nuclei the second-order quadrupolar interactions must also be considered, which cause a broadening and shift of the resonances. The additional quadrupolar shift σQS prevents the determination of the isotropic chemical shift. In addition, it is often difficult to determine the quadrupolar coupling constant and asymmetry parameter from the central transition (e.g. in 27Al MAS NMR spectra) since these mostly show only asymmetrically broadened unstructured Lorentzian lineshapes.

Samoson (1985) showed that by the evaluation of the satellite transitions and the central transition one can estimate both the quadrupole coupling parameters and the isotropic chemical shift. The estimation uses the characteristic that the quadrupolar shift for central transition and satellite transitions differs, since it is dependent on the magnetic quantum number m (Eqn. 30).

Since the quadrupolar shifts of the central transition and the satellite transitions are different (m = 1/2 and m = 3/2, 5/2) one can determine the isotropic chemical shift σiso by the quadrupolar shift ratio from Equation 30. The position of the centre of gravity of the transition is the sum of both shifts.  

formula
 
formula
The ratio of linewidths is determined by  
formula
The ratios for the different nuclear spins and transitions are listed in Table 2. Figure 11 shows a 27Al MAS NMR spectrum of corundum. The central transition is the strong intensive signal in the spectrum centre at about 0 ppm. The first satellite transition is characterised by the side band pattern from 2000 to –2000 ppm while the second satellite transition can be observed mainly by sidebands in the range of 2500–4000 ppm. Both transitions overlap in their sideband pattern between 3000–1800 ppm but they are separated due to the different quadrupolar shifts.

The quadrupolar shift for the central transition is half as large as the shift for the first satellite transition of an I = 3/2 nucleus, while eight times higher for an I = 5/2. The isotropic chemical shift can be determined by estimation of the weighted arithmetic mean value of both shifts. σiso is calculated for:  

formula
 
formula
The centre of gravity of the first satellite transition is determined by extrapolation of the nth sideband to the sideband with n = 0 because the central transition is so strong that it covers the low-intensity satellite transition.

Table 2.

Relative ratios of isotropic quadrupolar shift σQS and line width Δ for a satellite transition (m) and the central transition (CT) for different nuclear spins I (Samoson, 1985).

IσQS(m)/σQS(CT)Δ(m)/Δ(CT)
m = 3/2m = 5/2m = 7/2m = 3/2m = 5/2m = 7/2

3/2−2.000−0.889
5/2−0.125−3.5000.292−1.833
7/20.400−1.400−4.4000.622−0.511−2.400
IσQS(m)/σQS(CT)Δ(m)/Δ(CT)
m = 3/2m = 5/2m = 7/2m = 3/2m = 5/2m = 7/2

3/2−2.000−0.889
5/2−0.125−3.5000.292−1.833
7/20.400−1.400−4.4000.622−0.511−2.400
Fig. 11.

27Al MAS SATRAS NMR spectrum of corundum (α-Al2O3). The signal area is enlarged. The central resonance at 0 ppm is cut off.

Fig. 11.

27Al MAS SATRAS NMR spectrum of corundum (α-Al2O3). The signal area is enlarged. The central resonance at 0 ppm is cut off.

The quadrupolar coupling parameters can be determined in a similar way. Forming the difference of the central and first satellite transition quadrupolar shift and insertion of Equation 30 gives the quadrupolar coupling parameter CQE:  

formula
 
formula
The quadrupolar parameter CQE can be determined for  
formula
and  
formula

Dynamic angle spinning (DAS), double rotation (DOR), and multiple quantum magic angle spinning (MQMAS)

In the section on quadrupolar interaction it becomes clear that for quadrupolar nuclei with interactions of second order only limited resolution can be achieved by the MAS technique. Quadrupolar powder pattern result, where it is difficult to distinguish different signal components in the case of quadrupolar resonance superposition.

The reason for this second-order broadening can be located in the characteristics of the time-dependent quadrupolar interaction spin evolution. The spin coherency phase evolution φ(m,β,t) of a spin system under the influence of the quadrupolar interaction of terms of first and second order can be described as follows:  

formula
with vCS be the isotropic chemical shift, forumla (θ,φ) containing the isotropic quadrupolar shift (n = 0) and the anisotropic angle-dependent quadrupolar shift terms (n = 2, 4) with β, θ and φ describing the angles between principal axes system of the quadrupolar tensor and the external magnetic field. Pn(cosβ) contains the angle-dependent Légendre polynomials  
formula
 
formula
 
formula
and forumla are zero-, second and fourth-rank coefficients depending on the spin I and the order m of the transition:  
formula
 
formula
 
formula
From Equation 65 it is obvious that no spinning axes can be found where both Equations 67 and 68 are zero. A residual anisotropic broadening will still remain. The problem can be resolved by spinning the sample temporarily at spinning axis β1 for a time t1 and β2 for a time t2 and allow the spin system to evolve at these times in a two-dimensional experiment. The technique is called Dynamic Angle Spinning (DAS) (Mueller et al., 1990) and angles and times are chosen so as to simultaneously fulfil the following averaging equations  
formula
 
formula
The technique was developed and realised at the beginning of the 1990's. The probe head is constructed similar as a MAS probe head but has a precession tilting mechanism which is able to move the stator from one rotation angle into the second angle in a short period, while the magnetisation is fixed in the two-dimensional experiment. The evolution time t1 is consequently incremented and the FID is detected during t2. Typically 30–50 ms are needed to turn the spinning axis in the DAS experiment. This limits the DAS experiment to samples with spin-relaxation times T1 of 100–150 ms and longer. In addition, the experiment suffers from residual line broadening by remaining homonuclear dipolar interaction. An advantage is that the sample can be spun at high spinning frequencies.

Another method averaging out the angular dependence in Equations 72 and 73 is the double rotation method (DOR). A time modulation which occurs continuously at DOR by simultaneous sample spinning about two different angles leads to a time-averaging of both functions during a rotation period (Samoson et al., 1988; Wu et al., 1990). The experiment is realised by a “rotor in the rotor” design. The inner-rotor spinning axis is aligned at a fixed angle to the outer rotor spinning axis with β2 = 70°34', where Equation 68 is zero. The outer rotor spins at an angle of β1 = 54°44', where Equation 67 is zero. For an interference-free rotation of the “double rotor” it is important that the precession of the inner rotor about the outer rotor axis is torque-free (Wu et al., 1990). This condition is fulfilled if  

formula
where Jy and Jz are the moments of inertia of the outer rotor and ω1 is the rotation frequency of the inner and ω2 the rotation frequency of the outer rotor. The rotor system does work properly if the ratio of both rotation frequencies does not change. This means that an increase of the outer rotor speed implies an increase of the inner rotor speed until the inner rotor cannot increase its speed for technical reasons anymore (bearing). A further increase of the outer rotor speed leads to a crash of the inner rotor. Thus, for reasons of technical limitations the outer rotor can reach a maximum frequency of ω2/2π= 1000–1200 Hz, resulting in a lot of spinning sidebands, which make it difficult to distinguish sidebands and signals. Moreover, only the anisotropic terms are averaged, the isotropic quadrupolar shift still exists. Two signals with the same sum of isotropic quadrupolar and isotropic chemical shift would resonate at the same frequency and could be only distinguished at a different magnetic field.

The question arises if it is possible to achieve a similar averaging of second-order effects without application of complicated mechanical manipulations. This can be done by spinning at a fixed angle, usually the magic angle, where spins are allowed to evolve during a time t1 and t2 under the effect of two different multi-quantum transitions m1 and m2 and in such a way that the following conditions are fulfilled.  

formula
 
formula
The proposed method is called Multiple Quantum Magic Angle Spinning (MQMAS) (Frydman & Harwood, 1995; Medek et al., 1995). There is a clear analogy between the “spin-space” refocusing of MQMAS and the “spatial-space” refocusing of the DAS and DOR technique (compare Eqns. 72/73 and 75/76).

When these considerations are implemented into a practical 2D Fourier transform (FT) NMR experiment, the value of m2 becomes confined to 1/2, because only single-quantum excitations are visible in NMR due to the selection rule. The experiment can be done by a simple two-pulse experiment, where the first pulse excites the multi-quantum transitions and the second one converts them to single-quantum excitations (Fig. 12). The lower graphics shows the coherence scheme with the coherence order p and the coherence pathways.

Fig. 12.

MQMAS two-pulse sequence and coherence transfer pathways (Frydman & Harwood, 1995).

Fig. 12.

MQMAS two-pulse sequence and coherence transfer pathways (Frydman & Harwood, 1995).

For the –1/2 → 1/2 transition the coherence order is p = –1. Only these transitions are visible, and at the beginning of an experiment the coherence order is always zero.

It is important to know that several transitions of different coherence orders are excited by a radio frequency pulse. For an I = 3/2 nucleus, transitions with p = +3, +2, +1, 0, –1, –2, –3 are excited.

In order to filter the desired multiple-quantum transitions, in this case those with p = +3 and p = –3, the following theoretical relationship can be applied:

A preparation pulse with a phase of Δφ produces n-quantum coherences with a phase shift of nΔφ. That means for a triple-quantum transition (n = 3) with a phase Δφ the preparation pulse produces a signal with a phase shift of 3Δφ. Application of a phase cycle using this fact leads to the summation of triple-quantum resonances, but to a subtraction of all other transitions with different coherence order. The second pulse converts the not detectable multi-quantum resonance into a detectable single quantum resonance and always possesses the phase Δφ = 0°. Therefore, the following phase cycle is applied in a two-pulse MQMAS experiment as shown in Figure 12 (Frydman & Harwood, 1995): 

formula

Figure 13 shows a 27Al MQMAS NMR spectrum of Al- and F-rich phlogopite at 104.27 MHz. The 27Al MQMAS NMR experiments correlate peaks in the MAS dimension (F2) to signals in the isotropic dimension (F1), with shifts in this dimension being a linear combination of isotropic chemical and second order quadrupolar shifts. The individual signals in the spectrum can be separated and assigned to different aluminium environments. The phlogopite signal in the tetrahedral region at δ(F2) = 68.0 ppm and δ(F1) = 39.3 ppm can be clearly assigned. A second signal at δ(F2) = 59.0 ppm and δ(F1) = 35.0 ppm appears and is due to a kalsilite impurity phase (KAlSiO4, see also Fechtelkord et al., 2003) and represents the tetrahedral aluminium coordination in this tectosilicate. In the octahedral region, two signals dominate. The signal at δ(F2) = 6.0 ppm and δ(F1) = 6.8 ppm is assigned to the octahedral Al in phlogopite. The resonance of the corundum impurity phase occurs at δ(F2) = 11.8 ppm and δ(F1) = 9.0 ppm. In addition, a very narrow signal at δ(F2) = –3.5 ppm and δ(F1) = –1.1 ppm appears, which results from the potassium aluminium hexafluoride phase. In a normal one-dimensional MAS NMR experiment these resonances would superimpose to a complex pattern where the distinct resonances cannot be distinguished.

Fig. 13.

27Al MQMAS NMR spectrum at 104.26 MHz of an Al-rich phlogopite with composition K(Mg3–xAlx)[Al1+xSi3–xO10](OH)y,(F)2–y, with x = 0.8 and y = 0.5. Spinning sidebands are marked by asterisks (Fechtelkord et al., 2003).

Fig. 13.

27Al MQMAS NMR spectrum at 104.26 MHz of an Al-rich phlogopite with composition K(Mg3–xAlx)[Al1+xSi3–xO10](OH)y,(F)2–y, with x = 0.8 and y = 0.5. Spinning sidebands are marked by asterisks (Fechtelkord et al., 2003).

Combined structure determination by Rietveld powder refinements and solid state NMR spectroscopy

Solid state NMR can be a useful support in Rietveld powder refinement and structure solution. Compared to X-ray diffraction, solid state NMR gives details about the short-range order (local structure) while X-ray diffraction shows only the long-range order of a structure. Thus, in some cases solid state NMR is able to give crucial information that cannot be achieved by X-ray diffraction. The following three examples will show how NMR can help to understand structures.

Information from the 207Pb chemical shift interaction about the lead phosphate structure

Pure lead phosphate, Pb3(PO4)2 undergoes a structural phase transformation at T = 453 K from a paraelastic, high-temperature phase with space group to a ferroelastic, monoclinic, low-temperature modification with space group C2/c. The structure of the paraphase consists of PO4 tetrahedra located along the threefold inversion axis. The tops of the tetrahedra point out to each other. The Pb cations are located on two different crystallographic positions. The Pb2 atoms occupy Wyckoff position 6c (site symmetry 3m) and Pb1 cations occupy Wyckoff position 3a (site symmetry forumla). Both positions are located on the threefold inversion axis. The room temperature phase differs from the paraphase by displacements of all Pb atoms perpendicular to the threefold inversion axis of the paraphase along the binary axis of the ferrophase. Figure 14 shows structural details of the monoclinic unit cell of Pb3(PO4)2 at room temperature (Guimaraes, 1979). Ten oxygen atoms coordinate Pb2 sites while the Pb1 cations are located inside a twelve-fold coordination sphere of oxygen atoms (Keppler, 1970) with a nominal ratio of 2:1. Comparing the local environment of the two lead sites, Pb2 shows very anisotropic coordination of oxygen atoms, while the Pb1 cations are nearly spherically coordinated. In addition, the Pb1 coordination polyhedron is somewhat flattened and the Pb2 environment is stretched along the [111] axis. While Pb1 is centred in the oxygen six-ring plane, the Pb2 cation is located outside this plane. Both cations are shifted from the threefold inversion axis.

Fig. 14.

Monoclinic unit cell of Pb3(PO4)2 with spacegroup C2/c based on the structural data of Guimaraes (1979).

Fig. 14.

Monoclinic unit cell of Pb3(PO4)2 with spacegroup C2/c based on the structural data of Guimaraes (1979).

207Pb possesses the highest atomic number of all the I = 1/2 nuclei. The large electron shell gives rise to a chemical shift interaction which is very sensitive to changes in the Pb local structure. The chemical shift anisotropy (CSA) varies strongly and the isotropic chemical shift covers a wide frequency scale. Thus, the large spectral widths give rise to problems of uniform excitation (Neue et al., 1996; Sebald, 1994). In addition, the isotropic chemical shift shows a temperature-dependent behaviour (Neue et al., 1996; Mildner et al., 1995; Bielecki & Burum, 1995). MAS experiments yield line positions dependent on the rotational frequency or show additional broadening if a temperature gradient along the probe is present (Mildner et al., 1995; Bielecki & Burum, 1995). Although 207Pb is very sensitive to its local environment NMR studies on 207Pb are rare.

The 207Pb MAS NMR spectrum and the simulated CSA pattern of lead phosphate Pb3(PO4)2 are presented in Figure 15 (Fechtelkord & Bismayer, 1998). The superposition of two CSA patterns reflects two crystallographic Pb sites. The first signal near –1700 ppm (corresponding to Pb2) is broad, covering a range of nearly 1000 ppm indicating large chemical shift anisotropy. The other signal near –2800 ppm (corresponding to Pb1) is about 100 ppm wide. The linewidths of the two CSA MAS pattern sidebands differ. Two different 207Pb lineshapes are also present in the static experiment (Fig. 15). The anisotropic Pb2 coordination leads to a large chemical shift anisotropy, while the nearly spherical Pb1 coordination causes only small chemical shift anisotropy.

Fig. 15.

Experimental 207Pb static and MAS NMR spectrum with corresponding simulated spectra of Pb3(PO4)2 (Fechtelkord & Bismayer, 1998).

Fig. 15.

Experimental 207Pb static and MAS NMR spectrum with corresponding simulated spectra of Pb3(PO4)2 (Fechtelkord & Bismayer, 1998).

Table 3 contains the simulation parameters of the MAS and static NMR experiments. Accuracies were estimated by varying the simulation parameters until a distinct change in the standard deviation between experimental and simulated lineshape was observed. The chemical shift parameters reflect the local symmetry of the Pb polyhedra and match the crystal structure. The different isotropic chemical shifts for the MAS and static experiment of –2881 and 2888 ppm for Pb1 and –2017 and –2001 ppm for Pb2, respectively (Fechtelkord & Bismayer, 1998), correspond to the different Pb coordination numbers of 10 and 12 (Fayon et al., 1997). The integrated intensities of the two resonances give a ratio of nearly 2:1 for both experiments, in agreement with the site occupancies from the crystal structure. The chemical shift anisotropy δaniso = –1650 ppm (MAS, Table 3) for Pb2 is about eight times higher than that for Pb1 (δaniso = 201 ppm) and has a negative sign (Fechtelkord & Bismayer, 1998). Compared with other compounds the chemical shift anisotropy for Pb1 is extremely low (Neue et al., 1996). The lower chemical shift anisotropy of Pb1 reflects the highly symmetric coordination of the Pb1 site. The larger Pb2 shift perpendicular to the threefold inversion axis of 0.58 Å in comparison to Pb1 with 0.41 Å (Guimaraes, 1979) finds expression in the higher asymmetry parameter η. The 207Pb shifts estimated for the MAS experiments for the pure lead phosphate are in good agreement with data from MAS experiments of Fayon et al. (1997).

Table 3.

207Pb MAS and static chemical shift parameters obtained from experiments at 296 K for Pb3(PO4)2 (Fechtelkord & Bismayer, 1998).

Atom siteSpectrumδiso (ppm)δaniso (ppm)ηarea (%)

Pb1MAS-2881±1201 ± 50.25 ± 0.0535 ± 5
Pb1static-2888 ± 3210 ± 40.24 ± 0.0531 ± 5
Pb2MAS-2017 ± 3-1650 ± 500.4 ± 0.165 ± 5
Pb2static-2001 ± 25-1575 ± 250.40 ± 0.0569 ± 5
Atom siteSpectrumδiso (ppm)δaniso (ppm)ηarea (%)

Pb1MAS-2881±1201 ± 50.25 ± 0.0535 ± 5
Pb1static-2888 ± 3210 ± 40.24 ± 0.0531 ± 5
Pb2MAS-2017 ± 3-1650 ± 500.4 ± 0.165 ± 5
Pb2static-2001 ± 25-1575 ± 250.40 ± 0.0569 ± 5

Sodium cation dynamics in the high- and room temperature structure of nitrate cancrinite

Cancrinite belongs to the group of feldspathoids and is found in nature with the ideal chemical composition Na6Ca(AlSiO46CO3·2H2O (Strunz, 1978). The structure of natural (Ca,Na)CO3 cancrinite was first refined by Jarchow (1965). The periodic arrangement of ε-cages in this hexagonal structure forms one-dimensional twelve-membered ring channels along the c axis. Structural data about framework expansion as well as template and cation dynamics at high temperatures are rare. Hassan (1996) assumed from differential thermogravimetry and lattice parameter determinations at various temperatures that the expansion of the framework is mainly caused by the shift of sodium cations towards the six-membered ring plane of the ε-cage. The effect has been explained by electrostatic rejection of the cations after evaporation of the cage water molecules which previously shielded the sodium cations.

23Na NMR spectroscopy is ideal to study the sodium dynamics and the change of the local environment of the observed nucleus. At room temperature it is impossible to distinguish between the two crystallographic sodium sites in the 23Na MAS NMR spectra. 23Na static and MAS NMR experiments were carried out at high temperatures to obtain information about sodium coordination changes and to find the reason for the detection of only one single line although two different sites are known to exist. The Rietveld X-ray powder data refinement at T = 673 K was used to determine the correct framework structure and the cation positions. A comparison between quadrupolar interaction from the 23Na MAS NMR experiment and the theoretical electric field gradient that was calculated from the structural data by a simple point charge model was carried out.

Figure 16 shows some structural details of nitrate cancrinite at room temperature and 673 K. The structural plot at room and high temperature is based on our own powder structure data refined in spacegroup P63 (Fechtelkord et al., 2001). The hexagonal cancrinite structure is formed by layers of six-membered rings stacked in an ABA'B' sequence. The edges of these two-dimensional layers are occupied by silicon and aluminium in a strongly alternating manner. Two structural features of cancrinite are the ε-cages and the twelve-membered ring channels parallel to the c axis. The channel contains two nitrate groups and six sodium cations per unit cell. The Na2 sodium cations are located on the general crystallographic 6c position. The NO3 group occupies two sites, with nitrogen on the 2a position and occupancy of one-half. Na1 is located on the threefold axis and occupies the special Wyckoff position 2b in the ε-cage. The cation is positioned close to the six-membered ring window of the ε-cage. Water molecules coordinate the sodium atoms and are positioned on general position 6c.

Fig. 16.

. (a) Structure of nitrate cancrinite Na8[AlSiO4]6(NO3)2·2H2O based on the Rietveld structure refinement data of Fechtelkord et al. (2001) at T = 296 K and (b) structural details of the Na1 coordination sphere at T = 673 K (Fechtelkord et al., 2001).

Fig. 16.

. (a) Structure of nitrate cancrinite Na8[AlSiO4]6(NO3)2·2H2O based on the Rietveld structure refinement data of Fechtelkord et al. (2001) at T = 296 K and (b) structural details of the Na1 coordination sphere at T = 673 K (Fechtelkord et al., 2001).

Figure 17 shows the 23Na MAS NMR experiments at variable temperatures (Fechtelkord et al., 2001). At room temperature the 23Na MAS NMR spectrum shows a narrow asymmetric Lorentzian lineshape. It gets broader with increasing temperature and at T = 673 K the resonance exhibits the shape of a quadrupolar powder pattern. Cooling of the sample and recording the spectra at the same temperatures as during the heating period shows similar lineshapes for all temperatures (Fechtelkord et al., 2001).

Fig. 17.

23Na MAS NMR spectra of nitrate cancrinite at high temperatures. Temperatures are given in the figure (Fechtelkord et al., 2001).

Fig. 17.

23Na MAS NMR spectra of nitrate cancrinite at high temperatures. Temperatures are given in the figure (Fechtelkord et al., 2001).

Examining the static spectra in detail gives rise to the assumption that the total lineshape above 673 K is a superposition of two components. All static spectra show a similar behaviour during the heating cycle compared to the MAS spectra. Beginning at room temperature a broad Lorentzian lineshape exists until a temperature of 673 K is reached. Above 673 K two components seem to dominate the spectrum. After cooling the sample to room temperature a broad quadrupolar lineshape appears at the base of the Lorentzian lineshape. The quadrupolar pattern has an isotropic chemical shift of δiso = –16 ± 5 ppm, a quadrupolar coupling constant of CQ = 4.6 ± 0.1 MHz, and an asymmetry parameter of η = 0.0 ± 0.1 can be detected. The ratio of the two lineshape areas is 1:3. The broad quadrupolar pattern cannot be detected in the low-spinning MAS NMR spectra, since the broad and low intensive quadrupolar resonance is covered by spinning sidebands.

The static 23Na and MAS NMR spectra collected at 673 K and 773 K suggest the existence of two signal components corresponding to the two different Na sites. A simulation of the MAS NMR spectra is difficult due to the low spinning frequency, because the main signal contains less than half of the total central transition intensity while the residual intensity is distributed among the complicated sideband pattern caused. A solution of this problem and the extraction of reliable quadrupolar interaction parameters can be done by using the MAS and static NMR spectra at 673 K and 773 K for lineshape simulations (Fechtelkord et al., 2001). The estimated interaction parameters for both resonances from the MAS and static spectra should be similar at the same temperature. It should be also considered that the MAS sideband pattern contains more than one half of the total central transition intensity. Finally, the relative ratio of the two lineshapes should be in the same range for all simulations. Figure 18 shows the 23Na MAS NMR and static spectra at 773 K with their least-squares fit (Fechtelkord et al., 2001). The extracted quadrupolar parameters for T = 673 K are listed in Table 4. The estimated relative ratio of the two Na signal components is 1/3. Thus, the Na1 site with lower occupancy (Na1/Na2 = 1/3) can be attributed to the quadrupolar pattern with the lower asymmetry. The lower asymmetry of the lineshape allows a more precise estimation of the quadrupolar interaction parameters for the Na1 site than for Na2.

Fig. 18.

23Na MAS and static NMR high-temperature spectra of nitrate cancrinite at 773 K. Experimental spectrum, overall least-squares fit (simulated lineshape), Na2 quadrupolar pattern including spinning sidebands (component 1) and Na1 quadrupolar pattern including spinning sidebands (component 2) (Fechtelkord et al., 2001).

Fig. 18.

23Na MAS and static NMR high-temperature spectra of nitrate cancrinite at 773 K. Experimental spectrum, overall least-squares fit (simulated lineshape), Na2 quadrupolar pattern including spinning sidebands (component 1) and Na1 quadrupolar pattern including spinning sidebands (component 2) (Fechtelkord et al., 2001).

The Na2 cation shows a high asymmetry parameter of 0.7–0.8 (± 0.2) and a CQ between 1.4 and 2.1 (± 0.2) MHz. The high asymmetry and the strong broadening of the Na2 signal form lower the parameter accuracy. Furthermore, it is hard to separate sideband intensities from centre band intensities in the MAS NMR spectra. The experimental MAS NMR data show a poor signal-to-noise ratio and the signal forms give only little information about the correct position and width of the Na2 component. Considering the cancrinite structure, it is more likely that the cations in the cages (2 per unit cell) produce a quadrupolar lineshape with axial symmetry while the channel cations (6 per unit cell) should give rise to a quadrupolar pattern with higher asymmetry parameter.

Table 4.

Estimated experimental and theoretical calculated 23Na NMR quadrupolar parameters CQ and η for nitrate cancrinite at T = 296 K and 673 K (Fechtelkord et al., 2001).

TCation siteTheoreticalExperimental
CQ (MHz)ηCQ (MHz)η

296 KNa10.2 ± 0.50.0 ± 0.1n/an/a
Na21.2 ± 0.50.4 ± 0.1n/an/a
673 KNa14.2 ± 0.50.1 ± 0.12.3 ± 0.20.2 ± 0.2
Na21.7 ± 0.50.5 ± 0.11.4 ± 0.10.7 ± 0.2
TCation siteTheoreticalExperimental
CQ (MHz)ηCQ (MHz)η

296 KNa10.2 ± 0.50.0 ± 0.1n/an/a
Na21.2 ± 0.50.4 ± 0.1n/an/a
673 KNa14.2 ± 0.50.1 ± 0.12.3 ± 0.20.2 ± 0.2
Na21.7 ± 0.50.5 ± 0.11.4 ± 0.10.7 ± 0.2

n/a: not estimable

The previous investigations, however, do not answer the question whether the evaporation of the crystal water or local structural changes cause the strong quadrupolar interaction at temperatures above 673 K. To verify if structural changes initiate the change in the 23Na NMR spectra at temperatures above 673 K, Rietveld structure refinements were carried out at room temperature (296 K) and at 673 K. All refinements at room temperature and at 673 K were performed with the GSAS program package (Larson & van Dreele, 1985) in the hexagonal space group P63.

The difference between the room and high-temperature structure is mainly characterised in the thermal expansion of the framework at elevated temperature (Fechtelkord et al., 2001). The ε-cages experience nearly no change. In general, the Na1 ion shows the same distances to the framework oxygen atoms of the six-membered ring at both temperatures, which is in contrast to the assumption of Hassan (1996). He concluded from thermogravimetric and X-ray data that the Na1 cation is shifted towards the six-membered ring window. However, the Na2 site possesses larger distances to the other Na2 positions in the channel.

Estimation of the theoretical quadrupolar interaction using a simple point-charge model for both sodium sites at T = 673 K and 296 K for the given structure should give more details about the changes observed in the experimental spectrum (Koller et al., 1994). The atomic positions of framework and nitrate oxygen atoms as well as both Na sites (Na1 and Na2) were taken from the Rietveld refinements. Oxygen charges were estimated using the bond valence model of Brown and coworkers (Brown & Altermatt, 1985; Brown & Shannon, 1973). The calculated quadrupolar parameters are listed in Table 4 (Fechtelkord et al., 2001).

The theoretical values support the results from the experimental spectra very well. Hydrated and at room temperature both quadrupolar coupling constants for Na1 and Na2 are nearly equal in the limits of tolerance. A quadrupolar pattern cannot be observed, because an EFG distribution in cancrinite due to stacking order faults and cation motion is quite common. In contrast, the calculated values for Na2 increase only slightly from 1.15 MHz to 1.71 MHz and show a high asymmetry parameter of 0.5–0.6 which implies a similar coordination sphere of the Na2 site at high temperatures.

The quadrupolar coupling constant for Na1 increases from 0.71 MHz to 4.20 MHz at high temperatures. That differs from the experimental data with an estimated value of CQ = 2.4 ± 0.2 MHz. The strong increase for the Na1 quadrupolar interaction is caused as a result of dehydration at high temperatures which causes the subsequent loss of the electrostatic shielding effect of the water molecules. The difference between the experimental and theoretical results may be explained by a partial dynamic averaging of quadrupolar interaction by sodium dynamics. The isotropic displacement parameter Uiso in the powder structure refinements, which is an indicator for thermal dynamics, shows a strong increase for both sodium positions at 673 K. However, results from point-charge model calculations are only valid for static conditions and quadrupolar coupling constants cannot be estimated in the case of thermal sodium dynamics.

The presence of significant thermal dynamics is supported by the formation of a broad quadrupolar pattern during the cooling of the sample which slows down the thermal motion causing a stronger quadrupolar interaction. The dynamics from the fast motional to the static case passes an intermediate coalescence state that can be observed at 573 K in the static experiment (Fechtelkord et al., 2001).

Order and disorder of anions in carbonate and formate mixed sodalites

Most sodalites are synthesised under hydrothermal conditions in a basic aqueous sodium hydroxide solution. Sodalites containing carbonate as template could only be made by thermal treatment of nitrite sodalite in a carbon dioxide atmosphere (Buhl, 1991, 1993) or by oxidation of formate or acetate sodalite (Sieger et al., 1995).

The sodalite structure can be described by the space-filling array of a [4668]-truncated octahedron known as sodalite or β-cage. Each sodalite cage contains guest constituents like anions and/or water molecules. In the case of monovalent anions inside the sodalite, every β-cage contains one anion placed in the centre and four sodium cations at sites close to the six-membered ring openings, forming a tetrahedron. The unit cell contains two sodalite cages. Since there is a strong alternating order of silicon and aluminium atoms in the framework the cubic space group is forumla.

In the case of divalent anions like carbonate forumla the scenario is more complicated. Now only every second cage is occupied by an anion with the sum formula Na8[AlSiO4]6X(-2). The question of order and disorder of the anions inside the cages has to be discussed now. If the anions are distributed over the whole structure in a strong alternating order the space group symmetry will be P23. Sodalites with ordered divalent guest anions are usually called “noseans”. Randomly distributed anions over the whole structure cause an “average” structure and the higher symmetry of space group forumla would result, as in the case of monovalent anions.

Several structure investigations were performed on the question of order and disorder of sulphate in natural nosean and discussed in a controversial manner. Machatschki (1934) proposed a statistical sulphate distribution over the structure. Hassan & Grundy (1989) also assumed total disorder of sulphate and performed structural refinements in the high-symmetry space group forumla. However, additional reflections give rise to the existence of antiphase domain boundaries with lower space group symmetry of P23 inside the domains, a subgroup symmetry of forumla. Other authors (Saalfeld, 1959; Schulz & Saalfeld, 1965; Schulz, 1970) suggested a total ordering and lower space group symmetry. The aim of the current subchapter is to show that the structure of carbonate containing sodalite in the “as-synthesised” and the calcined product can be determined by combination of 1H, {1H} 13C CPMAS NMR, 23Na MAS NMR, 29Si MAS NMR, and high-temperature 1H and 23Na NMR spectroscopy studies with Rietveld powder structure refinements of the “as-synthesised” sodalite and the dehydrated calcined sodalite (Fechtelkord, 1999).

The “as-synthesised” product was synthesised hydrothermally in methanol using sodium methylate as base and sodium carbonate as guest anion. Afterwards the “as-synthesised” product was calcined at 873 K for 1–6 h. Examination of the X-ray powder diffraction pattern of the “as-synthesised” and the calcined sodalite showed important differences. The powder pattern of the “as-synthesised” sodalite shows only reflections corresponding to a random disordered distribution of carbonate inside the β-cages with spacegroup forumla. New reflections appear in the diffractogram of the calcined sodalite. The strongest additional reflection that can be observed is the 001 reflection. These additional reflections suggest an ordering process of the carbonate in the structure during heating, corresponding to the space group P23.

{1H} 13C CPMAS NMR and 1H MAS NMR experiments were applied for the identification of carbonate or organic residuals in the products. The {1H} 13C CPMAS NMR spectrum of the “as-synthesised” sample shows a resonance at 170.1 ppm with a shoulder at 172 ppm due to carbonyl carbon atoms (Fechtelkord, 1999). Methylen or methyl resonances were not present. The corresponding 1H MAS NMR spectrum in Figure 19 (Fechtelkord, 1999) shows a narrow signal at 8.9 ppm and superposition of two resonances at 4 ppm. The signal intensity at 8.9 ppm is about four times higher than that at 4 ppm. For the calcined product only a single signal at 172.2 ppm at the former position of the shoulder in the spectrum of the “as-synthesised” sodalite can be observed in the {1H} 13C CPMAS NMR spectrum. The signal at 8.9 ppm in the 1H MAS NMR spectrum of the calcined sodalite has vanished. The NMR spectroscopic data give evidence for the formation of formate during synthesis. The signal at 170.1 ppm in the {1H}13C CPMAS NMR spectrum and at 8.9 ppm in the 1H MAS NMR spectrum is assigned to the carbonyl atom and proton in formate. The results are in agreement with NMR spectroscopic studies of Sieger et al. (1995) and Fechtelkord et al. (1997). The instability of sodium methylate under pressure was first noticed by Tropsch & von Philipovic (1926), who observed the formation of formate from sodium methylate in a strong basic milieu under high pressure and temperature conditions. Calcination of the sodalite leads to oxidation of formate to carbonate, and carbon dioxide so that the corresponding signals in the {1H}13C CPMAS NMR spectrum and 1H MAS NMR spectrum vanish. The remaining signals can be assigned to carbonate (172.2 ppm) in the {1H}13C CPMAS NMR spectrum and to water (4.0 ppm) in the 1H MAS NMR spectrum (Fechtelkord, 1999).

Fig. 19.

1H MAS NMR spectrum of the “as-synthesised” sodalite (Fechtelkord, 1999).

Fig. 19.

1H MAS NMR spectrum of the “as-synthesised” sodalite (Fechtelkord, 1999).

The 29Si MAS NMR spectrum of the “as-synthesised” product shows only a single line at –87.8 ppm. The resonance shifts to lower ppm values after calcination, which can be interpreted as an increase of the unit cell volume. The 23Na MAS NMR spectrum of the “as-synthesised” sodalite shows a single asymmetric lineshape at –7.3 ppm and can be assigned to sodium sites in formate filled cages (Sieger et al., 1995). The mixed cage occupation with carbonate/formate causes a statistical distribution of the 23Na chemical shift and quadrupolar interaction parameters. Signals of sodium ions from carbonate-filled cages cannot be observed because of the dominance of formate-filled cages.

Calcination changes the 23Na MAS NMR lineshape (Fig. 20). A lineshape difference depending on the calcination time was not noticed (Fechtelkord, 1999). The resulting 23Na MAS NMR spectrum can be simulated by a superposition of a Lorentzian line at –13.0 ppm and a quadrupolar pattern with an isotropic chemical shift of δiso = 7.6 ppm, a quadrupolar coupling constant of CQ = 2.68 MHz and an asymmetry parameter of η = 0.2 (Fechtelkord, 1999). The estimated ratio of signal areas is 30:70. The two components are assigned to two different local sodium environments. One-half of all sodalite cages are filled with four water molecules while residual cages contain a carbonate anion. The Lorentzian lineshape is assigned to sodium cations coordinated by water molecules and thus these cations experience a low quadrupolar interaction. The quadrupolar pattern is due to sodium cations in carbonate filled cages. Surprisingly the estimated area ratio is 30:70 and not 50:50 as expected. Usually every cage should be occupied by four sodium cations. Nevertheless the ratio of areas obtained from the 23Na MAS NMR fit gives evidence for the existence of additional Na cations inside the carbonate-filled cages for reasons of charge balance. The confirmation of this assumption is found by Rietveld powder structure refinements.

Fig. 20.

Plot of the experimental 23Na MAS NMR spectrum of the re-hydrated calcined sodalite; overall fit (simulated lineshape), Lorentzian lineshape (component 1) and quadrupolar pattern (component 2) (Fechtelkord, 1999).

Fig. 20.

Plot of the experimental 23Na MAS NMR spectrum of the re-hydrated calcined sodalite; overall fit (simulated lineshape), Lorentzian lineshape (component 1) and quadrupolar pattern (component 2) (Fechtelkord, 1999).

The conversion of mixed formate-carbonate-filled sodalite to pure carbonate sodalite was investigated by high-temperature MAS NMR studies. The 1H MAS NMR spectra show a broad signal at 4.0 ppm at ambient temperature due to water and a formate signal at 8.9 ppm. At a temperature of 373 K the water signal loses intensity because of dehydration. The formate proton signal is now clearly visible (Fechtelkord, 1999). With increasing temperature the signal at 8.9 ppm becomes narrower and increases in signal height until a temperature of 773 K is reached. The narrowing process is explained by the increasing motion of the anion. A sudden collapse of the resonance is observed at 773 K and is interpreted as the beginning combustion of the formate anion to carbonate, carbon dioxide and water. The combustion of formate is completed at 873 K because no further proton signal is detected.

Rietveld structure refinements were carried out to explain the structural differences between mixed formate-carbonate sodalite and pure carbonate sodalite, to investigate the distribution ratio of both anions, and the grade of carbonate ordering using the GSAS program package (Larson & von Dreele, 1985).

The structure refinement for the mixed formate-carbonate sodalite was carried out in the cubic space group forumla. Since the type of cage filling has no influence on the initial positions of framework atoms, the typical positions for silicon, aluminium and framework oxygen were taken from many other structure refinements of sodalite. The NMR spectroscopic results indicate a mixed formate-carbonate filling inside the sodalite β-cages. It must be considered that cage filling takes place in two different ways. Every cage contains one anion and four sodium cations if the monovalent formate is present. The structure refinement of Sieger et al. (1995) served as a starting model. The formate oxygen OC2 was placed on general position 24i with a relative occupancy of 1/6 (0.894, 0.923, 0.021) and the carbon C2 was set on special position 12f (0.042, 0, 0) with the same relative occupancy. Restraints were applied for the C–O bond lengths (1.30 Å ± 0.05 Å) and O–C–O angles (120° ± 2°). The sodium atom Na2 was positioned on the threefold axis on position 8e (0.190, 0.190, 0.190) with full occupancy. The divalent carbonate anion cage filling is more complicated. In general, one cage contains forumla and the other four water molecules for reasons of charge balance. Two crystallographically different sodium sites result on the 8e position for both types of cage filling species. The carbonate carbon C1 was placed on the 2a position (0, 0, 0) with a relative occupancy of 0.5 and the carbonate oxygen atoms OC1, which show a twelve-fold disorder, on general position 24i (0.060, 0.948, 0.111) with a relative occupancy of 0.125 (Sieger et al., 1995). The sodium site Na1 of the carbonate cage is found on 8e (0.1859, 0.1859, 0.1859) with a relative occupancy of 0.5. The sodium site Na3 of the water containing cage is located on 8e (0.1504, 0.1504, 0.1504) with the same relative occupancy (Felsche et al., 1986). The water oxygen atom OH1 can be found on 3/8, 3/8, 3/8 with an occupancy factor of 0.5.

Preceding the refinement the occupancies of carbon, oxygen and sodium sites were correlated so that the occupancy decrease of two formate anions and the corresponding sodium cation leads to a substitution with one carbonate anion, four water molecules and the corresponding sodium positions. After a few refinement cycles it could be seen to emerge that a distinction of the Na1 and Na3 atom coordinates is not possible. Both sites were combined in sodium site Na1 with the apparent double occupancy. The displacement parameters of Na1 and Na2, Si1 and Al1, and C1 and C2 were restrained to be equal. The values of OC1 and OC2 were fixed at a value of uiso = 0.02.

The refinement converged with Rp = 0.0238, Rwp = 0.0344, and R(F2) = 0.0475. Atomic parameters are listed in Fechtelkord (1999). The carbon-oxygen distances for carbonate and formate are in the range of the ideal bond lengths. The estimated sum formula is Na8[AlSiO4]6(CO3)0.15(HCOO)1.7(H2O)0.6 and corresponds to the results obtained from thermogravimetry (Fechtelkord, 1999).

Structure refinement was also performed for the calcined fully carbonate containing sodalite. To simplify the refinement, the sodalite was dehydrated at 673 K and filled into a closed capillary afterwards to exclude water molecules in the refinement. The existence of the 001 reflection is only consistent with space group symmetry P23.

The question of full or partial order of carbonate over the structure has to be discussed now. The starting atomic positions were taken from Sieger et al. (1995). Silicon and aluminium were set on positions 6g and 6h. Two different framework oxygen atoms O1 and O2 exist in subgroup P23 on general position 12j. The carbonate atoms C1 and C2 were positioned in the centre of each β-cage (position 1a and 1b, respectively) with a starting occupancy of 0.5. The carbonate oxygen OC1 and OC2 are located on general position 12j (0.060, 0.948, 0.111; x + ½, y + ½, z + ½). The bond lengths (1.30 Å ± 0.05 Å) and angles (120° ± 2°) were restrained. Three sodium positions on (0.195, x, x), (0.8, x, x) and (0.7, x, x) (Wyckoff position 4e) were found by difference Fourier analysis after a first refinement of the framework atoms. The occupancies were set to 0.3333. In subsequent cycles the occupancies of Na, C, and OC were refined. It was noticed that occupancies of C2 and OC2 tend to zero. Full order of the carbonate anions inside the structure can be concluded and so C2 and OC2 were omitted in the next refinement cycles. The displacement parameters were refined in the last cycles.

The refinement converged with Rp = 0.0247, Rwp = 0.0359, and R(F2) = 0.0368. Atomic parameters are listed in Fechtelkord (1999). It is interesting that carbonate is principally coordinated by two sodium sites Na1 and Na3, while Na2 is coordinated by water as shown in Figure 21. The results are in agreement with Sieger et al. (1995), who refined a pure carbonate sodalite and found a third sodium site. In addition, the half width of the 001 reflection in our X-ray powder pattern is about one third larger than the average half width of all other reflections. A formation of domain boundaries during the calcination process could explain this deviation. For optimisation of charge balance one-half of the sodium cations of the free cage should be located partially in the carbonate containing cages. In addition the 23Na MAS NMR lineshape of the hydrated fully carbonate containing sodalite gives evidence for that assumption. From the simulations it is found that 70 % of the sodium cations coordinate carbonate and 30% coordinate water (Figure 20). To examine the question a final structure refinement of the hydrated calcined product was carried out. The atomic positional parameters were taken from the refinement of the dehydrated carbonate sodalite and an additional water oxygen position was placed on the threefold axis (Wyckoff position 4e; 0.38, x, x). Occupancies of Na and OH1 as well as the atomic parameters and the displacement parameters were refined.

Fig. 21.

Structural details of the β-cage of dehydrated carbonate sodalite containing the carbonate anion and the coordinating two different sodium sites (Fechtelkord, 1999).

Fig. 21.

Structural details of the β-cage of dehydrated carbonate sodalite containing the carbonate anion and the coordinating two different sodium sites (Fechtelkord, 1999).

The refinement converged with Rp = 0.0258, Rwp = 0.0345, and R(F2) = 0.0386. Atomic parameters are listed in Fechtelkord (1999). The Na1, Na2 and Na3 occupancies show clearly that the ratio of sodium coordinated by carbonate to sodium coordinated by water is 72:28 and is in agreement with the 23Na MAS NMR experiments.

Concluding remarks

The current chapter gave a short overview about today's NMR techniques and how NMR is able to support powder diffraction and structure refinement. However, a full insight into the state-of-the-art technique of solid state NMR spectroscopy and a listing of all supporting possibilities is impossible. Solid state NMR presents a wide-range method which can be included in several areas and is assumed to develop with great progress in the next years.

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Financial support by the Deutsche Forschungsgemeinschaft (DFG) is gratefully acknowledged. The author also thanks the Alexander von Humboldt foundation for a Feodor Lynen Research Fellowship.

Figures & Tables

Fig. 1

Chemical shift anisotropy lineshapes under static conditions for σaniso = 0, σaniso > 0, and σaniso < 0. The asymmetry parameter η is zero and η = 0.4.

Fig. 1

Chemical shift anisotropy lineshapes under static conditions for σaniso = 0, σaniso > 0, and σaniso < 0. The asymmetry parameter η is zero and η = 0.4.

Fig. 2

Dipolar interaction axes system with internuclear vector forumla and angle Θ between connecting axis and static magnetic field.

Fig. 2

Dipolar interaction axes system with internuclear vector forumla and angle Θ between connecting axis and static magnetic field.

Fig. 3

Quadrupolar lineshapes under static and MAS conditions with CQ > 0. The asymmetry parameter η is zero and η = 0.5.

Fig. 3

Quadrupolar lineshapes under static and MAS conditions with CQ > 0. The asymmetry parameter η is zero and η = 0.5.

Fig. 4

Coordinate system scheme and transformation operators for the MAS technique.

Fig. 4

Coordinate system scheme and transformation operators for the MAS technique.

Fig. 5

Solid echo pulse scheme. The first pulse is a 90° pulse. A 180° pulse follows after a time τ1.

Fig. 5

Solid echo pulse scheme. The first pulse is a 90° pulse. A 180° pulse follows after a time τ1.

Fig. 6

T1 relaxation time curve in half logarithmic scaling as a function of the reciprocal temperature of tetramethyl ammonium iodide. The Larmor frequency is v0 = 15 MHz.

Fig. 6

T1 relaxation time curve in half logarithmic scaling as a function of the reciprocal temperature of tetramethyl ammonium iodide. The Larmor frequency is v0 = 15 MHz.

Fig. 7

Cross-polarisation pulse scheme. The X nucleus is the observed nucleus and 1H the decoupled nucleus.

Fig. 7

Cross-polarisation pulse scheme. The X nucleus is the observed nucleus and 1H the decoupled nucleus.

Fig. 8

Theoretical REDOR curve for a 13C-15N two-spin system with a dipolar coupling constant of DHD = 1/Tr and one dephasing pulse in each half of a rotor cycle (P = 2) (Guillon & Schaefer, 1989).

Fig. 8

Theoretical REDOR curve for a 13C-15N two-spin system with a dipolar coupling constant of DHD = 1/Tr and one dephasing pulse in each half of a rotor cycle (P = 2) (Guillon & Schaefer, 1989).

Fig. 9.

Representative {19F} 27Al (left) and {1H} 27Al REDOR spectra (right) of two Al-rich phlogopites with composition K(Mg3-iAlx)[Al1+iSi3-iO10](OH)γ,(F)2-γ,. × and y are as indicated in the figure.

Fig. 9.

Representative {19F} 27Al (left) and {1H} 27Al REDOR spectra (right) of two Al-rich phlogopites with composition K(Mg3-iAlx)[Al1+iSi3-iO10](OH)γ,(F)2-γ,. × and y are as indicated in the figure.

Fig. 10.

{19F} 27Al REDOR curves (a) and {1H} 27Al REDOR curves (b) of an Al-rich phlogopite with composition K(Mg3–xAlx)[Al1+iSi3–xO10](OH)γ,(F)2–γ, with x = 0.8 and (a) y = 0,5, (b) y = 1,8 The integrated intensity ratios of the tetrahedral and octahedral regions of the spectra are plotted as a function of rotor cycles Nc.

Fig. 10.

{19F} 27Al REDOR curves (a) and {1H} 27Al REDOR curves (b) of an Al-rich phlogopite with composition K(Mg3–xAlx)[Al1+iSi3–xO10](OH)γ,(F)2–γ, with x = 0.8 and (a) y = 0,5, (b) y = 1,8 The integrated intensity ratios of the tetrahedral and octahedral regions of the spectra are plotted as a function of rotor cycles Nc.

Fig. 11.

27Al MAS SATRAS NMR spectrum of corundum (α-Al2O3). The signal area is enlarged. The central resonance at 0 ppm is cut off.

Fig. 11.

27Al MAS SATRAS NMR spectrum of corundum (α-Al2O3). The signal area is enlarged. The central resonance at 0 ppm is cut off.

Fig. 12.

MQMAS two-pulse sequence and coherence transfer pathways (Frydman & Harwood, 1995).

Fig. 12.

MQMAS two-pulse sequence and coherence transfer pathways (Frydman & Harwood, 1995).

Fig. 13.

27Al MQMAS NMR spectrum at 104.26 MHz of an Al-rich phlogopite with composition K(Mg3–xAlx)[Al1+xSi3–xO10](OH)y,(F)2–y, with x = 0.8 and y = 0.5. Spinning sidebands are marked by asterisks (Fechtelkord et al., 2003).

Fig. 13.

27Al MQMAS NMR spectrum at 104.26 MHz of an Al-rich phlogopite with composition K(Mg3–xAlx)[Al1+xSi3–xO10](OH)y,(F)2–y, with x = 0.8 and y = 0.5. Spinning sidebands are marked by asterisks (Fechtelkord et al., 2003).

Fig. 14.

Monoclinic unit cell of Pb3(PO4)2 with spacegroup C2/c based on the structural data of Guimaraes (1979).

Fig. 14.

Monoclinic unit cell of Pb3(PO4)2 with spacegroup C2/c based on the structural data of Guimaraes (1979).

Fig. 15.

Experimental 207Pb static and MAS NMR spectrum with corresponding simulated spectra of Pb3(PO4)2 (Fechtelkord & Bismayer, 1998).

Fig. 15.

Experimental 207Pb static and MAS NMR spectrum with corresponding simulated spectra of Pb3(PO4)2 (Fechtelkord & Bismayer, 1998).

Fig. 16.

. (a) Structure of nitrate cancrinite Na8[AlSiO4]6(NO3)2·2H2O based on the Rietveld structure refinement data of Fechtelkord et al. (2001) at T = 296 K and (b) structural details of the Na1 coordination sphere at T = 673 K (Fechtelkord et al., 2001).

Fig. 16.

. (a) Structure of nitrate cancrinite Na8[AlSiO4]6(NO3)2·2H2O based on the Rietveld structure refinement data of Fechtelkord et al. (2001) at T = 296 K and (b) structural details of the Na1 coordination sphere at T = 673 K (Fechtelkord et al., 2001).

Fig. 17.

23Na MAS NMR spectra of nitrate cancrinite at high temperatures. Temperatures are given in the figure (Fechtelkord et al., 2001).

Fig. 17.

23Na MAS NMR spectra of nitrate cancrinite at high temperatures. Temperatures are given in the figure (Fechtelkord et al., 2001).

Fig. 18.

23Na MAS and static NMR high-temperature spectra of nitrate cancrinite at 773 K. Experimental spectrum, overall least-squares fit (simulated lineshape), Na2 quadrupolar pattern including spinning sidebands (component 1) and Na1 quadrupolar pattern including spinning sidebands (component 2) (Fechtelkord et al., 2001).

Fig. 18.

23Na MAS and static NMR high-temperature spectra of nitrate cancrinite at 773 K. Experimental spectrum, overall least-squares fit (simulated lineshape), Na2 quadrupolar pattern including spinning sidebands (component 1) and Na1 quadrupolar pattern including spinning sidebands (component 2) (Fechtelkord et al., 2001).

Fig. 19.

1H MAS NMR spectrum of the “as-synthesised” sodalite (Fechtelkord, 1999).

Fig. 19.

1H MAS NMR spectrum of the “as-synthesised” sodalite (Fechtelkord, 1999).

Fig. 20.

Plot of the experimental 23Na MAS NMR spectrum of the re-hydrated calcined sodalite; overall fit (simulated lineshape), Lorentzian lineshape (component 1) and quadrupolar pattern (component 2) (Fechtelkord, 1999).

Fig. 20.

Plot of the experimental 23Na MAS NMR spectrum of the re-hydrated calcined sodalite; overall fit (simulated lineshape), Lorentzian lineshape (component 1) and quadrupolar pattern (component 2) (Fechtelkord, 1999).

Fig. 21.

Structural details of the β-cage of dehydrated carbonate sodalite containing the carbonate anion and the coordinating two different sodium sites (Fechtelkord, 1999).

Fig. 21.

Structural details of the β-cage of dehydrated carbonate sodalite containing the carbonate anion and the coordinating two different sodium sites (Fechtelkord, 1999).

Table 1.

Interaction Hamiltonians under application of a solid state echo (see Figure 5) before and after the 180° pulse during τ1 and τ2.

InteractionDuring τ1during τ2

Homonuclear dipolarforumlaforumla
Heteronuclear dipolarforumla
Chemical shiftforumlaforumla
InteractionDuring τ1during τ2

Homonuclear dipolarforumlaforumla
Heteronuclear dipolarforumla
Chemical shiftforumlaforumla
Table 2.

Relative ratios of isotropic quadrupolar shift σQS and line width Δ for a satellite transition (m) and the central transition (CT) for different nuclear spins I (Samoson, 1985).

IσQS(m)/σQS(CT)Δ(m)/Δ(CT)
m = 3/2m = 5/2m = 7/2m = 3/2m = 5/2m = 7/2

3/2−2.000−0.889
5/2−0.125−3.5000.292−1.833
7/20.400−1.400−4.4000.622−0.511−2.400
IσQS(m)/σQS(CT)Δ(m)/Δ(CT)
m = 3/2m = 5/2m = 7/2m = 3/2m = 5/2m = 7/2

3/2−2.000−0.889
5/2−0.125−3.5000.292−1.833
7/20.400−1.400−4.4000.622−0.511−2.400
Table 3.

207Pb MAS and static chemical shift parameters obtained from experiments at 296 K for Pb3(PO4)2 (Fechtelkord & Bismayer, 1998).

Atom siteSpectrumδiso (ppm)δaniso (ppm)ηarea (%)

Pb1MAS-2881±1201 ± 50.25 ± 0.0535 ± 5
Pb1static-2888 ± 3210 ± 40.24 ± 0.0531 ± 5
Pb2MAS-2017 ± 3-1650 ± 500.4 ± 0.165 ± 5
Pb2static-2001 ± 25-1575 ± 250.40 ± 0.0569 ± 5
Atom siteSpectrumδiso (ppm)δaniso (ppm)ηarea (%)

Pb1MAS-2881±1201 ± 50.25 ± 0.0535 ± 5
Pb1static-2888 ± 3210 ± 40.24 ± 0.0531 ± 5
Pb2MAS-2017 ± 3-1650 ± 500.4 ± 0.165 ± 5
Pb2static-2001 ± 25-1575 ± 250.40 ± 0.0569 ± 5
Table 4.

Estimated experimental and theoretical calculated 23Na NMR quadrupolar parameters CQ and η for nitrate cancrinite at T = 296 K and 673 K (Fechtelkord et al., 2001).

TCation siteTheoreticalExperimental
CQ (MHz)ηCQ (MHz)η

296 KNa10.2 ± 0.50.0 ± 0.1n/an/a
Na21.2 ± 0.50.4 ± 0.1n/an/a
673 KNa14.2 ± 0.50.1 ± 0.12.3 ± 0.20.2 ± 0.2
Na21.7 ± 0.50.5 ± 0.11.4 ± 0.10.7 ± 0.2
TCation siteTheoreticalExperimental
CQ (MHz)ηCQ (MHz)η

296 KNa10.2 ± 0.50.0 ± 0.1n/an/a
Na21.2 ± 0.50.4 ± 0.1n/an/a
673 KNa14.2 ± 0.50.1 ± 0.12.3 ± 0.20.2 ± 0.2
Na21.7 ± 0.50.5 ± 0.11.4 ± 0.10.7 ± 0.2

n/a: not estimable

Contents

GeoRef

References

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