Blind spheres of paramagnetic dopants in solid state NMR

Abstract

Solid-state NMR on paramagnetically doped crystal structures gives information about the spatial distribution of dopants in the host. Paramagnetic dopants may render NMR active nuclei virtually invisible by relaxation, paramagnetic broadening or shielding. In this contribution blind sphere radii r₀ have been reported, which could be extracted through fitting the NMR signal visibility function f(x) = exp(−ar₀³x) to experimental data obtained on several model compound series: La_1−xLn_xPO₄ (Ln = Nd, Sm, Gd, Dy, Ho, Er, Tm, Yb), Sr_1−xEu_xGa₂S₄ and (Zn_1−xMn_x)₃(PO₄)₂·4H₂O. Radii were extracted for ¹H, ³¹P and ⁷¹Ga, and dopants like Nd³⁺, Gd³⁺, Dy³⁺, Ho³⁺, Er³⁺, Tm³⁺, Yb³⁺ and Mn²⁺. The observed radii determined differed in all cases and covered a range from 5.5 to 13.5 Å. While these radii were obtained from the amount of invisible NMR signal, we also show how to link the visibility function to lineshape parameters. We show under which conditions empirical correlations of linewidth and doping concentration can be used to extract blind sphere radii from second moment or linewidth parameter data. From the second moment analysis of La_1−xSm_xPO₄³¹P MAS NMR spectra for example, a blind sphere size of Sm³⁺ can be determined, even though the visibility function remains close to 100% over the entire doping range. Dependence of the blind sphere radius r₀ on the NMR isotope and on the paramagnetic dopant could be suggested and verified: for different nuclei, r₀ shows a -dependence, γ being the gyromagnetic ratio. The blind sphere radii r₀ for different paramagnetic dopants in a lanthanide series could be predicted from the pseudo-contact term.

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Introduction

Nuclear magnetic resonance (NMR) has contributed to the study of paramagnetic systems, such as in structural biology,^1–5 battery materials,^6–8 characterization of pharmaceutical formulation,^9,10 polymers,^11,12 chemical shift thermometers^13,14 and luminescent materials.^15–23 The range of applications of paramagnetic NMR is surprising given that the strong electronic magnetic moment of the paramagnetic species interferes with the NMR measurement in different ways. Nevertheless, paramagnetic NMR may give information about structure,^2,12 dynamics^8,24 and the distribution of paramagnetic dopants in a host.^{17–22,25,26} In the latter case different approaches had been used namely relaxation times,^19,20,22 linewidths^17,18 and observed peaks areas^21,25 to relate doping homogeneity to the performance of these luminescent materials.

In solid-state NMR a distinction needs to be made between lanthanide atoms and transition metal atoms. The influence transition metal atoms exert on neighbouring NMR nuclei enables studies for example on battery materials^6–8 or metal organic framework compounds.²⁷ The computation of the influence on valence electrons is important and progress in the computation of paramagnetic shifts²⁸ has improved significantly in recent years. In contrast, lanthanides with few exceptions show hardly any influence from the valence shell and thus have been used for systematic experimental studies to identify contributions to the paramagnetic spin Hamiltonian.^28,29 Paramagnetic NMR of lanthanide containing compounds found various applications to luminescent materials.^15–21

A non-trivial problem to paramagnetic NMR is that NMR resonances of paramagnetic compounds may virtually vanish in the dead-time of the NMR spectrometer by relaxation, inhomogeneous broadening mechanisms or anisotropic susceptibility broadening.²⁹ Lineshape analysis as proposed by Van Vleck³⁰ in terms of moment-analysis or NMR line width comes with a visibility caveat. Nevertheless, this approach was proved successful in the characterization of several phosphors,^17,18 where it could be shown that, as expected by van Vleck, the linewidth depended linearly on the paramagnetic doping level x at low doping concentration. It should be noted that in the high doping regime a non-trivial dependence due to “exchange narrowing”^30–32 may lead to deviations from this simple behaviour. In case of luminescent materials, the doping homogeneity is of major importance to establish an optimal performance.^33,34 When luminescent ions are too close, energy can easily be transferred between them, leading to uncontrollable migration of energy in the dopant sublattice that can eventually be non-radiatively dissipated at so-called luminescence killer centres. This effect is usually referred to as concentration quenching^35,36 and has been related to the dopant distribution obtained from NMR lineshape analysis in a few cases.^17,18 Furthermore, it was evidenced by a microscopic investigation that the thermal quenching behaviour, i.e. the decrease of luminescence quantum yield for increasing temperatures,³⁷ is severely worsened for inhomogeneously doped materials due to areas with a locally more elevated doping concentration.³⁴

Instead of asking for the NMR properties of the visible paramagnetic signals, it may be interesting to ask for the fraction of NMR invisible signal. If a spherical regime around a paramagnetic centre is assumed, from which no NMR signals can be detected, it is possible to relate the signal loss to the size of this sphere of influence which is known under the name blind-sphere^38,39 or wipe-out radius.^40–42 Such radii are important in solution NMR^38,39 where paramagnetic constraints are used for structure solution, and in dynamic nuclear polarization (DNP) to estimate the zone which cannot be accessed by NMR^43–45 and for luminescent materials to study doping homogeneity.²¹

Spectroscopically the blind sphere can be explained by line broadening which leads to undetectable signal by standard experiments,³⁹ or signal shift⁴⁶ that puts the signal outside of spectral window. The origins of blind sphere may relate to (but are not limited to) the following contributions:³⁹ relaxation which may involve dipolar, Curie, contact and cross relaxation mechanisms, and hyperfine shift which contains contact and pseudo-contact parts.

Reported sizes of blind spheres have been mostly related to the solution NMR, for which Bertini³⁹ and co-workers have laid a foundation. Besides some attempts have been made to quantify the blind sphere radius of DNP polarizing agents.^43,47 In solid state, due to anisotropic interactions and more ambiguous estimation of correlation time τ_c and spectral density function J(ω), blind sphere sizes may differ and questions about the influence that a paramagnetic centre has on its environment remain open. To the best of our knowledge, for solid crystalline samples a systematic study on the sizes of blind spheres of different inorganic dopants has not been published before.

The target of this contribution is to relate the size of blind spheres of solid samples to other physical quantities, for example the gyromagnetic ratio of NMR nuclei and the effective magnetic moment of the paramagnetic ion by studying the blind sphere radii in lanthanide-doped solid solutions. Moreover, given that doping homogeneity of inorganic phosphors can be traced both via the visible signal by lineshape analysis and via the fraction of the invisible signal,²¹ it is natural to ask whether the blind sphere radius and linewidth or moment analysis can be linked, and if so under which conditions.

Experimental

The La_1−xLn_xPO₄ (Ln = Nd, Sm, Gd, Dy, Er, Ho, Tm, Yb) samples have been synthesized via a co-precipitation method: Ln₂O₃ (Nd₂O₃ was bought from chemPUR, the rest from smart elements, the purity is 99.999% for Dy₂O₃ and 99.99% for the rest) and La₂O₃ (chemPUR, 99.99%) were dissolved in nitric acid and later on mixed with NH₄H₂PO₄ (VWR chemicals) solution. The resulting precipitates were dried at 80 °C overnight, sintered in corundum crucibles at 1000 °C for 4 h.

Sr_1−xEu_xGa₂S₄ powders were obtained via a solid state synthesis method. Stoichiometric quantities of SrS (Alfa Aesar, 99.9%), Ga₂S₃ (Alfa Aesar, 99.99%) and EuF₃ (Alfa Aesar, 99.5%) were weighed and mixed in an agate mortar. A small amount (1 weight %) of NH₄F (Alfa Aesar, 98+%) was added as a fluxing agent. The mixtures were fired for two hours at 900 °C under a flow of forming gas (90% N₂, 10% H₂). The obtained powder was again lightly ground.

The (Zn_1−xMn_x)₃(PO₄)₂·4H₂O samples have been synthesized by a co-precipitation method: stoichiometric amounts of MnCl₂·4H₂O (ACROS organics, 99+%) and Zn(NO₃)₂·6H₂O (chemPUR, 98+%) were dissolved into water and mixed with an excess of NH₄H₂PO₄ (VWR chemicals) in aqueous solution. The precipitates were washed with water and ethanol and dried overnight.

Powder X-ray diffraction (XRD) measurements were performed on a Huber G621 diffractometer with Cu K_α1 radiation (λ = 0.15405931 nm) in transmission geometry. Diffractograms were extracted from the image files, which were obtained by scanning the photostimulable BaBrF:Eu²⁺ films with an image plate detector (Typhoon FLA 7000, λ = 650 nm), by home-written program ipreader-0.9. The Rietveld analysis was performed via the program TOPAS-Academic (TOtal PAttern Solution, by Coelho Software, V4.1).

The solid state NMR measurements were performed on a Bruker Avance II spectrometer at a magnetic field of 7.05 T. Magic angle spinning (MAS) was done with 4 mm pencil rotors at spinning frequencies of 10 kHz or 12.5 kHz with a home-built McKay probe head. The dead time delay was set to 15 μs. Quantification was assisted by a micro-balance (Sartorius MC5). The deconvolution of peaks and moment analysis were assisted with the program deconv2Dxy⁴⁸ (version 0.4). The NMR visibility fitting function²¹ for a homogeneously doped sample is defined as follows.

f(x) = exp(−ar₀³x) (1)

The wipe-out radius r₀ relates to the size of the blind sphere of a paramagnetic centre and the variable a is host-specific number (a = 4πN_hostUC/3V_UC where N_hostUC is the number of “dopable” sites in the unit cell and V_UC is the volume of the unit cell). For monazite⁴⁹ LaPO₄ the variable a amounts to 0.055 Å⁻³. The doping concentration x in this work is defined as the degree of substitution, which is dimensionless. The experimental NMR visibility f was calculated in the following way: first the molar peak areas P = A/n were calculated for all samples in one dopant series (A peak area, n amount of material). The visibility function f(x) and P(x) are related by normalization through the undoped sample, i.e. P(x) = f(x)·P(0). Therefore the experimental P values were fitted with the function P(x) to determine the free parameters r₀ and P(0). This approach gives equal weight to all measured points. The diagrams (Fig. 2, 4, 7 and 8) show P(x)/P(0) on the y-axis.

Results and discussion

The target of this contribution is to provide a better understanding of the blind sphere in solids. In order to achieve that, measurements have been performed on three model compounds series, Ln³⁺ (Ln = Nd, Sm, Gd, Dy, Er, Ho, Tm, Yb) doped monazite LaPO₄, Mn²⁺ doped hopeite Zn₃(PO₄)₂·4H₂O and Eu²⁺ doped SrGa₂S₄. Lanthanide doped xenotime YPO₄ and monazite LaPO₄ have been subject to several studies50,51 in terms of a determination of NMR parameters. Lanthanide dopants have the advantage that for a model study chemical interference is reduced to a minimum because of the low lying, unpaired f-shell electrons. To extract the sizes of blind spheres from measured data, approaches which relate the peak area and lineshape to the doping level are investigated. Finally, the obtained sizes are compared to establish relations between blind sphere and physical quantities, for example magnetic moment and gyromagnetic ratio.

Peak area and blind sphere

To judge the quality of obtained samples X-ray diffraction was applied. The synthesized samples were phase pure according to X-ray diffraction. Rietveld refinements of the diffraction pattern yielded lattice parameters which followed Vegard's law.⁵²Fig. 1 shows the lattice parameters of La_1−xSm_xPO₄ follow a linear dependence on the doping concentration x as an example. Such a fulfilment of Vegard's law is often considered as evidence for a homogeneously doped phase pure sample in the sense of a solid solution.

Fig. 1 Lattice parameters as a function of the substitution degree x in La_1−xSm_xPO₄, as determined by Rietveld refinement based on X-ray powder diffraction data. The dotted lines represent linear fits resulting in a/Å = 6.516 − 0.142·x, b/Å = 7.0794 − 0.183·x, c/Å = 8.294 − 0.232·x.

The single pulse ³¹P MAS NMR spectra offer information on the change of peak area upon increasing the doping level x. Based on the visibility function²¹f(x) = exp(−ar₀³x) blind sphere radii for trivalent dopants Ln³⁺ (Ln = Nd, Gd, Dy, Er, Ho, Tm, Yb) could be obtained (Fig. 2 and Table 1).

Fig. 2 Normalized visibility function f (circles with error bars) calculated from ³¹P MAS NMR data plotted against the substitution degree x in La_1−xLn_xPO₄, on a logarithmic scale. The dashed line corresponds to the fitting function f(x) = exp(−ar₀³x) with a = 0.055 Å⁻³ and r₀ = 5.5, 13.5, 12.5, 10.5, 10, 9 and 5.8 Å for Ln = Nd, Gd, Dy, Ho, Er, Tm and Yb, respectively.

Table 1 Radii of blind spheres r₀ obtained from ³¹P MAS NMR of La_1−xLn_xPO₄ sample series, the effective magnetic moments⁵³μ_eff in equivalents of the Bohr magneton μ_B and the number of unpaired 4f electrons N_unpaired for comparison; r₀ determined by peak area method except for “*” where an estimate from second moment analysis is reported (see main text)

Dopant ion	r 0/Å	μ eff/μB	N unpaired
Nd^3+	5.5	3.62	3
Sm^3+	0.45*	1.54	5
Gd^3+	13.5	7.95	7
Dy^3+	12.5	10.5	5
Ho^3+	10.5	10.5	4
Er^3+	10.0	9.55	3
Tm^3+	9.0	7.5	2
Yb^3+	5.8	4.4	1

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While in principle blind sphere radii can be determined from two points only, we fitted the visibility function to all points (experimental part). Because of the discrete nature of the pair distribution function in crystalline materials, radii determined this way become less reliable the smaller the blind-sphere radius (see discussion in ref. 21).

Line shape and blind sphere

While the visibility function f(x) = exp(−ar₀³x) proved to be a useful tool for the extraction of blind-sphere radii in the cases above and in case of Sr_1−xEu_xH₂,²¹ it could not be applied in case of La_1−xSm_xPO₄, because single pulse ³¹P MAS NMR (Fig. 3) recovers (Fig. 4) signals from all ³¹P atoms including those of the first coordination sphere around Sm³⁺, as can be seen from the visibility function (Fig. 4 in circles) which stays close to 100% for all doping concentrations x.

Fig. 3 Stack plot of ³¹P MAS NMR spectra of La_1−xSm_xPO₄ (x values are marked on the corresponding spectra, dashed line indicates the position of the peak of pure LaPO₄).

Fig. 4 Normalized visibility f(x) of the ³¹P MAS NMR signal of La_1−xSm_xPO₄. The circles represent the f(x) values calculated from whole peak area, which scatter around 100% (dashed line). The triangles represent the f(x) values from the sharp component which has a similar isotropic chemical shift value as the pure LaPO₄. The dotted line refers to the fitting function f(x) = exp(−ar₀³x) with a = 0.055 Å⁻³ and r₀ = 4.5 Å. Note that this r₀ does not follow the definition used in the rest of this contribution.

Because of line broadening, precise deconvolution of different environments is difficult for x > 0.1. Despite the large errors from peak deconvolution, an analysis was attempted to relate line width and doping concentration (ESI,† Fig. S1) as previously done on similar systems in literature,^17,18 however, a linear dependence could not be found. On the other hand a signal specific ³¹P visibility function f can be defined, which corresponds only to the area of the peak which has a similar shift and linewidth as the peak of pure host material LaPO₄ (Fig. 4, triangles). A radius of 4.5 Å could in principle be obtained this way which as expected covers the P-atoms in the first coordination sphere around the La atom.

A practical approach which works for any kind of lineshape is to calculate the second moment from the lineshape as suggested by Van Vleck³⁰ in the context of dipolar peak broadening on whole spectra, which does not rely on the separation of different environments. This dipolar broadening includes broadening by magnetic dipolar couplings because of the hyperfine coupling. In case of La_1−xSm_xPO₄ the second moment M₂/Hz² follows the doping concentration x in a linear way over a wide range (Fig. 5).

Fig. 5 Second moment M₂(x) as a function of the substitution degree x in La_1−xSm_xPO₄. The dotted line represents linear fit resulting in M₂/Hz² = 1.4 × 10⁴ + 1.07 × 10⁶·x.

To explain the observed linear relation, a brief discussion of line shape functions and line broadening mechanisms may be helpful. For an idealized free induction decay (FID) of a single nucleus the decay function can be expressed for example as a Gaussian or monoexponentially decaying function. The Fourier transformation of such a FID returns a Gaussian or Lorentzian line shape, respectively. When a resonance is broadened homogeneously, for example by relaxation i.e. by random oscillatory local field components at the Larmor frequency⁵⁴ or by lifetime broadening,²⁹ then the Lorentzian function is a good basis for the description of the line-shape in the frequency domain. On the other hand, inhomogeneous broadening,^55,56 which can be caused for example by magnetic field inhomogeneity, chemical shift anisotropy (CSA) or pseudo-contact term, produces a more complicated, often asymmetric lineshape, for which the description by van Vleck's moment approach is a suitable analysis tool. In van Vleck's moment approach, the dipolarly broadened spectral lineshape is decomposed into a sum of Gaussian functions. Hole-burning^57–59 experiments allow to distinguish inhomogeneous from homogeneous broadening.

A simple alternative to the hole-burning experiment is the 2D exchange spectroscopy (EXSY) experiment. EXSY experiments with zero mixing time show a sharp ridge on the diagonal in case of inhomogeneous broadening and a double Lorentzian type lineshape in case of homogeneous broadening. Note that the EXSY looks the same and behaves like the stimulated echo experiment. From the ³¹P MAS NMR 2D spectra (Fig. 6), it is clear that the ³¹P 2D NMR signal observed for Sm³⁺ doped LaPO₄ is typical for inhomogeneous broadening, while for Gd³⁺ homogeneous broadening is observed. The ³¹P NMR signals for the other measured Ln³⁺ doped samples, namely Nd³⁺, Dy³⁺, Ho³⁺, Er³⁺, Tm³⁺ and Yb³⁺, all show an inhomogeneous broadening behaviour (ESI,† Fig. S2).

Fig. 6 2D ³¹P MAS EXSY spectra (with zero mixing time) for La_0.997Gd_0.003PO₄ (left) and La_0.99Sm_0.01PO₄ (right), which depicted homogeneous and inhomogeneous type of line broadening, respectively.

In the following, a relation between the blind-sphere radius and the observed line-broadening is developed for simple cases.

The peak area which is the basis of the visibility function f(x) (see above) is related to the intensity of first point of the FID according to the integral theorem of the Fourier transformation.⁵⁶ Any loss in peak area should then be reflected by the decay of the FID during the dead-time delay t_de, either by relaxation or by coherent mechanisms. If spectral line-broadening is assumed to be following a simple Lorentzian or Gaussian function then the decay in the FID needs to be monoexponential or Gaussian, respectively. Thus for simple cases the blind-sphere radius and line-broadening follow simple analytical expressions for a given dead-time delay (see below).

A Gaussian type FID s_G(t) of an on-resonance signal is described with a linewidth parameter λ_G and the amplitude factor s_G,0.

s_G(t) = s_G,0 exp(−λ_G²t²) (2)

The corresponding spectral function I_G(ω) is obtained by a single-sided complex Fourier transformation.⁵⁶

The line width parameter λ_G is related to the full width half-maximum of the line-shape in Hz as .

Given the lineshape of a paramagnetically doped sample can be described with a simple Gaussian function then the peak area A_G for a given dead time t_de can be predicted on the basis of the integral theorem⁵⁶ of the Fourier transformation from the decay of the FID.

A_G = s_G,0 exp[−λ_G²t_de²] (4)

In order to derive the visibility function f(x) in terms of line-broadening λ_G(x), eqn (4) is plugged into the definition of the visibility function.

The amplitude factor s_G,0 of the doped and the pure host material should the equal, while the line width parameter λ_G(x) depends on doping concentration. The visibility function can also be expressed in terms of the blind sphere radii r₀, the number density parameter a and doping level x (eqn (1)). Therefore, the link between parameter λ_G and doping level x can be established as follows.

Assuming a negligible frequency difference between signals, the second moment^48,56 of the Gaussian type function is 2λ_G² and the second moment becomes M₂(x) = 2λ_G²(x). Eqn (6) can be rewritten in terms of the second moment.

A linear regression of the second moment M₂(x) yields the slope . From the latter the radius of the blind sphere r₀ can be estimated via a simple analytical equation.

This way the blind-sphere r₀ of Sm³⁺ doped La_1−xSm_xPO₄ was estimated to be 0.45 Å. The La to P and Sm to P distances in the monazite⁴⁹ structure of LaPO₄ and SmPO₄ are 3.2 Å and 3.1 Å, respectively, which means that all ³¹P nuclei are outside of the blind spheres of Sm³⁺ and thus visible. This is consistent with the observation the signal visibility function f(x) remains close to 100% (Fig. 4). In addition, ¹⁹F and ¹H NMR signals for SmF₃⁶⁰ and SmH₃^61,62 both have been reported to be visible, which supports the found small blind sphere size of Sm³⁺. Given the estimated blind sphere radius is even smaller than the Shannon radius⁶³ for Sm³⁺ (around 1 Å) it appears that Sm-compounds in general are good candidates for NMR studies, because no signal loss is expected. The second moment analysis as presented above can provide information on the size of the blind sphere, especially for paramagnetic dopants which have small blind spheres.

For Gd³⁺, the line broadening is mainly based on the homogeneous broadening mechanism which is relaxation dominated and follows a Lorentzian lineshape. Thus the FID can be described with a monoexponentially decaying function. The line width parameter λ_L(x) then has a slightly different relation to the blind sphere r₀, doping concentration x and deadtime delay t_de (see ESI†).

The λ_L(x) is related to the full width half-maximum in Hz as λ_L = 2πΔν^FWHM_L.

The relation between the size of blind sphere r₀ and the experimental results λ_L(0) (the line width of non-doped diamagnetic analog) and slope k_L value (from the linear fit of λ_L(x) against x) can be derived in a similar fashion as in the Gaussian case.

This way the blind sphere r₀ for the Gd³⁺ doped sample becomes 6.7 Å, which is much smaller than the estimated 13.5 Å from signal visibility method. This discrepancy can be explained by the inadequacy of simple functions to describe the FID. In case the spectral lineshape function becomes more complicated it is non-trivial to extract line-width parameters from the spectra and in those cases the visibility function f(x) provides an easier way to determine blind-sphere radii. As shown above line-shape analysis requires knowledge about the decay character of the FID in order to determine the blind sphere radius. The reported^17,18 linear relation of doping concentration x and linewidth is only expected in case of Lorentzian type spectral lineshapes. While the derivation of blind-spheres via a line-shape analysis suffers from a number of approximations including the assumption of a sharp transition of the visible to invisible nuclei, we believe that the scaling behaviour of the doping concentration x with respect to second moment and line-width, is the most important insight that is being conveyed by the above analysis.

Blind sphere radius dependence on the gyromagnetic ratio

In order to investigate the size dependence of the blind sphere on the gyromagnetic ratio γ, data obtained from the same paramagnetic ion but different NMR nuclei are compared (Table 2). In the following we assume that Eu²⁺ and Gd³⁺ are isoelectronic paramagnetic ions, and thus have equal blind sphere radii. The concept of the blind-sphere is based on the idea that the signals of all atoms inside the sphere vanish in the dead-time delay, i.e. nuclei which are situated on the blind sphere have a critical relaxation rate, a critical amount of line-broadening or paramagnetic shift. Critical relaxation rates R_1M and R_2M driven by a dipolar coupling mechanism with unpaired electrons or Curie nuclear spin relaxation are then proportional to γ²/r₀⁶,³⁹ while critical broadening through the pseudo-contact shift (PCS) relates to γ/r₀³.³⁹ Given that other influences, like the spectral density, are negligible, it can be concluded that the prefactors which relate to the critical broadening or relaxation rate would feature the same relation of gyromagnetic ratio γ to blind sphere radius r₀. In this case the blind sphere radii for different nuclei X and Y in the same compound should feature a cubic root dependence to the gyromagnetic ratio.

Table 2 The blind sphere radii r₀ values for different NMR nuclei obtained with the visibility method; γ is the nuclei gyromagnetic ratio; r₀(³¹P) is the blind-sphere radius of a virtual ³¹P nucleus converted from experimental r₀via the eqn (11)

Host	NMR nucleus	γ/10^7 rad T^−1 s^−1	a/Å^−3	Dopant	r 0/Å	r 0(^31P)/Å
SrH2	^1H	26.75	0.092	Eu^2+	17^21	12.6
SrGa2S4	^71Ga	8.18	0.025	Eu^2+	13	14.3
LaPO4	^31P	10.84	0.055	Gd^3+	13.5	—
Zn3(PO4)2·4H2O	^1H	26.75	0.051	Mn^2+	10	7.4
Zn3(PO4)2·4H2O	^31P	10.84	0.051	Mn^2+	7	—

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To test this hypothesis we compare blind-sphere radii from different nuclei in the same and in different compounds. The visibility method yielded a blind sphere radius from ⁷¹Ga NMR for Eu²⁺ in Sr_1−xEu_xGa₂S₄ (Fig. 7). This may be compared with a blind sphere for Eu²⁺ in Sr_1−xEu_xH₂ from ¹H NMR²¹ and with one for Gd³⁺ in La_1−xGd_xPO₄ from ³¹P NMR (Fig. 2). The values converted to the blind sphere viaeqn (11) of a virtual ³¹P nucleus are the same (Table 2) within approximated error limits. Given the low number of values in the comparison this is a weak indication that the blind sphere radii of lanthanide dopants may be abstracted from the host structure.

Fig. 7 Normalized NMR visibility function f(x) on a logarithmic scale for x for Sr_1−xEu_xGa₂S₄ (circles). The dashed line corresponds to the fitted function f(x) = exp(−ar₀³x) with a = 0.025/ų and r₀ = 13 Å.

In a second example the sphere radius of Mn²⁺ is detected by ¹H and ³¹P NMR in hopeite (Zn_1−xMn_x)₃(PO₄)₂·4H₂O (Fig. 8). Again the virtual radius of ³¹P converted from the radius applying to the ¹H nucleus agrees to the observed value within error limits (Table 2). Deviations from this fairly good agreement are expected especially in case of a Fermi-contact contribution^64,65 which however does not seem to be relevant here. Another explanation in case of a relaxation dominated blind sphere may be the change of the electronic relaxation mechanism at high doping concentrations which in principle could even lead to an increase visibility in the high concentration regime. In some cases of transition metal doping even at very high dopant concentration^64,65 the signal does not vanish. In the presented cases of lanthanide(III) doped monazite the only system where we have an indication of a relaxation dominated blind sphere radius is Gd(III). At 100% doping, i.e. pure GdPO₄ no intensity increase can be observed, in line with the presented interpretation.

Fig. 8 Normalized NMR visibility function f(x) on a logarithmic scale for x for (Zn_1−xMn_x)₃(PO₄)₂·4H₂O. f(x) data obtained from ¹H NMR are plotted as circles while those from ³¹P NMR as squares. The dashed lines feature the fitted functions f(x) = exp(−ar₀³x) with a = 0.051/ų, r₀ = 10 Å and 7 Å for ¹H and ³¹P, respectively.

Blind sphere radius dependence on the effective magnetic moment

In order to test the hypothesis that blind sphere radius depends on the effective magnetic moment, data obtained from the same host LaPO₄ and the same NMR nucleus (³¹P) but different paramagnetic dopants are compared. Relaxation and hyperfine shift are two possible origins of blind spheres, and are discussed resonances in La_1−xLn_xPO₄ are inhomogeneously broadened for all Ln³⁺ dopants except Gd³⁺, which indicates that relaxation is dominant in the case of doping by Gd³⁺ but not as important in case of other paramagnetic Ln³⁺ as is explained in detail in the following paragraph.

Relaxation

Dipolar, Curie-spin and Fermi-contact relaxation are in principle three possible relaxation contributions in paramagnetic systems which lead to homogeneous line broadening. For solid crystalline La_1−xLn_xPO₄, chemical exchange and stochastic reorientation are negligible, only vibrational motions exert efficient influence on the electronic relaxation. Based on the Solomon–Bloembergen–Morgan relaxation model,^66,67 the nuclear transverse relaxation time T₂ can be described as eqn (12) for lanthanide ions induced paramagnetic relaxation enhancement (PRE). Parameter A^FC is the Fermi-contact coupling and is the spin-dipolar coupling parameter.⁶⁷T_1e and T_2e are the longitudinal and transverse electronic relaxation times, respectively, which are used to approximate the corresponding electronic correlation times.⁶⁷ω_I and ω_S are the nuclear and electronic Larmor frequencies, respectively. J is the main total angular momentum quantum number. R is the distance between NMR nuclei and the unpaired electrons which belong to the paramagnetic ion and are assumed to be localized on the lanthanide ion (point-dipole approximation). The γ_I is the nuclear gyromagnetic ratio and g_J is the isotropic Born–Landé g-factor.

The first term in this equation is caused by Fermi-contact coupling and lacks a simple relation to the blind-sphere radius. The second term, called pseudo-contact term, is caused by a direct dipole–dipole interaction and has a distance dependence of R⁻⁶. At the critical relaxation rate (neglecting the Fermi contact contribution) the second term is proportional to b_JI²J(J + 1) or expressed with the effective magnetic moment (Landé formula) proportional to μ_eff²R⁻⁶. The electronic relaxation times in solids are driven by vibrational motions and differ by the lanthanide ion. Published ranges^39,67 of the electronic relaxation times show only minor differences for lanthanides Ln³⁺ (Ln = Sm, Nd, Yb, Tm, Er and Ho) with the exception of Gd³⁺ (see Table 8.6 in ref. 67) for which electronic relaxation rates are several orders of magnitude lower than for the other paramagnetic lanthanide ions, which causes efficient nuclear transversal relaxation and is consistent with the magnitude of the linewidth (few hundreds to kHz) observed from the EXSY (Fig. 6) and single pulse measurements. Note that the EXSY experiment with a very short mixing time behaves like a stimulated echo and provides information about transversal relaxation over the complete linewidth. For the other Ln³⁺ ions, the observed line broadening (Fig. 6 and ESI,† Fig. S2) is of the order of ten to a hundred Hz. This is much larger than the expected relaxation-induced line broadening which indicates relaxation not to be the predominant source of broadening. This interpretation is in agreement with the observation that the line broadening for Gd³⁺ is homogeneous while the other Ln³⁺ dopants generate inhomogeneous line broadening and explains the exceptional role of Gd³⁺ in the Fig. 9 and 10.

Fig. 9 Blind sphere radii for Ln(III) dopants of La_1−xLn_xPO₄ series, plotted against . The μ_eff and μ_B refer to effective magnetic moment⁵³ and Bohr magneton, respectively.

Fig. 10 Blind sphere radii for Ln(III) dopants of La_1−xLn_xPO₄ series, plotted against . |C_PCS| refers to the magnitude of electronic contribution from pseudo-contact shielding.^67,68 The dashed line features the fitting function with the coefficient of determination R² = 0.89.

Paramagnetic shift

Besides relaxation, another possible origin for blind sphere is the paramagnetic shift δ_param. The inhomogeneous line-broadening observed on the ³¹P resonances indicates that the paramagnetic shift is the relevant origin for the blind spheres of LaPO₄ doped with Ln³⁺ (Ln = Nd, Sm, Dy, Er, Ho, Tm, Yb), but not for LaPO₄ doped with Gd³⁺. Different mechanisms may cause a paramagnetic isotropic or anisotropic shift, which have been subject to a recent review.⁶⁷ In order to compare the relative size of paramagnetic shift in a lanthanide series Bleaney,⁶⁸ Golding and Halton⁶⁹ have developed an approach in which they separate the electronic term C from a coupling term. The size of the electronic term can be described by a real number and was tabulated by Pell et al.⁶⁷ for the contact shift C_con, the pseudo-contact shift C_PCS = g²J(J + 1)(2J − 1)(2J + 3) 〈J||α||J〉⁶⁷ and the shielding anisotropy C_SA (Table 3). Given that a critical broadening or shift exists (see above) which defines the blind-sphere radius r₀, it should be possible to obtain a linear relation between and the blind sphere radius r₀. In fact a good correlation is observed (Fig. 10), with the expected exception of Gd³⁺ (see above). For completeness we have also plotted the blind-sphere radius vs. (ESI,† Fig. S3) and (Fig. S4, ESI†). The observed excellent empirical correlation in case of the contact interaction lacks a good explanation, though. It should not be overinterpreted as it is caused by the drastic change of a single measurement, i.e. on La_1−xGd_xPO₄ for which relaxation is expected to have a big influence.

Table 3 Radii of blind spheres r₀ obtained from ³¹P MAS NMR of La_1−xLn_xPO₄ sample series, and corresponding Ln³⁺ hyperfine contributions.⁶⁷ |C_SA|, |C_con| and |C_PCS| refer to the electronic contribution to the paramagnetic shift from shielding anisotropy, contact and pseudo-contact term, respectively

Dopant ion	r ^A0/Å	|CSA|	|Ccon|	|CPCS|
Nd^3+	5.5	13.1	4.5	8.08
Sm^3+	0.45	0.7	0.063	0.94
Gd^3+	13.5	63	31.5	0
Dy^3+	12.5	113.3	28.5	181
Ho^3+	10.5	112.5	22.6	71.3
Er^3+	10.0	91.8	15.4	58.8
Tm^3+	9.0	57.2	8.2	95.3
Yb^3+	5.8	20.6	2.6	39.2

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To sum up, for Ln³⁺ both relaxation (in case of Gd³⁺) and the paramagnetic shift (in case of the other presented Ln³⁺) give an explanation for the origin of the blind sphere. The pseudo-contact term provides a simple explanation for the change of the blind-sphere radius in a lanthanide series and predicts a dependence for relaxation and a dependence for the paramagnetic shift, of which the latter is confirmed by the experiment.

Conclusions

In this work, sizes of blind spheres r₀ have been determined from NMR spectra by two methods: signal visibility decay function and lineshape analysis. A formula for the blind sphere radius could be derived, that relates the radii of blind sphere to the lineshape for a given dead time of the spectrometer, which allows to estimate blind-sphere radii even when the visibility remains close to 100% over the complete doping range.

The dependence of blind sphere radii on the effective magnetic moment of the dopant ions and on the nucleus give insight into processes leading to a blind-sphere and provide estimates for blind sphere radii upon switching the dopant in a lanthanide series or the nucleus. Blind-sphere radii up to 13 Å suggest that a significant amount of material may remain virtually invisible in NMR, while certain paramagnetic ions, like Sm³⁺, allow the detection of NMR signal even from atoms two bonds away from paramagnetic centre. These results are relevant for evaluating the paramagnetic doping homogeneity of inorganic phosphors and their optical performance and to estimate which nuclei around a paramagnetic centre can directly be observed, for example in dynamic nuclear polarization (DNP) NMR or by direct detection.

Conflicts of interest

There are no conflicts to declare.

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Footnote

† Electronic supplementary information (ESI) available: Numerical for the spectral analysis of LaPO₄:Sm, EXSY spectra for LaPO₄:Ln, ³¹P NMR relaxation data for LaPO₄:Ln. See DOI: 10.1039/c9cp00953a

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