Field-Dependent Proton NMR Relaxation in Aqueous Solutions of Ni(II) Ions. A New Interpretation

JOURNAL OF MAGNETIC RESONANCE, Series A 121, 108–113 (1996)ARTICLE NO. 0149

Field-Dependent Proton NMR Relaxation in Aqueous Solutionsof Ni ( II ) Ions. A New Interpretation

J. SVOBODA,* ,† T. NILSSON,† J. KOWALEWSKI,† P.-O. WESTLUND,‡ AND P. T. LARSSON§

†Division of Physical Chemistry, Arrhenius Laboratory, Stockholm University, S-10691 Stockholm, Sweden; ‡Physical Chemistry,Umea University, S-901 87 Umea, Sweden; and §STFI, P.O. Box 5604, S-114 86 Stockholm, Sweden

Received December 27, 1995; revised March 6, 1996

A new model is presented for nuclear-spin relaxation in para- spin relaxation was assumed to be due to distortional modu-magnetic transition metal complexes in solution, allowing the elec- lation of the zero-field splitting (ZFS) interaction. They alsotron-spin relaxation to be outside the Redfield limit. The novel carefully specified the validity limits of the theory (the elec-feature is that the transient zero-field splitting (ZFS), modulated tron-spin relaxation time must not be shorter than the correla-by distortions of the solvation shell, is allowed to be of rhombic tion time for the modulation of the relevant interaction).rather than cylindrical symmetry. The model, which assumes that

These conditions are sometimes denoted as ‘‘strong nar-the static ZFS is absent, is applicable to aqueous solutions of transi-rowing conditions’’ or Redfield limit [a slightly more gen-tion metal ions. The magnetic-field dependence of the proton spin–eral formulation of the strong narrowing conditions, in termslattice relaxation rate enhancement in aqueous solution has beenof interaction strengths and correlation times, has been dis-investigated, and calculations are presented for an S Å 1 systemcussed in a series of papers from our laboratories (4–6)] .such as Ni2/ (aq) , using different degrees of rhombicity of the

ZFS and different motional conditions (Redfield limit and slow- The set of simple equations, describing the enhancement ofmotion regime). The new model is also applied to fit the previously the nuclear-spin relaxation rates in the region where thereported data for the field dependence of proton relaxation in Bloembergen–Morgan theory is valid, is commonly calledaqueous solution of Ni(ClO4)2 at low pH [J. Kowalewski, T. Lars- the modified Solomon–Bloembergen (MSB) or Solomon–son, and P.-O. Westlund, J. Magn. Reson. 74, 56 (1987)] . The Bloembergen–Morgan (SBM) equations. Bloembergen andinclusion of rhombicity, motivated by recent theoretical work, pro- Morgan (3) issued a warning that the case of aqueous Ni(II)vides a model which performs as well as the earlier ad hoc model.

could possibly be beyond the strong narrowing conditions.q 1996 Academic Press, Inc.

Friedman et al. (7) reported additional experimental T1

data and a new theoretical treatment of aqueous Ni(II) ,allowing deviations from the strong narrowing conditions.

INTRODUCTIONHertz and Holz (8) published a comprehensive set of experi-ments, including variation not only of magnetic field but

Studies of proton nuclear magnetic relaxation as a functionalso of pH and temperature. Kowalewski et al. (9) used the

of magnetic field (nuclear magnetic relaxation dispersion, data of Hertz and Holz (8) for an acidic sample at elevatedNMRD) conducted in water containing simple paramagnetic temperatures, supplemented them with some measurementstransition metal ions have been an active area of research at higher fields, and interpreted the results in terms of afor several decades. In spite of the chemical simplicity of theory for nuclear-spin relaxation, valid under more generalthese systems, new experimental data and new interpreta- conditions for the electron-spin relaxation (4–6) ( i.e., withintions of old experiments keep appearing in the literature [ for the strong narrowing regime as well as outside of it, in thea review see, e.g., the book by Banci et al. (1)] . The subject so-called ‘‘slow-motion’’ regime). The same set of data hasof this paper, the aqueous solution of Ni(II) , was first stud- more recently been reinterpreted by Westlund et al. (10)ied by Morgan and Nolle (2) , who found no field depen- within a similar theoretical framework (and a somewhatdence for the proton spin–lattice relaxation time (T1) in the more general dynamic model for the lattice) and by Sharpfield range 0.05 to 1.4 Tesla (resonance frequency 2–60 (11) , who used still another theoretical model. Sharp’sMHz). Bloembergen and Morgan (3) explained this lack of ‘‘low-field approach’’ is based on the original work of Lind-field dependence in terms of the rapid electron-spin relax- ner (12) and is similar to the work of Bertini and co-workers,ation. They presented a simple theory, in which the electron- described in detail in the book by Banci et al. (1) . In all

this work, it has been assumed, for simplicity, that the zero-field splitting term in the spin Hamiltonian (1) can be de-* Current address: Faculty of Mathematics and Physics, Charles Univer-

sity, Ke Karlovu 3, CZ-121 16 Prague 2, Czech Republic. scribed by a cylindrically symmetric ZFS tensor.

1081064-1858/96 $18.00Copyright q 1996 by Academic Press, Inc.All rights of reproduction in any form reserved.

AID JMRA 0888 / 6j0f$$$101 07-08-96 06:13:13 maga AP: Mag Res, Series A

109PROTON RELAXATION IN NICKEL-ION SOLUTIONS

A different approach to electronic- and nuclear-spin relax- pole spin–lattice Hamiltonian: HDDIL Å ((01) i I 1

i T 10i . The

ation in aqueous Ni(II) solutions has recently been proposed lattice operator is a rank one tensor, obtained by contractingby Odelius et al. (13) . They used ab initio quantum-chemi- the electron-spin operator (rank one tensor) and the geome-cal calculations to estimate the variation of the zero-field try-dependent part of the dipole–dipole interaction (ranksplitting as a function of the geometry of the coordination two tensor) . The distance between the nuclear and the elec-shell. By combining this information with molecular-dynam- tron spin is assumed constant. The spatial orientation of theics (MD) simulations of the Ni(II) ions in aqueous solution, spin–spin axis defines the orientation of the molecular framethey were able to estimate the fluctuations of the ZFS and (M) with respect to the laboratory frame (L), whose z direc-to compute the relevant time-correlation functions (TCFs). tion is defined by the external magnetic field. The symbolAmong several important findings from that work, we wish sT denotes the thermal equilibrium density operator for thein particular to mention the observation that the splitting lattice; it should be noted that the lattice contains the classi-pattern of the triplet manifold did not conform to cylindrical cal degrees of freedom as well as the electron spin.symmetry of the ZFS tensor. In quantitative terms, the distri- A lattice of such a complicated nature can convenientlybution of the ZFS eigenvalues displayed three (and not two) be described, using the Liouville superoperator formulationpeaks. This observation provides us with a motivation to (14–16) , by a lattice Liouvillian LL . This superoperator canreconsider one of the assumptions of the previous models be expressed asfor spin relaxation in aqueous Ni(II) (9, 10) and to proposeanother reinterpretation of the 1987 data sets (9) . LL Å LS / LZFS / LR / LD, [4]

The outline of this paper is thus as follows. In the nextsection, we review briefly the relevant theory. Some compu- where the four terms correspond to the electron Zeemantational results are presented in the following section, which interaction, the zero-field splitting, and the reorientation ofalso contains the results of the least-squares fits of the new the complex and its distortion, respectively. The electronmodels to the experimental data in 0.091 molal Ni(ClO4)2 Zeeman Liouvillian is a commutator with the Zeeman Ham-solution (pH 0.1) at 51 and 717C. Finally, conclusions are iltonian, assuming an isotropic g tensor. The difference be-drawn in the last section. tween this study and the previous work is in the LZFS term,

which in the earlier studies was assumed to correspond to acylindrically symmetric ZFS tensor. Here, the tensor is al-THEORYlowed to have nonzero rhombicity. LZFS is thus a commutator

The theoretical formulation used in this paper follows with the Hamiltonian (in the laboratory frame):closely earlier work from our laboratories (4–6, 10) andwill therefore be presented very briefly. The enhancement HZFS Å ∑

n

(01) nS 20n f 2(L)

n ( t) . [5]of the nuclear spin–lattice relaxation rate (the paramagneticrelaxation enhancement, PRE) is due to dipole–dipole (DD)

S 20n are the electron-spin operators in the spherical tensorand Fermi contact (scalar, SC) interactions with electron

form and f 2(L)n ( t) are the spherical tensor components ofspin. The DD contribution usually dominates and can be

ZFS [the index (L) indicates the laboratory frame]. Theexpressed in terms of a spectral density functionZFS tensor components in the laboratory frame are explicitlyKDD

1 (v1) , taken at the nuclear Larmor frequency:time-dependent.

1T1

Å 02 Re[KDD1 (v1)] . [1]

The spectral density is the Fourier–Laplace transform of thetime-correlation function:

KDD1 (v) Å *

`

0

dtGDD1 (t)exp(0ivt) [2]

and

GDD1 (t) Å TrL{[exp(0iLLt)T 1

1sT]T 1

01}. [3]

FIG. 1. Limiting cases of the zero-field splitting of a triplet state: (a)T 1

1 (Å0T 1*01 ) is the lattice operator, formulated as a spheri- cylindrically symmetric ZFS, one doubly degenerate level and one nonde-generate level, (b) rhombic ZFS, three nondegerate levels.cal tensor component, in the expression for the dipole–di-


110 SVOBODA ET AL.

FIG. 2. The paramagnetic relaxation enhancement computed as a function of the magnetic field, using the SPR model. The diagrams a–d correspondto increasing pseudorotation correlation time and the motional conditions changing from the Redfield limit to the slow-motion regime for electron spinrelaxation (see text) . (a) tD Å 0.2167 ps (tDvD Å 0.1) , (b) tD Å 0.65 ps (tDvD Å 0.3) , (c) tD Å 2.167 ps (tDvD Å 1), and (d) tD Å 6.5 ps (tDvD

Å 3). The curves denoted I–IV in the four diagrams correspond to an invariant trace of the square of the ZFS tensor, and E /D ratios of 0, 0.1, 0.2,and 1

3, respectively.

The ZFS tensor is diagonal in its principal frame, denoted Dmn(V) are the Wigner rotation matrix elements. Equation[6a] is used in connection with the simple pseudorotation, SPR,by the superscript (P), and the components in the principalmodel and Eq. [6b] is utilized in the extended or coupledframe are assumed time-independent. The time dependencepseudorotation, CPR, approach (see below). The rhombicityof the f 2(L)

n components comes from the time-dependentof the ZFS tensor implies that both the f 2(P)

0 and f 2(P){2 elementstransformation of the coordinate system in one or two steps:

are different from zero. These spherical components in theprincipal frame are simply related to the more common energy-f 2(L)

n ( t) Å ∑m

f 2(P)m D 2

m ,n[VPL( t)] [6a]level splitting parameters, D and E , by f 2(P)

0 Å√2/3D ,

f 2(P){2 Å E . The rhombicity of the ZFS tensor for SÅ 1 implies,or

in terms of the energy-level pattern, the occurrence of threef 2(L)

n ( t) distinct levels, rather than the two present in the case of thecylindric ZFS tensor. The splitting patterns and their relationÅ ∑

mÅ0,{2

∑m=

f 2(P)m D 2

m ,m=[VPM( t)]D 2m=,n[VML( t)] . [6b]

to the parameters are illustrated in Fig. 1.



RESULTS AND DISCUSSION

We shall use two steps to present the results of the calcula-tions according to the models described above. First, weillustrate the effect of introducing the rhombicity of the ZFStensor by performing a set of calculations with differentdegrees of rhombicity (different E /D ratios) in differentmotional regimes. In the second step, we repeat the least-squares fitting of our old proton relaxation enhancement datafor the aqueous solution of Ni(ClO4)2 (9) , using the newmodels.

Figure 2 presents a series of NMRD profiles (the PREversus the magnetic field) , obtained using the simple pseu-dorotation (SPR) method. The molar ratio of water in theNi(II) hydration shell and in the bulk was arbitrarily set to0.001, the electronic g factor to 2.25, the distance betweenthe electron spin [treated as a magnetic point-dipole (5)]and the proton to 255 pm, and the rotational correlation timeFIG. 3. The paramagnetic relaxation enhancement computed as a func-

tion of the magnetic field, using the CPR model. The parameters are the to 50 ps. In each of the diagrams 2a–2d, the NMRD profilessame as those in diagram 2d (tD Å 6.5 ps or tDvD Å 3), corresponding are presented for a constant value of the pseudorotation cor-to the slow-motion regime for electron spin relaxation. The curves denoted relation time and the E /D ratio equal to 0, 0.1, 0.2, and 1

3I–IV correspond to an invariant trace of the square of the ZFS tensor, and( the maximum possible value) . For E Å 0, we assume D ÅE/D ratios of 0, 0.1, 0.2, and 1

3, respectively.3 cm01 ; for E x 0, the parameter D is adjusted so that thetrace of the square of the ZFS tensor, D 2 Å 2D 2 /3 / 2E 2

(17) , remains constant, independent of the change in theThe superoperators LD and LR are Markov operatorsrepresenting two rotational-diffusion processes: reorienta- E /D ratio. The rationale for this choice of the ZFS parame-

ters is that, in the simple Solomon–Bloembergen–Morgantion of the principal ZFS frame with respect to a molecule-fixed frame (distortion of the complex described as pseu- theory (1, 5) , the electron (and nuclear) relaxation rates

depend on tD and D, but not on the E /D ratio.dorotation) and reorientation of the molecule-fixed framewith respect to the laboratory frame ( rotation of the com- Diagrams 2a–2d differ in the assumed pseudorotation cor-

relation time. In Fig. 2a, we use a short tD Å 0.2167 ps,plex) . The former process corresponds to a variation ofVPM in Eq. [6b] , while the latter changes VML . The two which corresponds to the product tDvD (vD is the angular

frequency counterpart of the wavenumber quantity D definedprocesses are characterized by the rotational diffusion co-efficients DD and DR , respectively, or by correlation times above, vD Å 2pcD) equal to 0.1, well within the strong

narrowing or SBM limit specified by tDvD ! 1. In Figs.tD Å 1/6DD and tR Å 1/6DR . In the CPR model (10 ) ,the two reorientational motions are allowed to be corre- 2b–2d, the pseudorotation correlation times are tD Å 0.65,

2.167, and 6.5 ps, giving tDvD Å 0.3, 1, and 3, respectively,lated. In the SPR model (6 ) , the pseudorotation of theZFS tensor (and, consequently, the electronic relaxation) which corresponds to an increasing deviation from the strong

narrowing limit for the electronic relaxation, or to increas-and the spatial reorientation are assumed to be uncorre-lated; this may be problematic if tD is not much shorter ingly distinct slow-motion conditions. Because of the choice

made for the variation of the ZFS parameters, the diagramsthan tR . The assumption that the rotation of the complexhas no influence on the ZFS means that a single Euler in Figs. 2a and 2b contain only one curve each, correspond-

ing to the SBM limit. The variation of the NMRD profilesangle VPL transforms the principal ZFS frame into thelaboratory frame in Eq. [6a] of the SPR approach. The with the pseudorotation correlation time and the ZFS rhom-

bicity is different at high and at low field. All the curves inspectral density at the nuclear Larmor frequency is, inpractice, computed by setting up and inverting a matrix Fig. 2 display plateaus at low field. The level of the plateaus

decreases with increasing tD in Figs. 2a, 2b, and 2c. InM Å i (LL / v1 ) , expressed in a suitable basis set inLiouville space. Allowing rhombicity in the ZFS tensor simple terms, this trend corresponds to the increasing low-

field electron-spin relaxation rates when the modulation ofcorresponds, in a computational sense, to weakening theselection rules for the nonzero off-diagonal elements of the ZFS becomes slower. Under slow-motion conditions

(Figs. 2c and 2d), the low-field plateau level is also depen-the matrices LL or M . The CPR and SPR models differ inthe selection of the basis set, which is more complicated dent on the rhombicity—it decreases with increasing E /D

ratio. These results are in qualitative agreement with theand larger in the CPR model (10 ) . Both these models areused in this paper. findings of Sharp (18) .


112 SVOBODA ET AL.

TABLE 1Unconstrained Four-Parameter Least-Squares Fitting at 517C

SPR SPR CPR CPRMSB E Å 0 E Å D/3 E Å 0 E Å D/3

R (pm) 246 { 5 253 { 4 253 { 4 255 { 5 254 { 5tD (ps) 1.1 { 0.2 1.8 { 0.3 1.7 { 0.3 2.0 { 0.4 1.8 { 0.3tR (ps) 7.0 { 0.6 8.3 { 0.6 8.2 { 0.6 8.8 { 0.7 8.6 { 0.7D (cm01) 4.1 { 0.6 3.8 { 0.5 3.2 { 0.4 3.9 { 0.5 3.2 { 0.4s2 0.0008070 0.0007683 0.0008230 0.0007527 0.0008084

At high field, we can see in all the diagrams a local maxi- previous work (9, 10) , we assume that the rhombicity attainsthe highest possible value, E /D Å 1

3. There is also a physicalmum between about 10 and 60 Tesla. The increasing PREon the low-field side of the local maximum is related to reason for this assumption: the distribution of the eigenval-the electron-spin relaxation rates decreasing with increasing ues of the ZFS tensor in the quantum-chemistry/molecular-field. This trend occurs at lower fields in the slow-motion dynamics study (13) displayed three maxima. The interme-conditions. Increasing the rhombicity shifts the rise slightly diate eigenvalue was sharply peaked around zero, and theto higher fields. The decreasing PRE at the very high fields highest and the lowest eigenvalues were symmetrically dis-(actually not attainable with present magnet technology) is tributed around zero. We interpret this observation as indi-caused by the dispersion of the spectral density KDD

1 (v1) cating that the transient splitting of the triplet can meaning-when the nuclear Larmor frequency becomes greater than fully be represented by three equidistant levels. The spacingthe rate of modulation of the dipole–dipole interaction. This between the neighboring levels is 2E Å 2D /3; cf. Fig. 1.part of the curve is insensitive both to the rhombicity and The parameters from the least-squares fits [performed us-to the motional conditions. ing the program GENLSS (19) running on an IBM RISC

Using the coupled or extended pseudorotation method 6000 computer] are summarized in Tables 1 and 2, for the(CPR) results in diagrams very similar to those in Fig. 2. experiments at 51 and 717C, respectively. The results of theIn Fig. 3, we present the CPR counterpart of diagram 2d, SBM and SPR calculations with E Å 0 at 517C are identicalwhere the differences are largest. We can see that allowing to the data of Ref. (9) ; the corresponding CPR parameterscorrelation between the pseudorotation and the reorientation differ somewhat from the results of earlier work (10) . The(the cross correlation between the ZFS and the DD interac- earlier CPR data (10) were not based on least-squares fitting,tions) increases, for a given parameter set, the low-field but were rather a result of a set of sample calculations. Inplateau level of the NMRD profiles slightly, in agreement order to obtain convergence in all the fits at 717C, we foundwith earlier work (10) . it necessary to omit one of the points of the original data set

For comparison of the interpretative capability of the new (9) . The omission of this point resulted in rather substantialmodel with earlier work, using the SPR (9) and CPR (10) changes of some of the SBM and SPR (E Å 0) parametersmodels and assuming a cylindrically symmetric ZFS tensor, in Table 2 from those reported in the earlier work (9) .we report a series of least-square fits based on the same The least-squares-fitted parameters in each of the tablesexperimental data sets (aqueous proton PRE as a function differ significantly between the SBM results and the slow-of the magnetic field in 0.091 m solution of nickel(II) per- motion calculations. The parameters R , tD, and tR from thechlorate, low pH, and two temperatures, 51 and 717C). In four slow-motion calculations at each temperature are very

similar to each other. The parameter D differs little betweenorder to retain the same number of parameters used in our

TABLE 2Unconstrained Four-Parameter Least-Squares Fitting at 717C

SPR SPR CPR CPRMSB E Å 0 E Å D/3 E Å 0 E Å D/3

R (pm) 246 { 7 254 { 7 254 { 7 254 { 8 253 { 8tD (ps) 1.0 { 0.3 1.7 { 0.6 1.7 { 0.6 1.7 { 0.8 1.6 { 0.7tR (ps) 5.2 { 0.6 6.1 { 0.7 6.0 { 0.7 6.3 { 0.7 6.1 { 0.7D (cm01) 4.2 { 1.1 3.6 { 0.8 3.0 { 0.7 3.9 { 1.0 3.3 { 0.9s2 0.001300 0.001350 0.001386 0.001279 0.001303



the rhombicity, expressed as an E /D ratio, to the maximumpossible value of 1

3 [rather than to zero as in the earliercalculations (9, 10)] , and using the same number of least-squares parameters, does not reduce the variance of the fitcompared to the model with the ad hoc assumption of acylindrically symmetric ZFS. The reason for this may proba-bly be sought in the insensitivity of the NMRD curves (atleast under conditions of small deviations from the SBMlimit) to the fine details of the electron-spin relaxation mech-anism. Nevertheless, we believe that the new model maygive a better physical understanding of the underlying molec-ular processes.

ACKNOWLEDGMENTS

This work has been supported by the Swedish Natural Science ResearchCouncil. One of the authors (J.S.) expresses his gratitude to the Europeanand Swedish Tempus program authorities for providing a graduate-studenttraining scholarship within the framework of JEP 2327.

REFERENCES

FIG. 4. Results of the least-squares fits of the CPR model with E /D 1. L. Banci, I. Bertini, and C. Luchinat, ‘‘Nuclear and Electron Relax-Å 1

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tain also the variances of the fits, s 2 , which are only very6. P.-O. Westlund, N. Benetis, and H. Wennerstrom, Mol. Phys. 61,

weakly dependent on the computational procedure. The tem- 177 (1987).perature dependence of the data (revealed by a comparison 7. H. L. Friedman, M. Holz, and H. G. Hertz, J. Chem. Phys. 70, 3369of the two tables) is reasonable, in the sense that both corre- (1979).lation times decrease somewhat with increasing temperature 8. H. G. Hertz and M. Holz, J. Magn. Reson. 63, 64 (1985).while the interaction strength parameters (R and D) remain 9. J. Kowalewski, T. Larsson, and P.-O. Westlund, J. Magn. Reson.

74, 56 (1987).essentially temperature-independent. Graphical presenta-10. P.-O. Westlund, T. P. Larsson, and O. Teleman, Mol. Phys. 78,tions of one set of the new fits (CPR with E Å D /3) at the

1365 (1993).two temperatures are shown in Fig. 4. The best-fit curves11. R. R. Sharp, J. Chem. Phys. 98, 912 (1993).corresponding to the SPR model (as well CPR with E Å 0)12. U. Lindner, Ann. Phys. (Leipzig) 16, 319 (1965).are practically indistinguishable from the graphs presented13. M. Odelius, C. Ribbing, and J. Kowalewski, J. Chem. Phys. 103,in Fig. 4.

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14. L. T. Muus and P. W. Atkins (Eds.) , ‘‘Electron Spin Relaxation inCONCLUSIONS Liquids,’’ Plenum Press, New York, 1972.

15. J. Jeener, Adv. Magn. Reson. 10, 1 (1982).The new model for the distortional modulation of the16. R. R. Ernst, G. Bodenhausen, and A. Wokaun, ‘‘Principles of Nu-

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demic Press, New York, 1972.is applied to fit the previously reported NMRD data. Setting


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Field-Dependent Proton NMR Relaxation in Aqueous Solutions of Ni(II) Ions. A New Interpretation