Electron spin relaxation at low field

Abstract

The low field ESR lineshape and the electron spin–lattice relaxation correlation function are calculated using the stochastic Liouville theory for an effective electron spin quantum number S = 1. When an axially symmetric permanent zero field splitting provides the dominant relaxation mechanism, and when it is much larger than the rotational diffusion constant, it is shown that both electron spin correlation functions 〈S¹_n(0)S¹_n(t)〉 (n = 0,1) are characterized by the same relaxation time τ_S = (4D_R)⁻¹. This confirms the conjectures made by Schaefle and Sharp, J. Chem. Phys., 2004, 121, 5287 and by Fries and Belorizky, J. Chem. Phys., 2005, 123, 124510, based on numerical results using a different formalism. The stochastic Liouville approach also gives the paramagnetically enhanced nuclear spin relaxation time constants, T₁ and T₂, and the ESR lineshape function I(ω). In particular, the L-band (B₀ = 0.035 T) ESR spectrum of a low symmetry Ni(II)-complex with a cylindrical ZFS tensor is shown to be detectable at sufficiently slowly reorientation of the complex. The analysis shows that the L-band spectrum becomes similar to the zero-field spectrum with a electron spin relaxation time τ_S = (4D_R)⁻¹.

---

1. Introduction

The relaxation behavior of the electron spin system in transition metal complexes with an effective spin,¹S≥ 1, has important implications in basically two contexts.² The electron spin resonance (ESR) signals are strongly affected by the relaxation and under some circumstances the signals are so broad that they are not detectable in practice. Secondly, the electron spin provides a relaxation mechanism for nuclear spins in the vicinity. This effect is called paramagnetic relaxation enhancement, (PRE), and has been used successfully for more than fifty years³ to study molecular properties of transition metal complexes,^4,5 paramagnetic molecules in biological systems^6,7 and in the development of new contrast agents for magnetic resonance imaging.⁸

The dominant relaxation mechanism for electron spin systems with S≥ 1 is due to the zero field splitting (ZFS).¹ For octahedral hexa aquo complexes, [M²⁺(H₂O)₆], the ZFS interaction is zero by symmetry. However, there are fluctuations in the ZFS, caused by solvent collisions. For flexible paramagnetic complexes of low symmetry, the ZFS interaction has a fast fluctuating transient ZFS with a non-zero average. This residual ZFS is called the permanent ZFS, with a magnitude that changes through rotational diffusion. The collision induced distortions of the coordination sphere thus modulate the transient ZFS and are typically in the sub ps to ps time regime. These timescales of the transient ZFS interaction were recently demonstrated in an MD simulation of the fluctuating electric field gradient of the Gd(H₂O)³⁺₈ complex.⁹ The reorientation of the complex, on the other hand, is expected to take place on much a longer time scale of typically 30–100 ps. This difference in time scale makes it possible to decompose the ZFS Hamiltonian into permanent and transient ZFS terms. In recent developments of the PRE theory, both transient and permanent ZFS are taken into account.^10–13

We have previously presented a PRE formalism based on the stochastic Liouville equation (SLE).¹⁴ Within this formalism one can account for slow motion effects in the electron spin system. It is straightforward to generalize the formalism to explicitly describe the electron spin–lattice and spin–spin relaxation processes. In the present paper we focus on the electron spin relaxation properties of rigid paramagnetic complexes (S = 1), with only a permanent cylindrical ZFS interaction. We focus on the electron spin–lattice relaxation and the ESR line shape for slow tumbling complexes in the low field regime, where the Zeeman interaction is small compared with the permanent ZFS interaction. In this regime the traditional Bloembergen–Morgan theory⁵ is not valid.

The problem of interest in this work emerged from the work by Schaefle and Sharp¹⁵ and Fries and Belorizky.¹⁶ Their theoretical descriptions differ from the SLE approach, and based on their numerical results, the authors of these two papers make an interesting conjecture. In the low field limit, when the ZFS interaction (in rad s⁻¹) also dominates the rotational diffusion constant D_R, the spin relaxation time obeys the relation τ_S≈ (4D_R)⁻¹. The aim of this paper is to use the stochastic Liouville approach¹⁷ in the Fokker–Planck form¹⁹ to rationalize this result. We note that for the transverse relaxation, Lynden-Bell made a similar observation using the SLE approach already in 1971.¹⁸

The SLE formalism determines the orientation dependent spin density matrix operator, σ(Ω,t), from which several observables may be determined: (a) the ESR-lineshape function; (b) the electron spin–lattice correlation function, C₀(t); and (c) the spectral density determining the PRE nuclear spin relaxation rates. All these measurable quantities are given by different matrix elements of a Liouville matrix. This illustrates an important strength of the SLE formalism. The calculations of the electron spin–lattice relaxation at low field are compared with the results of Schaefle and Sharp¹⁵ and Fries and Belorizky.¹⁶ In particular, the SLE calculations of C₀(t) are obtained in the Fourier–Laplace space as C̃₀(ω) = ∫^∞₀C₀(t)e^iωt dt, and compared with the Fourier–Laplace of the analytical approximation obtained in ref. 16.

The influence of electron spin relaxation on the paramagnetically enhanced water proton spin T₁ NMR dispersion profile is often predominant. However, the electron spin relaxation time relation; τ_S≈ (4D_R)⁻¹, was not evident in the NMRD profiles first calculated.^13,14 It is thus interesting to note that the relation τ_S≈ (4D_R)⁻¹ was revealed in numerical calculations of electron spin–lattice relaxation in the time domain but no attempt was made to rationalize the observation.^15,16 Instead it turns out that the simple relation can be clearly explained within the SLE formalism. Different theoretical approaches may shed a different light on the same phenomenon.

A comparison between SLE¹⁴ and other simulation approaches^15,16 has recently been presented in the context of paramagnetically enhanced nuclear spin relaxation.²⁰ The low-field PRE data is complicated by the presence of a strong correlation between electron spin relaxation and molecular reorientation present in the electron spin dipole–nuclear spin dipole coupling. Thus using PRE data to study the electron spin relaxation may be complicated by this correlation effect. In fact, in some cases, it can contribute as much as 40% to the total relaxation enhancement.²¹ Consequently, the decomposition approximation¹² is a prerequisite for making a direct relation between electron spin relaxation processes and PRE data.

The paper is organized as follows: Firstly we review the Liouville theory and show how spin dynamic information is related to the actual Liouville matrix. Then, the numerical results are presented and discussed in terms of electron spin correlations functions at zero field and the ESR lineshapes at low and intermediate fields. From the Liouville matrix we show that the relation between the electron spin relaxation time τ_S at low fields and the reorientation diffusion constant, τ_S = (4D_R)⁻¹, is obtained from a perturbation treatment of the Liouville matrix. Finally, we show that an L-band ESR lineshape of a rigid low-symmetric and slow tumbling Ni(II)-complex (where the rhombic term E = 0) may give rise to a detectable narrow ESR spectrum.

2. The SLE formalism

The equation of motion of an electron spin system is,^18,19where σ(Ω,t) is the orientation dependent electron spin density operator. The Liouville (super)operator ([script L](Ω) ≡ [H(Ω),…]) is generated by the Hamiltonian H(Ω), here given in angular frequency units. The Liouville operator may be explicitly time dependent due to physical variables not explicitly included in the Hamiltonian. Several spectroscopy experiments give results in the frequency domain. Hence, it is convenient to consider the Fourier–Laplace transform of eqn (1) reading,¹⁸

σ(0) = i[[script L](Ω) − 1ω][small sigma, Greek, tilde](ω,Ω), (2)

where the transformed density operator is defined by [small sigma, Greek, tilde](ω) = ∫^∞₀σ(t)e^iωt dt, and the stochastic time dependent variables are denoted with Ω. The Hamiltonian comprises a Zeeman term and an axial symmetric permanent ZFS interaction,¹³Here, S⁽²⁾_n is a standard second rank electron spin operator and D⁽²⁾_0−n[Ω] a Wigner rotation matrix element with stochastic time dependent Euler angles, Ω, specifying the orientation of the principal frame of the permanent ZFS tensor (f₀ = √(⅔)D) relative to the laboratory fixed frame. When the SLE equation of motion describes a quantum mechanical spin system S imbedded in a “large” classical lattice, the classical reorientational degrees of freedom (Ω) may be described by a rotation diffusion operator, D_R∇²_Ω. In slowly reorienting spin systems the average spin density operator, σ(Ω,t), becomes a function of spin operators and reorientational degrees of freedom.¹⁸ The density operator [small sigma, Greek, tilde](ω,Ω) then satisfies:

[small sigma, Greek, tilde](Ω,ω) = [i[script L]₀−iω1 + i[script L]_ZFZ(Ω) −D∇²_Ω]⁻¹σ(Ω,0) ≡ [[scr M, script letter M]−iω1]⁻¹σ(Ω,0). (4)

In order to solve this equation, a complete basis set of operators is introduced which span the Liouville space, i.e. the electron spin system and the reorientational space. A convenient electron spin basis set is formed by the normalized electron spin operator |Σ,σ),^14,18,22where the ranks are Σ = 0,1,…,2S and components are σ = −Σ,−Σ + 1…Σ− 1,Σ. denotes a standard 3-j symbol and the ket–bra electron spin operator; |S,m〉〈S,n|; is formed as an outer product of spin eigenfunctions of S¹₀ and S⃑².

The molecular reorientational space is spanned by the symmetric top eigenfunctions.^13,18

A complete basis set of the Liouville space is then formed as a direct product of the two basis sets: {|Σ,σ) ⊗ |L,M)}. In the derivation of the low field lineshape function for S = 1, Lynden-Bell used a low field basis set which may be more convenient at B = 0.¹⁸ However, for the numerical calculations the high field basis set is also very useful. One may pass from the high field regime to the low field or ZFS regime just by increasing the dimension of the Liouville matrix to guarantee convergence.

The SLE equation of eqn (4) is expressed in this representation, where the expansion coefficients of the density matrix are {c̃_n(ω)}:

Here, [scr M, script letter M] refers to the Liouville matrix in the high field representation. The problem of solving the stochastic Liouville equation of motion has been transformed to find the inverse of the Liouville matrix, [[scr M, script letter M]−iω1]⁻¹.^13,14,23

3. The ESR lineshapes and spectral densities

Observable quantities are represented by different density matrix elements of the basis set {|Σ,σ) ⊗ |L,M)}. The electron spin density operator representing the transverse magnetization is characterized by the density matrix element |1,1)|0,0), and the ESR line shape function is obtained from the frequency dependence of one Liouville matrix element,The Fourier–Laplace transform of the electron spin–lattice correlation function C₀(t) is determined by the density matrix element |1,0)|0,0), through

The paramagnetically enhanced nuclear spin lattice relaxation rate is determined by a rather complex electron spin-nuclear spin dipole–dipole spectral density K^DD_1,−1(ω_I).^12–14,23 It is obtained from a 3 × 3 submatrix of the Liouville matrix [scr M, script letter M] defined by the basis operators {|1,p)|2,1 −p)} (p = 1,0,−1):

Here ℏ is the Planck constant divided by 2π, ω_I is the proton Larmor frequency, r_IS is the I–S interspin distance and γ_I, γ_S and μ₀ are the nuclear magnetogyric ratio, the electron magnetogyric ratio, and the permeability of space, respectively. The quantity, Tr{(S¹_qD²_0,1−q(Ω))*e^{−i[script L]τ}S¹_pD²_0,1−p(Ω)} is the composite lattice electron spin-reorientation correlation function.

When the spin dynamics and the reorientation dynamics are uncorrelated one can apply the decomposition approximation:

It is only in this limit that the spectral density K^DD_1,1(ω_I) is an explicit function of the ESR lineshape correlation function (cf. eqn (8)) and the electron spin–lattice spectral density C̃₀(t) (cf.eqn (9)),

However, when B₀≈ 0, the electron spin is trapped in a molecule-fixed principal frame of the ZFS-tensor. In the absence of an additional collisionally modulated ZFS tensor, and f₀τ_R≥ 1, the reorientational motion and the spin dynamics become strongly correlated. Consequently, the decomposition approximation, (cf.eqn (11)), is not applicable in this case. In PRE calculations of slowly reorienting low symmetric paramagnetic complexes it has been shown that the reduced spectral density, , was equal to unity in the whole low field region (f₀τ_R > 1).^13,14,23K̃ = 1, represents the case when the effective dipole–dipole correlation time is equal to τ_R≡ (6D_R)⁻¹. This result is independent of the electron spin quantum number S≥ 1.^13,14,23 We may estimate the correlation contribution by assuming for a moment that the spin relaxation and the molecular reorientation of the dipole–dipole correlation function are decomposed. Then the effective correlation time of eqn (11) may be defined as 1/τ_eff = 1/(τ_S(=[/]τ⁽²⁾_R) + 1/τ⁽²⁾_R) which yields K̃ = ⅗. Consequently, the correlation effect in the dipole–dipole correlation function contributes with ⅖ or 40%.

4. Numerical results

Schaefle and Sharp¹⁵ argued in favour of the direct simulation method by stating that “the density matrix theory (i.e., Redfield theory) of spin relaxation does not provide an appropriate description of the reorientational mechanism at low Zeeman field strength because the zero order spin wave functions are stochastic functions of time”. It should be clear by now that all three formalisms discussed²⁰ and correctly used, provide valid results. A practical advantage of the SLE formalism is that it also provides a basis for obtaining analytical results in different limiting cases.

For S = 1 and axial symmetry, Fries and Belorizky¹⁶ investigated the low field regime, B₀ = 0, using their direct simulation approach. The electron spin correlation function was calculated with ZFS interaction ranging from D (≡√([/])f₀) = 0.01, 0.1, 1.0 cm⁻¹ and a reorientational correlation time τ_R(≡6D_R⁻¹) = 100 ps. These illustrative cases correspond to the Kubo parameter f₀τ_R = 0.15, 1.54 and 15.4 which means that two cases correspond with slow-motion regime. The electron spin correlation function decays exponentially for D = 0.01 cm⁻¹, whereas for D = 0.1 and D = 1 cm⁻¹, it displays a damped oscillation. According to the authors of ref. 16, the electron spin correlation function can be fitted to the static limit expression with an exponential relaxation term reading:¹⁶

From the SLE calculations we obtain the electron spin correlation function in the Fourier–Laplace space. This means that SLE results may be compared with the Fourier–Laplace transform of eqn (12) reading:¹⁶

When f₀τ_R≫1, (ω_D is the frequency of the ZFS system with splitting D) and setting the electron spin–lattice relaxation time to τ_S = [/]τ_R, eqn (13) displays a dispersion at ω[/]τ_R≥ 1, and a Lorenzian peak centered at ω_D, which has a full width at half height equal to 2/[/]τ_R. Fig. (1a) displays the spin–lattice spectral density C̃₀(ω), which is obtained from the SLE theory (cf.eqn (9)) at low field, B₀ = 0.0035 T and τ_R = 100 ps. It may be compared with b), which displays the approximate spectral density eqn (13). In (c) the dispersion is shown, thus confirming that in the rigid limit, eqn (13) becomes identical with SLE. When the static field is increased to 0.35 T (X-band), the marked peak at ω_D almost disappears in the SLE calculation (not shown). In the limit of large ZFS interaction we could also confirm that for electron spin quantum number S = [/], S = [/] and S = 7/2, the electron spin relaxation time approaches [/]τ_R. Consequently, the analytical form of eqn (13) for the electron spin–lattice spectral density only conforms to the SLE calculations in the zero field limit when the condition, ω_Dτ_R≫ 1, also applies.

Fig. 1 The spectral density C̃₀(ω) is displayed for f₀ = 0.1 cm⁻¹ and τ_R = 100 ps. In (a) the SLE result (cf.eqn (9)) and in (b) eqn (13) and in (c) the function .

Fig. 2 displays the calculated ESR-lineshape function I(ω) for S = 1 corresponding to X-band, L-band and B≈ 0 together with the absorption lineshape for f₀ = 0.1 cm⁻¹ and the reorientational correlation time τ_R = 100 ps. Interestingly, the ESR- and the absorption lineshapes become very complex as we pass from X-band (0.35 T) in (2a), L-band (0.035 T) in (2b) to the zero field cases shown in (2c) and (2d). In Fig. 2d the reorientational correlation time is 500 ps and the central ESR signal now displays a nice Lorentzian. These findings may open up the possibility of recording the ESR signal of rigid low symmetric Ni(II) complexes. Generally Ni(II) complexes are not detectable at high fields because of their large ZFS interaction (i.e. >1 cm⁻¹). Consequently, for low field ESR spectra, such as L-band, we have H_ZFS≫H_Zeem and an ESR spectrum should be similar to a zero field spectrum. Fig. 3 displays a L-band ESR spectrum, calculated for f₀ = 1.0 cm⁻¹ and τ_R = 10 ns. It is centered at the ZFS frequency ω_D. In principle, when the ZFS is expected to be much larger than the Zeeman interaction, an otherwise unobservable ESR spectra may be detected if it is possible to slow down the reorientational dynamics of the reorientation modulated ZFS sufficiently to yield a narrow Lorenzian line shape. When f₀τ_R increases, the L-band ESR signal of Fig. 3 becomes more and more Lorenzian with a full at half height approaching .

Fig. 2 (a–d) The ESR spectra and the corresponding absorption lineshapes are displayed for S = 1. In (a) the X-band (B₀ = 0.33 T) and in (b) the L-band (B₀ = 0.035 T) spectra. In (c) and (d) the zero field spectra all calculated for the reorientation correlation time τ_R = 100 ps and the ZFS interaction f₀ = 0.1 cm⁻¹ except for (d) where τ_R = 500 ps.

Fig. 3 The L-band ESR and absorption spectra (S = 1) are displayed centered around the ZFS frequency ω_Q for the interaction f₀ = 1.0 cm⁻¹ and τ_R = 1.0 ns.

5. Why is τ_S = (4D_R)⁻¹ in the low field and slow tumbling regime?

A remarkable aspect of the low field spin relaxation for S = 1 is the simple result τ_S = (4D_R)⁻¹ (or, τ_S = [/]τ_R) for both T₁ and T₂ relaxation times. Such simple results should have a transparent physical interpretation. There are two aspects of the results that need to be explained. Firstly, we have the fact that the spin relaxation times are inversely proportional to the rotation diffusion constant. The second notable fact is the proportionality constant ¼. For rotational correlation times the proportionality constant is ½ for the relaxation of a first rank tensor and ⅙ for a second rank one.

In the zero field limit the electronic states remain unaffected in the molecular frame. This remains true as long as one does not consider additional relaxation mechanisms for the ZFS degrees of freedom. Such generalizations have been made listing a variety of methods.^24–27 In the model used in the present paper, as well as in ref. 15 and 16 it is only when the spin system is seen from the laboratory fixed frame that there are spin relaxation dynamics. This is due to the rotational diffusional motion. Consequently, the electron spin relaxation is directly determined by the rotational diffusion constant.¹⁸

It is more involved to see the origin of the proportionality factor ¼. The underlying mechanism can be traced back to the fact that the ZFS Hamiltonian leads to a coupling between second and first rank irreducible tensor operators. In her paper from 1971, Lynden-Bell gives the relevant matrix elements using ‘low field’ basis operators. Owing to the overall rotational symmetry of the combined ZFS and diffusion operators, the corresponding matrix is block-diagonal in the limit of zero magnetic field. For a cylindrical symmetric ZFS-tensor one obtains a 3 × 3 matrix.¹⁸ This matrix is decomposed into a ZFS(F) and a diffusion D component: M = F + D. Now F is diagonalized by a transformation T. In the slow reorientation regime, when the ZFS is much larger than D_R, the effect of the diffusion can be treated as a first order perturbation,

M⁻¹_ii = (T⁻¹MT)_ii≈ (T⁻¹FT)_ii + (T⁻¹DT)_ii, (14)

and an explicit calculation yields (T⁻¹DT)_ii = 4D_R for i = 1, 2, 3 while for D_ii = 6D_R.

6. Conclusions

In this paper we have demonstrated how the stochastic Liouville equation (SLE) is a powerful formalism to describe electron spin relaxation in the low magnetic field limit. It can be used to obtain numerical results for a general relation between the ZFS and D_R. We show that the analytical approximation for the 〈S¹₀S¹₀(t)〉 correlation function suggested by Fries and Belorizky¹⁶ is only valid for ZFS/D_R≫ 1. The relevant electron spin relaxation times are τ_S = 4D_R⁻¹ in this limit. This result has been derived analytically and the physical interpretation of the result has been given. An intriguing consequence of the calculation is that it suggests that one can obtain observable ESR signals in the low field limit. Normally a large ZFS results in very broad signals. The calculation further indicates that the refocusing of the signal occurs as the field becomes negligibly small and the rotation diffusion is slow. It requires further investigations to assess whether or not this conclusion remains valid in the presence of other relaxation processes for the zero field splitting energies and when the rhombicity E≠ 0.

Acknowledgements

We are grateful to Dr Michael Sharp for valuable linguistic revisions. This work was supported from the Swedish National Research Council.

References

1. A. Abragam and B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, 1970, Oxford Press, Dover, 1986.
2. NMR of Paramagnetic Molecules, Principles and Applications, ed. G. N. LaMar, W. DeW. Horrocks, Jr and R. H. Holm, Academic Press, 1973.
3. N. Bloembergen, J. Chem. Phys., 1957, 27, 572.
4. L. O. Morgan and A. W. Nolle, J. Chem. Phys., 1959, 31, 365.
5. N. Bloembergen and L. O. Morgan, J. Chem. Phys., 1961, 34, 842.
6. D. R. Burton, S. Forsen, G. Karlsrtöm and R. A. Dwek, Prog. Nucl. Magn. Reson. Spectrosc., 1979, 13, 1.
7. I. Bertini and C. Luchinat, NMR of Paramagnetic Molecules im Biological Systems, The Benjamin/Cummings Publishing Company, Inc, 1986.
8. S. Aime, M. Botta and E. Terreno, Gd(III) Based Contrast Agents for MRI, Adv. Inorg. Chem., 2005, 57 (Relaxometry of water–metal ion interactions, Elsevier).
9. M. Lindgren, A. Laaksonen and P.-O. Westlund, Phys. Chem. Chem. Phys., 2009 10.1039/b907099k.
10. L. Helm, Prog. Nucl. Magn. Reson. Spectrosc., 2006, 49, 45–64.
11. J. Kowalewski, D. Kruk and G. Parigi, NMR Relaxation in Solutions of Paramagnetic Complexes: Recent Theoretical Progress for S≥ 1, Adv. Inorg. Chem., 2005, 57 (Relaxometry of water–metal ion interactions, Elsevier).
12. P.-O. Westlund, in Dynamics of Solutions and Fluid Mixtures by NMR, ed. J.-J. Delpuech, John Wiley & Sons, Chichester, 1995, pp. 174–229.
13. J. Kowalewski, L. Nordenskiöld, N. Benetis and P.-O. Westlund, Prog. Nucl. Magn. Reson. Spectrosc., 1985, 17, 69–186.
14. N. Benetis, J. Kowalewski, L. Nordenskiöld, H. Wennerström and P.-O. Westlund, Mol. Phys., 1983, 48, 329.
15. N. Schaefle and R. Sharp, J. Chem. Phys., 2004, 121, 5387.
16. P. H. Fries and E. Belorizky, J. Chem. Phys., 2005, 123, 124510.
17. R. Kubo, J. Math. Phys., 1963, 4, 174; R. Kubo, Adv. Chem. Phys., 1969, 15, 101 , ed. K. E. Shuler, Wiley, New York.
18. R. M. Lynden-Bell, Mol. Phys., 1971, 22, 837.
19. J. Freed, Electron Spin Relaxation in Liquids, ed. L. T. Muus and P. A. Atkins, Plenum Press, 1972.
20. E. Belorizky, P. H. Fries, L. Helm, J. Kowalewski, D. Kruk, R. R. Sharp and P.-O. Westlund, J. Chem. Phys., 2008, 128, 052315.
21. N. Benetis, J. Kowalewski, L. Nordenskiöld, H. Wennerström and P.-O. Westlund, Mol. Phys., 1983, 50, 515.
22. D. M. Brink and G. R. Satchler, Angular Momentum, Oxford Univ. Press, 5th edn, 1971.
23. P.-O. Westlund, H. Wennerström, L. Nordenskiöld, J. Kowalewski and N. Benetis, J. Magn. Reson., 1984, 59, 91–109.
24. H. Eviatar, E. E. van Fassen, Y. K. Levin and D. I. Hoult, Chem. Phys., 1994, 181, 369.
25. N. Usova, P.-O. Westlund and I. Fedchenia, J. Chem. Phys., 1995, 103, 96.
26. K. Åman and P.-O. Westlund, Mol. Phys., 2004, 102, 1085.
27. P.-O. Westlund, J. Chem. Phys., 1998, 108, 4945.

---