Theoretical Aspects of the Cross Effect Enhancement of Nuclear

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Spectroscopy and Photochemistry; General Theory

Theoretical Aspects of the Cross Effect Enhancement of Nuclear Polarization under Static DNP Conditions Krishnendu Kundu, Akiva Feintuch, and Shimon Vega J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.8b03615 • Publication Date (Web): 13 Mar 2019 Downloaded from http://pubs.acs.org on March 16, 2019

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The Journal of Physical Chemistry Letters

Theoretical Aspects of the Cross Effect enhancement of Nuclear Polarization under Static DNP Conditions Krishnendu Kundu1, Akiva Feintuch1, Shimon Vega1*

1. Department of Chemical and Biological Physics, Weizmann Institute of Science, Rehovot-76100, Israel.

* Corresponding Author Email: [email protected]

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Abstract In this study, we perform quantum calculations of the spin dynamics of a small spin system including 9 coupled electrons and one nucleus placed in an external magnetic field and exposed to microwave (MW) irradiation. This is an extension of a previous work in which we have demonstrated on a system of 11 coupled electron spins the dynamics of the electron polarizations composing the EPR line during static dynamic nuclear polarization (DNP) experiments. There we have shown that the electron polarizations are determined by a spectral diffusion process, induced by the dipolar interaction and cross-relaxation. Additionally, we showed that a distinction had to be made between strong and weak dipolar coupled systems relative to the inhomogeneity of the EPR line with only the first behaving according to the Thermal Mixing DNP (with two electron spin temperatures) description. The EPR spectra in the weak and strong dipolar interaction cases show different types of spectral features. In the extended spin system, we again make a distinction between weak and strong electron-electron interactions and show that the DNP spectra for the two cases are different in nature, but that the DNP spectra can be derived in all cases from the EPR lineshapes using the indirect Cross Effect (iCE).

Table of Contents (TOC) Graphic

iCE eSD

iCE

TM


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Keywords Indirect Cross effect, Thermal Mixing, Cross Relaxation, DNP, Electron Spectral Diffusion



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Since the last decade, the applications of dynamic nuclear polarization (DNP) have a profound impact on signal enhancement experiments in NMR1,2 and MRI3,4. It is already more than sixty years ago that DNP was predicted by Overhauser5 and subsequently shown experimentally on metallic samples by Carver and Slichter6,7, known as the Overhauser-DNP effect. Shortly thereafter, DNP enhancements in non-conducting solids containing free radicals, exhibiting EPR lines with frequency widths smaller than the nuclear Larmor frequency, were reported by Jeffries8,9 and became known as the Solid Effect (SE). Later, DNP enhancements were observed in radical containing samples with EPR linewidths larger than the nuclear Larmor frequency, termed as the Cross Effect (CE)10,11. Following these observations and realizing that the DNP process is a multi-spin mechanism, a statistical theory was introduced based on spin temperature concepts, introduced by Redfield12 and extended by Provotorov to describe DNP13,14. Continuing in these lines the spin temperature description of the spin system for explaining DNP experiments was promoted and became known as Thermal Mixing (TM)15–20. A full description of TM can be found in a recently published book on “Essentials of Dynamic Nuclear Polarisation” 20. During the past few years, a variety of quantum mechanics simulations have been done on small spin systems to study the role of TM in DNP. For example, a recent work by Karavanov et al21 showed, for small spin systems, the interaction conditions that result in spin dynamics that coincide with the TM model, and in what sense it differs from the CE processes. In addition, De Luco et al. 22 and Caracciolo et al. 23 considered larger spin systems, demonstrating the conditions leading to TM relying on thermalization processes in open quantum systems. The DNP mechanism in solid solutions containing free radicals can, in general, be visualized in terms of a two-step process. At first, electron transitions composing the EPR spectrum of the free radicals are partially saturated by the MW irradiation field and the spectral diffusion process20,24–26. During SE-DNP process this saturation coincides with the DNP enhancement of the nuclei hyperfine coupled to the electrons. In the CEDNP case, we can make a distinction between the selective electron depolarization process caused by the MW and the DNP enhancement that is a result of this depolarization. In this case, the non-equilibrium electron 4 ACS Paragon Plus Environment

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polarization gradients over the whole EPR spectra result in a polarization of all nuclei that are hyperfine coupled to the free electrons in the sample27. This effect we have termed the indirect-CE (iCE), derived from the pioneering work of Wollan27,28, and can be used to derive the DNP enhancement. During the second stage of the DNP process, the polarizations of these localized groups of nuclei around the electrons are transferred to the rest of the (bulk) nuclei in the solid solution via the spin-diffusion process29,30. In a very recent publication31, we addressed the electron spin dynamics of the first stage of the DNP process. We did so, by performing quantum-based simulations of EPR spectra of small electron-spin systems exposed to MW irradiation. We discussed the requirements on the average magnitude of the electron-electron dipolar interactions for obtaining a well-defined electron spin temperature. Furthermore, we showed the need for zero-quantum cross-relaxation32–35 to explain experimental results of the lineshapes and the temporal dependence of the EPR spectra. In the present study, we address the DNP process and continue our earlier calculations, by simulating DNP spectra of nuclei coupled to the electrons in our small spin system. Here, we in particular aim to show the importance of the iCE mechanism for evaluating these DNP spectra. The motivation, to reinvestigate the DNP enhancement process by performing quantum simulations on small spin systems, was driven by a need to find theoretical indications for the validity of the iCE mechanism. During recent years we performed a variety of electron-electron double resonance (ELDOR) experiments at 3.34 T, on samples containing TEMPOL radicals at different concentrations and sample temperatures26,36. Sets of ELDOR spectra for different detection frequencies were used to reproduce the EPR spectra during MW irradiation. These experiments showed that the expected depolarization profiles, predicted by the TM description of the spin system, do not appear for the TEMPOL radical. To analyze our ELDOR data, we introduced a phenomenological model26 reconstructing the experimental ELDOR and EPR lineshapes, based on an electron spectral diffusion (eSD) mechanism. With this eSD model, we succeeded to reproduce the EPR lineshapes.



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Using these lineshapes, we relied in practice on iCE calculations to calculate DNP spectra that resembled the experimental DNP spectral lineshapes36. Thus, we used the two methods of calculation to analyze our experimental ELDOR, EPR and DNP results. The purpose of the present study is to verify, on the basis of quantum calculations, the validity of the iCE approach in spin systems with weak electron-electron interactions, in which the TM description is not sufficient, and with strong interactions, where the TM formalism does describe the spin behavior. To do so, we again consider a small spin system, but this time composed of nine electrons and one nucleus. After defining the interaction and relaxation parameters of this system we use the population rate equations derived from the master equation to calculate energy-population plots. In the strong dipolar interaction regime, we analyze these plots in terms of spin temperature concepts. Then we show the ELDOR and EPR lineshapes of these systems and finally show that the iCE model is able to predict the DNP spectra. The spin system, its Hamiltonian and the population rate equation In this section, we present the spin systems that are used to investigate the validity of the iCE mechanism. Choosing a system with interaction parameters, as similar as possible with parameters commonly found in real amorphous solid solutions of free radicals, we take into account that the average dipolar interactions between the electrons in a 20 to 40 mM radical solution is of the order of 0.6 to 1.2 MHz. Depending on the g-tensor parameters of radical, the resonance frequency spread of the electrons can vary between tens of MHz for trityl radicals to hundreds of MHz for nitroxides at 3.3T. To mimic this inhomogeneously broadened dipolar-coupled electron spin system, we have chosen for our calculations a hypothetical 10-spin systems. The systems consist of nine electrons, with different Larmor frequencies, ωe / 2π + (m − 5) × δω / 2π , where ωe / 2π = 95 GHz , δω / 2π = 10 MHz and m = 1,..,9 , and one nucleus that is hyperfine coupled to one of these electrons. The total frequency spread in this system is 100MHz , which is less than what we find in nitroxides at 3.3 T. We are limited in the number of electrons

during the simulations and in order to use electron dipolar interactions, which are in the megahertz regime, we 6 ACS Paragon Plus Environment

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must restrict ourselves to values of δω / 2π that do not quench these interactions. As in Ref 31, the nine electrons are positioned in an external field and are all dipolar coupled to each other with a strength equal to

xm,m ' D , m < m' = 1,..,9 , w , where 3 / 4 D represents an average dipolar frequency and xm,m' are randomly chosen scaling coefficients between − 1 / 2 and 1 and

∑

x

m< m ' m ,m '

= 0 . We will present “weak coupling” results

using D / 2π = 2 MHz , which is similar to the average dipolar interaction of a 40mM nitroxide solution. As x D for most electron pairs {m, m '} the value of m ,m '

is much smaller than the difference between their

2 frequencies, their average effective interactions for D / 2π = 2 MHz are thus proportional to 2 / 100 . For the

“strong coupling” we chose D / 2π = 6 MHz , which results in average effective interactions proportional to 62 / 100 ≈ 1.92 / 10 This difference can be compared with two samples with equal radical concentrations, but

with ERP linewidths that differ by about a factor of ten, such as TEMPOL and trityl. To simplify the system, and to perform quantum calculations in a reasonably short CPU time, we restricted ourselves to only single local nuclei coupled to one of the nine electrons. Sequential calculations of the spin dynamics of the system can then be performed, with the nucleus each time coupled to a different electron. Thus, each system has 210 eigenstates, which make it indeed possible to perform quantum spin dynamics calculations in a reasonable CPU time. These simple systems are relevant, because in real systems each free electronpolarizes the nuclei that are directly or, via the dipolar interaction, indirectly coupled via the hyperfine interaction30 . The computational method, introduced already in Ref 31, is shortly repeated mainly in order to emphasize the changes due to the presence of the nucleus. Despite the small size of our systems, we are able to show almost all ELDOR and EPR spectral features observed experimentally.



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We begin all calculations by defining the parameters determining the strength of the spin interactions. The MW rotating frame Hamiltonian, H RoF , of the 10-spin system contains the electron and nuclear Zeeman terms, the electron-electron dipolar interaction and in addition the hyperfine interaction: H RoF = H eRoF + H nRoF + H nRoF −e

∑ {(m − 5)δω + ∆ω }S

= H eRoF

e

z ,m

m 1...9

H

RoF n

+

∑x

m>m '

m,m '

D(3S z ,m S z ,m ' − S m ⋅ S m ' )

.

= −ωn I z

(1)

H eRoF 2 Amn S z ,mn I z + 0.5( Am+n S z ,mn I + + Am−n S z ,mn I − ) −n =

The off-resonance value ∆ωe , determined by the MW frequency ωMW , is defined as ∆ω=e (ωMW − ωe ) , where

ωe

the Larmor frequency of the electron m = 5 . The nuclear Larmor frequency is ωn and m = mn represents the electron coupled to the nucleus. The coefficients Amn and Am±n are the secular and pseudo-secular hyperfine coupling constants between electron mn and the nucleus. After constructing the matrix representation of H RoF , in the product basis set of the eigenstates of the electron operators S z , m and of the nuclear operator I z , we diagonalize this representation and obtain a set of 210 eigenstates and energies. Without the presence of the pseudo-hyperfine interaction terms, the eigenstates of ( H eRoF + H nRoF + H eRoF − n ) can be divided into manifolds of states, according to their total electron angular momentum z-component M e = −9 / 2,.....,9 / 2 and the nuclear z-component M n = −1 / 2,1 / 2 . The energies of the states belonging to

these groups of eigenstates,

{ϕ

M e ,i

,α n

}

i =1...iM e

{

and ϕ M e ,i , β n

}

i =1...iM e

can then be expressed in the form

ε i ( M e , α n ) = M e ∆ωe − 1/ 2ωn + δε i ( M e , α n ) ; ε i ' ( M e , β n ) = M e ∆ωe + 1/ 2ωn + δε i ' ( M e , β n ) ,

(2)

9   with iM e =   the number of states i = 1,..., iM e in each M e manifold. The values of δ ε i ( M e , χ n ) , with  Me + 9 / 2

χ n = α n , β n , are determined by the dipolar and secular hyperfine interactions and the shifts of the resonance frequencies of the different electrons. The addition of the off-diagonal elements of the pseudo-hyperfine 8 ACS Paragon Plus Environment

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interaction, connecting two subsets of α n and β n states, can after the diagonalization of H RoF results in a mixing

{

between pairs of states like ϕ M e ,i , α n ; ϕ M e ,i ' , β n

}.

To characterize these state mixings we make a distinction between weak mixing cases and strong mixing cases. The largest off resonance hyperfine matrix elements Am+n ϕ M e ,i , α n S z ,m I + ϕ M e ,i ' , β n , for a fixed Am+n value, are expected when the two electron states ϕ M e ,i

and ϕ M e ,i ' are about equal. Then the energy differences

between ε i ( M e , α n ) and ε i ' ( M e , β n ) are about equal to

ωn and with

Am±n

Theoretical Aspects of the Cross Effect Enhancement of Nuclear

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