Nuclear Spin Relaxation, Conversion and Polarization of Molecular

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Nuclear Spin Relaxation, Conversion and Polarization of Molecular Hydrogen in Paramagnetic Solvents Ernest Ilisca J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.9b01312 • Publication Date (Web): 06 Jun 2019 Downloaded from http://pubs.acs.org on June 7, 2019

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The Journal of Physical Chemistry

Nuclear Spin Relaxation, Conversion and Polarization of Molecular Hydrogen in Paramagnetic Solvents

Ernest Ilisca* Matériaux et Phénomènes Quantiques, Université Paris 7 Denis Diderot and CNRS UMR 7162, 75205 PARIS Cedex 13, France Abstract Recent NMR experiments have measured the conversion rate of hydrogen molecules dissolved in paramagnetic organic solvents. The registrated proton signals display negative peaks during an initial period. The present interpretation describes a polarization effect : the transient behaviour of the nuclear spins directed on the average along the magnetic electronic spins. It relies on a non‐equilibrium ortho drift and gives access to numerous parameters functions of the sample magnetic concentration. The most important ones are related to the negative transient nuclear magnetization : its building, maximum and life‐time. Important ratios are deduced : conversion versus relaxation characteristic times and collision versus sticking time intervals. The contact interaction is found stronger than the long range and fluctuating dipolar one for magnetically concentrated samples during the short sticking of the colliding partners, but weaker for diluted ones. Some information on the viscosity of the organic solvent is drawn out from the observed conversion delay. (*) [email protected]

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How effective are radio‐frequency magnetic pulses in measuring the irreversible transformation of one variety of hydrogen into another (ortho ↔ para) ? The following study is devoted to the interpretation of original series of NMR measurements of non‐ equilibrium H2 magnetizations in catalytic liquids, in which H2 molecules were diluted1. These experiments measured different kind of nuclear spin relaxations in paramagnetic solutions of organic solvents. Those that thermalize the spin orientations but conserve the spin momentum magnitude (the classical spin lattice relaxation) and those changing that momentum (the molecular conversion). The most important observation was that at the onset, upon insertion of the sample in the magnetic field of the spectrometer, the o‐H2 proton NMR registered signal appeared immediately as a negative peak. In the following the competition between the spin relaxation and the ortho drift is shown to be at the origin of a transient nuclear polarization. It illustrates how the relaxation and conversion processes are correlated in the context of non equilibrium mixtures of hydrogen molecules diluted in paramagnetic solutions of viscous solvents. 1.

Hydrogen Conversion and NMR Measurements

The hydrogen molecules are themselves quantum mixtures of different singlet and triplet nuclear spin manifolds, that might be considered as spin isomer varieties. Any sample of molecular hydrogen contains a mixture of ortho (o) and para (p) states that are characterized by different nuclear spin momenta (resp. I=1 and 0). These spin manifolds are associated to different rotational states (resp. odd and even), because of the fermion character of the protons. Consequently, the nuclear spin isomer energies differ by an order of magnitude of “far infra‐red frequencies”. The strong forbidding of the o‐p spontaneous transition and the weakness of the induced ones allow to conserve non‐equilibrium H2 mixtures during long times. The hydrogen conversion process is an irreversible relaxation of a hydrogen mixture, enriched in one isomer, towards its equilibrium concentration. The time scales for the hydrogen system to relax, vary from the age of the universe for isolated molecules to years for diluted gases, weeks or days for solid H2 at low temperature, days or hours for H2 diluted in insulating or semi‐conductor cages, hours or minutes in contact with diamagnetic species, minutes or shorter when interacting with efficient magnetic catalysts. The book of A. Farkas2 published in 1935, still an actual reference, formalized the distinction between the two families of conversion mechanisms, namely the chemical and the physical ones. The chemical conversion operates by a temporary molecular dissociation, followed by a subsequent recombination along 2


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statistical proportions, in accordance with the thermal conditions of the interacting systems. Differently the physical conversion relies on magnetic catalysts able to create inhomogeneous magnetic fields on atomic scale, that diphase the nuclear spin precession of the 2 protons, as first described by E.P. Wigner3. The subject of “singlet and triplet relaxations” and their inter‐relations must also be replaced in a wider context: conversion and relaxation rates were measured in fullerene cages and organic solvents by NMR4‐6, studied by infrared spectroscopies in Si and MOF cages7‐10, measured by electronic and optical devices for H2 adsorbed on surfaces11,12 for which new theoretical developments were applied13‐15. The important observation of the Canet’s group, that the o‐H2 proton NMR signal appeared immediately as a negative peak was explained by a temporary dissociation of the para‐H2 molecules during their collisions with the paramagnetic impurities. Although their general description of that novel finding is convincing, their model of transient chemisorption followed by a random desorption of o‐H2 created along thermal proportion is not likely for several reasons described in the last section 3c. The simple aim of that paper is to interpret the time‐evolutions of the nuclear magnetization, when para‐hydrogen molecules are diluted in a paramagnetic liquid solution at room temperature. It relies on a simple physical model where the hydrogen molecules and the magnetic impurities recurrently stick together for a short time and separate, without never losing their individualities. The second chapter displays the theoretical calculation of the polarization building, its recurrence and extinction. It describes the two possible time evolutions of the nuclear magnetization either directly towards a partial alignment along the magnetic field or through a temporary opposition. An ortho drift is shown responsible of the irreversible angular momentum flow and the resulting nuclear polarization is demonstrated first in the ideal case of a permanent contact of the colliding partners and thereafter delayed by the solvent. The third chapter interprets the NMR measurements. Several parameters are estimated by comparing the experimental and theoretical frameworks. The polarization transfers are related to the electron and nuclear thermal relaxations and discussed in terms of the magnetic concentrations, first by neglecting the contribution of the long range and random dipolar magnetic fields and then by taking them into account. The description of the solvent influence by detailed diffusion models is not undertaken, nor the calculation of the conversion absolute rates. At the contrary the information is extracted from the experimental measurements and the analysis concentrates on relative effects, particularly on the transient magnetic polarization. Finally a few experimental proposals are sketched. 3



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Figure 1. The nuclear spin‐rotation energy levels of the 𝐈

𝐉 hydrogen system.

The nuclear spin triplet (I=1) of a single ortho manifold (J=1) is surrounded by a pair of spin singlet (I=0) para ones, of lower (J=0) and higher rotational energies (J=2) about 600 cm‐1 apart. The magnetic nuclear spin splitting of the Ortho (I=1) manifold is represented amplified (although about 104‐105 smaller than the o‐p ones). Ortho states are indexed by their magnetic quantum number mi=i. (The weak proportion of molecules in the J=3 state of population ≪ 0.09 is neglected). Blue lines represent ortho‐para transitions whereas green ones are the ortho‐thermal bath transitions!

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2.


The Ortho Drift and the Nuclear Spin Polarization

a. The physical conversion The physical model relies on the integrity of the molecular electronic structure of the H2 molecule and assumes that no chemical bonding occurs during the collisions in the liquid (neither permanent nor temporary). Consequently the Pauli’s principle, applied to the electronic ground state, links the nuclear spin and rotational states11. The catalyst action is a symmetry‐breaking paramagnetic interaction that introduces an o‐p transition relaxing the hydrogen mixture towards its equilibrium concentration. The electro‐nuclear spin systems interact through hyperfine contact and dipolar forces that transfer angular momenta between the molecular rotational ones and the catalyst magnetic spins. The eigen‐energies of the Hydrogen nuclear system {I –J} is represented in Figure 1. The sum of the two protons spin momenta builds the total nuclear spin momentum of the H2 molecule: 𝐈 = 𝐈 + 𝐈 and its component I is quantized in the direction of the magnetic field. The two quantum numbers I= 0 and I=1 characterize the two different varieties para and ortho. The non‐magnetic (I=0) para manifold is splitted in two different rotational components: J=0 and J=2, about 600 cm‐1 apart. At room temperature, a para H2 mixture equilibrated at 300 K contains a proportion of molecules of about half‐half in the rotational states J=0 and J=2 (respective populations at T= 300K: 𝑝 0.513 and 𝑝

≅ 0.47 and that near equality: 𝑝

≅𝑝

≅

≅ 𝑝 𝑡 /2 remains at any time because the

thermal equilibrium is achieved much faster inside each variety than between the ortho and para varieties. The sum of their populations 𝑜 𝑡 and 𝑝 𝑡 : “𝑜 𝑡 + 𝑝 𝑡 = 1” represents the conservation of the hydrogen molecules. In most experiments the hydrogen is introduced in its para form1: p0=p(t=0)=1, and o0=o(t=0)=0. The new ortho molecules are mainly formed in the J=1 state and the ortho‐para concentration of the sample evolves slowly towards its equilibrium concentration at room temperature: (oe= ¾ ; pe= ¼). The observed phenomenon sensibility over different initial states (p0, o0), in particular the mixture p0=o0 = 0.5, will also be considered. The ortho nuclear spin populations, denoted 𝑛 (I=1, i=0, 1), remain proportional to the global ortho concentration: o(t) = ∑ 𝑛 , at any time t. These populations 𝑛 and the nuclear magnetization: m(t)= 𝑛 ‐ 𝑛 (in units of g μ are not in general at equilibrium. When they reach asymptotically their equilibrium: 𝑛 𝑜

𝑜

, (α = g μ H/kT , the resulting magnetization: 𝑚 = 𝑛

𝑛

, is directed along the magnetic field. For a typical experiment the ortho magnetization starts at

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m(t=0)=0, and evolves towards a positive value :

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. However experiments have shown that it might cross

over negative values! The electron spins (as well as the nuclear ones) are submitted to a magnetic field H. These Zeeman interactions are represented by the Hamiltonian: HZ gμ H. S ‐ g μ H. I . [μ (resp. μ are the Bohr (resp. nuclear) magneton and g resp. g the electron (resp. proton) Landé factors]. As well known, μ and μ are of opposite signs and therefore at equilibrium the electron and proton spins have opposite average directions. The amplification factor 𝛾 will denote the ratio of the electron and nuclear magnetic moments: 𝛾

=

. The electronic spin system {S} exchanges energy with the thermal reservoir and adjusts to the

temperature of the liquid, in relaxation times much shorter than the nuclear ones. Thus on the average, the electron spins are aligned in a direction opposite to the magnetic field. Each state 𝑆𝑚 has a population: 𝜋 𝜋 𝑆𝑚 ≅

, where 𝑆

2𝑆

1 and  = gμ H/kT , since at the experimental temperature gμ H ≪ kT.

Consequently the electron spin axial component is given by: 2 ∑

𝜋

in units of gμ .

The nuclear magnetic system {I} is under the competing influence of the isotropic spin‐lattice relaxation processes of rate R and time constant : T1=R‐1, the applied magnetic fields (static and pulsed) and the magnetic fields created by the paramagnetic impurities through the hyperfine interactions. The o‐p transitions, and their probabilities: 𝒫 , are defined by the sums of the transitions between each ortho state “i” and both para states J=0 and 2, as represented in the Figure 1. The o‐p corresponding probabilities: 𝒫 are products of orbital elements (electron spatial extension and molecular rotation) with spin ones (electron and proton): 𝒫

𝐾 Λ . The orbital elements 𝐾 are linear form of products: (i) geometrical form factors

(functions of the scatterings) (ii) rotational matrix elements (iii) spectral densities. Distinctly the hyperfine electronuclear matrix elements Λ

∑

α

,

𝝅 are linear forms of the electronic spin populations: 𝝅 .

Their coefficients satisfy two main properties i α

,

~𝑠

𝑆 𝑆

1 and thus all transition rates are

proportional to the square of the electron magnetic moment and ii the time‐reversal invariance: α α

,

,

for all m and i. The physical meaning of that invariance appears clearly by introducing the sum and the

difference of the transition probabilities 𝒫 : 𝒫 , 𝒫 𝒫 𝐘

𝐾 . Λ ( 𝜋

(𝜎 = 0 and +) 𝒫

, 𝜎 = 0, 𝐾 𝑠 and 𝒫

𝒫

𝒫

and 𝒫

𝒫

𝒫 . These probabilities

distinguish the even/odd components under time reversal; the even ones 𝒫

𝒫

2 𝐾 𝑠, being independent of the magnetic field. At the

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contrary, the odd magnetic one (𝜎 = ‐): 𝒫

𝒫

𝒫

𝛼𝐾 𝑠 is as expected proportional to the

electron magnetization and the magnetic field amplitude through the ratio:  = gμ H/kT, (as well as to the nuclear one: 𝛼 since  = 𝛾𝛼

𝛾

, with a ratio 𝛾 = 658). In the following it will be assumed that the

contact and dipolar interactions are not correlated and add their influences. The contact interactions are strong at short molecule‐catalyst distances (say smaller than 4 Å) and become completely negligible when the partners are apart. The dipolar ones are also strong at short distances but their influence is weakened by the molecular tumbling and their reorientation averages. At longer distances they involve more magnetic impurities and build collective long range random magnetic fields. b. The Master Equations The master equations describe the time evolution of the ortho populations n t under the combined influence of the ortho relaxation processes and the o‐p conversion ones. The relaxation rate R = T

and

the conversion probabilities 𝒫 (𝑖 ∈ 0, 1 ) are the essential ingredients of the detailed balance that describes the irreversible flow of the ortho populations towards equilibrium with the liquid surroundings. They differ greatly in magnitude : the conversion time op is observed to be much longer than the relaxation time T of the ortho magnetization. The 3 independent ortho populations functions of time n t are coupled by the 3 linear differential equations : = R ( 3n + o) + 𝒫 = R n – n

n 𝒫

–n

(1a)

(1b)

The o‐p transition probabilities 𝒫 and the relaxation rate R will be considered as parameters measured by the experiments. As concerns the variables, n t , it is more convenient to introduce the observables: the total ortho population o(t) = ∑ n t , and the nuclear magnetization : m(t) = n

n . They evolve towards

their equilibrium values 𝑜 and 𝑚 . By taking into account the much faster relaxation rate compared to the conversion one: R T1) during which the two forces: the conversion and relaxation ones compete. When the hydrogen flow introduced in the solution has been prepared in its para form, the initial nuclear magnetization vanishes: 𝑚 =𝑚 1

𝑀

= 0, since the nuclear magnetization is

always proportional to the ortho population. It converges towards the positive value: 𝑚

, (𝑀 = 0) after

a long duration (longer than op). The time evolution of the nuclear magnetization M(t) in that ideal case is represented in Figure 2. The two opposite cases where Γ is larger or smaller than 1 must be distinguished. Γ is denoted the polarization factor. It is the ratio of the two time scales 𝑇 and 𝜏 amplified by the 1, the magnetization m(t) increases towards its positive equilibrium

magnetons ratio 𝛾. If it is weak: Γ

value, whereas if it is sufficiently strong: Γ > 1, m(t) decreases first towards a negative value corresponding to a partial alignment along the electron spins and thus antiparallel to the applied magnetic field.

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Figure 2. Ideal Polarization of the Hydrogen Nuclear Magnetization and its deviation from equilibrium. The two possible time evolutions of the nuclear ortho magnetization correspond to different values of the polarization factor Γ. The right vertical axis represents the nuclear magnetization m(t), while the left one corresponds to its relative deviation from equilibrium : M(t)=

. When the initial

mixture is purely para and Γ > 1, the maximum of negative magnetization M1= Γ is reached after a short time t1 slightly larger than the spin‐relaxation one T1. It survives a long time t2 slightly shorter than the microscopic conversion one 𝜏 . When Γ null proportion of ortho‐H2 population 𝑜

. For an initial non‐

0), the polarization condition becomes difficult to satisfy. For

an ortho‐para mixture equilibrated at 77K (𝑜

0.5) the polarization condition Γ > 3 (

> 𝜏 , since most of the partners are far apart. In the sticking regions the polarization factor

is the previous one: Γ = Γ

whereas in the diffusive ones where the contact one is negligible, it is

possible to define a polarization factor : Γ = ξ Γ that links the magnetization evolution to the conversion drift. ξ is the ratio of the odd and even dipolar couplings: ξ

3𝐾

𝐾

2𝐾

Nuclear Spin Relaxation, Conversion and Polarization of Molecular

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