Constant-adiabaticity ultralow magnetic eld manipulations of parahydrogen-induced polarization application to an AA0X spin system Bogdan A. Rodin1 2James Eills3 4Rom an Picazo-Frutos3 4Kirill F. Sheberstov3 4Dmitry Budker3 4 5and

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Constant-adiabaticity ultralow magnetic ﬁeld manipulations of

parahydrogen-induced polarization: application to an AA0X spin system

Bogdan A. Rodin,1, 2 James Eills*,3, 4 Rom´an Picazo-Frutos,3, 4 Kirill F. Sheberstov,3, 4 Dmitry Budker,3, 4, 5 and

Konstantin L. Ivanov*1, 2

1)International Tomography Center SB RAS, Novosibirsk, Russia

2)Novosibirsk State University, Novosibirsk, Russia

3)Johannes-Gutenberg University, Mainz 55099, Germany

4)Helmholtz Institute Mainz, GSI Helmholtzzentrum f¨ur Schwerionenforschung, 55128 Mainz,

Germany

5)Department of Physics, University of California, Berkeley, CA 94720-7300

(Dated: 27 October 2022)

The ﬁeld of magnetic resonance imaging with hyperpolarized contrast agents is rapidly expanding, and

parahydrogen-induced polarization (PHIP) is emerging as an inexpensive and easy-to-implement method

for generating the required hyperpolarized biomolecules. Hydrogenative PHIP delivers hyperpolarized proton

spin order to a substrate via chemical addition of H2in the spin-singlet state, but prior to imaging it is typi-

cally necessary to transfer the proton polarization to a heteronucleus (usually 13C) in the molecule. Adiabatic

ultralow magnetic ﬁeld manipulations can be used to induce the polarization transfer, but this is necessarily

a slow process, which is undesirable since the spins continually relax back to thermal equilibrium. Here we

demonstrate constant-adiabaticity ﬁeld cycling and ﬁeld sweeping for optimal polarization transfer on a model

AA0X spin system, [1-13C]fumarate. We introduce a method for calculating constant-adiabaticity magnetic

ﬁeld ramps and demonstrate that they enable much faster spin-order conversion as compared to linear ramps

used before. The present method can thus be utilized to manipulate nonthermal order in heteronuclear spin

systems.

I. INTRODUCTION

Parahydrogen induced polarization (PHIP)1,2 is a

widely used method to enhance NMR signals. The source

of nonthermal spin order in PHIP experiments is the sin-

glet order of parahydrogen (pH2, molecular hydrogen in

the nuclear spin-singlet state). Although pH2does not

have a magnetic moment and is thus NMR-silent, upon

symmetry breaking (i.e. by rendering the two protons

chemically or magnetically inequivalent) the nonthermal

singlet order can be converted into observable NMR sig-

nals, which are strongly enhanced compared to those un-

der equilibrium conditions. The ﬁrst step for hydrogena-

tive PHIP is a catalytic hydrogenation reaction (addition

of H2to a suitable substrate, usually one with an unsat-

urated C-C bond). When the two pH2-nascent protons

occupy inequivalent positions in the reaction product the

symmetry is broken, and NMR signal enhancements can

be obtained. The magnetic interaction that induces sym-

metry breaking is typically a chemical shift diﬀerence, or

inequivalent J-couplings to a third nucleus.

A common step in PHIP is transferring nonthermal

spin order from the source spins – here the pH2-nascent

protons – to target spins of choice, which are more suit-

able for NMR detection for various reasons (longer relax-

ation times, higher spectral resolution, lower background

signals). A number of methods have been developed to

transfer the pH2spin order to various heteronuclei, via

rf pulse methods at high ﬁeld3–12, or through coherent

spin mixing under zero- to ultralow- ﬁeld (ZULF) NMR

conditions13–19. In the ZULF regime, Larmor frequen-

cies are small, and nuclear spins belonging to diﬀerent

isotopic species become “strongly coupled” – that is the

diﬀerence in Larmor frequencies becomes comparable to

the spin-spin couplings. Under these conditions, coher-

ent exchange of polarization among the spins becomes

possible.

A number of polarization-transfer techniques exploit-

ing ultra-low magnetic ﬁeld manipulations have been de-

veloped, for example: (1) performing the reaction with

pH2at ultralow magnetic ﬁeld to induce spontaneous po-

larization transfer18; (2) applying an adiabatic magnetic

ﬁeld cycle14–16 (FC), which is to perform the hydrogena-

tion reaction at high ﬁeld, nonadiabatically drop to ul-

tralow ﬁeld, and adiabatically return to high ﬁeld, and;

(3) applying an adiabatic magnetic ﬁeld sweep19 (FS),

which is to perform the hydrogenation at high ﬁeld, then

adiabatically reverse the magnetic ﬁeld passing through

zero ﬁeld.

All NMR methods using adiabatic variation of the

spin Hamiltonian are confronted with a common prob-

lem: adiabatic processes are by deﬁnition slow, and spin

relaxation can be signiﬁcant. Relaxation of hyperpolar-

ized samples is generally detrimental as it gives rise to

irreversible decay of the nonthermal spin order back to

thermal equilibrium. It is therefore desirable to use the

fastest possible adiabatic variation without disturbing

the adiabatic nature of the process20–23. Solutions have

been proposed such as “fast” adiabatic processes given

by optimal control theory24 or by varying the Hamilto-

nian ˆ

H(t) such that the eﬀective adiabaticity parameter

is constant at all times25. The latter approach, constant-

adiabaticity, is easy to implement and to adapt to speciﬁc

molecular cases.

arXiv:2210.14342v1 [physics.chem-ph] 25 Oct 2022

Na O

O Na

polarization

transfer

[1-13C]fumarate

J13 = 3.2 Hz

J12 = 15.7 Hz

J23 = 6.6 Hz

[1-13C]acetylene dicarboxylate [1-13C]fumarate

FIG. 1. The chemical reaction employed in this work to produce PHIP-polarized [1-13C]fumarate. In the inset the molecule

is shown with the J-couplings labelled.

In this work we demonstrate constant-adiabaticity ul-

tralow magnetic ﬁeld manipulations to transfer proton

singlet order into 13C magnetization in PHIP-polarized

[1-13C]fumarate. We form hyperpolarized fumarate by

chemical reaction of para-enriched hydrogen with an

acetylene dicarboxylate precursor molecule (see Fig. 1).

The protons are initially in the singlet state, and are

scalar-coupled to the 13C spin in the carboxylate posi-

tion (we work at natural 13C abundance). In the case of

[1-13C]fumarate, the J-coupling between the protons is

signiﬁcantly larger than the proton-carbon J-couplings;

this is referred to as the “near-equivalence” regime. As

a consequence, the proton singlet state is close to an

eigenstate, and signiﬁcant state mixing which allows for

polarization transfer occurs only at well-deﬁned mag-

netic ﬁelds, ±B(i)

LAC, corresponding to the i-th level anti-

crossings (LACs) of the spin system.19 Here we speciﬁ-

cally investigate two ZULF methods to perform polariza-

tion transfer: ﬁeld cycling, which uses a magnetic ﬁeld

variation from zero to Bmax, and ﬁeld sweeping which

uses a magnetic ﬁeld variation from −Bmax to Bmax. For

the case of [1-13C]fumarate, Bmax is a few µT, which is

considerably higher than the LAC ﬁelds, B(i)

LAC. For both

FC and FS experiments we derive constant-adiabaticity

magnetic ﬁeld proﬁles, B(t), and compare the perfor-

mance with linear (uniform) ﬁeld variations.

II. THEORY

A. Hamiltonian

The Hamiltonian of two protons, the Ispins (I1and

I2), and a 13C nucleus, the Sspin (S3), in an external

magnetic ﬁeld B(t) (aligned along the z-axis) is written

as:

H(t) = ˆ

HZ(t) + ˆ

HJ, (1)

where

HZ(t) = −B(t){γI(ˆ

I1z+ˆ

I2z) + γSˆ

S3z}, (2)

HJ= 2πJ12(ˆ

I1·ˆ

I2)+2πJ13(ˆ

I1·ˆ

S)+2πJ23(ˆ

I2·ˆ

S),(3)

and we set ~= 1 for simplicity. At high magnetic ﬁeld,

and given that J12 >|J13 −J23|, the eigestates of the

Hamiltonian (1) are approximately equal to those of the

ST Z (singlet-triplet-Zeeman) basis, which is deﬁned as:

ST Z ={S12,T12

+,T12

0,T12

−} ⊗ {|α3i,|β3i}.(4)

The singlet and triplet states of the proton pair are de-

ﬁned as:

S12= (|α1β2i−|β1α2i)/√2,(5)

T12

+1=|α1α2i,

T12

0= (|α1β2i+|β1α2i)/√2,

T12

−1=|β1β2i,

|αiand |βidenote the Zeeman spin states of an isolated

spin-1/2 nucleus with z-projection of +1/2 and –1/2, re-

spectively. The superscripts denoting the nucleus will be

dropped henceforth. When the proton-carbon couplings

are identical the eigenbasis is given exactly by eq. (4).

However, when J13 6=J23, the protons are magnetically

inequivalent which mixes the states, and the eigenbasis is

then denoted ST Z0. This is discussed in detail in Ref. 19.

By plotting the eigenvalues of ST Z0as a function of

magnetic ﬁeld as shown in Fig. 2, it can be seen that

there are a number of energy level-crossings. On close

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Constant-adiabaticityultralowmagneticeldmanipulationsofparahydrogen-inducedpolarization:applicationtoanAA0XspinsystemBogdanA.Rodin,1,2JamesEills*,3,4RomanPicazo-Frutos,3,4KirillF.Sheberstov,3,4DmitryBudker,3,4,5andKonstantinL.Ivanov*1,21)InternationalTomographyCenterSBRAS,Novosibirsk,Russia2)Novos...

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Constant-adiabaticity ultralow magnetic eld manipulations of parahydrogen-induced polarization application to an AA0X spin system Bogdan A. Rodin1 2James Eills3 4Rom an Picazo-Frutos3 4Kirill F. Sheberstov3 4Dmitry Budker3 4 5and

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