Measurement of Untruncated Nuclear Spin Interactions via Zero- to Ultra-Low-Field Nuclear Magnetic Resonance

0.93 MB·English

by  John W. Blanchard

MeasurementofUntruncatedNuclearSpinInteractionsviaZero-toUltra-Low-FieldNuclear
MagneticResonance
J. W. Blanchard,1,2,3 T. F. Sjolander,1,2 J. P. King,1,2 M. P. Ledbetter,4
E. H. Levine,2 V. S. Bajaj,1,2 D. Budker,3,4,5 and A. Pines1,2
1Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720
2Department of Chemistry, University of California at Berkeley, CA, 94720
3Helmholtz-Institut Mainz, Johannes Gutenberg University, Germany
4Department of Physics, University of California at Berkeley, CA, 94720-7300
5Nuclear Science Division, Lawrence Berkeley National Laboratory, Berkeley, CA, 94720
(Dated:October27,2015)
5
1
Zero-toultra-low-fieldnuclearmagneticresonance(ZULFNMR)providesanewregimeforthemeasure-
0 mentofnuclearspin-spininteractionsfreefromeffectsoflargemagneticfields,suchastruncationoftermsthat
2
donotcommutewiththeZeemanHamiltonian. Onesuchinteraction,themagneticdipole-dipolecoupling,is

t avaluablesourceofspatialinformationinNMR,thoughmanytermsareunobservableinhigh-fieldNMR,and
c
the coupling averages to zero under isotropic molecular tumbling. Under partial alignment, this information
O
isretainedintheformofso-calledresidualdipolarcouplings. Wereportzero-toultra-low-fieldNMRmea-
5 surementsofresidualdipolarcouplingsinacetonitrile-2-13Calignedinstretchedpolyvinylacetategels. This
2 representsthefirstinvestigationofdipolarcouplingsasaperturbationontheindirectspin-spin J-couplingin
theabsenceofanappliedmagneticfield.Asaconsequenceofworkingatzeromagneticfield,weobserveterms

] ofthedipole-dipolecouplingHamiltonianthatareinvisibleinconventionalhigh-fieldNMR.Thistechniqueex-
h pandsthecapabilitiesofzero-toultra-low-fieldNMRandhaspotentialapplicationsinprecisionmeasurement
p ofsubtlephysicalinteractions,chemicalanalysis,andcharacterizationoflocalmesoscalestructureinmaterials.
-
m
PACSnumbers:82.56.-b,82.56.Dj,76.60.-k,76.60.Jx
e
h
c
Nuclearspininteractionsareofsubstantialimportancefor As a proof of concept, we present in this Letter direct ob-
.
s many fields, including chemistry, quantum information pro- servationoftheuntruncatedresidualdipolarcouplingbetween
c
cessing, and precision measurement of fundamental symme- nuclearspinsinaweaklyaligningenvironmentintheabsence
i
s
tries. The most common technique for measuring such in- of an external magnetic field. Dipolar couplings have long
y
h teractionsisnuclearmagneticresonance(NMR),typicallyin been used in high-field NMR to provide structural informa-
p large magnetic fields in order to maximize signal via higher tioninadditiontothechemicalshift. Previousworkdemon-
[ nuclearspinpolarizationandsensitivityofinductivedetection stratedzero-fieldJ-spectroscopyofseveralsystemsforchem-

[1]. However,theonlytermsofthespin-couplingHamiltoni- icalanalysis[23–25]. Additionalinformationmayalsobeob-
3
v ans that may be observed in high-field NMR are those that tained from zero-field NMR spectra via application of weak
8 commute with the Zeeman Hamiltonian, which effectively magneticfields[4].Intheregimewheredipole-dipoleinterac-
6 truncates many interaction Hamiltonians that possess differ- tionscanbetreatedasaperturbationtothe J-coupling,zero-
7
ent symmetry. Recently, however, NMR experiments have toultra-low-field(ZULF)NMRallowssensitivemeasurement
5
beencarriedoutintheoppositeregimeofverysmallmagnetic ofthedipole-dipolecouplingtensor. However,directdipole-
0
. fields [2–5], taking advantage of advances in hyperpolariza- dipolecouplingsobservedinsolidsaretypicallyontheorder
1
tion [6–9] and new detection modalities [10–16], which of- oftensofkHz,substantiallylargerthan J-couplings,andco-
0
5 ferasignificanttimesavingscomparedtoearlierfield-cycling herenceandpopulationlifetimesareoftentooshortforcurrent
1 techniques [17, 18]. In zero- to ultra-low-field NMR (ZULF ZULFmethodology. Furthermore,alldipole-dipolecoupling
: NMR),thestrongestinteractions arethelocalspin-spincou- termsaveragetozeroinisotropicliquids[1]. Smaller,scaled
v
i plings, which involve coupling tensors that are of different couplingsareobtainedbyweaklyaligningthemoleculeofin-
X
symmetryfromtheZeemanHamiltonianandaremanyorders terestinanisotropicmedia,suchasliquidcrystals[26–28]or
r ofmagnitudesmallerinamplitude,thuspermittingthedirect stretched gels [29, 30], where molecular motion is partially
a
observation of nuclear spin interactions that vanish at high restricted, yieldingresidualcouplings. Suchtechniqueshave
magneticfields. Suchtermsthatareonlydirectlyobservable found widespread use in high-field NMR for structural mea-
in the absence of large magnetic fields include the antisym- surements of proteins and small organic molecules [31–34].
metricJ-coupling(ofimportanceformeasurementsofchiral- Here,weinvestigatetheeffectsofresidualdipole-dipolecou-
ity and parity violation), several terms of the direct dipole- plings(RDCs)onthezero-fieldspectrumofamodelXA spin
3
dipole coupling, and a number of as-yet-unobserved exotic system: acetonitrile-2-13C (13CH CN, where we have found
3
interactions such as those mediated by pseudoscalar (axion- it valid to neglect the 14N spin due to self-decoupling aris-
like) bosons [19, 20], which would lead to anomalous spin- ingfromfastquadrupolarrelaxation[35])alignedinstretched
spintensorcouplings[21,22],mostofwhichdonotcommute crosslinkedpolyvinylacetate(PVAc)gels.
withtheZeemanHamiltonian. Polyvinyl acetate (PVAc) polymer sticks containing be-
2
dipole-dipoleinteractionsis
(cid:88)
H =(cid:126) J I ·I
jk j k
j;k>j
µ (cid:88) γ γ (cid:104) (cid:16) (cid:17)(cid:16) (cid:17) (cid:105)
−(cid:126)2 0 j k 3 I ·rˆ I ·rˆ −I ·I , (1)
4π r3 j jk k jk j k
j;k>j jk
where(cid:126)isthereducedPlanckconstant,µ isthevacuumper-
0
meability, γ and γ are the gyromagnetic ratios of spins I
j k j
andI ,andr istheinternuclearvectorconnectingthespins.
k jk
In the case of isotropic liquids, the dipole-dipole interaction
term averages to zero. However, in aligned samples, such
asstretchedgels,themotionalaveragingofthedipole-dipole
termisincomplete. Forthesystemstudiedhere,theswelling
ofthepolymergelwithacetonitrilealongtheaxisoftheNMR
FIG.1. a)Schematicillustrationofthechangeinsymmetrythatoc-
cursduringuniaxialstretchingofthegelenvironmentduetoswelling tube leads to an orientational probability distribution of the
with acetonitrile-2-13C. The change in the order parameter is illus- solventmoleculesthatisslightlyanisotropic,withtheprefer-
trated by the different three-dimensional shapes. b) Energy level entialalignmentaxis(thedirector)determinedbytheswelling
structure of a partially ordered 13CH3 spin system. F is the total direction[29,30]. Thisaxisiscollinearwiththesensitivedi-
spinangular momentum, K isthe totalproton angularmomentum,
rectionofthedetector,andisdenotedz. Becauseoftherapid
andJ istheone-bond13C–1HJ-coupling. Solidarrowsindicate
CH rotation of the acetonitrile methyl group and the axial sym-
allowedtransitions,dashedlinesindicateforbiddentransitions.Note
metry of the alignment medium, the x and y components of
thatforthissystem,bothcouplingsarenegative.
ther vectorsareaveragedtozero,andthezcomponentsare
jk
scaledbythedegreeofalignment. Consideringtheseaverag-
ingeffectsonthesecondtermofEq. (1),theresidualdipolar
tween 1-6% v/v divinyl adipate (DVA) crosslinker were pre-
couplingHamiltonianis
pared in 5 mm NMR tubes as described in the Supporting
(cid:88) (cid:16) (cid:17)
Information. Anisotropic gels were prepared by adding ace- H =−(cid:126) D 3I I −I ·I , (2)
RDC jk j,z k,z j k
tonitrile to the tubes and allowing the polymers to swell for
j;k>j
2 weeks. Because the polymers were confined to the NMR
tubes in which they were cast, swelling was uniaxial and where
equivalent to stretching along the axis of the NMR tube. A
µ γ γ (cid:126)1(cid:68) (cid:69)
schematicrepresentationoftheprocessisshowninFig.1(a). D = 0 j k 3cos2θ −1 , (3)
jk 4π r3 2 jk
In order to maximize the ZULF NMR signal, the samples jk
were prepared using labeled acetonitrile-2-13C to which was
andwhereθ istheanglebetweentheinternuclearvectorand
added5%v/vdeuteratedacetonitrileforthepurposeofhigh- jk
theC -axisofacetonitrile.Wemayalsodefinethecouplingas
field NMR characterization. The molecular order parameter 3
being directly proportional to the molecular order parameter
[36,37]foracetonitrileinthestretchedgelenvironmentwas (cid:68) (cid:69)
S = 1 3cos2θ −1 [36,37],whereθ istheanglebetween
determinedbyanalyzingthequadrupolarsplittingofthedeu- zz 2 z z
the C -axis of acetonitrile and the laboratory z-axis and the
teriumresonance[26,38]usinga14.1TNMRspectrometer 3
time average provides a measure of the extent of alignment
with deuterium frequency 92.1 MHz (for additional details,
betweenthetwoaxessystems,suchthat
see Supporting Information). The value for the electric-field
gradient around the deuterium nuclei in acetonitrile was ob- µ γ γ (cid:126)1(cid:16) (cid:17)
tainedfromtheliterature[38]. Djk = 4π0 rj3k 2 3cos2φjk−1 Szz, (4)
jk
The ZULF NMR apparatus has been described previously
[4, 6, 39]. Samples were pre-polarized in a 2 T permanent where φ is the angle between r and the C axis, π/2 for
jk jk √ 3
magnet located outside of the magnetic shielding for ∼20 s
D andthetetrahedralangle(2arctan 2)forD .Because
HH CH
and then shuttled pneumatically to the zero-field region over the order parameter S may be readily extracted from high-
zz
0.5–1s. NMRsignalsweredetectedwithanatomicmagne-
fielddeuteriumNMRspectra,theonlyremainingparameters
tometer featuring a 0.6 × 0.6 × 1.0 cm3 87Rb vapor cell op-
forthedipole-dipolecouplingstrengthsaretheanglesanddis-
eratingat180◦C.Transientsignalswerecollectedover∼20s.
tancesthatdefinethemoleculargeometry,whichmaybeob-
ThespectrainFig.2aretheaverageofbetween256and1024
tainedfromtheliterature[40].
transients,andthespectrainFig.3aretheaverageof8tran- Itisworthpointingoutthedifferencebetweentheeffective
sients.
residualdipolecouplingHamiltonianinzerofieldandinhigh
The spin Hamiltonian in the presence of J-couplings and field. The zero-fieldheteronuclearcoupling term interms of
3
(a) Jiso/2π (b) 2xJiso/2π (c)
285
280
6% DVA
275
z)
5% DVA H 270
cy ( 265
4% DVA n
ue 260
q
e
2% DVA Fr 255
140
1% DVA 135
130
130 135 140 260 270 280 290 0 1 2 3 4 5 6
Frequency (Hz) Frequency (Hz) Molecular Order Parameter (/10-4)
FIG.2. Zero-fieldspectraofacetonitrile-2-13Cwithdifferentdegreesoforderingarisingfromtheconcentrationofthecross-linkerdivinyl
adipate (DVA). (a) K = 1 and (b) K = 3 peaks. (c) Peak positions as a function of molecular order parameter. The lines are calculated
2 2
transitionfrequenciesfromEqs.(8-14)withnofreeparameters.Theorderparameterforeachsamplewascalculatedfromhigh-fielddeuterium
quadrupolesplittingsusingliteraturevaluesfortheelectricfieldgradient[38]andthedipole-dipolecouplingstrengthswerecalculatedfrom
Eq. (4)usingbondlengthsfromRef. [40]. Solidsymbolsrepresentallowed∆m = 0transitions,opensymbolsrepresenttransitionswith
F
∆m =±1.Dashedlinesindicatepossibletransitionsthatarenotresolved.
F
the total 1H angular momentum K and the 13C angular mo- Inaddition,therearenominallyforbidden(assumingthatthe
mentumSis detectorisonlysensitiveinthez-direction)∆m =±1transi-
F
tionswithfrequency
H(het) =−(cid:126)D (3K S −K·S), (5)
CH z z
D
ω1,±1 = J − CH. (9)
ascomparedtothehigh-fieldcase, 0,0 CH 2
H(trunc) =−2(cid:126)D K S , (6) ForthetransitionsbetweenK = 3 states,thereareallowed
CH z z 2
transitionswithfrequencies
whereinatermoftheform
ω2,0 =2J +2D , (10)
(cid:126) 1,0 CH CH
HRDC −H(trunc) = 2DCH(K+S−+K−S+), (7) ω2,±1 =2J + 1(D +3D ), (11)
1,±1 CH 2 CH HH
is truncated because it does not commute with the high-field
andnominallyforbiddentransitions∆m = ±1withfrequen-
Zeeman Hamiltonian. The zero-field Hamiltonian is untrun- F
cies
cated, and thus includes this so-called “heteronuclear flip-
flop”termthatisinvisibletohigh-fieldNMR. 1
ω2,±2 =2J − (7D +3D ), (12)
In the regime where Djk (cid:28) Jjk, the residual dipolar cou- 1,±1 CH 4 CH HH
plings may be treated as a perturbation on the J-coupling, 1
ω2,±1 =2J + (5D −3D ), (13)
yieldingenergyshiftsthatarecalculatedintheSupportingIn- 1,0 CH 4 CH HH
formation, andsummarizedschematicallyinFig.1(b), along 1
ω2,0 =2J + (5D +9D ). (14)
withtheallowedtransitions. 1,±1 CH 4 CH HH
Because the observable in this experiment is the z-
magnetization M (t) ∝ Tr{ρ(t)(cid:80) I γ},thedetectablecoher- If the detector axis is not exactly aligned with the direc-
ences are those wzith ∆F = 0,±1j jaznjd ∆m = 0. An addi- tor/quantizationaxis, thenominallyforbiddentransitionsbe-
F
tional selection rule, ∆K = 0, arises in the case of equiva- comeobservable.
lentspins(e.g. themethylprotonsinacetonitrile)becauseK2 Zero-field spectra of acetonitrile-2-13C (13CH3CN) in
stretched polyvinyl acetate gels are shown in Fig. 2(a) for
commuteswiththeHamiltonian[4,39]. Itfollowsthatthere
is one allowed transition between K = 1 states, between the increasing values of the molecular order parameter. As the
|F = 0,mF = 0(cid:105) and |F = 1,mF = 0(cid:105)2states. Based on the orderparameterincreases,the K = 21 peakscorrespondingto
first-orderenergyshiftspresentedintheSupportingInforma- theorderedportionofthesamplesplit,whiletheK = 1 peak
2
tion,thistransitionhasfrequency correspondingtoexcessisotropicliquidexternaltothegelre-
mainsunchanged. Thelower-frequencypeakinFig. 2(a)cor-
ω1,0 = J +D . (8) respondstothe∆m = 0transitiondescribedbyEq.(8)and
0,0 CH CH F
4
thehigher-frequencypeakcorrespondstothe∆m =±1tran-
F
80 nT
sition described by Eq. (9). The magnitude and phase of the
∆m = ±1 peak are determined by the projection of the ini-
F
60 nT
tialspin-statepopulationontothetransversecomponentofthe
detectionoperator,andisthusasignatureofimperfectionsin
50 nT
theexperimentalconfiguration. Becausethe∆m =±1peaks
F
areconsistentlynarrowerthanthe∆m =0peaksinFig.2(a), 40 nT
F
it appears that the linewidth is affected by inhomogeneity in
thegelproducingadistributionoforderparametersandthus 30 nT
aspreadintransitionfrequenciesproportionaltoD .
CH
Figure2(b)showsfourK = 3 peaks,threefromthealigned 20 nT
2
acetonitrile-2-13C,andonefromtheisotropicliquid. Thetwo
10 nT
lower-frequencypeaksarisefromthe∆m =0transitionsde-
F
scribedbyEqs.(10-11)andthesmallhigher-frequencypeak 0 nT
correspondstothe∆m =±1transitiondescribedbyEq.(12).
F
TransitionscorrespondingtoEqs.(13-14)arenotresolved. 132 134 136 138 140 142
In high-field NMR, terms in the Hamiltonian that do not Frequency (Hz)
commutewiththeZeemanHamiltonianareneglected,dueto
their immeasurably small effect on the NMR spectrum. Due FIG.3. Acetonitrile-2-13C K = 1 peaksasafunctionofmagnetic
2
to the absence of a large Zeeman interaction, ZULF NMR fieldappliedorthogonaltothedirectoraxis,collinearwiththemag-
netometerpumpbeam.
providesspectroscopicaccesstoallspin-couplingterms[41].
Intraditionalhigh-fieldNMR, onlypartoftheheteronuclear
dipolarcoupling,H(trunc),fromEq.(6),yieldsmeasurableef-
azimuthalangletobeaveragedtozero. Ifthesetermsarenot
fectsinthespectrum. Byitself,thistermwouldyieldnoshift
of the |F = 0,m = 0(cid:105) ↔ |F = 1,m = 0(cid:105) transition. In averagedtozero(e.g. inabiaxialphase),theywillfurtherlift
F F
degeneracyandyieldadditionalpeaks[43].
Fig.2(a),theresidualdipolarcouplingsclearlyshiftthepeak
relativetotheisotropicliquidshowingtheobservationofthe We have also investigated the effect of applied magnetic
untruncatedHamiltonianofEq.(2),includingthecontribution fieldsonthespectrum,asshowninFig.3. Whentheeffective
ofEq.(7),aninteraction“invisible”totraditionalNMR.The detectionoperatoriscollinearwiththegeldirectoraxis,only
absenceoftruncationthatpermitsobservationofthistermhas the ∆mF = 0 transition is observed (corresponding here to a
also been demonstrated via the preparation of heteronuclear 10nTappliedfield). Asthefieldisincreased,however,rota-
spin-singletstatesinRef.[42]. tion of the nuclear spins mixes states with different mF, and
As shown in Fig. 2(c), the frequency shift varies linearly rotationofthealkalielectronspinselicitsachangeinthesen-
with the order parameter. The data closely match the simu- sitiveaxisofthedetector. Theoverallresultisthatasthefield
lated curves, which are calculated from the order parameter is increased, the vectors defining the detection operator and
usingEqs.(4)and(8-14). thequantizationaxisceasetobecollinear. Thisinturnleads
to the observation of ∆m = ±1 transitions, which become
Inthesituationwherethesensitiveaxisofthemagnetome- F
dominantabove50nT,atwhichpointtheeffectivedetection
ter is parallel to the director of the gel alignment, only the
∆m = 0transitionsaredetected. Ifthedetectionaxis(ordi- operatorhasbeenrotatedsubstantiallyawayfromthedirector
F
rectionofalignment)isrotated,the∆m =±1transitionsare axis.
F
alsoobservable,leadingtopeaksathigherfrequency,within- In the regime where the Zeeman interaction strength is on
tensitydependentontheinitialspinstatepopulationsandthe theorderoftheresidualdipolarcoupling,thepeakfrequencies
angle between the detection and alignment axes. The addi- in Fig. 3 do not vary linearly with the applied field strength.
tionalpeaksappearinFig.2becausethemeasurementswere Thisisbecausethedipole-dipolecouplingHamiltoniandoes
carriedoutusingamagnetometerconfigurationthatfeatureda not commute with the Zeeman Hamiltonian, and thus first-
rotatedaxisofdetectionduetoanon-zeroeffectivefieldatthe order perturbation is no longer sufficient to describe the sys-
Rbcell.Weattributethiseffecttoimperfectionsinthemagne- tem.
tometer configuration, potentially related to laser alignment, In conclusion, we have demonstrated the direct influence
AC Stark shifts, or a combination of the two. We point out of the heteronuclear dipole-dipole coupling “flip-flop” term
thattheeffectwasdiminishedafterexpandingthepumpbeam (whichisnotdirectlyobservableinthehigh-fieldregime)on
(thusdecreasingthelaserpowerdensity)andsubsequentlyre- ZULFNMRspectra.Theresultsareinagreementwithazero-
aligning the optics for the experiments in Fig. 3. It is worth free-parametermodelofresidualdipolarcouplingutilizinglit-
noting that the rotation of the detection axis is necessary for eraturevaluesforinternucleardistancesandtheelectricfield
the detection of the higher-frequency peaks only due to the gradientofacetonitrile,thelatterbeingusedtodeterminethe
axial symmetry of the system under study, which causes the order parameter via high-field deuterium NMR. The sub-Hz
termsofthedipolarcouplingHamiltonianthatdependonthe resolution of ZULF NMR may be of use for chemical and
5
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ACKNOWLEDGEMENT
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