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Transition-Selective Pulses in Zero-Field Nuclear Magnetic Resonance Tobias Fredrik Sjolander, Michael C.D. Tayler, Jonathan P. King, Dmitry Budker, and Alexander Pines J. Phys. Chem. A, Just Accepted Manuscript • DOI: 10.1021/acs.jpca.6b04017 • Publication Date (Web): 31 May 2016 Downloaded from http://pubs.acs.org on June 17, 2016

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The Journal of Physical Chemistry

Transition-Selective Pulses in Zero-Field Nuclear Magnetic Resonance Tobias F. Sjolander,

∗,†

Michael C. D. Tayler,

Budker,

†Department

∗,‡,k,§

‡, ¶

Jonathan P. King,

and Alexander Pines

†, §

Dmitry

†, §

of Chemistry, University of California at Berkeley, Berkeley, California 94720-3220, United States

‡Department

of Physics, University of California at Berkeley, Berkeley, California 94720-7300, United States

¶Magnetic

Resonance Research Centre, Department of Chemical Engineering and

Biotechnology, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK

§Materials

Science Division, Lawrence Berkeley National Laboratory, Berkeley California 94720-3220

kHelmholtz

Institute Mainz, Johannes Gutenberg University, 55099 Mainz, Germany

Abstract We

use

ability to selectively measure the data of interest by choosing out of hundreds of experi-

low-amplitude,

ultra-low

frequency

mental protocols.

4

Many of these protocols re-

pulses to drive nuclear spin transitions in zero

quire exciting only a select portion of the fre-

and ultra-low magnetic elds.

In analogy to

quency spectrum, using narrow-band pulses.

high-eld NMR, a range of sophisticated exper-

Frequency selectivity is implicitly assumed in

iments becomes available as these allow narrow-

the design of high-eld NMR pulse sequences,

band excitation.

As a rst demonstration,

as the resonance frequencies of dierent spin

pulses with excitation bandwidths 0.5-5 Hz

species are widely separated. Many techniques

are used for population redistribution, selec-

also rely on selectivity beyond dierentiating

tive excitation, and coherence ltration. These

spin species, to the extent of addressing in-

methods are helpful when interpreting zero -

dividual transitions.

and ultra-low-eld NMR spectra that contain a

the basis for many polarization-transfer meth-

large number of transitions.

ods (e.g. INAPT, Insensitive Nuclei Assigned

Introduction

by Polarization Transfer

), spin-spin correla-

10

) and sol-

Narrow-band pulses have yet to be explored in zero to ultra-low eld (ZULF) NMR where the leading elds are

chemical and structural composition of mat-

< 1µT. The ZULF regime

is characterized by spin-spin couplings dom-

High-eld chemical shifts probe the elec-

inating over the Zeeman interaction between

tronic environment, the magnitudes of spin-spin

spins and the external magnetic eld,

coupling constants correlate strongly with both

1114

con-

trary to traditional high-eld NMR where the

bond distances and bond angles, and geometric

Zeeman interaction is dominant.

constraints are given by rates of inter-nuclear

13

8,9

vent suppression techniques.

provide an abundance of information on the

cross-relaxation.

Selective irradiation is

tion experiments (e.g. SELCOSY

Nuclear magnetic resonance (NMR) techniques

ter.

47

Advantages

of ZULF NMR include longer coherence times

A strength of NMR is the

leading to mHz spectral resolution and the

ACS Paragon Plus Environment 1


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Page 2 of 11

ability to measure nonsecular spin interactions

cient to rotate nuclear spins eectively instan-

that are otherwise masked by high magnetic

taneously with respect to the time scale of free

elds.

Instrumentation for ZULF NMR de-

evolution, so that the result is a rotation about

tection includes magnetometers based on opti-

15,16

The

θi = γi Bt where γi th is the gyromagnetic ratio of the i spin, B is the eld strength, and t is the duration of the pulse.

use of an atomic magnetometer as a low-cost,

This method of manipulating spins has limita-

cryogen-free NMR spectrometer is described

tions. It is necessarily broadband with respect

in the literature.

to the ZULF spectrum and the range of avail-

cally pumped alkali vapors

1719

and supercon-

ducting quantum interference devices.

13,2427

the eld axis through angles

2023

Despite the absence of

chemical shifts in ZULF, chemically resolved

able operations depends on the ratios of the

NMR spectra are obtained.

of the spins involved.

For a molecule

γi s

For example, using DC

molecular ngerprint is a spectrum with char-

pulses one can perform a simultaneous θ ≈ 4π 1 13 rotation on H and θ = π on C, however a

acteristic frequencies given by the untruncated

θ ≈π

J -coupling

are typi-

many decoupling and refocusing sequences, is

cally in the frequency range of 0-1000 Hz and the linewidths achieved are routinely between

more challenging; using a single DC pulse it is 1 13 impossible to simultaneously rotate H and C

0.01 and 0.1 Hz, making ZULF NMR a high-

spins through the same angle, other than near-

resolution spectroscopic method for chemical

2π/3 multiples (e.g. θH ≈ 8π/3, θC ≈ 2π/3; γH /γC ≈ 3.97). Composite pulses have been

in the isotropic liquid phase in zero-eld, the

analysis.

Hamiltonian.

15,28,29

J -couplings

The ZULF technique extends to

rotation on both nuclei, as required by

35

spectroscopy in the presence of small perturb-

suggested

ing elds

and residual dipole-dipole interac-

is not a solved problem and generally speaking

which can increase spectral information

the control of multiple spin species is signi-

tions,

16

30,31

content.

cantly restricted.

In this paper we demonstrate frequency se-

Theory

lective pulses in the ZULF regime with a typical excitation bandwidth 0.5-5 Hz, detecting 87 the NMR signals with a sensitive Rb atomic magnetometer.

but performing arbitrary rotations

The

As an application we intro-

in

this

greater control over which transitions are ex-

This is concep-

cited.

tually similar to spin-tickling experiments per-

We write the liquid state zero-eld

Hamiltonian, in angular frequency units, as

formed in high-eld NMR, where connected transitions are used to identify the spin topol-

32,33

demonstrated

the zero-eld eigenstates and therefore allow

connected transitions between the ZULF en-

ogy and energy-level structure.

pulses

work drive spin populations directly between

duce an experiment that identies groups of ergy levels of a spin system.

resonant

HJ = 2π

Simplica-

X

Jij Ii · Ij ,

(1)

i,j>i

tion of ZULF NMR spectra into such groups assists with assignment and helps resolve am-

where the

biguity with interpretation, particularly in the

operators for groups of equivalent spins and the

case of molecules with large numbers of tran-

J

sitions. We demonstrate further selectivity by

eigenstates of

taking advantage of the selection rules for circu-

tum states denoted

larly polarized pulses

tal spin angular momentum quantum number

34

allowing us to address

I

are the total angular momentum

are scalar spin-spin coupling constants. The

HJ

are total spin angular momen-

|F, mF i,

where

F

is the to-

narrow splittings caused by small DC magnetic

and

elds

tion axis (for systems with more than two spins,

Prior to this work, ZULF NMR experiments

mF

is the projection on the spin quantiza-

additional quantum numbers are necessary).

36

have used DC magnetic eld pulses for exci-

The observable quantity is the total magnetiza-

tation and manipulation of the spin system.

tion along the sensitive axis,

Pulsed elds stronger than 100

µT

are su-


sˆ,

of the magne-

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tometer, represented by the operator

Os =

X

γi Ii · sˆ.

(2)

i

∆F = 0, ±1. The selection rules for mF are ∆mF = 0 ˆ = zˆ and ∆mF = ±1 if sˆ = x, ˆ yˆ. if s

This operator supports transitions with

The Hamiltonian for a pulse with frequency

ω

B is written as X HP (t) = cos(ωt) γi B·Ii ≡ cos(ωt)P. and amplitude

(3)

i Figure 1: Schematic of the experimental setup.

Evolution of the spin system under the total Hamiltonian

H = HJ + HP

The current amplier is used to generate strong

is analyzed in an

interaction frame where both

HJ

approximately time independent.

and

HP

DC pulses for broadband manipulations.

are

DAQ in conjunction with the 1 k Ω resistor is

To this end

used to generate the weak selective AC pulses

we consider the interaction frame Hamiltonian when irradiating close to a peak of frequency

and bias the static eld. The sensitive axis of

f

the magnetometer, denoted

and dene

˜

sˆ,

lies along the

axis of the NMR tube, perpendicular to the

˜ = ei ωf HJ t He−i ωf HJ t − ω HJ , H f where

The

plane dened by the pump and probe lasers (not

(4)

shown).

denotes the interaction frame. A more

detailed treatment of this process is given in the

similar to the setup used in Refs.

SI, where we also treat the case of a small static

Presently the apparatus employs a 5x5x8 mm

eld perturbing

HJ ,

but the result is that the

13,30,37. 3

to a time-independent Hamiltonian acting on a

(inner volume) uncoated glass cell with a small 87 amount of Rb metal and 700 Torr N 2 buer ◦ gas (Twinleaf LCC). The cell was kept at 180 C

two-level system with each 2x2 block given by

through resistive heating (40 kHz AC) using a

Hamiltonian above can be block diagonalized

˜ = H

Ω Pαβ Pβα −Ω

twisted wire pair wound around a ceramic pillar

,

(Shapal Hi-M Soft).

(5)

In order to generate and control the DC magnetic eld pulses we used a National In-

Ω is the total frequency oset of the transition from ω and the Pαβ are matrix elements of P in the eigenbasis of HJ . If the peak at f inwhere

struments 6229-DAQ card and a AE Techron LVC2016 linear amplier in conjunction with a Helmholtz coil (radius = 2.2 cm). The coil was

volves degenerate energy levels, Eq. 5 holds for

oriented to generate the eld along the sensitive

each block and the observed signal is the sum

axis of the magnetometer. The AC pulses were

of the solutions obtained for each one. Specic

generated by using three of the

conditions for when this is true are given in the

±10

V analog

outputs of the DAQ card directly, each output

SI.

is capable of driving up to 5 mA current and was

Experimental Methods

put in series with a 1 k Ω resistor. The current was fed into three orthogonal 10.5 cm diameter coils placed around the sample, normally used

87

Rb va-

for shimming the eld inside the magnetometer.

por cell magnetometer, operating in the spin-

The strength of the magnetic eld generated by

exchange-relaxation-free (SERF) regime

and

this process was calibrated by measuring the

congured for use as an NMR spectrometer,

proton Larmor precession frequency in a sam-

We detect ZULF NMR signals using a

19



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Page 4 of 11

ple of water, the measured frequencies ranged

pulse length (Rabi curves) as shown in Figs.

from 0.5 to 4 Hz.

2d/e was used to determine the length of the

The equipment connecting

to the various coils is powered from the mains,

pulses that gives maximum signal.

causing a small oscillating magnetic eld inside

signal is expected to be described by

the shields, observable in the detected spectra

sin(Pαβ t), 38

at integer multiples of 60 Hz.

mum signal for

Samples (∼150

µL)

Pαβ

were placed in standard

JCH S(tp ) ∝

The

and thus we expect to see maxi-

Pαβ t = π/2.

The evaluation of

may be simplied by noting that the opera-

5 mm NMR tubes and prepolarized in a 2 T

tor can be split into two parts, one proportional

permanent magnet before pneumatic shuttling

to the sum of the spin operators, and one pro-

(over

∼0.5

s) to a magnetically shielded region

(four layers of

µ-metal

portional to the dierence. The sum term com-

plus one layer of ferrite,

giving a residual eld of

mutes with

< 1 µG) for experiment

HJ ,

and therefore does not induce

F . Thus = [(γa Bz −

transitions between states of dierent

to control the shuttling and read in the mag-

the relevant matrix elements are Pαβ γb Bz )/2] hF 0 , mF | (Ia,z −Ib,z ) |F, mF i, which can

netometer response signal, the DAQ was pro-

be expanded in terms of Clebsch-Gordan coe-

grammed and controlled on a computer using

P Pαβ = [(γa Bz − γb Bz )/2] ma ,mb (ma − 0 ,m F F mb )CIFa ,m × CIF,m , where mF is cona ,Ib ,mb a ,ma ,Ib ,mb served and a and b refer to the proton and car-

and detection.

The DAQ card was also used

cients as

LabVIEW. A schematic of the setup is shown in Fig. 1.

bon spins.

Results and Discussion

ement predicts the 1 J transition to be driven tains two components, using the formula above the frequencies are 1.5 Hz and 1.3 Hz with a relative weighting of 1:2 respectively.

The zero-eld energy level diagram for the four

quencies is understood by noting that the line consists of several overlapping transitions, orig-

played in Fig. 2. There are two observable tran-

inating in the

sitions, one where the total angular momentum

F,

nents in the Rabi curve is equal to

F =2

and a

F =1/2

to

transition will have three.

DC eld on the order of mG applied at right angles to

Bz =0.94

sˆ;

such small eld perturbations have

been shown to be useful in ZULF spectroscopy,

mG) transi-

by splitting lines into multiplets that reveal the

tion frequencies. Since the bandwidth is much

quantum numbers involved.

less than the peak separation only the resonant

transitions.

sition will have one Rabi frequency but a

F =3

F =1

excitation in the presence of a small perturbing

The spectra

in Figs. 2b/c were recorded after applying se-

∆mF = 0

tran-

to

We now consider the possibilities of selective

or non-adiabatic switching to ZULF, aording

zˆ

F =3/2

to

sitions can be achieved using either a DC pulse

along the

F =0

a

stant. Simultaneous excitation of the two tran-

transition is excited.

Fmax rounded

down to the nearest integer. For example both

is the one-bond carbon-proton coupling con-

mG) or 2JCH (0.182

levels

gle. In general the number of frequency compo-

sˆ = zˆ, implying ∆mF = 0. The two transitions occur at frequencies JCH and 2JCH respectively, where JCH = 140.65 Hz

s,

mF = ±1

mum signal does not correspond to a single an-

to the detector axis,

JCH (0.167 s, Bz =0.94

and the

ments. Thus the pulse length that gives maxi-

We have

taken the spin quantization axis to be parallel

lective pulses, at the

mF = 0

and these transitions have dierent matrix ele-

changes from 0 to 1, and

the spectrum shown in Fig. 2a.

The 2 J

Rabi curve being composed of two dierent fre-

non-exchanging spins in the molecule, namely 13 the CH3 group (an AX 3 spin system), is dis-

another where it goes from 1 to 2.

However, the 2 J Rabi curve con-

at 1.5 Hz.

We demonstrate selective pulses in zero eld us13 13 ing [ C]-methanol ( CH3 OH) as an example.

quantum number,

This expression for the matrix el-

30

The experiments

are most conveniently analyzed in a coordinate

The pulses were applied

system where the spin quantization axis is along

axis in order to excite observable

the direction of the perturbing DC eld, so we assign the detection axis to

The excitation of the two transitions versus


ˆ. sˆ = x

From

Page 5 of 11

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Figure 2: Zero-eld NMR of [ consists of one peak at

JCH

13

C]-methanol. a: When excited by a strong DC pulse the spectrum

= 140.65 Hz and one peak at 2 JCH = 281.3 Hz. The amplier used

for pulsing generates strong 60 Hz overtones seen in the spectra. b/c: Using weak resonant pulses the transitions can be addressed separately and selectively.

The ampliers used for DC pulsing

are turned o, decreasing the 60 Hz noise. d: Signal amplitude versus pulse length, showing the

|0, 0i and |1, 0i states. Red crosses are experimental data points, the blue line corresponds to sin(Pαβ t) and the black dashes to a full numerical simulation. e: Driving the |1, mF i to |2, mF i transition. The blue line is now a weighted sum of sin(Pαβ t) for each mF coherent driving between the

as described in the text, giving two frequencies. energy level diagram on the right-hand side.

I

The two observed transitions are shown in the is the quantum number for total proton angular

momentum, which is conserved.

this view, observable transitions occur between states where

∆mF = ±1

(0.19 mG) is applied at 90

and the eect of the

◦

to

sˆ and the result-

ing spectrum contains two observable transi-

DC eld is to lift degeneracies between states of

J -coupling frequency, |0, 0i → |1, −1i and |0, 0i → ˆ simulDC pulse parallel to s

tions centered about the

mF , leading to splittings in the specThe ∆mF = +1 and ∆mF = −1 transi-

dierent

corresponding to

tra.

|1, +1i.

A strong

tions, in this case correspond to magnetization

taneously excites both transitions as shown in

rotating either clockwise or anticlockwise in the

Fig. 3a. Figure 3b shows the selective excitation

laboratory frame.

achieved by AC magnetic-eld pulses, rotating

This is in contrast to the

ˆ x

in the plane dened by

transitions correspond to magnetization rotat-

0.47 mG, frequency 222.15 Hz (giving a reso-

ing in the same sense, equal to the direction of

nance oset of

Larmor precession around the static eld.

s. The coupling matrix element

±0.52

and

yˆ:

high-eld-NMR case where all of the observable

amplitude

Hz), pulse duration 0.25

Pαβ /2π

evalu-

The use of two orthogonal pulsing coils to gen-

ates to 1.06 Hz using a Clebsch-Gordan expan-

erate a rotating magnetic eld allows us to se-

sion similar to the one used above, details in the

lect transitions with

∆mF = +1 or ∆mF = −1. 13 example using [ C]-formic

SI. Although the excitation bandwidth of this

Figure 3 shows an 13 acid (H COOH, an AX spin system, ignoring

pulse covers both transitions they may be ad-

the exchanging acidic proton). A weak DC eld

tion, eectively sidestepping the usual limit on

dressed selectively based on their sense of rota-



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Page 6 of 11

frequency selectivity. The signal magnitude vs. pulse duration for the higher-frequency peak is shown in Fig. 3c to be in quantitative agreement with both the theory and numerical simulation based on the parameters used. The decay of the signal at long pulse lengths may be attributed to relaxation or inhomogeneity of the eld. Selective irradiation may also be used for saturation or population inversion between two or more pairs of spin eigenstates.

A coherence

lter" based on this principle is demonstrated in Fig. 4 to edit the zero-eld NMR spectrum 15 13 13 13 15 of [ N, C2 ]-acetonitrile ( CH3 C N). The spectrum nominally contains a large number of observable transitions ( >32 dierent frequencies), but this number can be greatly reduced, and thus assignment of the spectrum made easier, by exciting transitions that share a common energy level.

The coherence lter is no more

than a dierence experiment requiring an even number of signal acquisitions to be performed. Every odd numbered acquisition the full spectrum is excited using a broadband DC pulse parallel to

sˆ = zˆ, taking the spin quantization sˆ as in our rst example (Fig.

axis parallel to

2). On even acquisitions, the broadband pulse

Figure 3: Ultra-low-eld NMR ( B0 =0.19 mG) 13 of [ C]-formic acid. a: The spectrum after ex-

is preceded by a selective AC pulse, applying the eld along the same axis as the DC pulse,

citation with a DC pulse contains two peaks of

which inverts the population dierence of a sin-

J -coupling splitting 2ν =

equal intensity centered about the

gle transition. The dierence between the odd

JCH = 222.15 Hz with B0 (γC + γH ). The corresponding transitions are |0, 0i → |1, 1i and |0, 0i → |1, −1i. b: Excitation using a weak near-resonant (= JCH ) rofrequency

and even spectra will contain only peaks corresponding to coherences belonging to the same spin manifold as the inverted transition. Other transitions will remain unchanged by the selec-

tating eld in opposite senses about the axis of the static eld.

tive inversion and therefore cancel when the dif-

The dashed blue trace is the

ference spectrum is made. The result of this l-

expected excitation prole, for the pulse used (|B|

=

tering may be interpreted in the following way:

0.47 mG and 0.25 s), dened by the

signal magnitude.

since magnetically equivalent spins will rotate

The inset shows the corre-

identically about the axis of an applied mag-

sponding energy level diagram. c: Signal magnitude for the

J +ν

pulse length.

The red crosses are experimen-

netic eld, their total angular momentum can-

transition plotted versus

not change during either the selective inversion 15 13 pulse or the DC pulse. In the case of [ N, C2 ]-

tal data, the black dashes to the result of a full

acetonitrile a conserved quantum number is the

numerical simulation, and the blue line corre-

I,

which has

and

I = 3/2)

total proton angular momentum,

sponds to the predicted signal magnitude when

two possible values ( I

solving for the evolution under Eq. 5, as shown

= 1/2

leading to two isolated spin manifolds. This is 13 also the case for [ C]-methanol, shown in Fig. 15 13 2, but for [ N, C2 ]-acetonitrile there is addi-

in the SI.

tional structure due to presence of more than


Page 7 of 11

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Figure 4: Spectral editing at zero-eld. The zero-eld spectrum of [

15

N,

13

C2 ]-acetonitrile is split

into three dierent frequency regions for convenience. Bottom blue trace: experimental spectrum recorded after applying a DC pulse in the

sˆ

direction, bottom green trace: Dierence spectrum

after selective inversion of the 155 Hz line (AC pulse, 0.47 mG, 1.08 s). Stars denote peaks that are still clearly visible in the dierence spectrum meaning they belong the

I

= 1/2 manifold. Top

Traces: Simulated spectra for the same conditions. The traces are oset for clarity and the peaks at 120 Hz, 240 Hz and 300 Hz are overtones of the 60 Hz line noise.

one

J -coupling.

when the selective inversion pulse is applied at

The zero-eld NMR spectrum

is partitioned into three distinct parts. The cor-

155 Hz.

responding signal frequencies are around the 1 13 one-bond H- C J -coupling frequency at 136

inating in the

Hz with

I = 3/2

I = 1/2,

It is expected that only peaks orig-

I

= 1/2 manifold to be visible

in the dierence spectrum. The persistence of

around 2J = 272 Hz with

the peaks in the region around 136 Hz is consis-

and close to 0 Hz arising from the in-

tent with population redistribution within the

ternal splitting of the two manifolds due to the

I

second carbon and the nitrogen. The selective

in the dierence spectrum around 272 Hz ( I =

inversion protocol presented here allows us to

3/2).

lter the observed NMR signal and display only

interesting, four peaks are observed in the dif-

peaks belonging to either the

I

= 1/2 or the

I

= 1/2 manifold. There are no observed peaks The region around zero Hz is the most

ference spectrum so these must belong to the proton spin

= 3/2 manifolds.

I

= 1/2 manifold.

Figure 4 shows both a simulation of this pro-

Figure 4 demonstrates another use for selec-

tocol and the experimental result for the case

tive inversion pulses. Note how in the simula-



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

tion the peak at

≈126Hz

Page 8 of 11

is not visible in the

Curie International Outgoing Fellowship Pro-

control spectrum, but is clearly apparent in the

gramme (author MCDT, project FP7-625054

dierence spectrum.

This indicates that be-

ODMR-CHEM). Contents of the work do not

fore the DC pulse there is no population dif-

reect the views of the university or the Euro-

ference between the two states involved.

pean Commission.

The

selective inversion pulse swaps the initial pop-

Supporting

ulation of one of the two states with a third

tude as a function of resonance oset and pulse

This is analogous to the technique

length.

of interchanging spin populations for signal en-

≈

39

Details regarding the simulations.

This material is available free of charge via the

In the experimental data, the peak at

Internet at

126 Hz also becomes signicantly enhanced.

http://pubs.acs.org/ .

References

However, the peak is also visible in the control spectrum, which we attribute to the sensitive axis of the magnetometer being slightly o-axis from

More detailed evaluation of matrix

elements.

hancement in NMR studies of quadrupolar nuclei.

Available:

Derivation of equations for the signal magni-

state so that the population dierence becomes nonzero.

Information

zˆ. 16

(1) Jameson,

C.

J.

Understanding

NMR

Chemical Shifts. Annual Review of Physi-

Conclusions

cal Chemistry

(2) Kumar, A.;

1996, 47, 135169.

Wagner, G.;

Ernst, R. R.;

In conclusion we have demonstrated selective

Wuethrich, K. Buildup Rates of the Nu-

J -spectra

clear Overhauser Eect Measured by Two-

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