Accepted Manuscript Heteronuclear J-coupling measurements in grossly inhomogeneous magnetic fields S. Mandal, Y.-Q. Song PII: DOI: Reference:

S1090-7807(15)00070-1 http://dx.doi.org/10.1016/j.jmr.2015.03.008 YJMRE 5624

To appear in:

Journal of Magnetic Resonance

Received Date: Revised Date:

12 February 2015 15 March 2015

Please cite this article as: S. Mandal, Y.-Q. Song, Heteronuclear J-coupling measurements in grossly inhomogeneous magnetic fields, Journal of Magnetic Resonance (2015), doi: http://dx.doi.org/10.1016/j.jmr.2015.03.008

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Heteronuclear J-coupling measurements in grossly inhomogeneous magnetic fields S. Mandala,b,∗, Y.-Q. Songb a

Case Western Reserve University, Cleveland, OH 44106, USA b Schlumberger-Doll Research, Cambridge, MA 02139, USA

Abstract It is difficult to measure chemical shifts in the small and inhomogeneous magnetic fields found in exsitu and single-sided NMR systems, such as those used for well-logging. However, it is still possible to obtain chemical information from J-coupling constants, which are independent of static field strength and temperature. We describe and analyze 1 H-13 C double-resonance pulse sequences that are suitable for measuring heteronuclear J-coupling in grossly inhomogeneous fields. We also present preliminary experimental results from a low-frequency fringe-field system. Keywords: J-coupling, inhomogeneous fields, ultra-broadband electronics 1. Introduction The indirect scalar coupling (J-coupling) constants between nearby nuclei are independent of field strength and temperature, and thus provide valuable chemical information even when the static magnetic field is too weak for chemical shifts to be measured [1, 2]. In addition, NMR spectroscopy at ultra-low field strengths (nT-µT) is of fundamental interest for understanding strongly-coupled spin networks; at very low fields, J-couplings can compete with the Zeeman interaction and even dominate the spectrum [3]. Such experiments require specialized instrumentation, including a pre-polarization system and detectors based on low-temperature superconducting quantum interference devices (SQUIDs) [4, 5]. Single-sided and inside-out NMR systems, such as those used for well-logging, have grossly inhomogeneous B0 fields in the 5-50mT range [6, 7]. These fields are strong enough for Zeeman interactions to be dominant and for coils to be useful as detectors, but too weak and inhomogeneous for chemical shifts ∗

Corresponding author Email addresses: [email protected] (S. Mandal), [email protected] (Y.-Q. Song)

Journal of Magnetic Resonance

March 23, 2015

to be resolved. There has been little prior work on measuring J-coupling constants at such intermediate field strengths, although Speier has proposed some experiments [8], and An et al. have patented several “J-editing” NMR pulse sequences for making such measurements in inhomogeneous fields [9]. These sequences are based on creating J-modulated spin echoes, which are also useful for a variety of common NMR methods such as J-resolved spectroscopy, INEPT, and INADEQUATE. However, in this case a single J-encoding period is used per scan to reduce sensitivity to pulse errors (rotation axis and flip angle) caused by the inhomogeneous field. The encoding period is then varied across multiple scans to build up the J-modulation curve, resulting in a two-dimensional experiment. J-coupling measurements are of significant interest for identifying molecular species in field applications of NMR. In particular, the spectrum of J-coupling constants provides information about the composition of hydrocarbons, such as crude oils. The single-bond heteronuclear J-spectrum of most naturally-occurring hydrocarbons can be well-modeled by two main components. The dominant component (typically 80% to 95% of the total signal) is generated by the C-H bonds found in saturated alipathic molecules, such as straight and branched-chain alkanes. These bonds are sp3 hybridized, and have J ≈ 125 Hz [10]. The second and smaller component (typically 5% to 20%) is generated by the C-H bonds associated with aromatic rings. These bonds are sp2 hybridized, and have J ≈ 160 Hz [11]. The amplitudes of these components are thus related to the weight fractions of saturated and aromatic molecules. These quantities, which are traditionally measured using SARA (Saturates, Aromatics, Resins, Asphaltenes) analysis techniques [12], are useful for characterizing complex hydrocarbons, such as crude oils. The coupling constants for other sp2 hybridized C-H bonds in hydrocarbons are similar to those from aromatic rings. However, those associated with internal C-C double bonds generally have slightly lower values (J ≈ 150 Hz) than from terminal double bonds (J ≈ 160 Hz). Molecules containing such C-C double bonds (alkenes) are rare in naturally-occurring hydrocarbons but are common in the synthetic base fluids used for creating oil-based drilling fluids. In particular, many drilling fluids contain internal olefins (IOs) or linear alpha olefins (LAOs) with internal and terminal double bonds, respectively [13]. Thus, the C-H single-bond J-spectrum also provides useful information for distinguishing between various natural and synthetic hydrocarbons that are encountered in the oilfield. In this paper we analyze the use of J-editing sequences for studying the composition of hydrocarbon mixtures in grossly inhomogeneous fields. We also present preliminary experimental results from a low-frequency fringe field system. 2

2. Theory 2.1. Basic pulse sequence Figure 1 shows a J-editing pulse sequence suitable for measuring heteronuclear J-coupling constants. This sequence is a generalized version of that originally proposed by An et al. [9]. It consists of N Jencoding periods to convert single-bond J-coupling between nuclei A and X into amplitude modulation of spin echoes, followed by NE CPMG refocusing cycles. No further modulation occurs during the CPMG, so all the echoes can be added together to increase the signal-to-noise ratio (SNR). These features makes the sequence suitable for use in grossly inhomogeneous fields, i.e., when sample volumes are power-limited and the effective rotation axes and flip angles of RF pulses vary considerably with position. In particular, small values of N reduce the effects of inaccurate pulses, relaxation, and diffusion on the modulated echoes, while large values of NE improve SNR. The operation of the sequence shown in Figure 1 is now described in more detail. Sensitive A spins (such as 1 H and

19

while X spins (such as

13

F) act as the source of magnetization and are also the subject of detection,

C) are flipped by π pulses during the initial N refocusing cycles to enable the

buildup of J-modulation. The location of the π pulses within each cycle is defined by a time delay τ that can be varied between scans. The effect of these pulses can be qualitatively understood using an extension of the vector model commonly used to describe uncoupled spin-1/2 systems [2]. The effect of J-coupling in this model is to split the magnetization vector into two components with identical magnitudes. The first rotates faster than the uncoupled magnetization vector on the x − y plane (by an amount +J/2), while the second rotates slower by the same amount. We assume that the π pulses only have enough bandwidth to invert one of the coupled spins (either A or X). This is usually an excellent approximation for heteronuclear couplings, since the Larmor frequency difference between dissimilar nuclei is usually ≫ ω1 . Thus π pulses on either channel invert the sign of the coupling term in the Hamiltonian, effectively flipping the labels on the two J-coupled vectors. The resultant evolution of the on-resonance magnetization during one refocusing cycle is shown in Figure 2. Note that we can ignore chemical shift evolution since it is refocused by the π pulses. It is evident from Figure that the angle θ made by either J-coupled vector with the refocusing axis (x-axis) at the end of a single cycle is given by

3

J-editing (N cycles)

CPMG (NE cycles)

180x

180x

90y

A (1H,19F) X (13C)

180

τ tE1

tE2

Figure 1: A J-editing pulse sequence with detection of proton (or fluorine) magnetization. The sequence consists of N J-editing cycles that encode the coupling constant into amplitude modulation of spin echoes, followed by NE CPMG refocusing cycles that allow multiple echoes to be added together to increase the signal-to-noise ratio (SNR). It is generally used in two-dimensional experiments in which the time delay τ is varied between scans to build up the J-modulation curve of the sample.

A

A B C

DE

y

B

F

y

x

y

C

x

x

180x A 180 D

X τ

tE1 2

τ

y

E

y

tE1 2 x

y

F

x

θ θ

x

Figure 2: Vector model of a single J-editing cycle for on-resonance spins, shown for the particular case of τ = tE1 /8. The red and green vectors rotate clockwise and anti-clockwise, respectively, relative to the uncoupled spins. The latter remain fixed along the x-axis.

4

2πJ θ = (τ ) − 2

(

tE1 −τ 2

)

2πJ + 2

(

tE1 2

)

2πJ = 2πJτ 2

(1)

The process continues for N cycles, increasing the angle to N θ. The refocused J-coupled signal is then proportional to the projection of both vectors onto the refocusing axis, i.e., cos(N θ) = cos(2πJN τ ). Thus, the magnetization is allowed to evolve due to J-coupling at a rate 2πJ during an encoding time of N τ . This process is analogous to the way molecular diffusion is encoded in diffusion-editing sequences [14], hence the term ‘J-editing’. The analysis is also valid when components with different J-coupling constants are present in the sample. Each component will evolve independently according to its own coupling constant, so the total signal can be found by superposition. The preceding analysis was only valid for on-resonance magnetization. The use of grossly inhomogeneous fields and power-limited RF pulses complicates the analysis, since each pulse will now excite a number of coherence pathways, with relative strengths that vary as a function of the resonance offset frequency [15, 16]. The number of pathways that contribute to the final echo increases rapidly with N , the number of J-encoding cycles. Moreover, the rate at which the coherence decays due to relaxation and diffusion also differs between various pathways. An advantage of the sequence shown in Figure 1 is that the echo-spacing tE1 is kept fixed, which simplifies data analysis by keeping such relaxation and diffusion effects constant with τ . However, random spin flips during the J-encoding periods may act as a source of additional N τ -dependent signal decay. We will ignore this possibility for simplicity. The effective J-coupling resolution of the sequence, given by ∆f ≈ 1/ (N τmax ) > 2/ (N tE1 ), can be increased by increasing the maximum J-encoding time N τmax , where τmax < tE1 /2. Diffusive attenuation can be reduced without affecting the resolution by decreasing tE1 while increasing N (the number of encoding cycles) in order to keep N τmax constant. The J-editing cycles are followed by a train of NE ≫ 1 refocusing pulses (CPMG), which refocus all heteronuclear J-couplings, thereby preventing further J-modulation. The resulting spin echoes can be added together to increase the SNR. However, this process also results in T2 -weighting of the different J-modulation components, i.e., possible distortion of amplitudes in the J-coupling spectrum. We will ignore this effect for simplicity, but it can be taken into account, if required, by storing all the CPMG echo amplitudes and inverting them into two-dimensional J − T2 maps [14]. Neglecting field inhomogeneity effects, the previous analysis shows that the asymptotic CPMG echo amplitude s(N τ ) is given by 5

s(N τ ) = a0 +

M ∑

ai cos(2πJi N τ ),

(2)

i=1

where ai is the fractional abundance of A-X bonds with coupling constant Ji , and M is the number of unique coupling constants. In addition, a0 is the background signal due to uncoupled A spins. The shape of this J-modulation curve can be analyzed to distinguish between different carbon groups, such as aliphatics and aromatics, that have significantly different J-coupling constants. It can also be used to compute the hydrogen index and hydrocarbon/water ratio of fluid samples. Since the possible values of J are well-known for hydrocarbon molecules, the usual experimental goal is to estimate the unknown amplitudes ai , i.e., the J-coupling spectrum. The latter is directly related to the molar weight of hydrogen atoms with specific J values that are associated with a molecular moiety. 2.2. TANGO excitation pulses Precession due to single-bond J-coupling can be used to distinguish between protons that are chemically bonded to NMR-active nuclei (such as

13

C or

15

N) and those that are not. Figure 3 shows clusters

of excitation pulses that use this principle. These sequences are known as TANGO (Testing for Adjacent Nuclei with a Gyration Operator) pulse clusters [17]. The TANGO-A sequence with τ1 = 1/(2J) behaves as a 90-degree (excitation) pulse for protons that are not chemically bonded to

13

C atoms, while acting

as a 360-degree (identity) pulse for bonded protons. In this case we will selectively observe protons bonded to

12

C (which has zero spin and is NMR-silent), while suppressing those bonded to

13

C. The

TANGO-B sequence, which differs from TANGO-A only in the phase of the final proton pulse, reverses the roles of these two groups. The same principle can be used to create pulse clusters for selective inversion. The latter are known as BIRD (BIlinear Rotational Decoupling) operators [18]. Our original J-editing sequence used a non-selective rectangular excitation pulse, as shown in Figure 1. We can replace it with the TANGO-B pulse cluster to suppress the unmodulated background signal from protons attached to

12

C atoms. The result is an increase in the fractional amount of

J-modulation. This provides increased robustness to multiplicative error sources such as drifts in temperature and preamplifier gain that are particularly troublesome during long experiments. However, it does not change the absolute amount of J-modulated signal, so the SNR set by additive error sources (such as thermal noise) remains unchanged. Most hydrocarbons contain a range of single-bond J-coupling constants. However, the TANGO-B 6

45y

180x

135y

1

H

τ1

(a) 13

180

τ1

C 45y

180x

135-y

1

H

(b) 13

τ1

180

τ1

C

Figure 3: TANGO pulse sequences for selectively exciting protons based on their adjacency to 13 C atoms: (a) non-adjacent protons are selected by TANGO-A, and (b) adjacent protons are selected by TANGO-B.

sequence only acts a perfect excitation pulse for one value of J (the one that satisfies the condition τ1 = 1/(2J)). A simple vector analysis of the pulse sequence for arbitrary values of the precession angle θ1 ≡ πJτ1 , i.e., for any value of the coupling constant, shows that the final magnetization produced by the cluster is given by

( ) ( ) ⃗ coupled = − sin2 θ1 xˆ + cos2 θ1 zˆ M ⃗ uncoupled = zˆ. M

(3)

As expected, all the coupled magnetization ends up along the x-axis when θ1 = π/2, i.e., τ1 = 1/(2J). In general, such perfect excitation is obtained whenever τ1 = n/(2J), where n = 1, 3, .... The uncoupled magnetization remains zero for other values of θ1 , but the amount of coupled magnetization along the x-axis decreases. As a result the amount of J-modulated signal also decreases. Hence the TANGO-B sequence with a fixed value of τ also acts as a “J-filter”, preferentially selecting spins that satisfy the condition τ1 = n/(2J) and suppressing the others. The value of τ1 can therefore be varied to enhance or suppress the J-modulation produced by different chemical groups. Figure 4 shows these filter functions for various values of n. Figure 4 shows that the amount of frequency selectivity increases with n. For example, by setting τ1 = 3/(2×160 Hz) = 9.375 ms we can substantially suppress the J-modulated signal from single-bonded 7

τ1 = n/(2×125 Hz)

(a)

τ = n/(2×160 Hz)

(b)

1

1

0.8

0.8

0.6

0.6 |Mx|

|Mx|

1

0.4

0.2

0.4 n=1 n=3 n=5

0 120

130

n=1 n=3 n=5

0.2

140 J (Hz)

150

0 120

160

130

140 J (Hz)

150

160

Figure 4: J-filtering functions produced by the TANGO-B excitation pulse for various values of the time delay τ1 . The value of τ1 was set to obtain perfect excitation (|Mx | = 1) for (a) J = 125 Hz, and (b) J = 160 Hz.

carbons (J ≈ 125 Hz) while emphasizing that from double-bonded and aromatic carbons (J ≈ 160 Hz). This behavior can be useful for detecting double bonds or aromatic rings, because it removes the (usually much larger) background signal produced by single bonds. However, signal loss due to relaxation and diffusion also increases with τ1 , thus limiting the maximum useful value of n. In addition, in practice it is not possible to completely suppress the uncoupled magnetization. The main cause is inaccurate flip angles of the RF pulses due to limited probe bandwidth and spatially inhomogeneous B0 and B1 , as described later. 3. Simulations The semi-classical vector model is widely used for simulating the dynamics of non-interacting spin1/2 nuclei. It can be easily extended to inhomogeneous magnetic fields by dividing the simulation region into small regions with constant B0 and B1 (isochromats), simulating the evolution of each isochromat independently, and adding the results. We used a similar approach for simulating our J-coupled twospin system in an inhomogeneous magnetic field. Instead of the vector model, we used the more general product operator formalism. The Hamiltonian for each isochromat in the system was written in a doubly-rotating frame, as follows:

8

Hf p = ∆ω1 Iz1 + ∆ω2 Iz2 + 2πJIz1 Iz2 Htot = Hf p + ω1,i [Ixi cos (ϕ) + Iyi sin (ϕ)] ,

(4)

where Hf p is its value during free precession, Htot is its value during an RF pulse of phase ϕ on channel i, and ∆ω1 and ∆ω2 are the resonant frequency offsets of the two spins. In addition, Ix1 , Ix2 , etc. denote product operators (represented as 4×4 matrices) formed from the single-spin operators I, Ix , Iy , and Iz . The evolution of the density matrix ρ was calculated in MATLAB (MathWorks, Natick, MA) by using its expm function, i.e., by direct matrix exponentiation: ρ(t + T ) = e−iHtot T ρ(t)eiHtot T ,

(5)

where Htot remains constant during the time period T , which can be an interval of free precession or an RF pulse. The observed signal on the i-th channel from each isochromat is given by

si (t) = Tr (ρ(t)Ixi ) .

(6)

The total signal is obtained by adding up the signals from all isochromats in the sample. We generally assumed a uniform distribution of resonant frequency offsets across the isochromats (corresponding to a constant B0 gradient) and no B1 inhomogeneity. This is a good approximation for the fringe field system used in our experiments. It should be noted that evolving the density matrix via direct matrix exponentiation is computationally intensive and must be replaced by other methods for large spin systems [19]. However, performance was acceptable for our two-spin problem, with simulations typically taking less than 15 seconds to run on a mid-range personal computer. We have plotted all simulated and measured echo shapes in normalized time units t/t180 , where t180 = π/ω1 is the length of a nominal 180◦ proton pulse. The relative weights of various coherence pathways, and thus the echo shapes themselves, are independent of the gradient strength when expressed on this time scale. Figure 5 shows simulated J-modulated echoes for the J-editing sequence assuming a constant B0 gradient and uniform B1 field. Plots are shown for various values of τ , the J-encoding time, and N , the number of J-encoding cycles. Specifically, the left, middle, and right plots use N = 1, 2, and 4, respectively, while keeping the number of CPMG refocusing cycles fixed at NE = 2. Thus they 9

show the asymptotic CPMG echo shape as a function of the number of J-editing cycles. We see that for N = 1 the echo shape is independent of τ to within a constant scaling factor. This is because only there is only one coherence pathway (the direct echo) that can refocus the magnetization after a single J-editing cycle. As a result, the relative amplitudes of the echo integrals, and hence the shape of the J-modulation curve, should be invariant with tacq . This expectation is confirmed by Figure 6(a), which shows that the simulated J-modulation curves for N = 1 are indeed invariant with tacq . However, Figure 5 shows that the echo shape changes with τ whenever N > 1. This makes it necessary to acquire the entire echo shape in order to obtain the correct relative echo amplitudes and avoid distortion of the J-modulation curve. This situation arises because now there are multiple coherence pathways (the direct echo and various stimulated echoes) that can refocus the magnetization after the J-editing cycles [15, 16]. The Fourier transform S(∆ω, τ ) of the time-domain echo s(t, τ ) is given by ∫

∞

s(t, τ )ej∆ωt dt.

S(∆ω, τ ) =

(7)

−∞

We define tacq to be the length of the rectangular signal acquisition window w(t), which is equal to one for |t| < tacq /2 and zero otherwise. The resulting echo integral s(τ ) is given by ∫

∫

−tacq /2

s(τ ) =

−∞

s(t, τ )dt = −tacq /2

−∞

s(t, τ )w(t)dt = Sa (0, τ ).

(8)

Here Sa (∆ω, τ ) ≡ S(∆ω, τ ) ∗ W (∆ω), where W (∆ω) = (tacq /2) sinc(∆ωtacq /(2π)) is the Fourier transform of w(t), and the ‘∗’ symbol denotes convolution. As tacq increases the quantity W (∆ω) approaches a delta function and s(τ ) → S(0, τ ). Thus, if we integrate the entire echo shape the echo integral s(τ ) becomes equal to that of the on-resonance component (at ∆ω = 0), which only has contributions from the direct echo pathway. In this case, we obtain the true J-modulation curve (the same as when N = 1) with no off-resonant effects. However, for smaller values of tacq this condition is no longer satisfied, and we expect some distortion of the resultant J-modulation curve. This expectation is confirmed in Figure 6. These plots show that the simulated J-modulation curves depend on tacq , and only approach the true curve when tacq ≫ t180 . However, as tacq increases the minimum usable echo spacing tE2 also increases, which reduces SNR. Fundamentally, this reduction in SNR arises because, for large values of tacq , only the on-resonance component of the echo remains after integration. Figure 5 also shows that the width of the echoes in the time domain increases with N . This means

10

Modulation angle JNτ, rad/(2π)

N=1

N=2

N=4

0.2

0.2

0.2

0.4

0.4

0.4

0.6

0.6

0.6

0

0.8

0.8

0.8

−0.5

1

1

1

−2

0 t/T

2

−2

180

0 t/T

2

−2

180

0.2

0.2

0.4

0.4

0.4

0.6

0.6

0.6

0.8

0.8

0.8

1

1

1

0 t / T180

2

−2

0 t / T180

0 t/T

2

180

0.2

−2

0.5

2

0.01 0 −0.01 −2

0 t / T180

2

Figure 5: Simulated echo shapes for the J-editing sequence as a function of the modulation angle JN τ in the case of a constant B0 gradient and uniform B1 . Plots are shown for various values of N , the number of J-encoding cycles. In each case the top and bottom plots show the real and imaginary components of the echo, respectively. All other parameters, such as the number of CPMG refocusing cycles (NE = 2) and tE2 , were kept constant across simulations.

11

(b)

0.5

0

−0.5

t

= 1.2t

t

= 2.4t

t

= 3.6t

acq acq acq

−1 0

t

acq

180 180

N=2

0.5

0

t

−0.5

acq

0.25 0.5 0.75 Modulation angle JNτ, rad/(2π)

180

1

= 1.2t

180

tacq = 2.4t180 t

180

= 4.8t

(c)

1

acq

−1 0

= 3.6t

180

tacq = 4.8t180

0.25 0.5 0.75 Modulation angle JNτ, rad/(2π)

1

Echo amplitude (normalized)

N=1 1

Echo amplitude (normalized)

Echo amplitude (normalized)

(a)

N=4 1

0.5

0

−0.5

t

= 1.2t

t

= 2.4t

t

= 3.6t

acq acq acq

−1 0

180 180 180

tacq = 4.8t180

0.25 0.5 0.75 Modulation angle JNτ, rad/(2π)

1

Figure 6: Simulated J-modulation curves for the J-editing sequence as a function of the modulation angle JN τ in the case of a constant B0 gradient and uniform B1 . Each plot shows the effect of tacq , the acquisition time. Different plots show the effect of N , the number of J-encoding cycles: (a) N = 1, (b) N = 2, and (c) N = 4. All other parameters, such as the number of CPMG refocusing cycles (NE = 2) and tE2 were kept constant across simulations.

that the refocusing bandwidth decreases, i.e., the pulse sequence refocuses a smaller range of offset frequencies. As a result, longer values of tacq are required to acquire the entire echo shape, causing SNR to decrease with N . The final detected echo shapes are also influenced by the CPMG pulse train. However, this effect is likely to be small, since the refocusing bandwidth of the CPMG is significantly larger than that of the J-editing cycles. Simulation results confirm that the CPMG refocusing pulses produce relatively minor changes in the overall echo shape. We also simulated the effect of using TANGO-B excitation pulses within the J-editing sequence in the presence of a constant B0 gradient and uniform B1 field. The acquisition time was set to a relatively large value (tacq = 6.74 × t180 ) to minimize off-resonance effects. Figure 7 shows that the expected Jfiltering effect is obtained even in grossly inhomogeneous fields. In particular, we can selectively remove J-modulation at 125 Hz by setting the TANGO delay τ1 = n/(2 × 160) sec. The removal is nearly complete for n = 5, in agreement with Figure 4. The figure also shows the simulated effects of such Jfiltering on the modulation curve of toluene, which contains a mixture of 125 Hz and 160 Hz components in a ratio of 3:5. The simulation confirms that using a TANGO delay of τ1 = 5/(2 × 160) sec = 15.6 ms removes most of the 125 Hz component but leaves the 160 Hz component unaffected. 4. Numerical experiments We now discuss methods for extracting the coupling spectrum from noisy J-modulation data. We shall consider the effects of two types of measurement noise: signal-independent (additive) and signaldependent (multiplicative). 12

(a)

(b)

J = 125 Hz 1

Original J−filtered (160 Hz, n = 5) 160 Hz component

0 −1 0

2

4

6

8

10

12

J = 160 Hz 1 0 −1 0

n=1 n=3 n=5

2

4 6 8 J−encoding time, Nτ (ms)

10

Echo amplitude (normalized)

Echo amplitude (normalized)

1

0.5

0

−0.5

−1 0

12

2

4 6 8 J−encoding time, Nτ (ms)

10

12

Figure 7: Simulated J-modulation curves using the TANGO-B excitation pulse. The TANGO time delayτ1 was set to n/(2 × 160) sec (n=1,3,5,...) to select J-modulation components around 160 Hz. (a) This plot shows that larger values of n provide larger amounts of filtering of J-modulation components near 125 Hz while leaving the 160 Hz components unaffected, as expected. (b) This plot shows the effects of such filtering (for n = 5) on the J-modulation curve of toluene, which contains a mixture of 125 Hz and 160 Hz components in a ratio of 3:5.

4.1. Additive noise: Two-component model We assume that the unmodulated background signal a0 has been removed by making a reference measurement with the 13 C pulse turned off. We also assume that the modulated signal has been ∑ normalized ( M i=1 ai = 1) for convenience. We now analyze the two-component case (M = 2) in some detail using numerical experiments. Synthetic data for this case was generated using an amplitude distribution that is typical for a crude oil or synthetic oil-based drilling fluid. Specifically, we generated modulation curves by assuming that 85% of the signal was generated by saturated single C-H bonds (J = 125 Hz), while the rest was generated by protons attached to aromatic rings or double-bonded carbon atoms (J = 160 Hz). We then contaminated each curve with additive white Gaussian noise (standard deviation = σ ) and fed it into a nonlinear curve-fitting routine. We used the interior point algorithms available within the well-known MATLAB Optimization Toolbox for this purpose. In particular, we used the function fmincon, which tries to find the minimum of a constrained nonlinear multivariable function. In this case the cost function to be minimized consisted of the mean squared error (L2 norm) between the model and the data. We also assumed that the vector of encoding times (N τ ) and the coupling constant values are known, so that the problem contains only M = 2 variables (the unknown non-negative amplitudes ai ) and hence converges rapidly. 13

We ran the optimization routine many times (typically 200 times) for a fixed set of parameters in order to obtain statistical information about the fitted amplitudes. The parameters were then varied. A typical example is shown in Figure 8 assuming a uniformly-spaced encoding vector (K = 32 steps, ∆τ = 0.2 ms between steps) and varying amounts of noise. The input amplitudes are shown as dashed lines in this plot. The points denote the fitted amplitudes, with error bars corresponding to ±1 standard deviation. We see that the mean estimation error is approximately zero, while the standard deviation increases monotonically with noise level. We also notice that the absolute estimation error is approximately equal for both components, resulting in a larger fractional error for the smaller component. (a)

(b)

1

0.5

0

−0.5

−1 0

1

0.8 Estimated amplitudes

J−modulation curve

Two−component model Single component (J = 125 Hz)

0.6

0.4

0.2

2

4 6 Encoding time Nτ (ms)

0

8

0

0.05

0.1 Noise level, σ

0.15

0.2

Figure 8: (a) Ideal, noiseless J-modulation curve for the two-component case with J = 125 and 160 Hz for K = 32 encoding steps and a step size of ∆τ = 0.2 ms, compared with a single component at 125 Hz. (b) Estimated J-coupling spectrum as a function of input noise level. The error bars correspond to ±1 standard deviation, while the ideal amplitudes (0.85 and 0.15) are shown as dashed lines.

We then found σest , the standard deviation of the amplitude estimation error per component, for various values of the maximum encoding time K∆τ . In this experiment we varied ∆τ while keeping the total number of encoding times fixed at K = 32. Figure 9 shows that the resulting estimation error is linearly proportional to the input noise level, i.e., σest σ = const for a given value of K∆τ . In addition, we see that the estimation error for a given amount of input noise initially decreases as K∆τ increases, but eventually saturates at a fixed value for K∆τ > 12.8 ms. This length of time is about half a cycle of the frequency difference (160125 = 35 Hz) that we are trying to resolve. However, we also note that the estimation error is already within 20% of its minimum when is only half of this value, i.e., for K∆τ = 6.4 ms. Thus the fitting algorithm only requires about one quarter of a cycle of the frequency 14

difference in order to satisfactorily resolve it. Such high performance cannot be obtained with purely linear frequency analysis methods that assume no prior knowledge of the spectrum, such as the Fourier transform. K = 32 steps 1

∆τ = 0.1 ms

J−modulation curves

0 −1 0 1

5

10

15

20

−1 0 1

5

10

15

20

25

∆τ = 0.4 ms

0 −1 0 1

5

10

15

0.1

25

∆τ = 0.2 ms

0

20

5

10 15 20 Encoding time Nτ (ms)

0.08

∆τ = 0.1 ∆τ = 0.2 ∆τ = 0.4 ∆τ = 0.8

0.06 0.04

25 0.02

∆τ = 0.8 ms

0 −1 0

K = 32 steps

(b)

Estimation error

(a)

25

0 0

30

0.05

0.1 Noise level, σ

0.15

0.2

Figure 9: (a) Ideal, noiseless J-modulation curves for the two-component case with J = 125 and 160 Hz for K = 32 encoding steps and various values of ∆τ , the step size. (b) Simulated amplitude estimation error per component as a function of the input noise level.

In the next experiment we varied the total number of encoding times K while keeping the maximum encoding time fixed. Figure 10 shows that the resulting amplitude estimation error steadily decreases as K increases. This result indicates that the algorithm uses the redundant information provided by additional points in the J-modulation curve to obtain a more accurate estimate, i.e., effectively averaging across multiple points. In order to confirm this hypothesis we fitted straight lines through the curves in Figure 10 to find the ratio σest /σ for each value of K. These results are also shown in Figure 10, and √ are well-described by a relationship of the form σest /σ ∝ 1/ K. Hence the fitting algorithm effectively averages over all K points in the J-modulation curve to obtain a more accurate estimate of the unknown variables (amplitudes). As a result, estimation error can be reduced either by adding more points to the modulation curve or by more averaging of a fixed number of points: both approaches are equivalent in terms of experimental time. 4.2. Additive noise: Three-component model The presence of acetylene (also known as ethyne) is often suspected during drilling operations. Jediting can identify acetylene because of its C-C triple bond: the associated C-H bond is sp hybridized 15

K∆τ = 12.8 ms 0.1

Estimation error

0.08

0.04

0.02

0.05

0.5 0.45

Simulation

K=8 K = 16 K = 32 K = 64

0.06

0 0

(b) Normalized estimation error, σest / σ

(a)

0.1 Noise level, σ

0.15

0.4 0.35 0.3 0.25 0.2 0

0.2

1/2

Fit, 1.4 / K

10

20 30 40 50 Total number of points, K

60

70

Figure 10: (a) Simulated amplitude estimation error per component for the two-component case (J = 125 and 160 Hz) as a function of the input noise level for a fixed value of maximum encoding time (K∆τ = 12.8 ms) and various values of K, the number of encoding steps. (b) Normalized amplitude estimation error per component for the two-component case (J = 125 and 160 Hz) as a function of K, the number of encoding steps. The maximum encoding time was kept fixed at K∆τ = 12.8 ms.

and has a large coupling constant (J ≈ 250 Hz). We simulated the presence of 10% acetylene by creating the following three-component model: J = 125, 160, and 250 Hz with amplitudes of 0.75, 0.15, and 0.10, respectively. The resulting J-modulation curve was then contaminated with varying amounts of noise and fitted to the signal model as described earlier. The results are shown in Figure 11. They are qualitatively and quantitatively similar to the two-component case. The amplitude estimation error is again evenly distributed across the three components, and also increases linearly with the input noise level. The signal model can be further generalized to account for more components (M > 3) and known experimental imperfections, such as signal offsets or temperature drifts. 4.3. Multiplicative noise NMR signal amplitudes are inversely proportional to the absolute sample temperature T . As a result, small temperature fluctuations ∆T cause the detected signal to vary as s(T + ∆T ) ≈ s(T ) (1 − ∆T /T ). Such fluctuations therefore act as a source of signal-dependent (multiplicative) noise that cannot be effectively removed by simple averaging. However, alternative techniques, such as homomorphic filtering [20] are suitable for removing such fluctuations. We studied the effect of multiplicative noise on the estimated amplitudes by generating noisy signals defined as s′ (N τ ) = (1 + n) × s(N τ ), where n is a zeromean Gaussian random variable with standard deviation σ, and s(N τ ) is the noiseless signal containing 16

0.1

0.8

0.6

0.4

0.2

0

K = 32 steps

(b)

1

Estimation error

Estimated amplitudes

(a)

0.08

∆τ = 0.1 ∆τ = 0.2 ∆τ = 0.4 ∆τ = 0.8

0.06 0.04 0.02

0

0.05

0.1 Noise level, σ

0.15

0 0

0.2

0.05

0.1 Noise level, σ

0.15

0.2

Figure 11: (a) Estimated J-coupling spectrum as a function of input noise level for the three-component case with J = 125, 160, and 250 Hz for K = 32 encoding steps and a step size of ∆τ = 0.2 ms. The error bars correspond to ±1 standard deviation, while the ideal amplitudes (0.75, 0.15, and 0.10, respectively) are shown as dashed lines. (b) Simulated amplitude estimation error per component as a function of the input noise level.

two J-components (M = 2) and an unmodulated background signal. We began by assuming that s(N τ ) had an amplitude distribution similar to natural hydrocarbons, with a large unmodulated background signal (a0 = 1) due to uncoupled protons, a small 125 Hz component (a1 = 0.011 × 0.85), and an even smaller 160 Hz component (a2 = 0.011 × 0.15). The simulated estimation error for the 160 Hz component is shown in Figure 12(a) for various amounts of multiplicative noise. We then simulated the effect of removing most of the background signal with a TANGO-B excitation pulse. Spin dynamics simulations in a gradient field suggest that about 99% suppression of this signal should occur if we use a relatively long echo acquisition window (tacq = 6.74 × t180 ). As a result, the estimation error for the 160 Hz component drops by almost two orders of magnitude (see Figure 12(a)). This is because most of the variability in the signal, which was being caused by fluctuations in the magnitude of the dominant background component, was removed by the TANGO-B excitation. This example demonstrates the effectiveness of such suppression techniques for reducing the effects of multiplicative noise. Estimation accuracy is further improved by using J-filtering to suppress the 125 Hz component, as shown in the figure. Similar results were obtained for signal models containing three modulated components. By contrast, the estimation error of a particular J-coupling component is independent of the magnitude of other components when the noise is additive, i.e., additive noise does not couple together 17

Normalized estimation error (percent)

Normalized estimation error (percent)

(a) 2

10

1

10

0

10

Normal TANGO TANGO and J−filtering

(b)

2

10

Normal TANGO TANGO and J−filtering

1

10

−2

10 Noise level, σ

−2

10 Noise level, σ

Figure 12: Simulated estimation error of a small J-modulated signal at 160 Hz (a2 = 0.011 × 0.15) in the presence of another modulated signal at 125 Hz (a1 = 0.011 × 0.85) and also a large unmodulated background signal (a0 = 1) for varying amounts of (a) multiplicative noise and (b) additive noise. Each plot shows the results of applying three J-editing pulse sequences: with 90y excitation, with TANGO-B excitation, and with both TANGO-B excitation and J-filtering (τ1 = 5/(2 × 160) Hz= 15.6 ms). The standard deviation of the Gaussian noise source is denoted by σ in all cases.

the estimation of different components in the J-spectrum. For example, Figure 12(b) shows that the simulated error for the 160 Hz component due to additive noise is unaffected by the presence of both a large unmodulated background signal and a 125 Hz component. Suppressing these components with TANGO-B excitation and J-filtering provides no benefits in this case. 5. Experiments We used an ultra-broadband low-frequency MR system for J-editing experiments in the fringe field of a 2 T superconducting magnet. The system, which is shown in Figure 13, operates between 100 kHz and 3 MHz with high sensitivity and no hardware modifications by using an un-tuned (non-resonant) sample coil and custom broadband transmitter and receiver electronics [21]. A commercial spectrometer (Kea, from Magritek) was used for pulse programming and data acquisition. The ultra-broadband nature of the electronics allows it to easily handle a wide range of RF frequencies, such as proton and carbon Larmor frequencies, and rapidly switch between them. The power amplifier uses four switches in an H-bridge configuration and is highly power-efficient. However, it is also nonlinear and incapable of simultaneously amplifying pulses at different frequencies. The Kea spectrometer has two independent low-power RF channels that were added together on 18

the broadband transmitter board with an op-amp-based adder circuit. This configuration allows us to generate phase-coherent RF pulses at two different frequencies. Spectrometer (Kea, Magritek)

Computer

Channel 1

Pulse sequence generator

Channel 2

Post-processing

(a)

GUI (Prospa)

ADC

(b) φ2

φ1

Buffer Vdd

Sample

φ1 φ2

VBB 1:N

J1

A

A

B

Bridge driver

Switch driver

C

Filter D Duplexer

Non-resonant transmitter

Feedback Non-resonant receiver

Figure 13: Simplified block diagram of the two-channel ultra-broadband low-frequency NMR system used in our fringefield experiments. The sample coil was a solenoid with ID = 2.7 cm and length = 8.4 cm, consisting of 42 turns of AWG-18 magnet wire wound with a uniform pitch of 2 mm. Its self-inductance was L = 14.7 µH after assembly inside a tight shield box. Figure reproduced from [21].

Our magnet was cylindrical in shape. We centered our sample coil along the z-axis of this cylinder, at a variable distance z from the end of the magnet. The fringe field at these locations is parallel to the z-axis, and its magnitude decreases with z. For values of z > 50 cm (corresponding to f0 < 2 MHz for protons) the decay is well-approximated as due to a dipole, i.e., Bz ∝ 1/z 2 . This dependence creates a strong static field gradient gz = −dBz /dz ∝ 1/z 3 in the sample, and also defines the on-axis thickness of the excited region, ∆z ≈ B1 /gz = ω1 /(γgz ). For typical operating frequencies between 1 and 2 MHz the static field gradient varies between 5.7 and 15.2 G/cm, respectively. At typical RF power levels this results in ∆z values of a few mm, which is much smaller than the diameter of the sample coil (5.5 cm). As a result, the sample appears to be much larger than the (power-limited) excited region. This situation is similar to that encountered in NMR well-logging. Hence our fringe-field setup emulates many key aspects of well-logging and other ex situ measurements, and is useful for rapid laboratory evaluation of the performance of J-editing (and other pulse sequences) in such environments. The B1 field generated by our non-resonant transmitter is inversely proportional to the impedance of the coil at the RF frequency, i.e., scales as B1 ∝ 1/ω0 = 1/(γB0 ). In addition, the gyromagnetic

19

ratio of carbon-13 nuclei is approximately four times smaller than that of protons. Thus B1 will be approximately four times larger for

13

C pulses than 1 H pulses at a given location, i.e., for fixed values

of z and B0 = B(z). As a result, the region ∆z ≈ B1 /gz affected by a

13

C pulse will also be four times

wider. As a result, 13 C pulses almost behave as ideal, on-resonance pulses with constant flip angle when we consider only the much smaller excited region defined by 1 H pulses. Figure 14(a) shows simulated J-modulation curves for methanol (J = 141 Hz) for N = 1, 2, and 4 editing cycles. Only the 3 methyl protons in the molecule are J-coupled to the carbon atom. Thus the expected modulation function is of the form a + b cos(2πJN τ ), where the offset b comes from the unmodulated hydroxyl proton and J = 141 Hz is the tabulated value for methanol. In the ideal case, with complete J-modulation and identical T1 and T2 for all protons, we have a = 0.25 and b = 0.75. Figure 14(b) shows measured J-modulation curves for a

13

C-enriched (99%) methanol sample (volume

= 5 ml) in the fringe field system at a proton Larmor frequency of 1.545 MHz. We used long acquisition intervals (tacq ≈ 8t180 ), which eliminates distortion of the measured J-modulation curves when N > 1. The resultant curves are in excellent agreement with simulations. They also fit the expected (ideal) modulation function 0.25+0.75 cos(2π141N τ ) very well, indicating that the methyl and hydroxyl protons have similar values of T1 and T2 . However, a small decrease in modulation amplitude is noticeable at long encoding times for N = 4. We are currently studying the reasons for this effect; it is not consistent with T2 decay. Figure 15 show measured asymptotic echo shapes produced by the J-editing sequence for various values of N , the number of J-editing cycles. We used the same

13

C-enriched methanol sample for this

experiment. The results are in good agreement with the simulated echo shapes shown in Figure 5. In particular, both simulated and measured echo shapes are invariant with the encoding time τ when N = 1, but vary considerably with τ when N > 1. We also used our non-resonant system to measure J-coupling in samples with natural abundance of 13

C (1.1%). These measurements are complicated by multiplicative noise sources, such as fluctuations

in the sample temperature, sample position, and receiver gain. In order to avoid this problem we incorporated reference scans (13 C pulses off) between normal scans (13 C pulses on), as shown in Figure 16. The measurement was run as a set of four nested loops. The two inner loops consisted of a pair of normal and reference scans nested within a four-part phase cycle. Initial saturation pulses were used to ensure identical amounts of polarization before each scan, thus eliminating the need for dummy scans. In 20

(b)

1

0.5

0 N=1 N=2 N=4 −0.5 0

0.25 0.5 0.75 1 Modulation angle JNτ, rad/(2π)

Echo amplitude (normalized)

Echo amplitude (normalized)

(a)

1

0.5

0

−0.5 0

1.25

N=1 N=2 N=4 Ideal

0.5 1 1.5 2 Modulation angle JNτ, rad/(2π)

Figure 14: (a) Simulated, and (b) measured J-modulation curves of methanol for various values of N , the number of J-editing cycles. Results have been normalized to the unmodulated echo amplitude (obtained with the

13

C pulse(s) off).

The experimental sample was 5 ml of 13 C-enriched (99%) methanol enclosed within two glass ampules (diameter = 0.9cm, length = 4cm). Experimental parameters included fH = 1.545 MHz, fC = 388.5 kHz, gz = 11.5 G/cm, NE = 500, tE1 = 8.5 ms, tE2 = 400 µs, t90 = 15 µs, tacq = 256 µs, and Navg = 4.

addition, the phase of the excitation pulse was reversed during the reference scan, thus inverting the unmodulated signal (shown in red). As a result only the modulated signal (shown in green) remained after adding up the echoes produced by the normal and reference scans. These scans were nested inside a loop over different values of the J-encoding time τ , and the outermost loop consisted of repeating the entire experiment to improve SNR. One disadvantage of using interleaved reference scans is increased experimental time. The amount of time required to perform a given number of scans doubles, and noise in the reference scans also adds to √ that in the normal scans, thus reducing SNR by a factor of 2 (in voltage units). As a result the number of scans needed to obtain a given amount of SNR doubles, increasing the total experimental time by a factor of 4. This factor can be reduced by averaging multiple scans to create the reference signal, or running reference scans more infrequently. Either modification, however, will increase the sensitivity to multiplicative noise sources. Figure 17 shows measured J-modulation curves for Fluorinert FC-43, a perfluorinated liquid used for cooling electronics. In this case we are measuring single-bond C-F coupling. The measured curve in both cases is well-fit by a single cosine with J = 265 Hz, in good agreement with typical values for such compounds. This measurement shows that we can reliably measure single-bond heteronuclear

21

N=1

N=2

N=4 400

0.2 0.4 0.6 0.8 1

Modulation angle JNτ, rad/(2π)

0.2 0.4 0.6

−2

0 t/T

2

0.5

2 −2

0.6

0 t/T

2

−2

−200 −2

180

0.2 0.4 0.6 0.8 1

0.4

0

1.5

180

0.2

200

1

0 2 t / T180

0 t/T

2

180

0.5

100 50

1 0

1.5

−50

2 −2

0 2 t / T180

−2

0 2 t / T180

Figure 15: Measured asymptotic echo shapes produced by the J-editing sequence for various values of N , the number of J-editing cycles. The real components of the echoes are shown in the top row, and the imaginary parts in the bottom row. The experimental sample was the same

13

C-enriched (99%) methanol as in Figure 14. Experimental parameters included

fH = 1.545 MHz, fC = 388.5 kHz, gz = 11.5 G/cm, NE = 500, tE1 = 8.5 ms, tE2 = 400 µs, t90 = 15 µs, tacq = 256 µs, and Navg = 4.

22

J-modulation scan Sat

Reference scan

N cycles

NE cycles

180y

180y

tE1

tE2

90x

Sat

90-x

N cycles

NE cycles

180y

180y

tE1

tE2

A (1H, 19F)

180

X

τ

(13C) tW

tW

Figure 16: Interleaved J-editing sequence, consisting of a normal and a reference scan. Each scan begins with a saturation pulse (labeled ‘Sat’), followed by a polarization interval tw . The phase of the excitation pulse is inverted during the reference scan, inverting the unmodulated NMR signal (shown in red). The modulated signal is shown in green.

J-coupling constants of natural abundance samples in inhomogeneous static magnetic fields.

Echo amplitude (normalized)

0 −0.2 −0.4 −0.6 −0.8 −1 Data Fit, J

−1.2

265

−1.4 0

2

4 6 J−encoding time (ms)

8

Figure 17: Measured J-editing function for Fluorinert FC-43. The sample was 25ml of FC-43 in a glass vial (diameter = 2.5cm, length = 5cm). Experimental parameters include fF = 1.45 MHz, fC = 388 kHz, gz = 11.5 G/cm, N = 2, NE = 1000, tE1 = 8.5 ms, tE2 = 400 µs, t90 = 16 µs, tacq = 256 µs, Navg = 80.

Figure 18(a) shows the measured J-modulation curves of benzene, toluene, and dodecane, while Figure 18(b) shows simulated curves for the same compounds in a uniform static field gradient. The experimental curves clearly show that both benzene and dodecane contain a single coupling constant (J ≈ 160 Hz and 125 Hz, respectively), while toluene contains a mixture of both these components. However, the amplitude of the experimental data is consistently lower than in the simulation, and also 23

appears to decrease gradually as the J-encoding time N τ increases. We are currently investigating the underlying reasons for these effects. (b)

0

−0.2

Echo modulation (percent)

Echo modulation (normalized)

(a)

−0.4 −0.6 −0.8 −1 Benzene Tolune Dodecane

−1.2 −1.4 0

2

4 6 8 J−encoding time (ms)

10

0

−0.5 −1 −1.5 −2

−3 0

12

Benzene Toluene Dodecane

−2.5

2

4 6 8 J−encoding time (ms)

10

12

Figure 18: (a) Measured J-editing functions for benzene, toluene, and dodecane, and (b) simulated functions for the same compounds in an uniform gradient. In each case, the sample was 25ml of liquid within a glass vial (diameter = 2.5cm, length = 5cm). Experimental parameters include fH = 1.45 MHz, fC = 365 kHz, N = 3, NE = 4000 (toluene and benzene) or 2000 (dodecane), tE1 = 8.5 ms, tE2 = 450 µs, t90 = 19 µs, tacq = 256 µs, Navg = 128.

We have also successfully performed a variety of J-editing experiments, including the estimation of aromatic and alkene content in complex hydrocarbon mixtures, background signal suppression, and J-filtering, using two other hardware setups: a high field (9 T) superconducting magnet and a 0.5 T permanent magnet. We do not discuss these experiments in this paper, which is focused on applications of J-editing in grossly inhomogeneous fields (such as that in the fringe field system). 6. Conclusion We have described 1 H-13 C double-resonance pulse sequences that are suitable for measuring heteronuclear J-coupling constants in grossly inhomogeneous fields. These sequences convert scalar coupling into amplitude modulation of a CPMG spin echo train. We have shown that off-resonance effects on the modulation curve can be minimized by acquiring and integrating over the entire echo shape. We have also shown that the large unmodulated background signal from uncoupled protons can be effectively suppressed by using TANGO-type excitation pulses. Finally, we have thoroughly analyzed the problem of numerically estimating the J-spectrum from modulation curves produced by these pulse sequences in the presence of both multiplicative and additive noise sources. 24

We have presented initial experimental results from a low-frequency fringe field system that validate the basic principles of J-editing. We are currently continuing these experiments. Particular goals include the demonstration of background suppression and J-filtering with TANGO excitation pulses, and estimating the aromatic content of naturally occurring hydrocarbons such as crude oils. We also plan to study the possible effects of T2 -weighting (caused by adding together a large number of spin echoes to improve SNR) on the estimated J-coupling spectrum. References References [1] E. Hahn, D. Maxwell, Spin echo measurements of nuclear spin coupling in molecules, Phys. Rev. 88 (1952) 1070–1084. doi:10.1103/PhysRev.88.1070. URL http://link.aps.org/doi/10.1103/PhysRev.88.1070 [2] C. P. Slichter, Principles of Magnetic Resonance, 3rd Edition, Springer, 1990. [3] B. Bl¨ umich, F. Casanova, S. Appelt, NMR at low magnetic fields, Chemical Physics Letters 477 (4) (2009) 231–240. [4] R. McDermott, A. H. Trabesinger, M. M¨ uck, E. L. Hahn, A. Pines, J. Clarke, Liquid-state NMR and scalar couplings in microtesla magnetic fields, Science 295 (5563) (2002) 2247–2249. [5] L. Trahms, M. Burghoff, NMR at very low fields, Magnetic resonance imaging 28 (8) (2010) 1244– 1250. [6] R. L. Kleinberg, A. Sezginer, D. Griffin, M. Fukuhara, Novel NMR apparatus for investigating an external sample, Journal of Magnetic Resonance (1969) 97 (3) (1992) 466–485. [7] M. D. H¨ urlimann, Well Logging, in: Encyclopedia of Magnetic Resonance (eMagRes), Wiley Online Library, 2012. [8] P. Speier, J-spectroscopy in the wellbore, U.S. patent 6,815,950 (2004). [9] L. An, Y.-Q. Song, K. Ganesan, Chemical properties of hydrocarbons measured using J-editing, U.S. patent 6,958,604 (2005). 25

[10] H. Molin, Use of J-modulated spin echoes for approximate estimation of one-bond C-H coupling constants in

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doi:10.1002/mrc.1260250711. URL http://dx.doi.org/10.1002/mrc.1260250711 [11] J. B. Lambert, E. P. Mazzola, Nuclear Magnetic Resonance Spectroscopy: An Introduction to Principles, Applications, and Experimental Methods, Prentice Hall, 2003. [12] T. Fan, J. S. Buckley, Rapid and accurate SARA analysis of medium gravity crude oils, Energy & Fuels 16 (6) (2002) 1571–1575. arXiv:http://dx.doi.org/10.1021/ef0201228, doi:10.1021/ef0201228. URL http://dx.doi.org/10.1021/ef0201228 [13] J. Neff, S. McKelvie, R. Ayers, Jr., Environmental impacts of synthetic based drilling fluids, Tech. Rep. OCS Study MMS 2000-064, prepared for MMS by Robert Ayers & Associates, Inc., U.S. Department of the Interior, Minerals Management Service (MMS), Gulf of Mexico OCS Region, New Orleans, LA (August 2000). [14] M. D. H¨ urlimann, L. Venkataramanan, Quantitative measurement of two-dimensional distribution functions of diffusion and relaxation in grossly inhomogeneous fields, Journal of Magnetic Resonance 157 (2002) 31 42. doi:10.1006/jmre.2002.2567. [15] M. D. H¨ urlimann, Diffusion and relaxation effects in general stray field NMR experiments, Journal of Magnetic Resonance 148 (2001) 367 – 378. [16] Y.-Q. Song, Categories of coherence pathways for the CPMG sequence, Journal of Magnetic Resonance 157 (1) (2002) 82–91.

URL http://www.sciencedirect.com/science/article/B6WJX-46K5DRY-9/2/bb64806ddeeadccaac3f2e [17] S. Wimperis, R. Freeman, An excitation sequence which discriminates between direct and long-range CH coupling, Journal of Magnetic Resonance (1969) 58 (2) (1984) 348 – 353. doi:10.1016/00222364(84)90227-0. URL http://www.sciencedirect.com/science/article/pii/0022236484902270 [18] J. Garbow, D. Weitekamp, A. Pines, Bilinear rotation decoupling of homonuclear scalar interactions,

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Chemical Physics Letters 93 (5) (1982) 504 – 509. doi:10.1016/0009-2614(82)83229-6. URL http://www.sciencedirect.com/science/article/pii/0009261482832296 [19] M. M. Veshtort, Numerical simulations in nuclear magnetic resonance: Theory and applications, Ph.D. Thesis, Massachusetts Institute of Technology (June 2003). [20] A. V. Oppenheim, R. W. Schafer, T. G. Stockham, Jr., Nonlinear filtering of multiplied and convolved signals, Proceedings of the IEEE 56 (8) (1968) 1264–1291. doi:10.1109/PROC.1968.6570. [21] S. Mandal, S. Utsuzawa, D. G. Cory, M. D. H¨ urlimann, M. Poitzsch, Y.-Q. Song, An ultrabroadband low-frequency magnetic resonance system, Journal of Magnetic Resonance 242 (2014) 113–125.

27

Highlights • • •

Heteronuclear J-coupling measurements in weak and grossly inhomogeneous magnetic fields. Analysis of measurement sensitivity and frequency resolution for hydrocarbon mixtures. Preliminary experimental results from a fringe-field setup.