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Dependence of nuclear spin singlet lifetimes on RF spin-locking power

Stephen J. DeViencea*, Ronald L. Walsworthb,c, Matthew S. Rosenc,d,e

aDepartment of Chemistry and Chemical Biology, Harvard University, 12 Oxford St., Cambridge, MA 02138

bHarvard-Smithsonian Center for Astrophysics, 60 Garden St., Cambridge, MA 02138

cDepartment of Physics, Harvard University, 17 Oxford St., Cambridge, MA 02138

dHarvard Medical School, 25 Shattuck Street, Boston, MA 02115

eMartinos Center for Biomedical Imaging, 149 Thirteenth St., Charlestown, MA 02129

*Corresponding Address:

Harvard-Smithsonian Center for Astrophysics,

MS 59, 60 Garden St., Cambridge, MA 02138

email: devience@fas.harvard.edu

Abstract

We measure the lifetime of long-lived nuclear spin singlet states as a function of the strength of the RF spin-locking

field and present a simple theoretical model that agrees well with our measurements, including the low-RF-power

regime. We also measure the lifetime of a long-lived coherence between singlet and triplet states that does not require

a spin-locking field for preservation. Our results indicate that for many molecules, singlet states can be created using

weak RF spin-locking fields: more than two orders of magnitude lower RF power than in previous studies. Our

findings suggest that in many biomolecules, singlets and related states with enhanced lifetimes might be achievable

in vivo with safe levels of RF power.

1. Introduction

The spin-lattice relaxation time T1is a limiting factor for a broad class of NMR experiments in which nuclear spin

polarization or order needs to be preserved or transported [1–4]. However, long-lived nuclear spin singlet states with

lifetimes up to 37 T1 have recently been measured in thermally-polarized samples [5–10]. These long-lived states

have been used to study slow processes such as diffusion, chemical exchange, and conformational dynamics in vitro

[11–13].

Nuclear spin singlet states are typically created from pairs of coupled spin-1/2 nuclei with equal or near-equal

resonance frequencies. The most well-known example is the H2molecule, which exists in two forms: para-H2, with

the singlet eigenstate |S0? = (| ↑↓? − | ↓↑?)/√2; and ortho-H2, with triplet eigenstates |T−? = | ↑↑?, |T0? = (| ↑↓

? + | ↓↑?)/√2, and |T+? = | ↓↓? [14]. In our notation ↑ represents a spin aligned with the applied magnetic field, B0,

while ↓ represents a spin that is anti-aligned. In the singlet state, the net nuclear spin is zero, and there is no net

magnetic dipole moment. Hence interactions with the environment are weak and the rate of interconversion between

singlet and triplet states is very slow, often much slower than the spin-lattice relaxation rate 1/T1. On the other

hand, the triplet states have non-zero magnetic moments and couple strongly with the environment. Relaxation

among the triplet states occurs on the timescale T1.

Recently, Levitt and colleagues demonstrated a general technique for the creation of singlet states for pairs of

magnetically inequivalent nuclei [7, 15]. Such inequivalent nuclear spins cannot form ideal, long-lived singlet states

as outlined above: the different local environments of the nuclear spins leads to rapid conversion to the triplet

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arXiv:1201.1482v1 [physics.chem-ph] 6 Jan 2012

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state and thus coupling to the environment. Nonetheless, as Levitt et al. showed, a properly designed RF pulse

sequence can prepare a singlet state, which is then preserved from triplet interconversion by the application of a

continuous resonant RF field. This “spin-locking” field forces the average Hamiltonian of the two nuclear spins to

be effectively equivalent. While this technique is applicable for a large variety of molecules, the large continuous

RF power employed in spin-locking experiments to date implies an RF specific absorption rate (SAR) that is likely

prohibitive for animal and human studies [16, 17].

In this paper, we report measurements of singlet state lifetime for a variety of organic molecules and as a function

of RF spin-locking field strength. We find that the measured RF power required to preserve a singlet state agrees

well with the predictions of a simple theoretical model with inputs from the molecule’s NMR spectrum. We also

present measurements of a singlet-triplet coherence with an extended lifetime that does not require the use of RF spin

locking for preservation. Moreover, our findings demonstrate that for many molecules of interest, singlet lifetimes

many times longer than T1can be achieved with much weaker RF spin-locking fields than have been used to date –

more than an order of magnitude smaller than in previous studies – leading to both an RF power and an SAR more

than 100 times lower. This result suggests that in vivo application of long-lived singlet NMR might be possible in

biomolecules with the appropriate properties, despite limitations imposed by RF SAR.

2. Theory

2.1. Singlet Relaxation Mechanisms

Many relaxation mechanisms are forbidden by symmetry from converting the singlet state to triplet states. For

example, magnetic dipole-dipole interactions between the singlet’s two spins cannot couple the antisymmetric singlet

state to the symmetric triplet states. Since this intra-pair interaction is often the dominant driver of relaxation,

the typical result is a singlet population with a lifetime TSmany times longer than the spin-lattice relaxation time

T1. Dipole-dipole interactions between the singlet spin pair and more distant spins can also lead to relaxation, but

the singlet is protected from dipolar fluctuations common to both singlet spins: i.e., in the far-field the net dipole

moment of the singlet is zero. Thus singlet-state relaxation must instead occur through differential interactions on

each spin of the singlet; these include chemical shift anisotropy (CSA), spin rotation (SR) due to collisions between

molecules, and magnetic quadrupole interactions with a third spin [18–21].

Since these three singlet relaxation mechanisms respond differently to the applied static magnetic field and temper-

ature, the dominant effect can be determined experimentally. CSA has a strong magnetic-field dependence with a

lifetime scaling as TCSA∝ 1/B2

as TSR∝ exp(E/kBT) ∝

scaling linearly with temperature in the extreme-narrowing regime (when molecular rotation rates are much greater

than the Larmor frequency). As demonstrated below, for the molecules used in the present study, the quadrupolar

mechanism dominates singlet-state relaxation, i.e., TS≈ TQ.

0[18]. Spin rotation collisions result in a lifetime scaling non-linearly with temperature

kBT[18]. Magnetic quadrupole interactions, on the other hand, produce a singlet lifetime

E

Magnetic quadrupole relaxation results from the two spins of the singlet interacting differently with a third spin.

This relaxation mechanism was modeled at high magnetic field by Tayler et al. [21], who derived an expression for

the enhancement of the singlet lifetime∗:

TS

T1

=

3b2

12

2

?

j>2

(b2

1j+ b2

2j− b1jb2j(3cos2φ1j2− 1))

.(1)

∗Ref.21 contains a typesetting error in which the summation has been taken over the whole expression rather than only the

denominator.

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∣T+〉,12

2

J

4

∣T0〉,J

4

b Strong scalar coupling

∣T−〉,−ν1−ν2

2

+J

4

∣S0〉,−3J

4

∣〉,12

2

∣〉,−12

2

∣〉,−1−2

2

∣〉,1−2

2

c Strong resonant

RF spin-locking

∣+〉,nJ

4

∣0〉,J

4

∣ϕ−〉,−νn+J

4

∣S〉,−3J

4

a Two weakly-coupled spins

W1

W1

W1

W1

W2

W0

W2

W1

W1

W'2

W'1

W'1

Figure 1: A system containing two spins with resonant transition frequencies ν1and ν2can be represented by four spin-pair eigenstates.

(A) For weak spin coupling, relaxation occurs via magnetic dipole-dipole interactions due to zero-, single-, and double-quantum transitions

(with rates W0, W1, and W2). (B) If the two spins are coupled strongly by scalar coupling (J >> ∆ν, the chemical shift splitting), the

spin-pair eigenstates are mixed into singlet and triplet states. The triplet states still interact via dipole-dipole interactions (solid lines),

but singlet-triplet transitions are forbidden (dashed lines). (C) Mixing of the spin-pair eigenstates can also be induced by a strong RF

spin-locking field (νn > 5∆ν), which produces a singlet state and three mixtures of triplet states. Singlet-triplet transitions are again

forbidden. The new triplet states exhibit new transition rates W?

1and W?

2.

Here spins 1 and 2 compose the singlet while j represents another nearby spin; bjk = γ2/r3

dipolar coupling strength between spins; and φ1j2is the angle between the vectors connecting 1 with j and 2 with

j. In principle, there is no limit to the singlet lifetime enhancement given the proper molecular geometry. However,

in practice other relaxation mechanisms gain importance if magnetic quadrupole relaxation is highly suppressed. In

a previous study, Equation 1 was found to agree well with measurements of singlet-state lifetimes using high RF

spin-locking power [21].

jkis a measure of the

2.2. RF-Power Dependence

A detailed theoretical analysis of the singlet’s lifetime during RF irradiation has been given by Pileio and Levitt,

who performed exact numerical calculations for the relationship between singlet lifetime and RF power [18]. Here,

we develop an approximate model that leads to a simple calculation of the measured singlet lifetime at a given

spin-locking frequency. Our model can easily be fit to measurements of singlet lifetime at a number of RF field

strengths so that the maximum singlet lifetime can be extracted.

A pair of spin-1/2 nuclei creates a system with four spin-pair eigenstates as shown in Fig. 1A: | ↑↑?,| ↑↓?,| ↓↑?, and

| ↓↓?. In general, there is mixing between the two eigenstates with no net z-component of spin (| ↑↓? and | ↓↑?) if the

spin states are degenerate, or if scalar coupling between the spins is strong compared with any difference between

the individual spin transition frequencies (ν1and ν2). As a result of such mixing, the spin-pair system is described

by one singlet state, |S0? = (| ↑↓? − | ↓↑?)/√2, and three triplet states |T−? = | ↑↑?, |T0? = (| ↑↓? + | ↓↑?)/√2, and

|T+? = | ↓↓?, with the energy levels shown in Fig. 1B. In this case, spin polarization cannot be transfered to the

singlet state from the triplet states via an RF pulse sequence because HRF|S0? = 0.

For many molecules of interest, however, chemical shifts induce a difference between spin transition frequencies

(∆ν = |ν1− ν2|) that is much larger than the scalar coupling, and hence there is little mixing of the bare spin-

pair eigenstates. In this case unitary transformations, via the RF pulse sequence shown in Fig. 2A, can transfer

initial thermal spin polarization to the singlet state with at most 50% efficiency by creating the singlet-enhanced

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90x 180x 90-y RF 90x Acquire

τ4

τ1 τ2 τ5

90x 90-y RF 90x Acquire

τ4

τ1 τ2 τ3 τ5

a

b

Figure 2: (A) The singlet-enhanced superposition state ρST is initialized via a 3-pulse preparation sequence. (B) A long-lived coherence

(LLC) between the singlet and triplet states ρLLC, as well as mixtures of the singlet and LLC states, are initialized via a 2-pulse

preparation sequence. Both preparation sequences are followed by application of the RF spin-locking field and a signal acquisition pulse.

superposition state [15]

ρST=|T0??T0| − |S0??S0|

=| ↑↓??↓↑ | + | ↓↑??↑↓ |.

(2)

(3)

Similarly, the RF pulse sequence shown in Fig. 2B can transfer initial thermal spin polarization into a long-lived

coherence between the singlet and triplet states with density matrix

ρLLC=|S0??T0| + |T0??S0|

=| ↑↓??↑↓ | − | ↓↑??↓↑ |.

(4)

(5)

A similar long-lived coherence has been previously studied as a way to extend T2[22, 23].

For both ρST and ρLLC, population in the singlet state will be rapidly interconverted with the triplet states on a

timescale ∼ 1/∆ν, providing strong coupling to the environment and thus rapid relaxation to the thermal state.

However, a strong on-resonance RF field has been shown to be effective for maintaining the singlet-enhanced super-

position state ρST [7, 15]. When the spin-locking RF field is set to the average resonant transition frequency of the

two spins, the Hamiltonian in the bare spin-pair basis becomes

Here νnis the effective spin nutation frequency due to the RF field, which drives single-quantum spin transitions.

Also, we assume scalar coupling is weak and neglect its contributions to the Hamiltonian.

H = h

−ν1+ν2

νn

2

νn

2

0

2

νn

2

νn

2

0

0

ν2−ν1

2

0

νn

2

νn

2

ν1−ν2

2

νn

2

νn

2

ν1+ν2

2

. (6)

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Diagonalizing this Hamiltonian yields four spin-locked eigenstates given by

|φ+? =1

2sinθ(| ↑↑? + | ↓↓?) +1

+1

2(| ↑↓? + | ↓↑?)

|φ0? =1

1

√2cosθ(| ↑↑? + | ↓↓?)

+1

√2sinθ(| ↑↓? − | ↓↑?)

|φ−? = −1

+1

2(| ↑↓? + | ↓↑?).

2cosθ(| ↑↓? − | ↓↑?)

(7)

√2(| ↑↑? − | ↓↓?) (8)

|φS? = −

(9)

2sinθ(| ↑↑? + | ↓↓?) −1

2cosθ(| ↑↓? − | ↓↑?)

(10)

The mixing angle θ is controlled by the ratio of the spin nutation frequency to the chemical shift splitting:

θ = arctan2νn

∆ν

(11)

At very large nutation rates (νn>> ∆ν), i.e., high RF spin-locking power, the spin-locked eigenstates simplify to

|φ+? =1

2(| ↑↓? + | ↓↑? + | ↑↑? + | ↓↓?)

=1

2(|T−? + |T+?)

|φ0? =1

|φS? =1

|φ−? =1

=1

2(|T−? + |T+?).

√2|T0? +1

(12)

√2(| ↑↑? − | ↓↓?) =

1

√2(|T−? − |T+?) (13)

√2(| ↑↓? − | ↓↑?) = |S0?

(14)

2(| ↑↓? + | ↓↑? − | ↑↑? − | ↓↓?)

√2|T0? −1

(15)

Note that the spin-locked singlet state |φS? corresponds to |S0? in this limit of large spin nutation (i.e., large RF

spin-locking field), whereas the three spin-locked triplet states are each mixtures of eigenstates |T0?,|T+? , and |T−?.

In this case, the initial state ρST is described well by Equation 2. The singlet, |S0?, is well-protected by the RF

spin-locking field, and after a short initial period during which the triplet states equilibrate, the remaining |S0?

component relaxes exponentially with the characteristic time predicted by Equation 1.

In the high-RF-power regime, the long-lived coherence ρLLCis a sum of coherences containing |φ+?, |φ−?, and |φS?,

which experience decoherence due to both dipole-dipole interactions and inhomogeneities in the RF spin-locking

field.

If instead very small RF spin-locking power is applied (νn<< ∆ν), the singlet component of ρSTrapidly interconverts

with the central triplet state, |T0?. When no RF power is applied, ρSTis a zero-quantum coherence that precesses in

the xy-plane, with a lifetime up to 3.25 T1if inter-pair dipole-dipole interactions are the sole relaxation mechanism

[24]. The addition of a small amount of RF power quickly decreases the lifetime of the ρST coherence because the

RF field efficiently drives single-quantum transitions but creates very little long-lived singlet component.

In the low-RF-power regime, the long-lived coherence ρLLCis well-described by Equation 5 as a population difference

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