Cross-polarization phenomena in the NMR of fast spinning solids subject to
adiabatic sweeps
Sungsool Wi, Zhehong Gan, Robert Schurko, and Lucio Frydman
Citation: The Journal of Chemical Physics 142, 064201 (2015); doi: 10.1063/1.4907206
View online: http://dx.doi.org/10.1063/1.4907206
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THE JOURNAL OF CHEMICAL PHYSICS 142, 064201 (2015)
Cross-polarization phenomena in the NMR of fast spinning solids
subject to adiabatic sweeps
Sungsool Wi,1,a) Zhehong Gan,1 Robert Schurko,2 and Lucio Frydman1,3,a)
1
National High Magnetic Field Laboratory, Tallahassee, Florida 32304, USA
Department of Chemistry and Biochemistry, University of Windsor, 401 Sunset Avenue, Windsor N9B 3P4,
Ontario, Canada
3
Department of Chemical Physics, Weizmann Institute of Sciences, 76100 Rehovot, Israel
2
(Received 24 November 2014; accepted 15 January 2015; published online 9 February 2015)
Cross-polarization magic-angle spinning (CPMAS) experiments employing frequency-swept pulses
are explored within the context of obtaining broadband signal enhancements for rare spin S = 1/2
nuclei at very high magnetic fields. These experiments employ adiabatic inversion pulses on the
S-channel (13C) to cover a wide frequency offset range, while simultaneously applying conventional
spin-locking pulse on the I-channel (1H). Conditions are explored where the adiabatic frequency
sweep width, ∆ν, is changed from selectively irradiating a single magic-angle-spinning (MAS) spinning centerband or sideband, to sweeping over multiple sidebands. A number of new physical features
emerge upon assessing the swept-CP method under these conditions, including multiple zero- and
double-quantum CP transfers happening in unison with MAS-driven rotary resonance phenomena.
These were examined using an average Hamiltonian theory specifically designed to tackle these
experiments, with extensive numerical simulations, and with experiments on model compounds.
Ultrawide CP profiles spanning frequency ranges of nearly 6 · γB1s were predicted and observed utilizing this new approach. Potential extensions and applications of this extremely broadband transfer
conditions are briefly discussed. C 2015 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4907206]
I. INTRODUCTION
Frequency-swept radiofrequency (rf) pulses have long
been used in nuclear magnetic resonance (NMR) spectroscopy and imaging (MRI) for achieving uniform inversion or
excitation profiles over a wide range of frequency offsets.1–13
Reasons driving the use of these schemes include decreasing
the sensitivity to B1-inhomogeneity effects and achieving a
more efficient coverage of wide bandwidths when faced with
limited peak powers. Frequency sweeps are usually achieved
by modulating the rf phase with time and are often implemented in an “adiabatic” mode, where the angle between the
spin magnetization and the effective field remains constant.
Such adiabatic pulse schemes have been utilized in solid-state
NMR studies of nuclei possessing broad anisotropic frequency
spans; adiabatic CPMG-like pulse schemes utilizing WURSTshaped sweeps have proven particularly useful for obtaining wideline NMR spectra of both spin-1/2 and quadrupolar nuclei.14–19 Based on these features, a technique dubbed
broadband adiabatic-inversion cross-polarization (BRAINCP)20 has recently been developed. BRAIN-CP can cover a
broad range of Hartman-Hahn (HH) matching conditions,21,22
and deliver superior wide-line spectra of rare nuclei under
static conditions. Lying at the core of BRAIN-CP is an adiabatic inversion pulse, which replaces the monochromatic spinlock pulse of the rare spin S-channel in conventional CP.
This leads to sequential fulfillment of the Hartman-Hahn
a)Authors to whom correspondence should be addressed. Electronic ad-
dresses: sungsool@magnet.fsu.edu and lucio.frydman@weizmann.ac.il
0021-9606/2015/142(6)/064201/16/$30.00
matching conditions, as the effective field strength of the RF
sweeping on the rare S-nuclei, matches the spin-lock field
of the polarizing I-spin. BRAIN-CP can also be combined
with WURST-based CPMG schemes for further improving
the sensitivity; broadband CP NMR spectra of static powders,
including 119Sn, 195Pt, 199Hg, 39K, 14N, and 35Cl species, were
thus demonstrated.20,23–25
Under moderate field strengths, conventional CP utilizing rectangular26,27 or ramped28 pulses fulfills most of the
needs for obtaining spectra from common spin-1/2 nuclei like
13
C, 15N, or 31P subject to magic-angle-spinning (MAS).29–32
However, the advent of new magnet technologies expected
to yield platforms operating at fields ≥25 T,33,34 call for the
development of CP methods that cover broader frequency
bandwidths under fast MAS rate and reasonable rf power
levels. For instance, the ∼250 ppm range characteristic of 13C
chemical shifts will span in excess of 95 kHz at the 36 T
magnet currently being built at the National High Magnetic
Field Laboratory (NHMFL) in Tallahassee, Florida.34 This
highlights the need to develop efficient, broadband CP techniques suitable for covering ultrawide spectral windows and
accommodating high MAS spinning rates also for these spin1/2 species. The broadbandness of the BRAIN-CP method
makes this a natural candidate to be explored. As will be
discussed below, however, many of the simplifying assumptions underlying BRAIN-CP are no longer applicable under
MAS, where interferences arise between the adiabatic frequency sweep, the spin-lock pulse, and the sample spinning
modulation. Herein, an analysis of these effects is presented; in
particular, the interferences arising between BRAIN-CPMAS
142, 064201-1
© 2015 AIP Publishing LLC
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Wi et al.
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and a number of rotary resonance (RR) conditions are discussed in detail. New potential forms for eliminating or avoiding these interferences were explored both numerically and
theoretically with average Hamiltonian analyses, and experimentally corroborated by 13C solid-state NMR experiments on
model amino acid compounds. Overall, this development can
deliver proton-enhanced, distortion-free spectra under ultrafast spinning rates, over a frequency-offset window that exceeds the low-γ rf field γB1, by nearly a factor of 10.
II. THEORETICAL BACKGROUND
A. Adiabatic inversions in static and spinning
powders
An essential component of this study is the adiabatic
inversion pulse. Although the detailed properties and working
mechanisms of adiabatic inversion pulses have been amply
described,1,4,6–8 we briefly summarize their properties using a
WURST pulse as a prototype that is applied on an anisotropic
spin-1/2 nucleus under static and spinning conditions. This
pulse has an amplitude- and phase-modulated waveform that
sweeps a range of frequencies linearly. In the usual rotating
frame, its associated Hamiltonian will be given by1
Hrfs = −ω1s · A (t) · Sx cos ψ [t] + S y sin ψ [t] .
(1)
Here, ω1s = 2πν1s = −γs B1s is the rf-field maximum amplitude, A(t) is therf’s amplitude envelope that we assume as
1 − cos40 πt/t p (although other even powers ofthis WURST
modulation or odd powers with absolute value cosn πt/t p
would be equally valid), Sx and Sy are the transverse spin
angular momentum operators, and the time-dependent phase

ωeS = 2πνeS =
Ωs + ωcsa −
profile ψ (t) is given by
ψ (t) =
(2)
where ψ0, ∆ω = 2π∆ν, and t p are an arbitrary initial phase, the
bandwidth of the adiabatic frequency sweep, and the duration
of WURST pulse, respectively.
The inversion properties of a frequency-swept adiabatic
pulse can be conveniently viewed in the frequency-modulated
(FM) frame that is rotating synchronously with ψ (t).1,4,8 The
spin Hamiltonian H(t) defined in Eq. (1) can be transformed in
this FM frame into a Hamiltonian H ′ (t) given by U H (t) U −1
d
U −1, where U = exp [iΨ (t) Sz]. Thus, if a S-spin that
+ iU dt
precesses at a Larmor frequency (+ isotropic chemical shift)
ω0 = 2πν0 is irradiated by an adiabatic pulse, the resultant FM
Hamiltonian is


dΨ (t)
Sz − ω1s 1 − cos40 πt/t p Sx ,
H ′(t) = − Ωs + ωcsa −
dt
(3)
where Ωs = ω0 − ωrf is the center offset of the sweep, ωcsa is
the chemical shift anisotropy (CSA), and dψ (t) /dt is the RF’s
instantaneous offset frequency given in the FM frame as
dψ (t)
∆ω
∆ω
= ω p (t) =
.
t−
dt
tp
2
(4)
The non-secular, Coriolis term in Eq. (3) can be safely disregarded. It follows that a suitable value of ω1S then leads to the
inversion of an initial state Sz, via a Sz → +Sx → −Sz trajectory
along the perimeter of a semicircle drawn on the Sz − Sx plane
of the FM frame. The effective frequency of nutation during
this process is
∆ω
∆ω
t+
tp
2
Finally, the effects of MAS can be accounted for in Eq. (5)
by imparting onto the CSA a time dependence ωcsa

= 2k=−2,k,0 ak eikω r t Sz, where the {ak (δcsa, η)} coefficients
have the usual dependence on the Euler angle set transforming
the tensor parameters δcsa, η defined in the CSA principal axes
system, into the rotor frame.35
Given the emphasis of this work on spin-1/2 nuclides
at high fields, we consider the inversion properties of such
swept pulse when applied on a site S subject to a CSA
under fast MAS. Shown in Fig. 1 are the simulated inversion
profiles of the Sz operator, arising for on-resonance sweep
conditions for such isolated site as a function of different ∆ν
values upon spinning at a MAS rate of νr = 60 kHz. The
inversion behavior evidenced is unusual even for narrowband ∆ν < νr sweeps, in the sense that Sz inverts not only
around the centerband (CB) frequency position being swept
but also around sideband frequency positions separated by
∆ω 2 ∆ω
t −
t + ψ 0,
2t p
2
2
2
+ ω1S 1 − cos40 πt/t p
.
(5)
±νr , ±2νr , etc. The inversion frequency width of each frequency band is identical to the sweep bandwidth as long as
the centerband sweep and the pseudo-sweeps at the sidebands do not overlap—even if a complete inversion profile
is obtained only for the centerband. A frequency span of
maximum inversion is, therefore, obtained for ∆ν = νr (Fig.
1(b)). If νr < ∆ν < 2νr , an inverted region from the centerband will overlap with those of the neighboring sidebands,
resulting in regions of double (i.e., no net) inversion due to
the destructive interference among adjacent inversion band’s
(Figs. 1(c) and 1(d)). When ∆ν ≥ 2νr (Figs. 1(e) and 1(f)),
sweeps at sidebands positioned at multiple nνr harmonics (n
= 0, ±1, ±2, . . .) will interfere with one another, and eventually the possibility of fully inverting Sz is lost throughout the
entire frequency range.
This ⟨Sz⟩ behavior is also reflected onto the spectra that
can be collected after the application of adiabatically swept
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FIG. 1. Sz inversion profiles calculated upon varying the span ∆ν of a WURST pulse for an anisotropic site under a MAS rate ν r = 60 kHz. An isolated ensemble
of spin-1/2 nuclei (δ csa = 400 ppm, η = 1, ν 0 = 150 MHz) irradiated at on-resonance by a rf field of ν 1S = 20 kHz, was considered. ∆ν was varied as 30 (a), 60 (b),
80 (c), 110 (d), 200 (e), and 300 kHz (f), always for a t p = 10 ms sweep pulse length. Dashed vertical lines indicate the positions of the site’s CB and spinning
sidebands. The red central double arrow (∆ν, CB) refers to the actual extent of the sweeps; horizontal blue arrows are virtual replicas centered at off-resonance
positions separated by the spinning rate. Notice the differences arising when ∆ν is chosen to be smaller than νr; the shape and width of the inversion profile are
identical to the isotropic Sz state expectation within the center-sweep region ((a) and (b)); the maximum width of the inversion profile is achieved when ∆ν = ν r
(b). When ν r > ∆ν > 2ν r , interferences occur between the center and sideband inversion profiles, resulting in the decrease of the successfully inverted central
region ((c) and (d)). When ∆ν ≫ 2ν r , the frequency regions of interference exceed the |2ν r | window, resulting in the disappearance of Sz’s central inversion
profile ((e) and (f)).
pulses. Figure 2 shows such spectral simulations, assuming
that after the adiabatic sweeps spectra were read by application
of ideal 90◦ “read” pulses of infinite bandwidth. Under static
conditions, the inversion frequency ranges that can be expected for on- or off-resonance adiabatic sweeps, reflect oneto-one correspondence with the frequency ranges that were
swept (see Figs. 2(a) and 2(b) for spectra of a spin-1/2 nuclide
with δcsa = 60 kHz). For the static cases, inverted spectral
lineshapes are obtained for whichever ∆ν region is chosen:
selective inversions for ∆ν < δcsa, and full powder inversions
if ∆ν ≥ δcsa. Upon subjecting the same powder to spinning (νr
= 30 kHz), however, the response is different. When either the
centerband (Fig. 2(c)) or a single spinning-sideband (Fig. 2(d))
is selectively irradiated with a narrow ∆ν value (∆ν < 2νr ),
completely inverted, distortion-free MAS sideband manifolds
are obtained even if ∆ν < δcsa. This is in agreement with
previously reported results.14–16,36 Irradiation at the centerband
behaves slightly more ideal than the sideband-irradiation case
even if ∆ν > 2νr ; for instance, a reasonable inversion is obtained for the centerband-irradiation case in Fig. 2(c) even with
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FIG. 2. Inverted powder spectra calculated for an anisotropic site of spin-1/2 nuclei (δ csa = 400 ppm; η = 1; ν 0 = 150 MHz) upon varying the ∆ν of the swept
pulse under static ((a) and (b)) and MAS ((c) and (d)) conditions (ν r = 30 kHz)—in all cases followed by an ideal 90◦ “read” pulse. Adiabatic pulses were
applied on-resonance ((a) and (c)) and +30 kHz off-resonance ((b) and (d)). The ∆ν values used are shown on the left and indicated by red horizontal lines. All
pulses were swept for t p = 10 ms with ν 1S = 20 kHz. Following the hard read pulse, the free evolutions were digitized for 8.192 ms (data points: 4096; dwell
time: 2 µs) and Fourier transformed to yield the spectra. The spectrum shown on the bottom of each column was computed without inversion.
∆ν = 100 kHz. Distortions clearly result when a moderate ∆ν
value is used at a sideband irradiation at Ωs = +νr . Eventually,
phase-distorted MAS spectra result when sufficiently large ∆ν
values are employed regardless of centering the irradiation at
zero or at νr , because of the multiple interferences that occur
between the MAS sideband modulations and the frequency
sweep of the adiabatic pulse.
B. BRAIN cross polarization under MAS: Basic
features
The capability demonstrated in Fig. 2 of obtaining distortion-free, broadband spectral inversion regions under MAS by
incorporating an adiabatic pulse with a relatively narrowband
(∆ν < 2νr ) swept irradiation, suggests that this low-power
route could serve for developing a broadband version of the
CPMAS NMR experiment. Shown in Fig. 3(a) is the prototype
pulse sequence employed for exploring this. In this BRAIN
CPMAS sequence, I-spin transverse magnetization is spinlocked during a time t p , while simultaneously irradiating the
S-channel with a WURST pulse over this CP mixing time.
The expectation is that, as in the static BRAIN case, any Smagnetization accrued over the course of this CP period will
follow the adiabatic inversion that the swept pulse creates, ending up aligned along the Sz axis at the completion of the mixing
time. A 90◦ “read” pulse would then convert this accrued Sz
magnetization into a detectable transverse signal.
To simplify the description of this BRAIN-CPMAS
method, we consider first the outcome of “brute force” numerical density matrix calculations for an isolated I (1H)-S (13C)
spin pair under fast (65 kHz) MAS. In these simulations,
the offset frequency of the I-channel was ignored (Ω I = 0),
and that of the S-channel was set to either ΩS = 0, ΩS /2π
= νr /2, or ΩS /2π = νr . The first aim of these calculations
was identifying the optimal I and S rf-field strengths ν1I , ν1S
enabling a transfer; to do so we followed a series of criteria
that included fulfilling the so-called zero-quantum (ZQ) and
double-quantum (DQ) Hartman-Hahn matching conditions29–32 arising under fast MAS. These are conditions satisfying νes ± ν1I = k νr (k = ±1 or ±2), where νeS and ν1I are
the effective rf frequencies for the S- and I-channels, and the
positive/negative combinations refer to the ZQ/DQ conditions,
respectively. Among the many possibilities for satisfying these
matching conditions, our choice was to select the smallest
possible ν1I rf fields for each case, while employing a ν1S
value that is larger than 0.675 ∆ν/t p to ensure that the Spowder undergoes an adiabatic passage.37 For instance, given
our arbitrary choices νr = 65 kHz and ν1S = 45 kHz, optimal
ν1I values were found at ν1I = 17 (DQ1: k = 1; ν1S + ν1I
≈ νr ), 76 (DQ2: k = 2; ν1S + ν1I ≈ 2νr ), 108 (ZQ−1: k = −1;
ν1I − ν1S ≈ νr ), and 175 kHz (DQ−2: k = −2; ν1I − ν1S ≈ 2νr ),
for the various DQ and ZQ matching conditions. The ranges
of optimal ν1I and ν1S values found by this approach were
found experimentally accurate within a ±5 kHz range, see
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FIG. 3. (a) BRAIN-CPMAS scheme
including a phase-modulated WURST
pulse applied along the S-channel
together with a standard spin-lock I
scheme. A 90◦ “read” pulse is required
after the CP mixing time to convert
the Sz-polarization obtained from the
CP process into detectable signal and
I -decoupling is applied during the signal
detection. The pulse phases used in the
sequence were: φ 1 = x, x, y, y, −x, −x,
−y, −y;
φ 2 = y, y, −x, −x, −y, −y, x, x;
φ R X = x, −x, y, −y, −x, x, −y, y. Illustrated in (b) are the x- (blue) and y-components (red), modulation phase [ψ (t)],
pulse
amplitude
given as ω 1S 1 − cos40
πt /t p , and instantaneous frequency
[dΨ(t)/dt] of the rf-field all for
parameters ∆ν = 10 kHz, t p = 10 ms,
and Ψ0 = 0◦.
supplementary material.38 This robustness is not surprising as
in swept CP experiments, it is not the ν1S that defines the HH
transfer, but rather the changing effective rf frequency νeS given
by the rf-field strength and by the instantaneous frequency
offset dψ (t) /dt coming from the adiabatic sweep.
With these assumptions set, an initial input density matrix
of the simulation was provided as Ix, and the forward and
backward polarization transfers between I- and S-channels
were recorded by monitoring Ix (blue), Sx (green), and Sz
(red) signals for the entire CP mixing time. Figures 4(a)–4(d)
illustrate the behavior of these spin polarizations, calculated
as a function of time for the aforementioned νr , ν1S , and
ν1I values. I-S CP transfers clearly occur in most of these
cases, as evidenced by signal losses of the spin-locked Ispin magnetization and by signal gains for the various S-spin
polarization components. The transverse S-spin generated by
the CP appears to oscillate strongly in the rotating frame, as
they follow the adiabatic sweep pathway of the WURST pulse.
Still, in all instances, Sx (and Sy) end up going to zero as the
bulk of the S-magnetization ends up as ±Sz at the conclusion
of the adiabatic sweep (t = t p ). CP polarization build-up along
the S-channel can be viewed more clearly in the FM frame. All
of the S-spin magnetizations begin at zero, develop along the
Sz − Sx plane, and end up at ±Sz at the end of the sweep as
the I → S transfers take place in the FM frame. A spherical
plot is included in each case in Fig. 4 to visualize the buildup
of S-spin polarization accrued on the Sz − Sx plane in the FM
frame.
Analysis of Fig. 4 reveals that the I → S transfer occurs at
specific times, when the ν1I and νeS fields satisfy a DQ or ZQ
CP condition νeS ± ν1I = ±k νr (k = ±1 or ±2). Among the
three offset values tested, the best CP efficiency was observed
for ΩS /2π = νr /2, under the νeS + νe I = νr DQ matching
condition. In the previously treated BRAIN CP static case,17
these MAS-derived conditions did not exist, and polarization
transfer occurred only when |νeS − ν1I | = 0. Moreover, in those
static cases, the transferred polarization would always end up
along the −Sz axis at the conclusion of the sweep. This is
different from the behavior shown in Fig. 4, which predicts that
under fast MAS the net CP signal gain at t p is not necessarily
aligned with −Sz, but can also be along +Sz—depending on
the CP mode and the offset-frequency range considered. In
the particular cases treated in Figs. 4(a)–4(d), the end-point
magnetization is −Sz when the DQ mode is operational at
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FIG. 4. Spin dynamics of BRAIN-CPMAS calculated numerically for an isolated, dipole-coupled I -S pair (I =1H; S=13C) for different ν 1I spin-locking
fields. The dipolar coupling constant is 4 kHz; S-spin δ csa = 300 ppm (η = 1, ν 0 = 150 MHz). The rf pulse strength, frequency sweep width, and duration
were ν 1S = 45 kHz, ∆ν = 110 kHz, and t p = 10 ms, respectively. Calculations were carried out for a MAS rate ν r = 65 kHz, starting from a transverse Ix
magnetization and monitoring the rapid, time-dependent transients of Ix (blue), Sx (green), and Sz (red) in the rotating frame. The effective rf-field strengths of
the spin-lock pulse employed were (a) ν 1I = 17 kHz (ν eS +ν e I = ν r ); (b) ν 1I = 76 kHz (ν eS +ν e I = 2ν r ); (c) ν 1I = 108 kHz (ν e I −ν eS = ν r ); (d) ν 1I = 175 kHz
(ν e I −ν eS = 2ν r ), favoring, respectively, the DQ1 (a), DQ2 (b), ZQ−1 (c), and ZQ−2 (d) HH matching conditions. The carrier frequency of the S-channel was
placed on-resonance (first column), off-resonance at +ν r /2 (second column), and off-resonance at +ν r (third column). In each case, the effective S-spin
polarization formed during the CP process is also displayed in an accelerated FM frame, as spherical plots adjacent to each I x , S x , and Sz trajectories.
ΩS /2π = 0 and ΩS /2π = +νr /2, or when the ZQ mode is
operational at ΩS /2π = +νr . The opposite is the case when
the DQ mode is operational at ΩS /2π = +νr or when the ZQ
mode is operational at ΩS /2π = 0 and ΩS /2π = +νr /2. This
means that the sign of the S-spin magnetization at the end of
the rf sweep acts as an indicator of the CP mechanism/mode,
as well as of MAS-driven RR effects39,40 which, in the presence
of CSA, will efficiently invert an Sz spin-locked magnetization
when νeS matches ±νr or ±2νr .
The extension of BRAIN CP to rotating solids has revealed some new and interesting behavior in terms of adiabatic
inversion and spin polarization transfer; we now turn to a
more thorough theoretical description of this swept-rf CPMAS
experiment, and give careful consideration to the incorporation
of the RR effects.
III. ANALYTICAL DESCRIPTION OF THE
BRAIN-CPMAS NMR UNDER ADIABATIC SWEEPS
A. The frequency-swept CPMAS Hamiltonian
For the sake of clarifying the behavior summarized in
Fig. 4, we consider a case where the I (1H) spins are locked on
resonance (i.e., ΩI = 0), and the S-spins are devoid of CSA. For
the pulse scheme in Fig. 3(a), the Hamiltonian of an isolated
I-S dipolar pair under MAS in the double-rotating frame is
given by
H = −ω1I I x − ΩS Sz + Hrfs + 2b (t) Iz Sz,
(6)
where ω1I (= γ I B1I ) is the rf field strength applied along Ispin, Hrfs is the adiabatic rf-Hamiltonian given in Eq. (1), and
2b (t) Iz Sz is a heteronuclear dipolar coupling interaction which
under MAS is modulated as32
b (t) =
2
k=−2,k,0
bk eikω r t ,
(7)
where the {bk } coefficients have the known Euler angle dependencies in the I-S dipolar tensor defined in a principal axes
system to the rotor frame.32
As mentioned, an adiabatic sweep can be visualized in
a FM frame rotating synchronously with the instantaneous
sweep phase ψ (t). Assuming for simplicity a square-shaped
rf amplitude profile, ω1S , the corresponding rf Hamiltonian
HrfS is
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

−ω1S eiΨ(t)Sz e−iΨ(t)Sz Sx eiΨ(t)Sz e−iΨ(t)Sz
sin (θ S [t 0]) = ω1S /ωeS (t 0)
d  iΨ(t)Sz −iΨ(t)Sz
e
e
= −ω1S Sx + ω p (t) Sz.
(8)
dt
Since all other terms in Eq. (6) commute with Sz, the total
Hamiltonian in the FM frame is
−i
HFM = −ω1I I x − ΩFM
S (t) Sz − ω1S Sx + 2b (t) Iz Sz,
(9)
where ΩFM
S (t) = Ω S − ω p (t), and ω p (t) is the instantaneous
frequency of the FM frame, as defined in Eq. (4).
The Hamiltonian in Eq. (9) possesses dual time modulations, coming from the adiabatic frequency sweep and from
the νr-driven MAS modulations. Under the fast spinning rates
and slow adiabatic sweeps of interest in this study, an average
Hamiltonian over the time scale of a rotor period may be
approximated by assuming a quasi-static offset value ωp for a
rotor period. This assumption will be valid as the rotor periods
considered are in the 10-30 µs ranges, whereas the adiabatic
pulse duration t p ≈ 10 ms. Then, by approximating the ωp (t 0)
term in ΩFM
S as quasi-constant for time intervals t = t 0 to t
= t 0 + 2π/ωr , a rotational transformation into a doubly tilted
frame where all rf fields lie in parallel to the z ′ axes,32,41 leads
to
HT (t) = −ω1I Iz − ωeS (t 0) Sz + 2 sin θ S (t 0) b (t 0) I x Sx
− 2 cos θ S (t 0) b (t) I x Sz.
(10)
In this equation, the effective rf field strength along the tilted
z ′-axis of S is given by

2
2
+ ΩFM
ωeS (t 0) = ω1S
S (t 0)


2
1 = ω1S *1 + 2 ΩS − ω p (t 0) +
ω1S
,
(
) 2 1/2


∆ω ∆ω
1
−
= ω1S 1 + 2 ΩS +
t 0  , (11)
2
tp


ω1S
and the tilt angle θ S (t 0) that relates the tilted z ′-axis to the
original z-axes in the FM frame is given by
cos (θ S [t 0]) = ΩS − ω p [t 0] /ωeS (t 0) ,
(12)
(
) 2 −1/2


∆ω ∆ω
1

−
= 1 + 2 Ω S +
t 0 
. (13)
2
tp


ω1S
The last term in Eq. (10), possessing spin operators I x Sz,
can safely be dropped because it does not contribute to the
CP signal transfer between I- and S-spin. Then, by utilizing a single-transition operator notation, the relevant terms in
Eq. (10) can be rewritten into32,42
HT (t) = −ω∆ (t 0) Iz2,3 − ωΣ (t 0) Iz1,4
+ sin θ S (t 0) b (t) I x2,3 + I x1,4 ,
(14)
where ω∆ (t 0) = ω1I − ωeS (t 0), ωΣ (t 0) = ω1I + ωeS (t 0), and
Iξs, t (ξ = x, y, or z; s, t = 2, 3 or 1, 4) are single-transition
operators defined by
Iz2,3 =
Iz − Sz
,
2
I x2,3 =
I+ S− + I− S+
,
2
Iz1,4 =
Iz + Sz
,
2
I x1,4 =
I+ S+ + I− S−
,
2
I+ S− − I− S+
I+ S+ − I− S−
,
I y1,4 =
(15)
2i
2i
acting in either the {|2⟩ , |3⟩} or the {|1⟩ , |4⟩} subspaces. Given
that the single transition operators commute with one another,
HT (t) in Eq. (14) can be written as the direct sum of two
separate pseudo 2 × 2 matrices
I y2,3 =
HT2,3 (t) = −ω∆ (t 0) Iz2,3 + b (t) sin θ S (t 0) I x2,3
(16)
HT1,4 (t) = −ωΣ (t 0) Iz1,4 + b (t) sin θ S (t 0) I x1,4.
(17)
and
These last two equations will determine the ZQ and the DQ
CP processes, respectively, for a rotor period. Obtaining an
average Hamiltonian over a rotor period from these doubly
tilted frame expressions
will result by applying an additional
Up = exp ikωr t Izs, t transformation onto the MAS Hts, t Hamiltonians32
d −1
U
dt p
(18)
(
) 2 −1/2 



1
∆ω ∆ω
× I xs, t cos(kωr t) + I ys, t sin(kωr t) ,
1 + 2 Ωs + 2 − t t 0 
ω1S
p


(19)
H̃Ts, t = Up HTs, t Up−1 − iUp
producing

(
) 2  1/2



1
∆ω ∆ω
  Izs, t
H̃Ts, t = −(ω1I − kωr ) + mω1S 
1
+
Ω
+
−
t
s
0 
2


2
t
ω
p

1S

 
+
2
k=−2
bk e
ikω r t 

where m is +1 for the ZQ process and −1 for the DQ CP process. Zeroth-order average Hamiltonians over a rotor period
can finally be calculated from these expressions by taking an
integral of a time t r = ω2πr ,
⟨ H̃Ts, t ⟩av
1
=
tr
t r
H̃Ts, t dt,
(20)
0
leading to
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⟨ H̃Ts, t ⟩av
Wi et al.
J. Chem. Phys. 142, 064201 (2015)

(
) 2 1/2
(
) 2 −1/2




1
bk 
∆ω ∆ω 
1
∆ω ∆ω 


s, t


−
−
= −(ω1I − kωr ) + mω1S  1 + 2 Ωs +
t   Iz + 1 + 2 Ωs +
t 
I+s, t ,
2
t
2
2
t


ω
ω
p
p


1S
1S



 
(21)
where k = ±1 and ±2, I+s, t = I xs, t + iI ys, t , and t 0 has been generalized for simplicity to a suitable time t.
time points for different modes that fall into the time window
of the swept S-pulse are indicated in Fig. 5.
B. Fulfilling the HH matching conditions over
the course of a sweep
C. Swept CPMAS and the onset of CSA-driven rotary
resonance effects
The time-progressive ZQ and DQ HH matching conditions of the swept CPMAS experiment can be determined
by inspecting the coefficients of the Iz2,3 and Iz1,4 terms
in Eq. (21). A ZQ−k HH transfer event will occur when
the instantaneous S-pulse offset ω p (t) fulfills an “antimatching” CP condition, νeS (t) − ν1I + k νr = 0 (k = −1 or
−2). Likewise, a DQk HH transfer occurs at a time t when
a matching CP condition νeS (t) + ν1I − k νr = 0 (k = 1 or
2) is met. Because of the time-progressive nature of the
ω p (t) term, only a few instants will satisfy these matching
conditions during the entire CP mixing time. This is illustrated
in Figure 5(a) which shows ZQ−k and DQk HH matching
curves calculated for a variety of ΩS /2π offsets frequency
under fast MAS (νr = 65 kHz). Highlighted arrows in these
curves indicate the time points at which the various DQk and
ZQ−k matchings are satisfied. Notice that, because of ωes (t)’s
∥ω p (t)∥ dependence (Eq. (11)), DQ- or ZQ-CP matching
conditions can be met at two different transient times t 1
and t 2 throughout the course of the sweep. Depending on
∥Ωs ∥, these time points may or may not fall within the
0 ≤ t ≤ t p period and hence may or may not be physically
relevant to the CP process. If ΩS /2π = 0, these t 1 and t 2 points
will be mirror images of each other across the center point of
the symmetric frequency sweep. If an off-resonance frequency
contribution is considered, these t 1 and t 2 time points will
shift uniformly for the DQ1 conditions to earlier times when
ΩS < 0 and to later times when ΩS > 0. Conversely, if the
frequency offset is in the range of |νr | /2 < |ΩS |/2π < |νr |,
only a single contact time will be found in the time period
0 ≤ t ≤ t p —with the other appearing at either t > t p
(ΩS > 0) or at t < 0 (ΩS < 0). For offset-frequencies in the
|ΩS |/2π > |νr | range, both CP-contact time points of DQ1
disappear from the physically meaningful 0 ≤ t ≤ t p window.
Moreover, if parameters are optimized to produce an efficient
DQ1 mode in the central ΩS ≈ 0 spectral region, other modes
such as DQ2, ZQ−1, and ZQ−2 will not contribute to the CP
signal transfer within the physically meaningful 0 ≤ t ≤ t p
mixing interval. Similar considerations can be derived for
the remaining DQ±2, ZQ±1, and ZQ±2 transfer modes. One
interesting property following from Eq. (11) is that if CP
transfer times are met for ΩS /2π at certain t i s, those for
−ΩS /2π will be met at t p − t i . For instance, the transfer times
found for ΩS /2π = 32.5 kHz during a 10 ms sweep (Fig. 5)
are 6.44 and 9.47 ms for DQ1 and 1.72 ms for ZQ−1; those
found at ΩS /2π = −32.5 kHz are then at 3.56 and 0.53 ms
for DQ1 and 8.28 ms for ZQ−1. Other relevant CP transfer
An important aspect mentioned earlier concerns the complex sign behavior of the S-polarizations arising at the conclusion of these swept CPMAS experiments. These complexities
are illustrated again in Figs. 5(c) and 5(d), which show the
time-dependent Ix (blue), Sx (green), and Sz (red) transients
involved in the CPMAS signal transfer dynamics in the presence and absence of S-spin CSA. The effects of introducing
this anisotropy are two-fold. On one hand, the powder-angle
dependence introduced by the CSA originates a spread of the
CP transfer instant around the various time points that were
found analytically in Fig. 5(a) without considering CSA. This
spread in the timings of a spin-packet’s actual HH transfer
reflects the contributions of the S-spin anisotropic frequencies,
which may add a shift to the effective offset-frequency term.
Examination of Figs. 5(c) and 5(d), however, reveals that in
the presence of CSA, a more significant phenomenon begins
to affect the swept-CP MAS experiment: this is the presence
of ±Sz → ∓Sz inversions, introducing in turn reversals in the
outcome of the overall CP profile. These reversals are associated to MAS-driven RR effects.39,40
When a shift anisotropy is modulated by a MAS process
of spinning rate νr, a phenomenon known as RR takes place,
which dephases a magnetization that has been spin-locked
by a field ∥ν1S ∥ = νr, 2νr.43–45 The reason for such dephasing
can be visualized in a tilted rotating frame where the spinlocking rf takes the role of a Zeeman-like field, and the CSA
time modulation would be akin to an oscillating transverse
field tilting the magnetization away of the ν1s spin-lock. In
the constant-rf scenario where all such RR effects have been
analyzed so far, the strength of the CSA-imparted precession is
orientation-dependent and leads, over a powdered sample, to
a dephasing of the spin-locked magnetization. The frequency
swept case hereby considered, by contrast, is different: given
the relatively slow sweep rates being considered, the effective
field that is now relevant for defining the rotary resonance
effect will “sweep” adiabatically through the ∥νeS ∥ = νr, 2νr
RR conditions. Because of this adiabatic character, the “transverse” CSA field will effectively be able to affect an inversion
of the spin-locked magnetization for all orientations, leading
to a homogeneous inversion for the full powder, devoid of
dephasing.
With RR as a new mechanism capable of morphing a spinlocked magnetization into an “anti-spin-locked” state (and
vice versa), one needs to consider how will this factor into
the signs of the S-spin polarizations that may have formed
throughout the swept CPMAS process. Fig. 5(b) facilitates
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Wi et al.
J. Chem. Phys. 142, 064201 (2015)
FIG. 5. CP transfer and RR inversion during swept BRAIN CPMAS (νr = 65 kHz) experiments. (a) DQ1 (solid red line), DQ2 (dashed red line), ZQ−1 (solid
magenta line), and ZQ−2 (dashed magenta line) matching conditions at the specified ΩS offsets are shown, by the time points (zero crossing conditions) at which
CP signal transfers occur. (b) Curves showing the positions that satisfy the RR condition ν eS (t) = ν r , occurring during the CP mixing time in the presence of
an S-spin (δ csa = 120 ppm, η = 0). ((c) and (d)) Time propagations of an initial Ix state (blue) and of the CP-enhanced Sx (green) and Sz (red) polarizations,
recorded with and without the inclusion of CSA effects. Notice the sign inversion of the CP-enhanced S-components occurring at the time points at which a RR
condition is satisfied—but only when CSA is included.
visualization, by illustrating the time points that satisfy the
νeS(t) = νr RR condition for a series of frequency offsets and
parameters like those hitherto utilized for calculating the swept
CPMAS matching conditions. Highlighted in Fig. 5(b) are the
RR time points that fall in the 0 ≤ t ≤ t p range. Notice then
that if a S-spin component has been generated by CP, it will
be inverted by the CSA if this creation happened prior to a
RR-driven S-spin event, but will remain spin-locked otherwise.
Notice that for some offsets (e.g., the fourth and fifth columns
in Fig. 5), RR occurs before the HH transfer events, resulting
in non-inverted CP signals. Thus, curves calculated with (Fig.
5(c)) and without (Fig. 5(d)) including the S-spin CSA are
identical. For other ΩS/2π offsets, the RR inversion occurs
after transfer events like DQ1, resulting in an inversion of the
CP signals. These features are clearly visible from the timedependent Sx and Sz transients in Fig. 5(c). Moreover, for
the on-resonance case, two time points may satisfy the RR
condition; since two inversions are the same as no inversion,
no RR-driven effects on the final CP signal.
This RR inversion effect can be generalized to arbitrary
offset-frequencies. It then follows that, in general, the sign
of Sz obtained upon sweeping around a particular Ωs, will
be 180◦ out-of-phase to that obtained when sweeping around
−Ωs. Thus, in the presence of CSA, the Sz states calculated
at ±Ωs would be placed on the opposite hemispheres in the
FM frame. This RR inversion effect takes place even when
a small magnitude of CSA is present—of the kinds that are
effectively averaged out from a spectrum at the MAS rates here
considered. Still, their effects on the final state of the spinlocked magnetizations can be remarkable.
D. Propagating the spin ensemble throughout
the BRAIN-CPMAS transfer
The Hamiltonian in Eq. (21) enables the propagation of
the spin density matrix throughout the course of the CP sweep
time. In particular, the value taken by Sz in the FM frame for
the whole time period 0 to t p , upon starting from a ρ (t = 0)
= Iz = Iz14 − Iz23 state, is32,42


†
⟨Sz⟩ t p = trace Iz1,4U1,4 t p Iz1,4U1,4
tp


†
−trace Iz2,3U2,3 t p Iz2,3U2,3
tp
(22)
( 
Us, t (t p ) = T̂ exp −i
(23)
with
0
tp
)
dt ⟨ H̃Ts, t ⟩av (t) ,
where T̂ is the Dyson time-ordering operator that determines
the ordering of terms
in the
integration. In principle, the
Izs, t and I+s, t terms in H̃Ts, t av do not commute, and therefore Eq. (23) should be evaluated numerically in a stepwise
manner. However, considering the working mechanism of
a sweep for which the adiabaticity of the pulse employed
fulfills dθ s (t) /dt ≪ 1, perturbations in θ S (t) can be neglected
and the angle between the spin magnetization and the effective field assumed constant. In such case, an approximate
average Hamiltonian over the whole mixing time, H̃Ts, t av

t
= t1p 0 p H̃Ts, t av (t)dt may be obtained by integrating Izs, t and
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I+s, t terms separately.8 The whole CPMAS process then yields
⟨ H̃Ts, t ⟩av = [kωr + m(Π0 + Π1 − ω1I )]Izs, t +
bk
Π1 I+s, t ,
ω1S
(24)
where m = −1 or +1 for ZQ- and DQ-HH processes, respectively,
(
)
(

∆ω
∆ω ) 2
1 
2
 ΩS +
ω
+
Ω
+
Π0 =
S
1S
2∆ω 
2
2


(
(

∆ω )
∆ω ) 2 
2

(25)
− ΩS −
ω1S
+ ΩS −
2
2 

and
 2
2 

∆ω


2


Ω
+
ω1S + ΩS + ∆ω
ω1S
S


2 +
2
.
Π1 =
(26)
ln 

2∆ω 


2

 Ω − ∆ω + ω2 + Ω − ∆ω 
S
1S
2
2

 S
Equation (24) is the equivalent of a zeroth-order average
Hamiltonian for the whole CP process. According to it, a ZQdriven signal transfer occurs if an anti-matching CP condition, Π0 + Π1 − ω1I + kωr = 0 (k = −1 or −2), is met, and
a DQ-driven process occurs if a matching condition Π0
+ Π1 + ω1I − kωr = 0 (k = 1 or 2) is satisfied. A description
of how Eqs. (24)–(26) were obtained is summarized in the
supplementary material.38 These expressions revert to the
known ZQ-HH mode at ωeS − ω1I + kωr = 0 (k = −1 or −2)
and to the DQ-HH mode at ωeS + ω1I − kωr = 0 (k = 1 or 2),
if sweeps are not considered.
Given the Hamiltonian in Eq. (24), the spins’ evolution
operator can be evaluated as42
(
)
(
)
Us, t (t p ) exp −it p ⟨ H̃Ts, t ⟩av = exp iφ s, t Izs, t exp iθ s, t I ys, t
(
)
× exp −i β s, t Izs, t exp −iθ s, t I ys, t exp −iφ s, t Izs, t .
(27)
Equation (27) represents a rotation in {|s⟩ , |t⟩} subspace
through β s, t about an axis with polar angles (θ s, t φ s, t ). Given
this rotation, the S-spin polarization Sz in the FM frame is then
2 1,4
cos θ
+ sin2 θ 1,4 cos β 1,4
⟨Sz⟩ t p =
2
2 2,3
cos θ
+ sin2 θ 2,3 cos β 2,3
−
, (28)
2
√
s, t
where ωeff
= β s, t /t p = Γ s, t describes the spin dynamics of
the process, sin2θ s, t =
b 2k Π12
2 Γ s, t
ω 1S
its geometry, and
Γ s, t = (Π0 + Π1 − mω1I )2 + 2mkωr (Π0 + Π1 − mω1I )
2
+ k 2ωr2 + b2k Π12/ω1S
.
(29)
While encompassing the details of the CPMAS dynamics,
these equations still do not factor in the rotary resonance effects
described earlier and which, as mentioned, may significantly
alter the final ⟨Sz⟩ expectation value. Consider, for instance,
a sweep centered at an on-resonance Ωs = 0 position. In such
case, a DQ1-derived contribution occurring at t 2 > t p /2 may
have a phase that lags by π behind the polarization generated
at t 1 < t p /2. These two DQ1 buildups will add destructively,
resulting in partial ⟨Sz⟩ cancellation. The extent of this destructive interference will be maximal for the ΩS = 0 on-resonance
position, for which the CP events at t 1 and t 2 are placed
symmetrically about the FM frame’s equator. By contrast,
other Ωs offsets and CP modes, such as the ZQ ones, do
not experience destructive signal interferences because they
happen only once for the |ΩS | > 0. The frequency-swept CP
enhancement will thus be most effective at off-resonance irradiation points. To account for these RR effects on the CP
dynamics, one can write Eq. (28) as
)
(

2 1 2,3
2 2,3
⟨Sz⟩ t p = ε sin θ sin
ω tp
2 eff
(
)
1 1,4 
− sin2 θ 1,4 sin2 ωeff
(30)
tp ,
2
where ε is an a posteriori factor inserted to reflect the offsetdependent RR inversion effect; ε is +1 when Ωs ≥ 0, and −1
when Ωs < 0.
Figure 6 shows the Ωs -dependent CP profile expected
upon introducing this ε factor, within the average Hamiltonian
framework. The same profile is also presented for conventional
Hartmann-Hahn under the same νr = 65 kHz MAS rate for
comparison; both of these plots are calculated for varying ΩS
and ν1S , under a fixed ν1I (= 17 kHz) and νr . Under these
conditions, the optimal CP modes that are operational in both
approaches are the ZQ and the DQ ones. Notice how, as the
offset-frequency increases from zero to |±ΩS|, the optimal ν1S
of DQ and ZQ modes in each CP method moves so as to counterbalance the increase in the offset-frequency; eventually, for
both methods, the matching conditions disappear from the plot
window when the ν1S value becomes negative. Also worth
remarking is the fact that, due to the occurrence of the RR
inversion for ΩS ≤ 0, the lower half of the offset-frequency
profiles attain reversed intensities in BRAIN-CPMAS, vis-àvis those in a conventional CPMAS experiment. Finally, notice
that because the swept method attains matching conditions in
a time-progressive manner, the optimal ν1S value for each CP
mode contains a contribution from the instantaneous frequency
offset dψ (t) /dt, and is therefore different from the optimal ν1S
of the corresponding conventional CPMAS mode.
IV. MATERIALS AND METHODS
A. Numerical simulations
All the frequency-swept simulations that were described
above resulted from numerically considering the effects of the
Hamiltonian in Eq. (1). Utilizing in-house programs written
in Matlab® (The Mathworks, Inc.), the performance of the
frequency-swept CPMAS scheme was examined numerically
by calculating the time propagation of the density matrix using
piecewise (2 µs) time increments in the RF amplitude, rf phase,
and—when applicable—rotor position. An isolated anisotropic
chemical site was considered for examining the inversions
of longitudinal magnetizations, the lineshapes of static and
spinning powders, as well as when considering the RR inversion effect. These simulations nulled the offset frequency of I
(Ω I = 0), and ΩS was set to either 0, ΩS /2π = νr /2, or ΩS /2π
= νr . The actual CP process was examined by considering
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FIG. 6. Theoretical 2D-(Ω S, ν 1S ) maps of the DQ- and ZQ-CP modes of BRAIN-CPMAS and conventional CPMAS methods, calculated using the average
Hamiltonian theory in the text at ν 1H (= 17 kHz) and ν r (= 65 kHz). The optimal ν 1C = 37.2 kHz and 48 kHz satisfying BRAIN- and HH-CPMAS’s DQ1
modes, respectively, were used for both plots for comparison. The frequency sweep width and the duration of the S-pulse are ∆ν = 110 kHz and t p = 10 ms.
Calculations were carried out by considering a single S (13C) site (δ csa = 40 ppm, η = 0.3), and an isolated S-I (13C-1H) dipolar pair with bIS = 23 kHz.
an isolated I-S spin pair in the presence of a suitable MASmodulated dipolar Hamiltonian. Powder averaging calculations of the S-spin’s CSA as well as of the I-S dipolar coupling
interactions were carried out by considering the 6044 and 1154
crystal orientations of the ZCW’s Euler angle sets46 for the
static and spinning cases, respectively, assuming coincident
dipolar and CSA tensors.
B. Experimental
This study’s predictions were examined experimentally
on powdered samples of [2-13C]Ala, [U-13C]Ala, and [U13
C]Gly, focusing on conventional and BRAIN 1H-13C CPMAS experiments. Amino acids were purchased from Cambridge Isotope Laboratories (Andover, MA) and used without
further treatment except for [U-13C]Ala, that was dissolved in
water and recrystallized by slow evaporation. About 2 mg and
7 mg of each sample were packed into 1.3 mm and 2.5 mm
MAS rotors, respectively. All experiments were carried out at
room temperature, on 11.7 T and 14.1 T magnets equipped
with Bruker Avance consoles operating at 1H frequencies of
500.23 MHz and 600.92 MHz, respectively (13C frequencies
of 125.80 and 150.45 MHz, respectively). A 2.5 mm Bruker
MAS NMR probe was used at 11.7 T for obtaining a moderate
MAS rate, νr = 25 kHz; a 1.3 mm Bruker MAS NMR probe
at 14.1 T provided an “ultrafast” MAS rate, νr = 65 kHz. The
1
H and 13C 90◦ pulses used were 2 µs for all experiments. The
swept rf pulse shapes were constructed numerically on the
basis of WURST utilizing Matlab scripts, and subsequently
fed into the NMR console for out-clocking. The shape of
these pulse schemes was akin to the ones employed in the
20
static BRAIN-CP
experiments:
an amplitude pulse profile
40
ν1s 1 − cos πt/τp , and an offset-frequency profile undergoing a linear sweep with time between 0 to t p . The number
of digital data points employed for encoding these WURST
pulse shapes was 2000. Suitable frequency sweep windows
(∆ν) and CP mixing times (t p ) for the duration of the WURST
pulse were also used, after scouting the ranges of 40–400 kHz
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J. Chem. Phys. 142, 064201 (2015)
FIG. 7. Experimental ΩC -dependent CP profiles of BRAIN- and conventional CPMAS methods measured on 2-13C Alanine at MAS rates of ν r = 25 kHz (a)
and ν r = 65 kHz (b) with individually optimized ν 1C and ν 1H values: for BRAIN CPMAS ν 1H = 17 kHz and ν 1C = 45 kHz in the case of ν r = 65 kHz, and
ν 1H = 42 kHz and ν 1C = 20 kHz in the case of ν r = 25 kHz. For CPMAS, ν 1H was 112 kHz (rectangular) and 148 kHz (ramped) in the case of ν r = 65 kHz,
and 44 kHz (rectangular) and 82 kHz (ramped) in the case of ν r = 25 kHz. The same ν 1C (= 45 kHz) was used as the BRAIN-CPMAS method in each case of
MAS rate, and spectra were collected in ∆Ω S = 1.5 kHz (10 ppm) increments to simulate the effects of sites with varying offsets. Included in (c) are expanded
versions of (b) to better view the central ΩC range, with the BRAIN inset’s negative peaks multiplied by −1 in order to better appreciate the CP dip.
and 2–14 ms, respectively. For νr = 65 kHz, CP-optimal
pulse parameters were found with ∆ν = 110 kHz (< 2νr );
◦
t p = 10 ms; ψ0 = 0 . When νr = 25 kHz, these parameters
were ∆ν = 48 kHz, t p = 10 ms, ψ0 = 0◦. The optimal 13C and
1
H rf power conditions of these swept experiment for a given
MAS rate were found experimentally by sweeping both rf
channels independently, while keeping the 13C offset slightly
off-resonance (e.g., 20 ppm) to avoid the aforementioned RRdriven signal intensity cancellation effects occurring for an
on-resonance sweep. When νr = 65 kHz, the optimal 13C rf
frequency was found at 45 ± 5 kHz with ν1H = 17 kHz, as
predicted by the DQ1 condition ν1H = νr − νeC . In a separate
set of experiments employing νr = 25 kHz, optimal 1H rf
frequencies were set at ν1H = νr + νeC = 42 ± 5 kHz and
ν1H = 2νr + νeC = 65 ± 5 kHz; the optimal 13C rf frequency
was set at 20 ± 5 kHz as found from an independent frequency
sweep.
For comparison, conventional CPMAS experiments were
also carried out with independent optimizations, employing
either square-shaped or ramped (90%-110%) spin-lock pulses
along the 1H channel while simultaneously applying a rectangular spin-lock pulse on the 13C. The mixing time used in
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J. Chem. Phys. 142, 064201 (2015)
FIG. 8. Comparisons between the experimental and theoretical ΩC -dependent CP profiles observed for 2-13C Alanine from the BRAIN- (a) and conventional
CPMAS (b) experiments measured at a MAS rate ν r = 65 kHz. The theoretical profiles in (a) and (b) show cross-sections taken at the optimal ν 1C values
from Figs. 6(a) (ν 1C = 37.2 kHz) and 6(b) (ν 1C = 48 kHz) that match the DQ1 mode of each case at the central region, with ν 1H = 17 kHz and ν r = 65 kHz
(tensor parameters employed in both simulations were: b IS = 23 kHz; δ csa = 40 ppm, ηCSA = 0.3). Experimental CPMAS profiles are taken from Fig. 7(b). The
theoretical profile shown in (a) with a red line depicts the theoretical BRAIN-CPMAS profile expected when neglecting CSA-driven RR effects. Both the
experimental and simulated spectra of the BRAIN-CPMAS case shown in (a) are the same DQ1 mode. However, the experimental HH CPMAS spectra that were
compared to the simulated DQ1 mode in (b) correspond to ZQ1 (rectangular; ν 1H = 112 kHz, ν 1C = 48 kHz) and ZQ2 (ramped; ν 1H = 148 kHz, ν 1C = 48 kHz)
modes that were found independently by experimental optimizations.
these CPMAS experiments was 2 ms for both modes, based
on optimizations. All CP spectra were acquired by co-adding 4
transient signals with 5 s of recycling delay. SPINAL-6447 proton decoupling was used during the direct acquisition period,
with a 100 kHz decoupling power.
V. RESULTS
The main aim of the experiments performed was to verify
the various phenomena introduced in Figs. 4–6, and in particular the offset dependence of the BRAIN CPMAS profile at
the fast spinning rates and large range of offsets that would
be relevant at high fields. Towards this end, the wide-band
profiles of BRAIN and conventional CPMAS methods were
measured for model amino acids while varying 13C’s offsetfrequency (Ωs ) under optimal ν1H and ν1C conditions. The
CPMAS’ performances were evaluated between −400 and 500
ppm 13C offset ranges when νr = 25 kHz, and between −1000
and 1100 ppm when νr = 65 kHz.
Frequency-swept spectra arising from BRAIN CPMAS
experiments on 2-13C-Ala under moderate (νr = 25 kHz) and
fast (νr = 65 kHz) MAS rates are shown in Figs. 7(a) and 7(b),
respectively. A single vertical line in each profile corresponds
to a CP spectrum measured at a particular 13C frequency offset
(spectra were in fact free from CSA-derived spinning sidebands because of the fast MAS rates employed). Also included
in Figs. 7(a) and 7(b) for comparison are the offset dependent
profiles of CPMAS experiments based on the conventional
spin-lock pulse along 13C channel, while employing either a
rectangular or a ramped pulse block on the 1H channel. In
these HH experiments, the same ν1C value as in the swept CP
experiment was used, while employing an optimal ν1H that was
found by sweeping the latter in the 0–200 kHz range.
It follows from Fig. 7 that at moderate MAS rates (νr
= 25 kHz), the swept CP method does not possess an advantage over the conventional HHCP methods. When νr = 65 kHz,
however, the swept CP method provides a superior capability
for polarizing a far wider range of offsets—-over 285 kHz in
this case. For the central frequency region, the best CP modes
correspond to DQ1, ZQ1, and ZQ2 for the BRAIN-, rectangular
HH-, and ramped HH-CPMAS experiments, respectively. The
width of the DQ1 mode in the BRAIN-CPMAS profile is |2νr |,
which reaches to 130 kHz when νr = 65 kHz. This is far wider
than what is achieved by the ramped HH-CPMAS method. The
signal intensity of the swept CP method is comparable to that
of rectangular HH-CPMAS—even if for all cases, the ramped
HH-CPMAS method provides the strongest signal intensity
for frequencies that are close to the on-resonance position. An
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J. Chem. Phys. 142, 064201 (2015)
FIG. 9. Influence of the MAS rate ν r (a) and of the adiabatic frequency sweep range ∆ν (b), on the ΩC -dependent 13C polarization profile in BRAIN-CPMAS.
Experiments were carried out on U-13C Glycine and spectra show both the CO and Cα peaks, highlighted in red and blue colors, respectively. Variable spinning
rates of ν r = 50, 55, 60, and 65 kHz were tested in (a) while maintaining ∆ν/t p at 70 kHz/10 ms. Variable ∆ν of 400, 110, 70, and 30 kHz were swept for
t p = 10 ms in (b), while maintaining the MAS rate at ν r = 65 kHz. Notice that due to the fast spinning rates, the ΩC -dependent CP profiles are invariant to the
variations in ∆ν/t p (b).
interesting feature displayed by all the swept CPMAS profiles
is the presence of an overall anti-symmetry about Ωs ≡ 0; this
is due to the afore-mentioned RR inversion effects that occur
when the matching condition νeC = νr is met. Fig. 7(c) shows
this with a magnified view of the central portion of Fig. 7(b),
with the negative intensity portion in the BRAIN-CPMAS
profile multiplied by −1 to better compare its overall intensity
profile to the HH-CPMAS results. It is worth remarking that
similar RR effects were also observed with other types of
adiabatic pulse schemes, such as hyperbolic secants.8
It is also valuable to compare the experimental frequency
profiles that arise from these swept CPMAS experiments,
against the predictions stemming from analytical simulations
based on using the average Hamiltonian theory developed in
Sec. III D. For this we take 1D slices at specified ν1C values
(i.e., along vertical) from the 2D ν1C − Ωs plots in Fig. 6, and
compare these with experimental BRAIN- and HH-CPMAS
profiles. Shown in Fig. 8(a) is the 1D slice calculated analytically by employing ν1H = 17 kHz, 37 kHz ≤ ν1C ≤ 42 kHz,
and νr = 65 kHz, compared against the experimental BRAINCPMAS profile measured on 2-13C Ala and already introduced
in Fig. 7(b). Shown in Fig. 8(b) against an experimental CP
profile is the simulated Ωs -dependent CP profile expected
under HH match an optimum ν1C = 48 kHz, calculated for
ν1H = 17 kHz and νr = 65 kHz. As can be seen from Fig. 8(a),
the CP model introduced earlier together with the RR inversion
explains the overall shape and frequency positions of the Ωs dependent CP profile.
It is worth concluding by showing the dependences of the
swept CP profile on the MAS rate, as well as on the width
of the adiabatic frequency sweep ∆ν. U-13C Gly was used
for monitoring these profiles for the CO and Cα sites, in the
−500 ppm to +700 ppm frequency range. Figure 9(a) shows
profiles measured for the sites by varying the MAS rate from
50 to 65 kHz in 5 kHz step under a fixed adiabatic sweep rate,
∆ν/tp = 70 kHz/10 ms. Marked by red arrows at ΩC/2π = 0
and at ±νr are the frequency positions where the RR inversion
occurs and the width of the DQ1 mode ends for each site,
respectively. The width of the DQ1 mode for each site increases
linearly as the MAS rate increases from 50 kHz to 65 kHz.
Figure 9(b) demonstrates the negligible effect of ∆ν on CP
profiles at these fast MAS rates (νr = 65 kHz): the overall
shape as well as signal intensity profile is identical in all cases.
VI. DISCUSSION AND CONCLUSIONS
In principle, the classical square-pulse scheme can provide arbitrarily broadband CPMAS transfer profiles, provided
sufficient rf power can be employed. However constraints of
various kinds—arcing, sample power deposition, low-γ values,
available rf field strength—-impose limitations to the spinlocking capabilities of the standard CPMAS method. The
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present study explored the possibility to polarize large bandwidths at low rf amplitudes, by employing adiabatic passage
schemes covering a wide frequency range of ΩS -offsets. While
adiabatic passage variants of the Hartmann-Hahn experiment
have been previously proposed,48,49 a distinction should be
introduced between those experiments sweeping adiabatically
through an optimum match via fixed-frequency amplitude
modulations of the rf, and BRAIN-CP methods sweeping over
wide ranges of rf frequencies. The latter not only modulate
the effective rf field amplitude but also achieve far wider
matching offsets than those dictated by rf strength considerations. On exploring such approach under MAS, similarities
but also significant differences were noted vis-à-vis the static
BRAIN CP approach that we have recently introduced. Most
salient among the new, MAS-induced features, are the needs
to contend with both νr -dependent DQ and ZQ matching
conditions during a sweep, and with new rotary resonance phenomena. The matching conditions of the ZQ- and DQ-CPMAS
modes in the swept rf scheme could be accurately predicted
by an average Hamiltonian model specifically developed to
analyze the BRAIN-CPMAS experiment. Under fast spinning
rates, this formalism predicted the ZQ−1,−2 transfers at νeS − ν1I
= −k νr , and the DQ+1,+2 transfers at νeS + ν1I = k νr , where k

2
+ (ΩS /2π + ∆ν/2 − ∆νt/t p )2. Iden= 1 and 2 and νeS = ν1S
tical transfer events could occur twice at t 1 and t 2 because of the
quadratic dependence of νeS on the effective offset-frequency.
S polarizations created at these different times, however, can be
differently affected by RR effects. In particular, partial signal
cancelations appeared if CP matching conditions appeared on
both sides of the Bloch sphere’s equator. This phenomenon
leads to substantial cancelations at Ωs offsets that are close to
the zero frequency position; by contrast, CP modes obtained
at other frequency offsets possess only a single CP contact
point within the entire 0 ≤ t ≤ t p mixing time, and do not
experience these partial signal cancelations. Apart from this
distinction, it is interesting to note that the ensuing BRAINCPMAS provides signal enhancements that are as strong as
those of the rectangular CPMAS—even though transfers in
the former occur only at a few time points over the contact
instead of during the entire CP mixing period, and even though
they require nearly an order-of-magnitude lower powers. This
efficiency may be partly due to the fact that in the swept CP
method, polarizations generated at a time point t do not go
back and forth between the two spin reservoirs as the matching
condition is satisfied only at a transient time point.
The various predictions and phenomena derived in this
study on the basis of time-averaged Hamiltonian theories and
of numerical simulations, coincided well with experiments on
model samples. As part of these experiments, an enhancement
profile spanning in excess of 285 kHz (1900 ppm at 14.1 T) was
measured with the new swept CPMAS method on 13C-labeledAla—an unprecedented frequency span, particularly remarkable in light of the weak rf fields ν1I , ν1S that were involved.
Even the width of the single DQ1 transfer mode, the most practically useful mode of the BRAIN-CPMAS method, reached
∼130 kHz; ca twice as wide as the whole window experimentally obtained by the ramped HH-CPMAS method (75 kHz).
This transfer mode displays a dip in its polarization enhance-
J. Chem. Phys. 142, 064201 (2015)
ment at the central position of the sweep; although unwanted,
this dip is fairly sharp and can either be placed at a position that
is a priori known to be devoid of resonances (e.g., the ∼100
ppm region in a protein NMR experiment), or be compensated by the acquisition of multiple scans with slightly shifted
frequency sweeps. Moreover, the magnitude of the central
DQ1 mode transfer in the swept method becomes larger as
the MAS rate increases. Thus, BRAIN-CPMAS promises to
become an ever more attractive alternative over conventional
HH approaches, for obtaining broadband CP transfers of nuclei
possessing large chemical shift dispersions and anisotropies
and subject to even higher MAS rates. In particular, for the
magnetic field strength of 36 T being planned for the NHMFLs
the series connected hybrid (SCH) magnet33,34 where bandwidth profiles of common nuclei such as 31P are expected to
reach 370 kHz, BRAIN-CPMAS could become a method of
choice. Further uses could arise in moderate-field cryogenic
MAS experiments, where the use of Helium gas prevents
the application of strong rf pulses without arcing. Further
experiments aimed at exploiting these phenomena, including
CPMAS NMR studies of rare nuclei possessing large chemical
shift dispersions and anisotropies and methods relying on twodimensional correlations, will be reported in an upcoming
publication.
ACKNOWLEDGMENTS
This work was supported by the National High Magnetic
Field Laboratory through the National Science Foundation
Cooperative Agreement (No. DMR-0084173) and by the State
of Florida. This research was also funded by the Israel Science
Foundation Grant No. 795/13, by ERC Advanced Grant No.
246754, and by the generosity of the Perlman Family Foundation.
1E.
Kupce and R. Freeman, J. Magn. Reson., Ser. A 115, 273 (1995).
R. Bendall, J. Magn Reson., Ser A. 112, 126 (1995).
2M.
3T.-L. Hwang, M. Garwood, A. Tannús, and P. C. M. van Zill, J. Magn. Reson.,
Ser. A 121, 221 (1996).
Baum, R. Tycko, and A. Pines, Phys. Rev. A 32, 3435 (1985).
J. Ordidge, M. Wylezinska, J. W. Hugg, E. Butterworth, and F. Franconi,
Magn. Reson. Med. 36, 562 (1996).
6J. Shen, J. Magn Reson Ser B. 112, 131 (1996).
7D. Rosenfeld, S. L. Panfil, and Y. Zur, J. Magn. Reson. 129, 115 (1997).
8M. Garwood and L. delaBarre, J. Magn. Reson. 153, 155 (2001).
9M. S. Silver, R. I. Joseph, and D. I. Hoult, J. Magn. Reson. 59, 347 (1984).
10M. S. Silver, R. I. Joseph, and D. I. Hoult, Phys. Rev. A 31, 2753 (1985).
11J. Pfeuffer, I. Tkac, I.-Y. Choi, H. Merkle, K. Ugurbil, M. Garwood, and R.
Gruetter, Magn. Reson. Med. 41, 1077 (1999).
12H. Barfuss, H. Fischer, D. Hentschel, R. Ladebeck, A. Oppelt, and R. Wittig,
NMR Biomed. 3, 31 (1990).
13A. Abragam, Principles of Nuclear Magnetism (Oxford University Press,
Oxford, 1961).
14K. Riedel, C. Herbst, J. Leppert, O. Ohlenschläger, M. Görlach, and R.
Ramachandran, J. Biomol. NMR 35, 275 (2006).
15G. Kervern, G. Pintacuda, and L. Emsey, Chem. Phys. Lett. 435, 157 (2007).
16A. Pell, G. Kervern, L. Emsley, M. Deschamps, D. Massiot, P. J. Grandinetti,
and G. Pintacuda, J. Chem. Phys. 134, 024117 (2011).
17L. A. O’Dell and R. W. Schurko, Chem. Phys. Lett. 464, 97 (2008).
18A. W. MacGregor, L. A. O’Dell, and R. W. Schurko, J. Magn. Reson. 208,
103 (2011).
19L. A. O’Dell and R. W. Schurko, J. Am. Chem. Soc. 131, 6658 (2009).
20K. J. Harris, A. Lupulescu, B. E. G. Lucier, L. Frydman, and R. W. Schurko,
J. Magn. Reson. 224, 38 (2012).
21S. R. Hartmann and E. L. Hahn, Phys. Rev. 128, 2042 (1962).
4J.
5R.
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.76.50.5 On: Mon, 27 Apr 2015 08:36:16
064201-16
Wi et al.
J. Chem. Phys. 142, 064201 (2015)
22A.
35M.
23K.
36R.
Pines, M. G. Gibby, and J. S. Waugh, J. Chem. Phys. 59, 569 (1973).
J. Harris, S. L. Veinberg, C. R. Mireault, A. Lupulescu, L. Frydman, and
R. W. Schurko, Chem. Eur. J. 19, 16469 (2013).
24R. W. Schurko, Acc. Chem. Res. 46, 1985 (2013).
25B. E. G. Lucier, K. E. Johnston, W. Xu, J. C. Hanson, S. D. Senanayake, S.
Yao, M. W. Bourassa, M. Srebro, J. Autschbach, and R. W. Schurko, J. Am.
Chem. Soc. 136, 1333 (2014).
26E. O. Stejskal, J. Schaefer, and J. S. Waugh, J. Magn. Reson. 28, 105 (1977).
27J. Schaefer and E. O. Stejskal, J. Am. Chem. Soc. 98, 1031 (1976).
28R. L. Cook, C. H. Langford, R. Yamdagni, and C. M. Preston, Anal. Chem.
68, 3979 (1996).
29M. Sardashti and G. E. Maciel, J. Magn. Reson. 72, 467 (1987).
30R. Wind, S. F. Dec, H. Lock, and G. E. Maciel, J. Magn. Reson. 79, 136
(1988).
31B. H. Meier, Chem. Phys. Lett. 188, 201 (1992).
32X. Wu and K. W. Zilm, J. Magn. Reson., Ser. A 104, 154 (1993).
33I. R. Dixon, M. D. Bird, and J. R. Miller, IEEE Trans. Appl. Supercond. 16,
981 (2006).
34I. R. Dixon, T. Adkins, H. Ehmler, W. S. Marshall, and M. D. Bird, IEEE
Trans. Appl. Supercond. 23, 4300204 (2013).
M. Maricq and J. S. Waugh, J. Chem. Phys. 70, 3300 (1979).
Siegel, T. T. Nakashima, and R. E. Wasylishen, J. Magn. Reson. 184, 85
(2007).
37A. Tal and L. Frydman, Prog. Nucl. Magn. Reson. Spectrosc. 57, 241 (2010).
38See supplementary material at http://dx.doi.org/10.1063/1.4907206 for
brief description.
39A. G. Redfield, Phys. Rev. 98, 1787 (1955).
40Z. Gan, D. M. Grant, and R. R. Ernst, Chem. Phys. Lett. 254, 349 (1996).
41D. E. Demco, J. Tegenfelt, and J. S. Waugh, Phys. Rev. B 11, 4133 (1975).
42M. H. Levitt, J. Chem. Phys. 94, 30 (1991).
43F. G. Oas, R. G. Griffin, and M. H. Levitt, J. Chem. Phys. 89, 692 (1988).
44M. H. Levitt, T. G. Oas, and R. G. Griffin, Israel J. Chem. 28, 271 (1988).
45Z. Gan and D. M. Grant, Chem. Phys. Lett. 168, 304 (1990).
46V. B. Cheng, H. Henry, J. Suzukawa, and M. Wolfsberg, J. Chem. Phys. 59,
3992 (1973).
47B. M. Fung, A. K. Khitrin, and K. Ermolaev, J. Magn. Reson. 142, 97 (2000).
48S. Hediger, B. H. Meier, N. D. Kurur, G. Bodenhausen, and R. R. Ernst,
Chem. Phys. Lett. 223, 283 (1994).
49S. Hediger, B. H. Meier, and R. R. Ernst, Chem. Phys. Lett. 240, 449
(1995).
This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:
132.76.50.5 On: Mon, 27 Apr 2015 08:36:16