Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–50

Contents lists available at SciVerse ScienceDirect

Solid State Nuclear Magnetic Resonance

0926-20

doi:10.1

n Corr

E-m1 Th

journal homepage: www.elsevier.com/locate/ssnmr

Pulse-transient adapted C-symmetry pulse sequences

Johannes Weber, Marten Seemann, Jorn Schmedt auf der Gunne n

Ludwig-Maximilians-Universitat, Department Chemie, Butenandtstr. 5-13 (Haus D), 81377 Munchen, Germany

a r t i c l e i n f o

Article history:

Received 3 January 2012

Received in revised form

28 February 2012Available online 13 March 2012

Keywords:

NMR

Solid-state

Pulse transients

C-sequences

40/$ - see front matter & 2012 Elsevier Inc. A

016/j.ssnmr.2012.02.009

esponding author.

ail address: gunnej@cup.uni-muenchen.de (J.

e NMR probes heads here are considered to

a b s t r a c t

In solid-state NMR, pulse sequences make often use of pulses which are short compared to the

recovery time of the probe head. Especially, rotorsynchronized dipolar recoupling experiments under

magic-angle-spinning conditions can profit from the use of very high pulse amplitudes which in turn

will reduce the length of the individual pulses. In this contribution we show that C-symmetry based

pulse sequences used for double-quantum filtering experiments can strongly be influenced by pulse

transients. We analyze the origin of pulse transients and show that the quadrature component can be

minimized by cable-length variation which causes a mutual cancellation of probe-external and internal

contributions. We implement and test a model to investigate the influence of pulse-transients by

numerically exact calculations of the spin-density matrix allowing for composite pulses consisting of

slices which are short compared to the circuit recovery time-constant. Moreover we introduce a phase-

tuned C-element, which can be applied to g-encoded experiments from the C-symmetry class, to

reconstitute an almost ideal performance of the sequence. We have validated the modified transient-

adapted pulse sequences theoretically on the basis of numerically exact calculations of the spin-

dynamics. While comparably easy to apply the scheme proved also robust in practical application to15N, 13C and 31P double-quantum filtered experiments and leads to a significantly increased conversion

efficiency.

& 2012 Elsevier Inc. All rights reserved.

1. Introduction

Pulse transients occur in any real NMR spectrometer and referto the transient difference between the intended pulse-shape andthe pulse-shape which finally manipulates the nuclear spin-statesin the sample coil of the NMR probe head. For a simple seriesRLC circuit pulse transients become more pronounced the longerthe rise-/recovery-times of the probe circuit tR, which is givenby [1,2]

tR ¼2

Do¼

2Q

o0ð1Þ

where Do is its band-width, Q the quality factor [2] of the probe1

and o0 ¼ffiffiffiffiffiffiffiffiffiffiffi1=LC

pits natural frequency. Pulse transients can make

the real pulse deviate from the intended pulse-shape both inpulse amplitude and pulse phase, which generally leads to adegraded performance of pulse sequences.

The NMR community has long been aware of pulse transients.There are special probes for experiments, like the CRAMPS experi-ment [3], which are known to be sensitive to pulse transients.These probes are designed to have a lower quality factor Q

(Q ¼o0=Do) in order to shorten the rise-time tR of the electroniccircuits. However this is bought with a decreased signal-to-noise

ll rights reserved.

Schmedt auf der Gunne).

be series RLC circuits.

ratio which is correlated to the quality factor Q [2,4]. Thus mostcommercial probes follow the design principle of a high Q value.Pulse transients are known to potentially influence pulse sequenceswith long cyclic pulse trains [1,5] and need to be taken into accountif the individual piece-wise constant slices of a multiple-pulsesequence are short compared to tR. Clearly, modern sequencesdesigned through optimal control [6], double-frequency sweeps[7] or different windowless symmetry-based pulse sequences [8]are influenced much stronger by pulse transients than the simplespin-echo experiment or short windowed pulse-sequences likeBABA [9] because of fast amplitude and phase modulations beingused. The influence of pulse transients has been investigated fordifferent pulse sequences in detail. For the CRAMPS-type experi-ments it was shown that transients can be reduced [1,3,5,10,11] tosuch an extent that they are no longer the line-width limiting factor.For symmetry based experiments [8,12] the influence of pulse-transients [10,13,14] seems to depend on the actual conditions ofthe experiment. Here we want to investigate the influence of pulsetransients on specific C-symmetry based pulse sequences namelydouble-quantum filtered pulse sequences for which pulse transientscan have a strong influence [14,15].

How can the influence of pulse transients on pulse sequencesbe modeled in the computer? Any discontinuity in pulse phase,amplitude and frequency will cause pulse transients. For a probesolely consisting of linear passive electronic elements like capacitors,coils and resistors the time dependence can be predicted usinglinear time-invariant system (LTI) theory [16], which neglects

J. Weber et al. / Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–50 43

however important non-linear elements like power amplifiers anddiodes of the duplexer present in the signal pathway. Pulse sequenceswith a constant pulse frequency which show several discontinuitiesin pulse phase or/and pulse amplitude can then simply be describedby a superposition of the system response in the time domain causedby the individual discontinuities. Alternatively, an elegant solution isto apply the transient behavior to the intended pulse-shape byswitching from a time based description to another domain whichis generated by the Laplace transform [16,17]. Here we stick to adescription in the time domain. Several functions have beendescribed which are supposed to give the response of a simplesystem of a NMR probehead to a single pulse discontinuity[1,5,13,18]. These functions describe the response of a real systemwith a limited number of parameters.

Analytical functions to describe the response function to asingle discontinuity, split the pulse-transient into different terms/contributions, which can be given a physical meaning for simplenetworks. Components which have the same pulse phase as thesteady-state current to which they relate are called in-phase

contributions [1,5,18], while components which are off in pulse-phase are called quadrature contributions [1,5,18]. Moreover thereare contributions related to the ‘‘real’’ turn-off or turn-on pulse-phase, i.e. not the pulse-phase which is defined relative to thereceiver phase. The latter contribution may be neglected for ourprobe heads (Q 450) however because it scales with 1=ð4Q Þ [5,18].

Experimentally, the operator has several options at hand toinfluence pulse transients. It was soon realized that a number ofmethods exist to suppress the quadrature contribution, for exampleby tuning the system [19] with special tune-up sequences [20–22],by damping or removing reflections in the transmission lines to theprobe head with the help of circulators/isolators [3], high-powerattenuators or through varying the cable length.2 The in-phasecontribution may be influenced through the rise-time, i.e. thebandwidth, of the probe head. This may be achieved passively bylowering the Q-value simply by adding a resistor to the series RLCcircuit or more efficiently by switching between a high Q valueduring the acquisition of the free induction decay and a low Q valueduring the pulse [23–25]. Moreover active-feedback circuits [17]have been shown to almost completely cancel the effect of pulsetransients for both the in-phase and quadrature component.

The purpose of this contribution is to study the influence ofpulse transients on double-quantum filtered pulse-sequences, toimplement a simple scheme to model the effect of transients onmultiple-pulse sequences in a computer and to show how theperformance for double-quantum filtered experiments (PostC7[12,26]) can be reestablished both in theory and experimentwithout hardware modifications except for cable-length changes.Variations of the cable-lengths will be used to investigate underwhich circumstances cable-length effects will help to remove thequadrature component to pulse transients.

2. Pulse transients

2.1. Drive current and coil current

Real pulses on NMR spectrometers differ in many respectsfrom what an ideal radio-frequency pulse is supposed to look like.Transient deviations which are absent for indefinitely long pulsesare in short often termed ‘‘pulse transients’’. Pulse transients areexpected for any RLC tank circuit and hence for any NMR probe-head. Since sample magnetization is affected by the linearlypolarized oscillating magnetic field caused by the coil current

2 Personal communication by Hans Forster (Bruker Biospin).

I(t), the task is essentially to describe the transient deviations ofthe coil current relative to the current expected from the drivingvoltage. We will assume that the magnetic field Brf of a pulse isproportional to the coil current.

In the following description we restrict ourselves to multiplepulse sequences which do no involve a change of transmitterfrequency otr. If necessary, frequency changes can be described inthe same framework with the help of phase ramps. Because today’sspectrometers work on a digital basis, it is convenient to describepulses piecewise constant function of phase f, current amplitude A

and frequency otr , alternatively with the complex current ampli-tude [27] a ¼ ðareþ iaimÞ, where are, aim, A and f are real.

IðtÞ ¼ A cosðotrtþfÞ ¼ RefAeifeiotrtg ¼ Refðareþ iaimÞeiotrtg ð2Þ

The complex notation with the imaginary unit i2 ¼�1, hasthe advantage that it is easy to extend to multiple-pulse sequences,because it allows the superposition of sinusoidal functions withdifferent phases and amplitudes (see below). The variables are, aim,A and f can be functions of time, while otr is time independent.

In the following part this notation will be used to describepulse phase and amplitude as a function of time.

2.2. Modeling pulse transients

Different models have been proposed in the literature [1,18,28]to describe the time dependent response of phase and amplitudeafter a single junction/discontinuity between two indefinitely longpulses m and n.

Here we chose a model related to the one by Mehring andWaugh [1,5], which is based on the LTI analysis of a tank circuitclose to its natural oscillation frequency o0 [1]. Our simplifiedversion describes transients of a series LCR network with only twoparameters: a recovery time constant tR as defined in Eq. (1)and an ‘‘electronic’’ offset frequency oeoff , which is the differencebetween the ‘‘free ringing’’ [5] (or resonance frequency [1]) ofthe circuit or and the transmitter frequency otr of the radiofrequency (rf) pulse. This model neglects effects of the turn-offand turn-on phase as explained in the Introduction section. Thecomplex current amplitude a

totalðtÞ at the time t may be decom-

posed into (i) a decaying, transient contribution Td influenced bythe previous segment with phase fm and amplitude Am and (ii) arising, contribution TrþAn expðifnÞ of the current segment whichconsists of a pulse transient Tr with phase fn and amplitude An

and the non-transient contribution of the active pulse An expðifnÞ,where t refers to a time point after the discontinuity at time tmn inphase and/or amplitude, i.e. t4tmn.

atotalðtÞ ¼ Tdðfm,Am,tmn,tÞþTrðfn,An,tmn,tÞþAn expðifnÞ ð3Þ

The rising and the decaying contribution can be split up into apart in-phase with the pulse-phase and a part out of phase by p=2(in quadrature).

Tdðfm,Am,tmn,tÞ ¼ Am exp �t�tmn

tR

� �cosðft�tmngoeoff ÞexpðifmÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

in phase

þAm exp �t�tmn

tR

� �sinðft�tmngoeoff Þexp i fmþ

p2

n o� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

quadrature

ð4Þ

Trðfn,An,tmn,tÞ ¼ �An exp �t�tmn

tR

� �cosðft�tmngoeoff ÞexpðifnÞ|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}in phase

�An exp �t�tmn

tR

� �sinðft�tmngoeoff Þexp i fnþ

p2

n o� �|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

quadrature

ð5Þ

J. Weber et al. / Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–5044

Finally we get

atotalðtÞ ¼ exp �

t�tmn

tR

� �expðift�tmngoeoff Þ

�½Am expðifmÞ�An expðifnÞ�þAn expðifnÞ ð6Þ

The observed transient behavior is calculated from the real part ofthe complex current a

totalðtÞ. As expected transient contributions

(first term in Eq. (6)) vanish if fm�fn ¼ 0 and Am�An ¼ 0. Thenon-zero factor ½Am expðifmÞ�An expðifnÞ� thus is what drives apulse-transient into existence.

Note that the original Mehring–Waugh model predicts oscilla-tions in amplitude and phase (Fig. 1) [28] which are absent inrecently proposed models [13,18]. Despite the chosen modelbeing closely related to the original Mehring–Waugh model wedeliberately use it to describe pulse transients in more compli-cated networks. The price we pay is that the probe parameters tR

and oeoff loose their physical meaning and become empiricalparameters to describe a real probe, including filters, amplifierand duplexer.

2.2.1. Pulse transients in multiple-pulse sequences

The above model described for a single discontinuity will nowbe extended to the description of transients in a multiple-pulsesequence, that consists of k pulses. Again, a complex current

volta

ge/a

.u.

time/μs

phas

e/de

gree

(4μs)45° (4μs)225° (4μs)135° (4μs)315°

1 75

840

1

0

0°

100°

200°

300°

12 16

time/μs840 12 16

2 3 4 6 8

Fig. 1. Phase and amplitude transients of a windowless eight-pulse sequence

measured with a pickup coil close to the sample-coil of a probehead (see text for

details); dashed lines refer to a probe tuned to the minimum of the scattering

parameter spectrum S11ðoÞ; solid and dotted lines refer to a the probe tuned and

matched with an offset of 0.41 MHz and 0.91 MHz, respectively.

atotalðtÞ is to be calculated which describes the amplitude and

phase during the last pulse (t4tkðkþ1Þ). The transient is thenexpected to be a simple superposition of transients caused by anydiscontinuity in phase and/or amplitude prior to the time t forwhich the complex current is to be calculated, i.e.

atotalðtÞ ¼ Ak expðifkÞþ

Xk�1

p ¼ 1

½Tdðfp,Ap,tpðpþ1Þ,tÞ

þTrðfpþ1,Apþ1,tpðpþ1Þ,tÞ� ð7Þ

2.3. Validation of the transient model

How well is the simplified Mehring–Waugh model able todescribe the pulse transients in a real spectrometer? Is it possibleto describe phase and amplitude transients well enough topredict their effect onto a complex pulse-sequence numerically?To this end we have validated the model in different ways:

(A)

Measurement with a small pickup coil positioned close to thesample coil.

(B)

Fitting of experimental curves with the presented model and (C) Numerical calculation of the coil current neglecting all non-

linear effects caused e.g. by diodes and amplifiers.

(A) The current in the sample coil is relatively big, hence it willinduce a voltage into a small pick up coil [2] positioned nearbywhich serves as a measure of the coil current and can be detectedwith the help of an oscilloscope. Ideally pulse transients can bemonitored experimentally with high time resolution with a pickup coil. However care has to be taken with respect to distortionswhich are caused by cables, mixers, filters and the oscilloscopeitself used for the detection of the induced voltage. Significantdistortions are already observed due to stray capacities if the pickup coil is placed close to the grounded outer probehead tube.

For this reason we have used an open probehead with removedouter tube, a 6–9 pF passive probe connected to a 500 MHzoscilloscope at 1 MO and a straight 6 cm copper wire as antenna.We have tested the bandwidth of the rf generator of the NMRconsole and the bandwidth of the oscilloscope by feeding the lowvoltage signal directly into the oscilloscope. The observed pulsetransients have a duration of approximately 50 ns. If the linear rftransistor amplifier and a 30 dB resistor are added, the transientsdo not become significantly longer. This setup results in a datasetof voltage versus time. The conversion from voltage-time to adataset phase-amplitude-time is an under-determined problem. Inorder to obtain experimental values for current phase and amplitudeversus time, we have chosen the dwell time of the oscilloscope forvalidation as 2p=ð4otrÞ, so one experimental data point every 901will be collected. Since phase and amplitude of the coil currentchange slowly in comparison with the time scale set by 2p=otr , thedata set can be converted to a series of phase and amplitude valuessampled in steps of 2p=ð4otrÞ using Eqs. (8) and (9). Therein a singlepair of phase and amplitude values at the time t is calculated fromthree experimental values of the real part of the current Refa

totalðtÞg

measured at t, t�1=ð4otrÞ and tþ1=ð4otrÞ.

fffiarctanRe a

totalt� 1

4otr

� �n o�Re a

totaltþ 1

4otr

� �n o2Refa

totalðtÞg

0@

1A�otrt ð8Þ

Affi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRefa

totalðtÞg2þ

Re atotal

t� 14otr

� �n o�Re a

totaltþ 1

4otr

� �n o2

0@

1A

2vuuut ð9Þ

Since this data analysis averages over a time period of 1=ð2otrÞ,processing artifacts are to be expected when changes are no

J. Weber et al. / Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–50 45

longer slow compared to the oscillation frequency otr . In ourexperience artifacts can be neglected for changes slower than1=ð4otrÞ.

Experimentally we have investigated the transient behaviorwith a windowless pulse train of eight concatenated pulsesð4 msÞ4512ð4 msÞ22512ð4 msÞ13512ð4 msÞ31512ð0:4 msÞ4512ð0:4 msÞ13512

ð0:4 msÞ22512ð0:4 msÞ3151 (notation: ðpulse lengthÞphase) for the fol-lowing three cases, with the condition that the transmitterfrequency and the frequency of the NMR transition are the same.

1.

TabPar

fitti

o

0

þ

þ

The probe was tuned and matched, so that the scattering-parameter spectrum S11ðoÞ possesses a minimum at thetransmitter frequency otr .

2.

As in 1 but minimum at otrþ0:41 MHz 2p. 3.

time/μs

phas

e/de

gree

volta

ge/a

.u.

0°

100°

200°

300°

1612840

1

As in 1 but minimum at otrþ0:91 MHz 2p.

Due to the increased detuning of the probe more power is beingreflected by the probe. A pickup-coil was used to monitor thesample-coil current (Fig. 1). Significant changes are observeddepending on the amount of detuning which indicates that on atypical commercial spectrometer the generation of pulse transi-ents is dominated by the probehead and not by the rf generationand amplification. The transients have short time constants tR

below 1 ms. Clearly, oscillations of phase and amplitude arepresent, which are predicted by the chosen simplified Mehring–Waugh model. Since the driving force of a transient is the factor½Am expðifmÞ�An expðifnÞ� in Eq. (3), it depends on the phase shiftat the junction. Therefore the amplitude transients starting at thebeginning of the second and fourth pulse look alike.

Transients may constructively or destructively interfere withthe intended ideal pulse and may diminish the effective ampli-tude of a pulse. If strong pulses during a pulse sequence areneeded to control interactions like the dipole–dipole coupling thestrength of the rf field, i.e. its amplitude, should not fall below acertain value, because the evolution of the spin dynamics wouldthen be dominated by the internal interactions and not the pulse.

To observe a cumulative effect of transients, the pulse trainfinishes with four pulses of short duration compared to the circuitrecovery time tR. Since the pulse phase of the four pulses has beenchosen such that it is shifted by 901 to the phase of the pulsebefore, any deviation in the amplitude plot between the amplitudetransients starting at time points 5–8 (Fig. 1) can be ascribed tothe cumulative effect of several transients as expressed in Eq. (7).

(B) In order to validate the transient model (Eq. (3)) theexperimental transient curves were fitted with tR and oeoff asthe only free parameters (see Table 1). Corrections for linearchanges in phase and amplitude were applied to the experimentaldata prior to fitting. We obtain good agreement between fittedand experimental curves (Fig. 2), especially for small offsetfrequencies oeoff , which are typical for experimental conditions.Deviations can be explained by the contributions of non-linearelectronic elements in the oscilloscope, preamplifier or the poweramplifiers. As previously shown [19] a probe deliberately mis-matched and tuned off-resonance offers full control of the offsetfrequency oeoff . Consequently tuning offset and offset frequencycorrelate (see Table 1).

le 1ameters tR and oeoff at a transmitter frequency of 83.333333 MHz obtained by

ng the response of a probe to a pulse sequence as shown in Fig. 1.

R�otr (MHz) tR ðmsÞ oeoff

2p (kHz)

(on resonance) 0.49 �172

0.41 0.30 þ607

0.91 0.26 þ1184

When the rise-time tR is known, the frequency oeoff can bedetermined experimentally with a pulse-sequence of consecutiveð18002180180Þn-FID pulses. In a plot of signal amplitude versus n asinusoidal modulation is observed. Its modulation frequency is afunction of oeoff . If oeoff is zero, the frequency of the sinusoidalmodulation is zero and no signal is observed. This sequence mayalso be used to find a suitable cable-length so that oeoff is zero.

(C) The transient response of an arbitrary network of passivelinear components can be simulated with programs like GNUCAP[28,29]. For simulations we used a simplified circuit diagram of ourdouble-resonance probe, taking into account only the X-channel(Fig. 3) and assuming it is directly grounded on one side of thesample coil Lsample. The 1H-frequency trap still remains in theX-branch. Fixed inductances and capacities can be estimated fromthe used hardware, while the matching inductance Lmatch, the tuningcapacity Ctune and the resistance R were chosen such that theexperimental spectrum of the scattering parameter is reproducedas good as possible by the numerically calculated one (diagram notshown).

For the calculation of the transient behavior we assumed aperfect voltage source at 50 O impedance. Calculation for a pulsesequence of two short concatenated pulses allows to study theeffect of pulse transients (Fig. 4). We get almost perfect overlapbetween the fitted transient model (Eq. (3)) and calculated curveseven at big tuning offsets. The determined parameters tR andoeoff have similar values as the ones found in the experiments.For the probe tuned on-resonance at 83.333333 MHz the modelparameters tR and oeoff =2p are 0:40 ms and �23 kHz, respec-tively, which have the same order of magnitude as the experi-mental values measured with a pick up coil on an open probe.

time/μs

01612 80 4

Fig. 2. Phase and amplitude transients of a windowless eight-pulse sequence

measured with a pickup coil close to the sample-coil of a open probehead (grey

solid line, see text) of a probe tuned on the magnitude of the scattering parameter;

the dashed line shows the best fit using Eqs. (7)–(9).

CtuneLsample

Ltrap

Ctrap

Lmatch

Rcoaxialconnector

Fig. 3. Simplified equivalent circuit diagram for the X-channel branch of the used

double-resonance NMR probe, which was used for transient calculations.

time/μs

phas

e/de

gree

ampl

itude

/a.u

.

0

0.01

0.02

0.03

0

100

200

300

0 1 2 3 4 5

0 1 2 3 4 5

Fig. 4. Calculated pulse transients from the electric response of the circuit

described in Fig. 3 to a two-pulse sequence with 2:4 ms length and shifted by

1801; upper graph: phase, lower graph: amplitude.

probe headpulse P reflection R1

sample coilcurrent

point of im

pedance

mismatch

pulsesource

reflection R2

t

I

t

I

t

I

t

I

Fig. 5. Reflections illustrated for a simplified circuit including a probehead.

J. Weber et al. / Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–5046

We were able to verify [1] that a good estimate for tR is availablefrom the probe band-width Dn measured at the �3 dB line.

The second model parameter, the offset frequency oeoff , isfound to be sensitive to cable-length effects in our setup. Cable-length effects are not expected in a system where the componentsamplifier, cables and duplexer have a characteristic impedance ofexactly 50 O. Numerical simulations indicate that in such an idealsystem the parameter oeoff is much smaller than what we observein our system. Therefore we took into account other sources thanonly the probe-internal electronic circuits [3], namely the impe-dance mismatch in the duplexer and the power amplifier. Theimpedance mismatch caused by imperfect coaxial cables wasfound to be negligible. Numerical simulations show how reflec-tions which originate at a specific point of impedance mismatchcause quadrature transients which depend on the cable lengthlcable between the point-of-impedance-mismatch to the probe

(Fig. 5): The ideal pulse P reaches the probe after passing thoughthe cable. The probe initially will reflect part of the pulse becauseit has a finite rise-time tR. Therefore one part of the wave isconsumed by the probe circuits and a second part R1 is reflected.When R1 propagates back through the cable it is partiallyreflected at the point-of-impedance-mismatch, which causesanother pulse package R2 traveling in the direction of the probeagain. When R2 has reached the probe it will have picked upa phase shift fcable of 2 � lcable=lrf , where lrf � ð2p � cÞ=otr is thewavelength of the radio-frequency pulse and c the speed of light.With the typical cable lengths found in a commercial spectro-meter, all reflections and the original pulse are superimposedtypically within less than 0:04 ms to form the sample coil current,which indicates that the simplified Mehring–Waugh model is stilla good approximation of the transient behavior of the NMRpulses. If the reflections at the point-of-mismatch are bigger thanthe probe-internal quadrature transients [5], then a suitable cablelength lcable can be found so that the quadrature transients andoeoff become zero.

As expected, detuning the probe to þ0.41 MHz and þ0.91 MHzoff the transmitter frequency affects the parameter oeoff =2p (Table 1)which is shifted to higher values by 0.41 and 0.92 MHz, respectively.Note that oeoff can be controlled by the frequency to which the probeis tuned. In order to calculate arbitrary multiple-pulse sequencesnumerically exact we have implemented Eq. (7) as a Tcl scriptso it can be used with SIMPSON [30]. We conclude that thesimplified model (Eq. (3)) with two parameters tR and oeoff issufficient to describe the calculated results even at moderatetuning or matching offsets.

3. Transients and C-based pulse-sequences

In general, a robust pulse sequence is thought to be insensitivetowards unwanted interactions and insensitive towards pulseerrors. Pulse transients are often considered as negligible, which –depending on the probe – may be true. However during a cyclicpulse sequence even small pulse-non-idealities can lead tosignificant errors. As an example, we have chosen the popularpulse sequence PostC7 [12,26] which – under the chosen condi-tions (vide infra) – should provide close to ideal suppression ofthe chemical shift interaction and very efficient double-quantumexcitation [14]. All the presented results refer to 31P double-quantum coherence generation under magic-angle-spinning oncrystalline Ag7P3S11 however similar experiments have beenperformed for 15N, 13C and 1H (not shown).

3.1. PostC7 as an example for a C-symmetry sequence

PostC7 [26], alias C712 with a Post-C element, is a g-encoded

pulse sequence to convert zero-quantum (ZQ) to double-quantum(DQ) coherence, which allows a DQ filter with a theoretical

J. Weber et al. / Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–50 47

efficiency of approx. 73% [8]. Non-g-encoded pulse sequenceshave only 50% theoretical efficiency [8]. Pulse sequences from thesame family of CNn

n pulse sequences are able suppress the chemicalshift interaction reliably to measure even small dipole–dipolecouplings [12,14,26]. They consist of a pulse block used for two-spinDQ coherence excitation, followed by a pulse block to reconvert tozero-quantum coherence and they are terminated by a single read-pulse with 901 flip-angle after which the FID is detected (see Fig. 6).Both DQ excitation and reconversion use the same rotor-synchro-nized multiple-pulse sequence from the C-symmetry library [8],whose appearance is determined by the three symmetry numbersN¼7, n¼2, n¼ 1 (for C71

2), the C-element C¼ 900236018022700

(the Post-C element, notation: flip anglephase) and a supercycle whichrepeats the C-cycle with all phases shifted by 1801 [31], which is alsocalled p-shifted cycle [32]. The duration of excitation tDQ ,exc andreconversion tDQ ,recon can be controlled by repeating the PostC7sequence several times. Similar sequences from the C-symmetrylibrary use the same C-element and supercycle, but differentsymmetry numbers N, n and n. A cogwheel Cog12(2,5,6;0) phasecycle [14,33] is used to select the coherence order pathwaysp¼ f0-2-0-�1; 0-�2-0-�1g.

It has been noted before [14] that pulse transients influence theDQ filtered intensity EDQ (defined as DQ filtered peak area overpeak area of a direct excitation spectrum under similar conditions)

36018090φ

+2+1

0−1−2

p

CNnν

π/2

FID

time

phase−tuned C−element

τDQ,exc

2700

τDQ,recon

Fig. 6. Scheme of a transient-adapted CNnn sequence with a phase tunable

C-element; the coherence transfer pathway indicates which coherence orders p

are selected by the phase cycling scheme.

x

y

−1−1−1

0

1

0 1

Fig. 7. Projections of simulated single-spin magnetization trajectories onto the XY-pla

complete on-supercycled C-cycles; ideal (left), including pulse transients (middle), inc

were performed for a 2-spin system with a dipole coupling ndip of �436 Hz, an tR of 0

0.47931; note that the second and third cycles for phase-adapted PostC7 overlap with

of PostC7, which becomes evident from a plot of EDQ as a functionof the nutation frequency nrf (¼radio frequency amplitude, definedas the inverse of the length of a pulse with 3601 flip angle).Consequently, also DQ build-up curves deviate from the expectedideal behavior. Long conversion periods are necessary to determineweak couplings, which makes the experiment sensitive to smallpulse errors. As can be seen in Fig. 8 pulse transients seriouslyreduce the conversion efficiency and even may cause negativevalues for build-up curves which have been acquired with asymmetric protocol (tDQ ,exc ¼ tDQ ,recon) [14]. Neither of these effectsis desired. Because the numerically exact spin-dynamics simula-tion (Fig. 8) took into account only the homonuclear through-spacedipole–dipole interaction, we conclude, that the chemical shift isnot necessary for pulse transients to spoil the performance ofPostC7.

How will pulse transients influence the PostC7 sequence?C-sequences are built on the premises that the pulses form acycle. The magnetization of a single spin-1/2 which is subject to aPostC7 sequence should therefore move on a trajectory [34]which repeats after each full C-cycle (see Fig. 7). This is no longertrue when pulse transients are considered. The effective pulsefield at the sample no longer consists of piecewise definedfunctions of constant pulse phase and amplitude but consistsrather of continuous functions which have a non-trivial influenceon the single-spin magnetization trajectory. Numerical simula-tions indicate that the C-cycle looses its cyclicity when pulsetransients are active which causes the observed degradation ofthe pulse-sequence (see Fig. 6).

Quadrature transients are under the control of the spectro-meter operator. With the help of the above mentioned pulsesequence ð18002180180Þn-FID and a line-stretcher we adjustedthe cable length lcable to make the quadrature transients vanish(oeoff ¼ 0). The DQ excitation curve improves a little bit but stillsignificant deviations from the expected response are observed(figure not shown). We conclude that pulse transients may causehigh losses in simple DQ experiments.

3.2. g-Encoded transient adapted C-sequences

In the following we describe how g-encoded C-sequences canbe modified to significantly reduce the effect of pulse transients.The simple idea is to use a modified C-element which tries tocompensate the pulse transients in the sense that a single-spinpolarization vector in a Bloch picture performs rotations close toan ideal trajectory without pulse transients. A consequence of this

x x1 10 0−1

ne of the rotating frame during a PostC7 sequence for an evolution up to three

luding pulse transient but using transient-adapted PostC7 (right); the simulations

:49 ms and an oeoff =ð2pÞ of �68.7 kHz; the phase z in phase-adapted PostC7 was

the first cycle, while the trajectories for ordinary PostC7 sequence fan out.

τDQ/s

E DQ

0

0.2

0.4

0.6

0.8

1

−0.20 0.002 0.004 0.006 0.008 0.010

τDQ/s0 0.002 0.004 0.006 0.008 0.010

Fig. 8. Double-quantum filtered intensity EDQ for a supercycled PostC7 sequence as a function of the conversion time tDQ ; simulations (right): ideal behavior from a

numerical simulation without pulse transients (solid black line), including pulse transients (dashed curve), including pulse transient but using transient-adapted PostC7

(dotted curve); experiment (left): obtained by PostC7 (filled squares), obtained by transient-adapted PostC7 (open squares); almost perfect agreement between simulated

and experimental curve of transient-adapted PostC7 can be obtained by including a monoexponential damping factor for the simulations (not shown).

ζ/°

E DQ

−3−1

0

1

−2 −1 0 1 2 3

Fig. 9. Double-quantum filtered intensity EDQ for a phase-adapted supercycled

PostC7 experiment as a function of the phase z (‘‘phase-tuning plot’’) with a

constant conversion time tDQ of 2 ms; experimental values (filled circles) versus

simulated curves (solid line) at a sample spinning frequency of nr ¼ 10 kHz; the

simulation was performed for a 2-spin system with a dipole coupling ndip of

�436 Hz and tR of 0:49 ms; the offset frequency oeoff and the scaling factor for EDQ

were adjusted to obtain an overlap with the experiment data points.

J. Weber et al. / Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–5048

approach is that experiments done with the transient adaptedC-sequences can be analyzed as usual. For this purpose the Post-Celement is replaced by a ‘‘phase-tuned’’ element, C¼ 90z23601802

2700, whose phase z is optimized once for a given setup. It sharessome similarities with the ‘‘phase-tuned’’ R-element which hasbeen proposed by Carravetta et al. [13] in order to compensate forchemical shift effects. In our case z typically takes optimal valuesbetween �31 and þ31. Optimization is achieved by finding thevalues of z for which the intensity of the DQ filtered signal ismaximum. A plot of the DQ-filtered intensity against phase z(‘‘phase-tuning-plot’’) is displayed in Fig. 9. The same shape isobserved for all experiments we have performed (1H, 15N, 13C, and31P), also in higher magnetic fields like 13C2-glycine in a magneticfield of 11.74 T (see Supporting information, Fig. S1 and S2) andalso for a numerical simulation on the basis of the transientformula described above. Simulations with varying offset frequencyoeoff indicate that oeoff causes a significant shift of the optimumphase z and also a small change in conversion efficiency. For thisreason it is a good idea to minimize oeoff by cable-length effects asdescribed above. With the described phase-tuned C-element wehave been able to improve the DQ filtered intensity in some cases

from 0% to reasonable values (15N NMR of NH4Cl), in cases of 31PNMR we often find twice the amount of signal intensity afteroptimization depending on the rise time of the probe. On the otherhand, 1H NMR experiments performed on the same probe, howeveron a separate channel dedicated to the 1H frequency, hardly showany influence of quadrature pulse transients, possibly because it iseasier to match impedance over the complete line of used hardware(amplifier – preamplifier – probe) than for an ‘‘X’’ channel experi-ment which has to cover a wider frequency range. Consequentlyfewer reflections contribute to the generation of quadrature tran-sients. Moreover we have experimentally verified that the approachalso works for other double-quantum sequences from the C-sym-metry library (e.g. C73

1, C941, C115

1, etc.) [35]. It should be noted thatthe sensitivity of g-encoded C-sequences towards pulse-transientsincreases the longer the conversion times tDQ and the higher thenecessary radio-frequency amplitudes nrf are. Therefore transient-adapted PostC7 shows a sharper maximum in the phase-tuning plotat 20 kHz than at 10 kHz (Fig. 9) sample spinning frequency.

The impact of a phase-tuned C-element is visualized with thehelp of single-spin polarization trajectories (Fig. 7). The phase-tuned C-element restores the cyclicity of the full C-cycles, so thatthe trajectories of several consecutive C-cycles exactly overlap andthe relevant single-spin interaction, the chemical shift, is sup-pressed. We note that the simulated and the experimental phase-tuning plot are in reasonable agreement which indicates that thedesign of pulse-transient compensated pulse sequences on thebasis of the above-mentioned formula is possible in general.Simulations indicate that the 1801-supercycle on a full C-cyclereduces the influence on transients, but the supercycle is not ableto compensate pulse-transients as the phase-tuned C-element is.

4. Experimental

4.1. NMR

The 31P NMR experiments were carried out on a Bruker Avance-IINMR spectrometer equipped with a commercial 2.5 mm double-resonance (H/X) MAS-NMR probe. The magnetic field strength was4.7 T corresponding to a resonance frequency of nð31PÞ ¼ 81:1 MHz.Samples were rotated within zirconia spinners. A commerciallyavailable pneumatic control unit was used to limit MAS frequencyvariations to a 2 Hz interval for the duration of the experiment. Forcrystalline Ag7P3S11 the spinning frequency nMAS was set to 10 kHzwhen not mentioned otherwise. During the PostC7 sequence this

J. Weber et al. / Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–50 49

required pulse nutation frequencies nrf of 70 kHz. A saturation combwas used to erase the phase memory of the spins, since the usedrepetition delays of 10–32 s were too short with respect to spin-lattice relaxation times. The saturation comb was used in front ofevery scan and typically consisted of 10–20 pulses with a 901 flip-angle and a delay of 50 ms. Radiofrequency pulses were generatedon an SGU600 board of the Avance-II console and amplified througha Bruker BLAX500 transistor amplifier (6–365 MHz bandwidth, classAB). The reflections at the output of the power amplifier proved tobe important for controlling the quadrature component of the pulsetransients.

The numerical simulations of the spin-dynamics were doneusing the SIMPSON NMR interpreter by Nielsen and co-workers[30,36]. Powder averages were chosen according to the Zaremba–Conroy–Wolfsberg scheme [37] with a number of 1760 orienta-tions (88a-, b-angle-pairs times 20g-angles) or better, to achieveconvergence of the powder average. The integration time step forthe DQ simulations was chosen as 1/357th of the shortest rf-unitin the pulse sequence, i.e. 10 ns or smaller. Numerically exactpulse transient simulations were performed with a procedureimplementing the formula described Eq. (7) with the programSIMPSON in the scripting language Tcl (compatible with version8.4 and 8.5). The transient procedure allows the calculation ofcyclically used pulse blocks and used a cut-off delay of 6 ms,which is the longest period up to which a discontinuity wasassumed to have an influence on pulse amplitude and pulse phaseafter its occurrence.

4.2. Electronics

The parameters for the equivalent circuit diagram in Fig. 3 werechosen as follows: Ctune ¼ 22:11 pF, Lmatch ¼ 8:95 nH, Ctrap ¼ 18:2 pF,Ltrap ¼ 34:8 nH, Lsample ¼ 114 nH, R¼ 0:435 O. The resistance R wasdetermined by adjusting R until an experimental scattering para-meter spectrum S11ðoÞ of the probe measured on a network analyzeris reproduced by simulation. The band width was determined fromthe �3 dB line in S11ðoÞ-spectrum [2]. Coarse cable length adjust-ments were realized with self-assembled coaxial-cables using iden-tical type-N connectors which were crimped onto RG214 cables ofvariable length. Fine adjustments were implemented with the help ofa trombone-type line-stretcher (Microlab/FXR ST-05N) at almostconstant impedance of 50 O.

The sample coil currents were monitored through mutual-inductive coupling a small pickup coil (approx. 6 cm copper wire)connected with a passive probe (HP 1160A) connected to a digitaloscilloscope (Hewlett Packard Infinium 54280A oscilloscope,500 MHz, 2 GSa/s).

Simulations of scattering parameter spectra were achieved withthe program ViPEC [38]. Calculations of transients were executedwith the program GNUCAP [29, version 0.35]. Scattering parametermeasurements were performed with an Agilent 8712ES RF vectornetwork analyzer.

5. Conclusions

In this contribution we have shown that C-symmetry based pulsesequences used for double-quantum filtering experiments undertypical experimental conditions are influenced by pulse amplitudeand phase transients. Based on a simple model for the description ofpulse transients we were able to develop a phase-tuned C-elementwhich can be applied to g-encoded C-sequences to reconstitutean almost theoretical performance. The chosen rather simpleapproach shows a tremendous increase in the experimentallyobserved double-quantum efficiencies which may make it use-ful for double-quantum filtered experiments where C-sequences

have their domain, i.e. small dipole–dipole couplings and bigshielding tensors. Especially for small sample rotor diameters(r2:5 mm) and low Larmor frequencies we have found that notthe pulse power but the pulse transients were the bottle neck forthe performance of the pulse sequences which can be overcomewith phase-adapted supercycled C-sequences.

Acknowledgments

Financial support by the Deutsche Forschungsgemeinschaftthrough the Emmy-Noether program (SCHM1570/2-1) is grate-fully acknowledged.

Appendix A. Supporting information

Supplementary data associated with this article can be foundin the online version at doi:10.1016/j.ssnmr.2012.02.009.

References

[1] J.D. Ellett, M.G. Gibby, U. Haeberlen, L.M. Huber, M. Mehring, A. Pines,J.S. Waugh, Spectrometers for multiple-pulse NMR, Adv. Magn. Reson. 5(1971) 117–176.

[2] J. Mispelter, M. Lupu, A. Briguet, NMR Probeheads for Biophysical andBiomedical Experiments: Theoretical Principles & Practical Guidelines,Imperial College Press, London, 2006.

[3] R. Prigl, U. Haeberlen, The theoretical and practical limits of resolution inmultiple-pulse high-resolution NMR of solids, Adv. Magn. Opt. Reson. 19(1996) 1–58.

[4] D.I. Hoult, R.E. Richards, The signal-to-noise ratio of the nuclear magneticresonance experiment, J. Magn. Reson. 24 (1976) 71–85.

[5] M. Mehring, J.S. Waugh, Phase transients in pulsed NMR spectrometers, Rev.Sci. Instrum. 43 (1972) 649–653.

[6] C.T. Kehlet, A.C. Sivertsen, M. Bjerring, T.O. Reiss, N. Khaneja, S.J. Glaser,N.C. Nielsen, Improving solid-state NMR dipolar recoupling by optimalcontrol, J. Am. Chem. Soc. 126 (2004) 10202–10203.

[7] D. Iuga, H. Schafer, R. Verhagen, A.P.M. Kentgens, Population and coherencetransfer induced by double frequency sweeps in half-integer quadrupolarspin systems, J. Magn. Reson. 147 (2000) 192–209.

[8] M.H. Levitt, Symmetry-based pulse sequence in magic-angle spinning solid-state NMR, Encyc. NMR 9 (2002) 165–196.

[9] M. Feike, D.E. Demco, R. Graf, J. Gottwald, S. Hafner, H.W. Spiess, Broadbandmultiple-quantum NMR spectroscopy, J. Magn. Reson. A 122 (1996) 214–221.

[10] L. Bosman, P. Madhu, S. Vega, andE. Vinogradov, Improvement of homo-nuclear dipolar decoupling sequences in solid-state nuclear magnetic reso-nance utilising radiofrequency imperfections, J. Magn. Reson. 169 (2004)39–48.

[11] M. Leskes, P.K. Madhu, S. Vega, Proton line narrowing in solid-statenuclear magnetic resonance: new insights from windowed phase-modulatedLee-Goldburg sequence, J. Chem. Phys. 125 (2006) 124506.

[12] Y.K. Lee, N.D. Kurur, M. Helmle, O.G. Johannessen, N.C. Nielsen, M.H. Levitt,Efficient dipolar recoupling in the NMR of rotating solids. A sevenfoldsymmetric radiofrequency pulse sequence, Chem. Phys. Lett. 242 (1995)304–309.

[13] M. Carravetta, M. Eden, O.G. Johannessen, H. Luthman, P.J.E. Verdegem,J. Lugtenburg, A. Sebald, M.H. Levitt, Estimation of carbon–carbon bondlengths and medium-range internuclear: distances by solid-state nuclearmagnetic resonance, J. Am. Chem. Soc. 123 (2001) 10628–10638.

[14] J. Schmedt auf der Gunne, Distance measurements in spin-1/2 systems by 13Cand 31P solid-state NMR in dense dipolar networks, J. Magn. Reson. 165(2003) 18–32.

[15] J. Schmedt auf der Gunne, Effective dipolar couplings determined by dipolardephasing of double-quantum coherences, J. Magn. Reson. 180 (2006)186–196.

[16] T. Frey, M. Bossert, Signal- und Systemtheorie, first ed., Teubner, Stuttgart,Leipzig, Wiesbaden, 2004.

[17] Y. Tabuchi, M. Negoro, K. Takeda, M. Kitagawa, Total compensation of pulsetransients inside a resonator, J. Magn. Reson. 204 (2010) 327–332.

[18] A.J. Vega, Controlling the effects of pulse transients and RF inhomogeneity inphase-modulated multiple-pulse sequences for homonuclear decoupling insolid-state proton NMR, J. Magn. Reson. 170 (2004) 22–41.

[19] G.C. Chingas, Overcoupling NMR probes to improve transient response,J. Magn. Reson. 54 (1983) 153–157.

[20] D.P. Burum, M. Linder, R.R. Ernst, A new ‘‘tune-up’’ NMR pulse cycle forminimizing and characterizing phase transients, J. Magn. Reson. 43 (1981)463–471.

J. Weber et al. / Solid State Nuclear Magnetic Resonance 43–44 (2012) 42–5050

[21] W.K. Rhim, D.D. Elleman, L.B. Schreiber, R.W. Vaughan, Analysis of multiplepulse NMR in solids. II, J. Chem. Phys. 60 (1974) 4595–4604.

[22] R.W. Vaughan, D.D. Elleman, L.M. Stacey, W.K. Rhim, J.W. Lee, A simple, lowpower, multiple pulse NMR spectrometer, Rev. Sci. Instrum. 43 (1972) 1356.

[23] J.J. Spokas, Means of reducing ringing times in pulsed nuclear magneticresonance, Rev. Sci. Instrum. 36 (1965) 1436.

[24] K.E. Kisman, R.L. Armstrong, Coupling scheme and probe damper for pulsednuclear magnetic resonance single coil probe, Rev. Sci. Instrum. 45 (1974) 1159.

[25] A.S. Peshkovsky, J. Forguez, L. Cerioni, D. Pusiol, RF probe recovery timereduction with a novel active ringing suppression circuit, J. Magn. Reson. 177(2005) 67–73.

[26] M. Hohwy, H.J. Jakobsen, M. Eden, M.H. Levitt, andN.C. Nielsen, Broadbanddipolar recoupling in the nuclear magnetic resonance of rotating solids: acompensated C7 pulse sequence, J. Chem. Phys. 108 (1998) 2686–2694.

[27] H.H. Meinke, F. Gundlach, Taschenbuch der Hochfrequenztechnik I: Grundlagen,fifth ed., Springer, Berlin, 2009.

[28] T.M. Barbara, J.F. Martin, J.G. Wurl, Phase transients in NMR probe circuits,J. Magn. Reson. 93 (1991) 497–508.

[29] A.T. Davis, An overview of algorithms in gnucap, in: Proceedings of the 15thBiennial IEEE University/Government/Industry Microelectronics Symposium,2003, pp. 360–361.

[30] M. Bak, J.T. Rasmussen, N.C. Nielsen, SIMPSON: a general simulation programfor solid-state NMR spectroscopy, J. Magn. Reson. 147 (2000) 296–330.

[31] M. Hohwy, C.M. Rienstra, C.P. Jaroniec, R.G. Griffin, Fivefold symmetrichomonuclear dipolar recoupling in rotating solids: application to doublequantum spectroscopy, J. Chem. Phys. 110 (1999) 7983–7992.

[32] P.E. Kristiansen, M. Carravetta, W.C. Lai, M.H. Levitt, A robust pulse sequencefor the determination of small homonuclear dipolar couplings in magic-anglespinning NMR, Chem. Phys. Lett. 390 (2004) 1–7.

[33] M.H. Levitt, P.K. Madhu, C.E. Hughes, Cogwheel phase cycling, J. Magn. Reson.155 (2002) 300–306.

[34] M.H. Levitt, Spin Dynamics: Basics of Nuclear Magnetic Resonance, Wiley, 2001.[35] A. Brinkmann, M. Eden, M.H. Levitt, Synchronous helical pulse sequences in

magic-angle spinning nuclear magnetic resonance: double quantum recou-pling of multiple-spin systems, J. Chem. Phys. 112 (2000) 8539–8554.

[36] T. Vosegaard, A. Malmendal, N.C. Nielsen, The flexibility of SIMPSON andSIMMOL for numerical simulations in solid- and liquid-state NMR spectro-scopy, Monatsh. Chem. 133 (2002) 1555–1574.

[37] H. Conroy, Molecular Schrodinger equation. VIII. A new method for theevaluation of multidimensional integrals, J. Chem. Phys. 47 (1967)5307–5318.

[38] J. Rossouw, A. Jansen, Version 3.2.0, 2003 /http://vipec.sourceforge.net/S.