This journal is© the Owner Societies 2016 Phys. Chem. Chem. Phys., 2016, 18, 9167--9175 | 9167

Cite this:Phys.Chem.Chem.Phys.,

2016, 18, 9167

Theory for spiralling ions for 2D FT-ICR andcomparison with precessing magnetizationvectors in 2D NMR

Akansha Ashvani Sehgal,abc Philippe Pelupessy,abc Christian Rolandodef andGeoffrey Bodenhausen*abc

Two-dimensional (2D) Fourier transform ion cyclotron resonance (FT-ICR) offers an approach to mass

spectrometry (MS) that pursuits similar objectives as MS/MS experiments. While the latter must focus on

one ion species at a time, 2D FT ICR can examine all possible correlations due to ion fragmentation in

a single experiment: correlations between precursors, charged and neutral fragments. We revisited the

original 2D FT-ICR experiment that has hitherto fallen short of stimulating significant analytical

applications, probably because it is technically demanding. These shortcomings can now be overcome

by improved FT-ICR instrumentation and computer hard- and software. We seek to achieve a better

understanding of the intricacies of the behavior of ions during a basic two-dimensional ICR sequence

comprising three simple monochromatic pulses. Through simulations based on Lorentzian equations,

we have mapped the ion trajectories for different pulse durations and phases.

Introduction

Two-dimensional nuclear magnetic resonance (NMR) has inspiredthe development of 2D techniques in several more or less relatedfields like infrared (IR) spectroscopy,1 electron spin resonance(ESR),2 ion cyclotron resonance (ICR),3 electron spectroscopy(ES),4 and femtosecond optical spectroscopy.5 So far, most ofthese 2D techniques have not achieved an impact comparable tothe revolution that occurred in NMR. Two-dimensional Fouriertransform ion cyclotron resonance mass spectroscopy (2DFT-ICR-MS) was initiated nearly 30 years ago through a jointeffort of MS and NMR groups.3 This work was inspired by two-dimensional exchange spectroscopy (2D EXSY), also known asnuclear Overhauser effect spectroscopy (NOESY), which is widelyused to either obtain information on chemical reactions thatinvolve sites which differ in chemical shifts, or on the migrationof magnetization due to cross-relaxation.6 As the inspiration for2D ICR comes from 2D NMR, we shall develop simple models to

explain the behaviour of ions in ICR in analogy to the magnetiza-tion vectors used in NMR, in view of exploring their similaritiesand differences.

Both EXSY and NOESY NMR experiments can be understoodin terms of classical magnetization vectors that obey Bloch’sequations. These vectors follow trajectories that are confinedto a sphere with unit radius. Relaxation causes the vectorsultimately to return to the north pole of Bloch’s sphere, whichtherefore acts as an ‘attractor’. The curvature of this sphere isresponsible for many non-linear effects in NMR, which arepronounced when using pulses with large nutation angles, butcan be neglected if the perturbations are weak. The trajectoriesof ions in 2D ICR also exhibit a linear response if the perturbationsare weak, but these trajectories are not necessarily confined to thecentre of the ICR cell. As a result, concepts such as phase-cyclescannot be readily transferred from NMR to ICR.

The unparalleled success of 2D NMR has been a source ofinspiration for 2D ICR, but also a source of some conceptualmisunderstandings. There are indeed close similarities between2D ICR and 2D EXSY. In either case, the magnetic field leads tocircular motions: in NMR the sense of precession of the magneticmoments is determined by the sign of the gyromagnetic ratio,while in ICR it is the charge of the ions that determines the senseof cyclotron motions.7,8 On the one hand, 2D EXSY and 2D NOESYallow one to correlate pairs of chemical shifts that are character-istic of the environments of nuclei before and after a chemicalreaction or cross-relaxation process. On the other hand, 2D ICRallows one to correlate pairs of cyclotron frequencies that are

a Ecole Normale Superieure-PSL Research University, Departement de Chimie, 24,

rue Lhomond, 75005 Paris, France. E-mail: Geoffrey.Bodenhausen@ens.frb Sorbonne Universites, UPMC Univ Paris 06, LBM, 4, Place Jussieu,

75005 Paris, Francec CNRS, UMR 7203 LBM, 75005 Paris, Franced Univ. Lille, CNRS, USR 3290, MSAP, Miniaturisation pour la Synthese l’Analyse

et la Proteomique, 59 000 Lille, Francee Univ. Lille, CNRS, FR 3688, FRABIO, Biochimie Structurale & Fonctionnelle des

Assemblages Biomoleculaires, 59 000 Lille, Francef Univ. Lille, CNRS, FR 2638, Institut Eugene-Michel Chevreul, 59 000 Lille, France

Received 29th January 2016,Accepted 3rd March 2016

DOI: 10.1039/c6cp00641h

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characteristic of precursor and fragment ions before and after amodification of their mass-to-charge ratio due to losses of neutralfragments, protonation, etc. These modifications can be inducedby collisions with neutral molecules or by irradiation with infraredor electron beams.

In its most basic form, two-dimensional exchange spectro-scopy (2D EXSY), uses a sequence comprising three pulses:b�t1�b0�tm�b00�t2, as shown in Fig. 1. The interval t1 is knownas the evolution time, tm the mixing interval, and t2 the detectionperiod. The amplitude of an rf pulse can be expressed in terms ofan angular frequency, o1 = �gB1, and a ‘nutation angle’ or ‘flipangle’ given by the pulse area, b = o1tp =�gB1tp. Thus the three rfpulses are characterized by b = o1tp, b0 = o1

0tp0 and b00 = o1

00tp00.

For ICR, the three-pulse sequence can be denoted asP1�t1�P2�tm�P3�t2. The three pulses can be characterizedby the energy they confer to the ions through acceleration (orwithdraw through their deceleration). This energy is determinedby the product of the pulse durations T1, T2 and T3 and theiramplitudes E1, E2 and E3. In ICR, there is no universallyaccepted expression for the area of a pulse Ai = TiEi, i.e., forthe product of its duration Ti and its amplitude Ei.

In NMR, the basic sequence mentioned above can be tracedback to early work on ‘stimulated echoes’.9 Part of the magne-tization, said to be ‘transverse’, precesses in a plane perpendi-cular to the applied magnetic field B0, both in the evolutioninterval t1 and in the detection interval t2. On the other hand,only ‘longitudinal’ magnetization that is parallel to the appliedmagnetic field B0 needs to be considered in the mixing intervaltm. In ICR, if we neglect so-called ‘magnetron motions’, themotion of the ions is limited to a plane perpendicular to theapplied magnetic field B0, so that one cannot speak of anylongitudinal component. In NMR, the frequency range, Do0 isusually of the order of a few tens or hundreds of parts permillion (ppm), centred on the Larmor frequency of the nucleiunder investigation. All three rf pulses can have the samemonochromatic carrier frequency, provided their rf amplitudeis sufficient to cover the required frequency range, i.e., providedo1 = �gB1 4 Do0. By contrast, ICR frequencies can range froma few kHz to many MHz. To cover such large bandwidths, oneusually has to employ frequency-swept ‘chirp’ pulses.

In their most common forms, 2D EXSY or NOESY use threeequal nutation angles b = b0 = b00 = p/2. By analogy, the original2D ICR experiments used three pulses with the same durationsT1 = T2 = T3, the same amplitudes E1 = E2 = E3 and hence the

same ‘pulse areas’ Ai = TiEi. If 2D EXSY experiments areperformed using three pulses with identical flip angles,b = b0 = b00 = p/2, the fate of the magnetization can be readilydescribed. The first pulse converts the longitudinal magnetiza-tion Mk

z of a site k into a transverse component Mkx, the second

pulse b0 has the opposite effect and generates a t1-dependentlongitudinal component Mk

z(t1), while the remaining transversecomponents Mk

x and Mky can be suppressed by applying pulsed

field gradients (PFGs) or by phase cycling (vide infra). Thelongitudinal component can be partly transferred from a sitek to a site l, i.e., from Mk

z(t1) to Mlz(t1), through exchange or cross-

relaxation, and finally reconverted by the third pulse b00 from Mlz

into Mlx. This transverse term induces a signal in the detection

interval t2. Unlike many other 2D NMR experiments that involvea transfer of coherence, and therefore require a quantum-mechanical treatment, most applications of 2D EXSY andNOESY can be discussed purely in classical terms (except forsome artifacts known as ‘J-peaks’ that are due to the unwittingexcitation of zero-quantum coherences). In NMR, this meansthat the Bloch equations are sufficient, without resorting todensity operators, so that each spin Ik (k = 1, 2,. . .K) can beassociated with a classical magnetization vector Mk = (Mk

x, Mky,

Mkz). Relaxation can be treated in simple phenomenological

terms. The transverse components Mkx and Mk

y decay with a timeconstant T2, while the longitudinal component Mk

z returns toits equilibrium with a time constant T1. If relaxation can beneglected, the trajectory of a vector Mk in the course ofmultiple-pulse experiments can be described on the surfaceof what is commonly referred to as Bloch’s sphere. Note thatthe curvature of this sphere is responsible for the non-linearityof the response of the magnetization in NMR. As we shall seebelow, there is no analogy in ICR for such non-linear effects.

Although this does not appear to be common practice, onecan modify 2D EXSY NMR by using pulses with small flip anglesb0 = b00 { 901, say 151, while the first pulse may remain b = 901without loss of generality. This can be used to map longitudinalrelaxation pathways in systems with scalar-coupled spins or toobtain z-filtered COSY spectra.10–13 Although 2D EXSY withsmall flip angles remains rather confidential in NMR, it offersstraightforward analogies with 2D ICR. Indeed, by using onlysmall nutation angles, the excursions of the vectors Mk arelimited to the vicinity of the north pole of Bloch’s sphere.The precession of a magnetization vector leads to a circulartrajectory around the north pole in a plane that is parallel to theequatorial plane. If one focuses attention on this plane andneglects the curvature of the sphere, the non-linearity of theresponse can be ignored, so that NMR becomes analogous toICR in this linear regime.

While this analogy has lead to the realization that 2D EXSYcould simply be ‘carried over’ from NMR to ICR, it has alsogiven rise to some misunderstandings. In NMR, regardlesswhether the description is limited to the polar region of Bloch’ssphere or not, the north pole acts as an ‘attractor’. Deviationsfrom the north pole only arise if there are non-vanishingtransverse magnetization components. Not only do thesecomponents decay through transverse T2 relaxation, but they can

Fig. 1 (a) Sequence for 2D exchange spectroscopy (EXSY) with a notationcommonly used in NMR, i.e.: b�t1�b0�tm�b00�t2 with nutation angles(or ‘flip angles’) b, b0 and b00. (b) 2D FT-ICR sequence P1�t1�P2�tm�P3�t2

with three pulses P1, P2 and P3 of lengths T1, T2 and T3.

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be suppressed either by applying pulsed field gradients (PFG), i.e.,by deliberately degrading the homogeneity of the field B0 duringa brief interval, or by phase-cycling. A phase-cycle consists incombining several experiments with the same sequence of pulsesand the same intervals, but where the phase fi of the ith pulsein the sequence is incremented from one scan to the next,i.e., fi = f0

i + kiDfi for the kth scan in a series of scans with ki =0, 1, 2,. . .,(Ki � 1) complementary experiments that are identicalexcept for the phases of the pulses that are incremented in stepsDfi = 2p/Ki. In the common case where Ki = 2 and hence Dfi = p,a phase-cycle boils down to a simple two-step phase alternation.Regardless of the nutation angles b and b0, such a two-step cyclecan be applied either to the first or second pulse in 2D EXSY,combined with alternating addition and subtraction of the resultingsignals (which for Krec = Ki = 2 can be achieved by alternating thereceiver phase frec = kDfrec with k = 0, 1 and Dfrec = 2p/Krec = p).This leads to the cancellation of transverse magnetizationcomponents in the mixing interval tm. Such methods can beunderstood in terms of ‘coherence transfer pathways’, wheretransverse magnetization components correspond to coherencesof order p = �1, while longitudinal magnetization componentshave order p = 0.14 Unfortunately, it is not possible to extend theseconcepts to ICR, where one cannot make a distinction betweenlongitudinal and transverse components. As a result, the ideasunderlying phase-cycles cannot be transferred from NMR to ICR.

As mentioned above, the excitation of the magnetization inNMR only remains in the linear regime if the nutation anglesare small so that sin(b) E b. This means that the magnetizationremains confined to the vicinity of the north pole of the Blochsphere. In ICR, the response to excitation is always linear,regardless of the areas Ai = TiEi of the pulses, as long as theions are not ejected from the cell. If we neglect ‘magnetronmotions’, the trajectories of the ions are limited to a planeperpendicular to the applied magnetic field B0. However, as weshall show below, the trajectories of ions in 2D ICR are notnecessarily centred in the middle of the ICR cell.

In the now widely accepted parlance of 2D NMR, the evolu-tion interval t1 = n1Dt1 is incremented in N1 steps with n1 =0, 1, 2,. . .,(N1 � 1), while the increment Dt1 determines thespectral width Dn1 = 1/Dt1 in the F1 domain, also known as n1 oro1 domain. The maximum duration tmax

1 = (N1 � 1)Dt1 deter-mines the digital resolution dn1 = 1/tmax

1 = Dn1/(N1 � 1) in the F1

domain. Likewise, the signal is observed in the detectioninterval t2 by taking N2 samples at intervals Dt2 that determinethe spectral width Dn2 = 1/Dt2 in the F2 domain, also known asn2 or o2 domain. The duration tmax

2 = (N2� 1)Dt2 determines thedigital resolution dn2 = 1/tmax

2 = Dn2/(N2� 1) in this domain. Thesame principles apply to 2D FT-ICR spectra. A 2D spectrum canbe represented by a contour plot of intensities as a functionof two frequencies F1 and F2 (also known as n1 and n2 or o1 ando2), which are obtained by Fourier transformation of thesignals recorded as a function of two time variables t1 and t2.The position of each peak is specified by two frequencyco-ordinates. By convention, most 2D NMR spectra are plottedso that o1, which represents the evolution in the ‘indirect’ t1

dimension, appears along the vertical axis, while the ‘direct’

dimension o2 usually plotted along the horizontal axis. In ICR,it has become fashionable to speak of ‘parent’ and ‘fragment’dimensions rather than ‘vertical’ (F1) and ‘horizontal’ (F2) dimen-sions. A 2D FT-ICR spectrum allows one to map arbitrarycorrelations between precursor and fragment ions withoutany a priori information about the samples.

The general scheme for obtaining a 2D FT-ICR spectrum isshown in Fig. 2a. The evolution interval t1 is sandwichedbetween a first ‘excitation’ pulse P1 and a second ‘encoding’pulse P2. The sequence, P1–t1–P2, is collectively referred as the‘encoding sequence’. In an ICR cell, we wish to monitor theconversion of a precursor ion Ik into a fragment ion Il duringthe fragmentation interval tm, which plays a similar role as themixing time tm in NMR. The fragmentation may be induced byan electron beam, for example by electron capture dissociation(ECD),15 by electron detachment dissociation (EDD),16 byelectron-induced dissociation (EID),17 or by electron transferdissociation (ETD),18 by infrared light in infrared multiphotondissociation (IRMPD).19 Alternatively, ion fragmentation maybe brought about by blackbody infrared radiative dissociation(BIRD)20 or by collisions with neutral molecules in collision-induced dissociation (CID).21 After the fragmentation intervaltm, a third ‘monitoring’ or ‘detection’ pulse P3 is applied, andthe signals are recorded during the detection period t2.

The evolution interval t1 is incremented stepwise and thesignals are recorded as a function of t2 for each value of t1. After2D Fourier transformation, one obtains a 2D spectrum asshown schematically in Fig. 2b. In ICR, the conversion of afraction fkl of precursor ions Ik into fragment ions Il leads to across-peak at coordinates (o1 = Ok, o2 = Ol), with an amplitudeproportional to the fraction fkl. Fragmentation leads to a changein mass from mk to ml or to a change in charge from qk to ql, andhence to a jump in the ICR frequency from a ‘parent frequency’

Fig. 2 (a) Detailed view of a 2D ICR sequence where the time axis issequentially labelled 1, 2, etc. to distinguish the beginning and end ofvarious intervals. (b) 2D ICR spectrum for a vanishingly short fragmentationinterval tm = 0. The signals evolve with the same frequency in both t1 and t2

intervals, so that one only sees a single diagonal peak, o1 = o2 = Ok. (c) Ifsome of the ions break up during the fragmentation interval tm, the parentsignals appear at o1 = Ok and the signals of the fragments or daughter ionsat o2 = Ol, in this example with Ol 4 Ok.

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o1 = Ok that is proportional to qk/mk to a ‘daughter frequency’o2 = Ol that is proportional to ql/ml. Note that one can haveOk o Ol when ions break up into smaller fragments or acquire agreater charge, and Ok 4 Ol, when ions form larger adducts orlose part of their charge. Such frequency jumps can be readilycharacterized by 2D ICR.

Frequency-swept ‘chirp’ pulses are usually applied to excite ionsin Penning traps of commercial ICR instruments.22–24 In our ICRinstrument (which is quite different from NMR instruments inthis respect), chirp rf pulses sweep through a relevant range offrequencies from an initial frequency oi to a final frequency of in Ndiscrete steps Do of duration DT, so that the total pulse duration isT = NDT. Usually it can be assumed that the kth ion is only affectedby the rf irradiation in the kth interval and the frequencies of theadjacent steps are too far off-resonance to influence the motion ofthe kth ion. The actual evolution interval t1 between the two pulsesof the encoding sequence is therefore equal to the duration betweenthe end of the kth block of the first ‘excitation’ pulse and the startof the kth block of the second ‘encoding’ pulse, as shown in Fig. 3.Thus the interactions of ions with chirp rf pulses can be describedas if one used only simple monochromatic pulses. We shall there-fore describe the behaviour of a single ion excited by a sequence ofmonochromatic pulses, and we shall neglect the effects of ion–ioninteractions. Although 2D FT-ICR has the potential to become auseful method for chemical analysis, only few studies have beenpublished so far, such as the paper by Guan and Jones on the theoryof 2D FT-ICR.25 Despite its elegant description of the ion trajectories,the assumption that the rf pulses should be long turned out to bedetrimental to sensitivity. In recent work from our laboratory,26 wehave shown that 2D ICR spectra obtained with shorter pulsedurations have higher signal-to-noise ratios. Such spectra are lesscluttered by harmonics so that their interpretation is facilitatedwithout requiring sophisticated ‘de-noising’ algorithms.26,27 Herein,we describe the behaviour of ions when the pulses are short.

TheoryEquations of motion for 2D FT-ICR

In an ICR cell, the ions are subjected to a static magnetic field-

B0 = (0,0,B) and to a perpendicular electric field EðtÞ��!

¼ Ex; 0; 0ð Þ.

A precursor ion Ik of mass mk and charge qk moves on a circulartrajectory in a plane perpendicular to the magnetic field. Themotion is determined by the Lorentz equation:

mkdvk!dt¼ qk~E þ qk vk

!� ~B� �

(1)

If there is no applied electric field, the cyclotron frequency is:

ok = (qkB)/mk (2)

We shall neglect ‘trapping’ motions along the z-axis, andassume that the magnetic field is homogeneous across theICR cell. We shall also neglect ion–ion Coulomb repulsions,although Chen and Comisarow have shown that these interactionslead to a dispersion of ion clouds, thus affecting the phases of theions and the radii of their trajectories.28

The electric field in ICR is usually applied in the formof frequency-swept ‘chirped’ radio-frequency (rf) pulses. Tosimplify the discussion, we shall discuss monochromatic rfpulses which are ‘on resonance’, i.e., with a carrier frequency ok

that matches the cyclotron motion of the parent ion Ik duringthe first two pulses of lengths T1 = T2 = T with amplitudesE1 = E2 = E (intervals 0–1 and 2–3 in Fig. 2):

Ex = E sin(okt) (3)

The components vkx and vky of the parent ion Ik can easily bedetermined by substituting eqn (2) to solve eqn (1):

dvkx

dt¼ qkEx

mkþ okvky (4a)

dvky

dt¼ �okvkx (4b)

Solving eqn (4a) and (4b), we get:

vkx ¼ vkx0cos oktð Þ þ vky0sin oktð Þ þ Eqktsin oktð Þ2mk

(5a)

vky ¼ vky0 cos oktð Þ � vkx0 sin oktð Þ þ Eqktcos oktð Þ2mk

� Eqk sin oktð Þ2mkok

(5b)

In our 2D FT-ICR experiments, we used ‘chirp’ pulses comprisingN discrete steps with a duration DT on the order of a few ms. Insuch cases, the trajectories of the ions during DT fall short ofcompleting a full orbit along an Archimedes spiral. Therefore,none of the terms of the eqn (5) can be neglected. In the followingsections, analytical solutions are obtained for the velocity compo-nents and the position coordinates of an ion at the end of thesecond pulse, i.e., at the beginning of the fragmentation interval.

(a) Excitation pulse. At the end of the first pulse (t = T1 in thescheme of Fig. 2), the velocity of the ion is obtained by solvingeqn (5) for the initial condition where nkx0 = nky0 = 0. We assumethat all ions are initially located in the centre of the ICRcell. The corresponding x and y coordinates with respect to

Fig. 3 Encoding sequence of the 2D ICR scheme showing the first twochirp pulses. For the sake of illustration, the pulse is broken down intoblocks corresponding to frequencies stepped from oi to of. The blackblock represents the frequency ok corresponding to the kth ion.

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the centre of the ICR cell are obtained by integrating theirrespective velocity components:

vkx1 ¼EqkT sin okTð Þ

2mk(6a)

xk1 ¼Eqk sin okTð Þ � okT cos okTð Þf g

2mkok2

(6b)

vky1 ¼EqkT cos okTð Þ

2mk� Eqk sin okTð Þ

2mkok(7a)

yk1 ¼Eqk 2cos okTð Þ þ okT sin okTð Þ � 2f g

2mkok2

(7b)

(b) Evolution interval. During the evolution interval t1, theelectric field is switched off (Ex = 0) so that only the phasesevolve:

vkx2 ¼Eqk

4mkok� cos ok T þ t1ð Þ½ � � cos ok T � t1ð Þ½ �f

þ 2okT sin ok T þ t1ð Þ½ �g(8a)

xk2 ¼Eqk

2mkok2cos okt1ð Þ sin okTð Þ � okT cos ok T þ t1ð Þ½ �f g

(8b)

vky2 ¼Eqk

2mkokokT cos okTð Þ cos okt1ð Þf

� sin okTð Þ½cos okt1ð Þ þ okT sin okt1ð Þ�g(9a)

yk2 ¼Eqk

4mkok2cos ok T þ t1ð Þ½ � � cos ok T � t1ð Þ½ �f

þ 2okT sin ok T þ t1ð Þ½ � þ 4 cos okTð Þ � 4g(9b)

(c) Second pulse. The trajectories can be calculated inanalogy to the first pulse, using initial velocities that are equalto the final velocities of eqn (8a) and (9a). The two first pulsesthat bracket the evolution time t1 and together make up the‘encoding sequence’, are chosen to be exactly similar, withequal pulse durations, electric field amplitudes, and areas(T1 = T2 = T, E1 = E2 = E, A1 = A2 = A). Inserting these valuesinto eqn (5), we obtain at the end of the second pulse (which isalso the beginning of the fragmentation interval tm):

vkx3 ¼Eqk

4mkokcos ok 2T þ t1ð Þ½ � � cos okt1ð Þf

þ 2okT sin okTð Þ þ sin ok 2T þ t1ð Þ½ �f gg(10a)

xk3 ¼Eqk

4mkok22 sin okTð Þ � sin okt1ð Þ þ sin ok 2T þ t1ð Þ½ �f

� 2okT cos ok 2T þ t1ð Þ½ � þ cos okTð Þf gg(10b)

vky3 ¼Eqk

4mkoksin okt1ð Þ � 2 sin okTð Þ � sin ok 2T þ t1ð Þ½ �f

þ2okT cos okTð Þ þ cos ok 2T þ t1ð Þ½ �f gg(11a)

yk3 ¼Eqk

4mkok22okT sin ok 2T þ t1ð Þ½ � þ sin okTð Þf gf

þ cos ok 2T þ t1ð Þ½ � � cos okt1ð Þ þ 8 cos okTð Þ � 8g(11b)

At the beginning of the fragmentation interval tm, the positionof an ion with respect to the centre of the cell is given by:

~R�� �� ¼ xk3

2 T ; t1ð Þ þ yk32 T ; t1ð Þ

� �1=2(12)

Thus the position of the ion depends on the pulse durationT and the evolution interval t1. Only if the ratio of the pulseduration T divided by the ion’s period Tk = 1/ok is larger than 10,i.e., when T/Tk 4 10, can the last term of eqn (5b) be neglected,thereby giving the distance from the centre of the cell, in agreementwith the expression given by Guan and Jones.25

~R�� �� / 2ð1þ cos½o T � t1Þð Þ�f g1=2 (13)

(d) Fragmentation interval. The trajectories during the frag-mentation interval tm can be calculated as before:

vkx4 ¼Eqk

4mkokcos ok 2T þ t1 þ tmð Þ½ � � cos ok t1 þ tmð Þ½ �f

� cos ok T � tmð Þ½ � þ cos ok T þ tmð Þ½ �

þ2okT sin ok 2T þ t1 þ tmð Þ½ �f þ sin ok T þ tmð Þ½ �gg(14a)

xk4 ¼Eqk

4mkok2sin ok 2T þ t1 þ tmð Þ½ � þ sin ok T � tmð Þ½ �f

þ sin ok T þ tmð Þ½ � � sin ok t1 þ tmð Þ½ �

� 2okT cos ok 2T þ t1 þ tmð Þ½ � þ cos ok T þ tmð Þ½ �f gg(14b)

vky4 ¼Eqk

4mkoksin ok t1 þ tmð Þ½ � � sin ok T � tmð Þ½ �f

� sin ok T þ tmð Þ½ � � sin ok 2T þ t1 þ tmð Þ½ �

þ 2okT cos ok 2T þ t1 þ tmð Þ½ � þ cos ok T þ tmð Þ½ �f gg(15a)

yk4 ¼Eqk

4mkok28 cos okTð Þ � cos ok t1 þ tmð Þ½ � � cos ok T � tmð Þ½ �f

þ cos ok 2T þ t1 þ tmð Þ½ � þ cos ok T þ tmð Þ½ �

þ 2okT sin ok 2T þ t1 þ tmð Þ½ � þ sin ok T þ tmð Þ½ �f g � 8g(15b)

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Trajectory mapping

The x and y coordinates calculated above describe the trajectoryof the kth ion with a cyclotron frequency ok during the entirecourse of the experiment. In a 2D experiment, the ion acquiresa phase j = okt1 at the end of the evolution interval t1. Oursimulations show trajectories of ions with phases varying inthe interval as 0 o j o 2p. We mapped the trajectories for‘matched’ pulses where the pulse length matches the precessionof the parent ion Ik with frequency ok, so that okT1 = okT2 = 2pnwith integer n = 1, 2,. . . The trajectories for n = 1 are shown inFig. 4 for a doubly charged ion of substance P. The case ofmatched pulses has been discussed previously in the literature.25,29

When the phase okt1 acquired by the ion at the end of the evolutioninterval t1 is a multiple of 2p, both pulses contribute to excite the iononto a trajectory with a large radius. If the phase at the end of t1 is

an odd multiple of p, i.e., if okt1 = (2k + 1)p, the ion is ‘de-excited’back to the centre of the ICR cell by the second pulse.

For ‘non-matched’ pulses (okT1 = okT2 a 2pn), the pulsedurations T1 = T2 do not match the precession period. Weconsidered the case where okT1 = okT2 o 2p (Fig. 5). Only whenthe excitation pulse is matched does the ion travel through aninteger number of complete circles about the centre of the cellin the interval t1, so that the trajectories remain centred. For non-matched pulses, the ion’s trajectory is no longer centred in themiddle of the cell after the excitation pulse. The displacement of thecentre of ion’s trajectory from the centre of the cell is given by

-

dt1:

~dt1 ¼E0qk

mkok21� cosðokTÞf gj (16)

At the end of the encoding sequence, i.e., at the beginning of thefragmentation interval tm, the displacement of the centre of theion’s trajectory is exacerbated if the pulses are not matched, pulledin the direction of the applied electric field. The ion orbits incontracting or enlarging circles, with a centre shifted from thecentre of the cell, as described by eqn (17), which is twicethe displacement with respect to the centre of the cell duringthe evolution period.

~dtm ¼2E0qk

mkok21� cos okTð Þf g j (17)

Fig. 4 Trajectories for matched pulses with durations T so that okT = 2p.The ion’s frequency is ok = 200 kHz, corresponding to the doubly chargedion of substance P in a field of 8.78 T, the evolution interval is taken tobe t1 = 10 ms, the excitation and encoding pulses have durations T1 = T2 =5 ms and the pulse voltage is 2666 V. The pulse area remains constant. Bluebullets represent the ions, red bullets the centre of the trajectories duringthe evolution and fragmentation intervals.

Fig. 5 Trajectories for short excitation and encoding pulses with durationsso that okT1 = okT2 o 2p. The excitation and encoding pulses have durationsT1 = T2 = 4 ms. The remaining conditions are the same as in Fig. 4. Blue bulletsrepresent the ions, red bullets the centres of the trajectories during theevolution interval, and black bullets the centres of the trajectories during thefragmentation interval.

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The actual position of the ion with respect to the centre of thecell,

-

R, is the vector sum of the displacement of the centre ofthe ion’s orbit,

-

d, and the position of the ion with respect to thecentre of the shifted orbit, -

r. The variation of the distance of theion from the centre of the cell during the fragmentation period-

Rtm=

-

dtm+ -

rtmis shown in Fig. 5. For matched pulses, i.e., when

-

dtm= 0, the variation of the distance observed for phases

p o j = okt1 o 2p mirrors the variation for 0 o j = okt1 o p.But in the case of shifted trajectories, the ions never come backto the centre of the ICR cell during the fragmentation interval tm.The ions do however pass periodically near the centre of the cell,depending on the phase j = okt1 that is acquired at the endof the evolution period. On our FT-ICR spectrometer, the rffrequencies during the steps DT within the chirp pulses do notcorrespond exactly to the frequencies of the ions, so that ionsexcited by a chirp pulse do not experience ideal matchedexcitation. Thus for most of the ions, the trajectories will bedisplaced.

The ion is required to be near the centre of the cell forfragmentation by cell-centred techniques to be effective. Forpulses that are matched in frequency, the ion returns to thecentre of the cell, provided the evolution interval is such thatthe ion acquires a phase j = okt1 = p. For non-matched pulses,as shown in Fig. 6, the ions do not return to the centre of thecell and the efficiency of fragmentation will depend on theoverlap of the fragmenting beam with the shifted trajectoriesof the ions. For example, the laser beam of IRMPD techniqueexhibits a Gaussian profile at the centre of the ICR cell andthe fragmentation thus will not be performed with maximumyield, if the ion’s trajectory is not within the breadth ofthe beam.

The displacements of the ICR orbit with respect to the centreof the cell are very similar to so-called ion-locking experiments,where the centre of an ion’s motion is deliberately displacedand the radius of an ion’s orbit is increased or decreased byapplying phase shifts of the rf fields.30,31 The shifts that areobserved due to a non-matched encoding sequence can beutilized to develop 2D ion-locking experiments based on similarprinciples as 1D ion-locking experiments. Thus the lack of ionsthat can be fragmented because they are shifted far away fromthe centre of the cell can be manipulated to advantage byselectively sequestering the ions and observing them with othercombinations of pulses.

Conclusions

When optimizing pulse sequences using frequency-swept‘chirp’ pulses, the best results were obtained when using thesmallest possible duration for each frequency step (DT = 0.5 ms,as opposed to 20 ms for standard excitation pulses). We initiallybelieved that the phase difference j = okt1 between the ion andthe second pulse varies only with the cyclotron frequency ok

and t1, so that if the ions would be in-phase with the secondpulse P2, they would be propelled onto a higher radius, whereasif the ion is out-of-phase with respect to P2, it would be broughtback to the centre of the cell.24 However, this only occurs whenthe ions are excited by pulses which have rf frequencies that areexactly matched with the cyclotron frequency of the ion. Wesimulated the ion trajectories leading as far as the fragmentationinterval tm by solving equations of ion movements in an idealcell. For monochromatic pulses that are on-resonance with the

Fig. 6 Variation of the distance of ions with respect to the centre of the cell during the fragmentation interval (a) for matched pulses, (b) for non-matched pulses. The colour code represents the phase j = okt1 that the ions acquire at the end of the evolution interval. The bottom and top figuresindicate the variations for phase changes from 0 to p and p to 2p respectively.

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9174 | Phys. Chem. Chem. Phys., 2016, 18, 9167--9175 This journal is© the Owner Societies 2016

ion’s frequency, we observed that during the first pulse the iongains in kinetic energy, moving in spirals away from the centre.The movement of the ion during the subsequent evolutioninterval t1 follows an orbit which is off-centre, unless the productT1ok of the duration T1 of the initial excitation pulse and thefrequency ok of the ion Ik is exactly a multiple of 2p. Hence, thesecond pulse P2, even when it is perfectly out-of-phase withrespect to the ion, fails to bring the ion back exactly to the centreof the cell, which is essential for performing fragmentation byIRMPD and other cell-centred techniques. In this study, weanalysed the motions of ions in the ICR cell in the course ofthe three-pulse scheme of 2D FT-ICR. By mapping the iontrajectories, we found that the centre of the trajectories of theions exhibit a displacement with respect to the centre of the cellthat varies periodically with the durations of the first two pulses.The shift of the centre leads to off-centred trajectories of the ionsduring the fragmentation period. These shifted trajectoriesmay lead to a loss in efficiency of cell-centred fragmentationtechniques. However, if the pulses are of short duration, theoverlap of the laser beam and the precursor ions is not compro-mised. Another advantage of using pulses of short duration is thedecrease of intensities of harmonic signals that appear at multi-ples of the basic frequencies. Pulses with small durations thusoffer a simple way to improve 2D ICR spectra without requiringany sophisticated ‘de-noising’ techniques.

Appendix

Glossary for ICR adapted from the work of van Agthoven et al.32

Encoding sequence: the sequence which encodes thefrequencies of the precursor ions prior to fragmentation whichis normally composed of an ‘excitation’ pulse, an evolutioninterval, and an ‘encoding’ pulse.

Excitation pulse: first pulse (designated by P1) of the pulsesequence, which excites all precursor ions equally.

Evolution interval (previously referred to as frequencyencoding period): regularly incremented interval, designatedas t1, between the two pulses of the encoding sequence, duringwhich all precursor ions rotate at their cyclotron frequency.

Encoding pulse: second pulse (designated by P2) of the pulsesequence, which modifies the radius of the trajectories dependingon the phase of the ions.

Fragmentation period: fixed interval between the encodingpulse and the detection pulse during which precursor ions arefragmented.

Detection pulse (previously referred as observe pulse): thirdpulse (designated as P3) of the pulse sequence, which excitesboth precursor and fragment ions prior to detection.

Abbreviations

2D FT-ICR-MS Two-dimensional Fourier transformation ioncyclotron resonance mass spectroscopy

NMR Nuclear magnetic resonance

Acknowledgements

The authors thank the Centre National de la Recherche Scien-tifique (CNRS, France), the Defi Instrumentation aux limites,the Agence Nationale pour la Recherche (ANR, France) (grant2010FT-ICR2D and grant Defi de tous les savoirs 2014, ONE_SHOT_FT-ICR_MS_2D) and the European Research Council(ERC, advanced grant ‘Dilute para water’) for financial support.

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