Journal of Magnetic Resonance 248 (2014) 105–114
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Rotation operator propagators for time-varying radiofrequency pulses in NMR spectroscopy: Applications to shaped pulses and pulse trains Ying Li a,1, Mark Rance b, Arthur G. Palmer III a,⇑ a
Department of Biochemistry and Molecular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032, United States Department of Molecular Genetics, Biochemistry and Microbiology, University of Cincinnati, 231 Albert Sabin Way, Medical Sciences Building, Cincinnati, OH 45267-0524, United States b
a r t i c l e
i n f o
Article history: Received 22 June 2014 Revised 31 August 2014 Available online 22 September 2014 Keywords: Rotation operator Euler angle Perturbation expansion Shaped pulse CPMG
a b s t r a c t The propagator for trains of radiofrequency pulses can be directly integrated numerically or approximated by average Hamiltonian approaches. The former provides high accuracy and the latter, in favorable cases, convenient analytical formula. The Euler-angle rotation operator factorization of the propagator provides insights into performance that are not as easily discerned from either of these conventional techniques. This approach is useful in determining whether a shaped pulse can be represented over some bandwidth by a sequence s1–R/(b)–s2, in which R/(b) is a rotation by an angle b around an axis with phase / in the transverse plane and s1 and s2 are time delays, allowing phase evolution during the pulse to be compensated by adjusting time periods prior or subsequent to the pulse. Perturbation theory establishes explicit formulas for s1 and s2 as proportional to the average transverse magnetization generated during the shaped pulse. The Euler-angle representation of the propagator also is useful in iterative reduction of pulse-interrupted-free precession schemes. Application to Carr–Purcell–Meiboom–Gill sequences identifies an eight-pulse phase alternating scheme that generates a propagator nearly equal to the identity operator. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Numerous techniques in NMR spectroscopy utilize radiofrequency (rf) pulses in which the carrier frequency, amplitude, and/or phase are time-dependent. Examples include shaped pulses, shifted laminar pulses, adiabatic pulses, and pulse-interruptedfree-precession sequences [1]. For an isolated spin, in the absence of scalar couplings and ignoring relaxation, evolution of the density operator during such a pulse scheme is described by
rðtÞ ¼ U r ð0ÞU1
ð1Þ
in which r(t) is the density operator, t is the length of the pulse, and U is the propagator given by
Z t U ¼ T exp i xx ðt0 ÞIx þ xy ðt0 ÞIy þ Xðt0 ÞIz dt0
ð2Þ
0
T is the Dyson time-ordering operator, xx(t) and xy(t) are the timedependent rf amplitudes with x- and y-phase, respectively, and X(t) ⇑ Corresponding author. Fax: +1 212 305 7932. E-mail address: [email protected] (A.G. Palmer III). Present address: Department of Chemistry, University of Louisville, 2320 South Brook Street, Louisville, KY 40208, United States. 1
http://dx.doi.org/10.1016/j.jmr.2014.09.001 1090-7807/Ó 2014 Elsevier Inc. All rights reserved.
is the time-dependent resonance offset of the nuclear spin under consideration. The two most common approaches for calculating the evolution of the density operator under the propagator are average Hamiltonian theory (AHT) [2,3], which has the advantage of yielding analytically tractable results in many instances, and direct numerical integration, which has the advantage of high accuracy. In the latter case, the integral is expressed as a summation using a time step Dt sufficiently short that the exponential of the sum can be expressed as the product of individual rotations during each time step [1]. The rotation operator formalism of Sanctuary and coworkers [4] and Siminovitch [5,6] provides an alternative approach and is the subject of the present work. In this method, the propagator given in Eq. (2) is expressed as:
U ¼ exp ½icðtÞIz exp ½ibðtÞIx exp ½iaðtÞIz
ð3Þ
Thus, evolution of the density operator is calculated as the product of three individual rotation operations by the Euler angles a(t), b(t), and c(t). As this approach makes clear, any time-varying rf pulse train can be represented as an x-rotation flanked by two (not necessarily identical) z-rotations. The initial z-rotation is inconsequential for an excitation pulse applied to equilibrium z-magnetization, so that any excitation pulse is described by an x-rotation followed by
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114
a z-rotation. Inversion of the initial equilibrium density operator is described simply by cos[b(t)]. The present work employs the rotation operator formalism to characterize shaped and other time-dependent rf pulse schemes commonly employed in modern NMR spectroscopy of biological macromolecules. The off-resonance effects of a square 90° pulse are known to be approximated by a sequence s–90°x–s, in which s = 2sp/p and sp is the on-resonance pulse length [1,7]. Recent work has used numerical methods to approximate the off-resonance effects of shaped pulses by a pulse rotation preceded and/or followed by a time delay [8]. As Eq. (3) indicates, such relationships arise whenever a(t) and/or c(t) are proportional to the resonance offset of spins within the bandwidth of the shaped pulse and such dependence can be required as part of the optimization of shaped pulses [9].
1
bðtÞ ¼ cos
" # 1 jf ðtÞj2 1 þ jf ðtÞj2
ð10Þ
cðtÞ ¼ Arg½f ðtÞ and by construction, a(0) = c(0), b(0) = 0, and f(0) = 0. The Riccati equation also can be converted to a second-order linear homogeneous equation by making the substitution:
d ln yðtÞ 1 ¼ x ðtÞf ðtÞ dt 2
ð11Þ
to yield:
2
d yðtÞ dt
2
d ln x ðtÞ dyðtÞ 1 þ iXðtÞ þ jxðtÞj2 y ¼ 0 dt dt 4
2. Theory
If x(t) is written as |x(t)|exp[i/(t)], then:
The functions a(t), b(t), and c(t) are found by differentiating Eqs. (2) and (3) with respect to time. The derivative of Eq. (2) is simply:
d yðtÞ
dU ¼ i xx ðtÞIx þ xy ðtÞIy þ XðtÞIz U dt
2
dt
2
dU dcðtÞ dbðtÞ Iz U þ ¼ i exp ½icðtÞIz Ix exp ½ibðtÞIx exp ½iaðtÞIz dt dt dt daðtÞ exp ½icðtÞIz exp ½ibðtÞIx Iz exp ½iaðtÞIz þ dt daðtÞ dcðtÞ dbðtÞ Iz þ Ix cos½cðtÞ þ Iy sin½cðtÞ þ ðIz cos½bðtÞ ¼ i dt dt dt ð5Þ Iy sin½bðtÞcos½cðtÞ þ Ix sin½bðtÞsin½cðtÞ U
ð13Þ and the term in curly braces can be regarded as a time-dependent resonance offset of the affected spin. A phase-shifted pulse is treated by subtracting the phase shift / from a(t) and adding / to c(t). Once Eqs. (12) or (13) are solved, either analytically or numerically, to obtain b(t) and c(t), a(t) is obtained by direct integration of Eq. (7a). The resulting values for t equal to the pulse length are used in Eqs. (1) and (3). For clarity in the following, the pulse length will be denoted sp, so that a(sp), b(sp), and c(sp) are the quantities of interest. For amplitude-modulated pulses, d/(t)/dt = 0 and x+(t) = x (t) = x(t), and Eq. (13) becomes:
2
Equating the terms in Eqs. (4) and (5) that are proportional to the same operator Ix, Iy, or Iz yields a system of equations:
ð6Þ
dbðtÞ ¼ xx ðtÞ cos½cðtÞ þ xy ðtÞ sin½cðtÞ dt
ð7bÞ
ð7cÞ
Eqs. (7b) and (7c) are a pair of coupled differential equations for b(t) and c(t). As first demonstrated by Zhou et al. [4] in the context of NMR, defining:
bðtÞ exp½icðtÞ 2
ð8Þ
enables Eqs. (7b) and (7c) to be combined to give the Riccati equation:
2 df ðtÞ 1
1
¼ xx ðtÞ ixy ðtÞ f ðtÞ þ iXðtÞf ðtÞ þ xx ðtÞ þ ixy ðtÞ dt 2 2 1 1 2 ¼ x ðtÞf ðtÞ þ iXðtÞf ðtÞ þ xþ ðtÞ 2 2 where x±(t) = xx(t) ± ixy(t). From the definition in Eq. (8),
d ln xðtÞ dyðtÞ 1 2 þ iX þ x ðtÞy ¼ 0 dt dt 4
ð14Þ
2
d yðtÞ 2
iX
dyðtÞ 1 2 þ x1 y ¼ 0 dt 4
ð15Þ
yðtÞ ¼ eiXt=2 fcosðxe t=2Þ iðX=xe Þ sinðxe t=2Þg ð7aÞ
f ðtÞ ¼ tan
2
with solution:
daðtÞ ¼ xx ðtÞ sin½cðtÞ xy ðtÞ cos½cðtÞ = sin½bðtÞ dt
dcðtÞ ¼ XðtÞ xx ðtÞ sin½cðtÞ xy ðtÞ cos½cðtÞ = tan½bðtÞ dt
dt
dt
These equations can be rearranged to give:
d yðtÞ
For a square pulse with xx(t) = x1, xy(t) = 0, and X(t) = X. Eq. (14) reduces to:
xx ðtÞ ¼
d ln jxðtÞj d/ðtÞ dyðtÞ 1 þ i XðtÞ þ jxðtÞj2 y ¼ 0 dt dt dt 4
ð4Þ
The derivative of Eq. (3) is
dbðtÞ daðtÞ cos½cðtÞ þ sin½bðtÞ sin½cðtÞ dt dt dbðtÞ daðtÞ xy ðtÞ ¼ sin½cðtÞ sin½bðtÞ cos½cðtÞ dt dt dcðtÞ daðtÞ XðtÞ ¼ þ cos½bðtÞ dt dt
ð12Þ
ð16Þ
Substitution into Eq. (11) gives:
f ðtÞ ¼ sin h tanðxe t=2Þ
1 þ i cos h tanðxe t=2Þ 1 þ cos2 h tan2 ðxe t=2Þ
ð17Þ
in which h = tan1(x1/X) is the tilt angle and xe ¼ X2 þ x21 the effective field. Using Eqs. (7a) and (10) yields:
1=2
is
aðsp Þ ¼ tan1 cos h tanðxe sp =2Þ / h
i 2 bðsp Þ ¼ cos1 cos2 h þ sin h cosðxe sp Þ
cðsp Þ ¼ tan1 cos h tanðxe sp =2Þ þ /
ð18Þ
The pulse phase / has been added in Eq. (18) to give the most general result. Eqs. (17) and (18) have been reported previously [4]. The propagator for a square pulse is then: "
U¼
cosðxe sp =2Þ icos hsin i/ ie sin hsin xe sp =2
xe sp =2 iei/ sin hsin xe sp =2 cosðxe sp =2Þ þ i cos hsin xe sp =2
#
ð19Þ
ð9Þ
Eqs. (12)–(14) assume that the rf field is described by an explicit function. In many cases, the pulse shape is defined by a list of N paired values giving the amplitude and phase (or xx and xy
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114
components) of the x1 field applied for a time interval Dt, with sp = NDt. The propagator U of Eq. (1) can be determined by iterative application of Eq. (19) for each step in the pulse shape. The resulting propagator is expressed as [10]
2
3 i½aðsp Þcðsp Þ=2 i½aðsp Þþcðsp Þ=2 e cos½bð s ie sin½bð s p Þ=2 p Þ=2 5 U¼4 i aðs Þcðsp Þ=2 ie ½ p sin½bðs Þ=2 ei½aðsp Þþcðsp Þ=2 cos½bðs Þ=2 p
p
ð20Þ and the parameters a(sp), b(sp), and c(sp) are obtained directly from this matrix.
The NMR spectra of H2O in a doped water sample (0.1 mg/ml GdCl3, 0.1% 13CH3OH, 1% H2O, 99% D2O; Cambridge Isotope Laboratories, CDLM-5010) were recorded using a pulse sequence 90/°– CPMGk–90/°–Grad–90x°–acquire, in which the first two 90° pulses have / = x or y phase, k is the number of CPMG blocks, and Grad is a z-gradient pulse to dephase transverse magnetization. Each CPMG block consisted of 8 repetitions of the element scp/2–180°–scp/2; the 180° pulse separation scp = 1 ms. The phases of the 180° pulses were set as described in Section 4. The carrier frequency for the first two 90° pulses and the CPMG blocks was varied between experiments from 0 to 10 kHz in 250 Hz steps. The last 90° pulse was applied on-resonance with the H2O signal. Pulses were applied with x1/2p = 10 kHz. The first and last delays in the CPMG sequences were shortened by 2p/s90, in which s90 is the 90° pulse length, to account for phase evolution during the first and second 90° pulses. Each experiment was recorded using 8 scans, a recycle delay of 8 s, a spectral width of 25 ppm, and 8192 complex data points. Experiments were conducted at T = 300 K on a Bruker 500 MHz DRX NMR spectrometer with a triple-resonance tripleaxis gradient probe. 4. Results and discussion The Euler-angle decomposition provides a representation of the propagator that is particularly convenient for NMR spectroscopy, in which z-rotations assume critical prominence. The Euler-angle decomposition is used in the following to establish general properties of amplitude-modulated shaped pulses and repetitive pulse trains, such as the Carr–Purcell–Meiboom–Gill scheme. In particular, the results demonstrate when z-rotations induced by various pulse schemes can be compensated by adjustment of time delays flanking the pulse sequence or minimized by phase alternating schemes. When Eq. (14) cannot be solved analytically, perturbation theory can be used to determine the limiting properties of a particular rf sequence. Near resonance, the solution to Eq. (14) is assumed to have the form:
yðtÞ ¼ y0 ðtÞ þ ey1 ðtÞ þ e2 y2 ðtÞ þ
ð21Þ
in which e = X/x1 and x1 is the maximum strength of the rf field. Substituting for y(t) and equating terms of the same order in e transforms Eq. (14) into a system of equations:
d2 yn ðtÞ dt 2
d ln xðtÞ dy0 ðtÞ dt dt
þ x ðtÞy0 ¼ 0
d ln xðtÞ dyn ðtÞ dt dt
ðtÞ þ 14 x2 ðtÞyn ¼ ix1 dyn1 dt
1 4
2
ð22Þ ðn P 1Þ
The first equation is simply the result for an on-resonance pulse and has the solution y0(t) = cos[d(t)/2], in which:
dðtÞ ¼
Z
t 0
xðt Þdt 0
0
dun ðtÞ dv n ðtÞ yþ ðtÞ þ y ðtÞ ¼ 0 dt dt dun ðtÞ dyþ ðtÞ dv n ðtÞ dy ðtÞ dy ðtÞ þ ¼ ix1 n1 dt dt dt dt dt
ð23Þ
ð24Þ
Substituting for y(t) in Eq. (11) and solving for f(sp) to second-order in e yields:
ie e2 dðtÞ Sðsp Þ S2 ðsp Þ tan 2 sin dðsp Þ 2dðsp Þ ð25Þ þSðsp ÞCðsp Þ þ Gðsp Þ
f ðsp Þ tan
3. Experimental methods
d2 y0 ðtÞ dt 2
is the instantaneous accumulated rotation angle. The solutions of Eq. (22) for n P 1 have the form yn(t) = un(t)y+(t) + vn(t)y(t), in which y±(t) = exp[±id(t)] and un(t) and vn(t) satisfy the equations:
dðsp Þ 2
1þ
in which,
Sðsp Þ ¼ x1 Cðsp Þ ¼ x1
Z sp
sin½dðtÞdt
Z0 sp
cos½dðtÞdt
ð26Þ
0
Gðsp Þ ¼ x21
Z sp Z 0
t0
0
sin½dðt 0 Þ dðtÞdt dt
0
S(sp) is the integrated y-magnetization over the duration of the rf pulse and d(sp) is the nominal rotation angle of the pulse. Notably, the integrals in Eqs (25) and (26) depend only on the pulse shape and not on the resonance offset. The second-order expansion gives:
tan cðsp Þ eSðsp Þ= sin dðsp Þ e2
cos½bðsp Þ cos dðsp Þ þ Sðsp ÞCðsp Þ sin dðsp Þ þ Gðsp Þ sin dðsp Þ 2 Sðsp Þ cos dðsp Þ ð27Þ and the next correction to tan[c(sp)] is of order e3. If a(sp) and c(sp) are proportional to X over some bandwidth near resonance for a particular rf pulse, then the pulse can be treated as a sequence sa–R/[b(sp)]–sc, in which R/(b) is a rotation by an angle b around an axis with phase / in the transverse plane, sa = a(sp)/X and sc = c(sp)/X. Using tan[c(sp)] c(sp) near resonance yields,
sc
1 sin dðsp Þ
Z sp
sin½dðtÞdt
ð28Þ
0
When the average y-magnetization is zero, then c(sp) = 0 and no phase correction is needed following the pulse. An analogous expression for sa is obtained by time-reversing the pulse:
sa
1 sin ~dðsp Þ
Z sp
sin½~dðtÞdt
ð29Þ
0
in which
~dðtÞ ¼
Z sp
xðt0 Þdt0
ð30Þ
sp t
The above results suggest that all amplitude-modulated p/2 pulses will have a(sp) and c(sp) proportional to X over at least some bandwidth |X/x1| 1 and allow direct calculation of sa and sc as integrals over the rf field amplitude function. Far from resonance, a perturbation solution for f(t) is directly obtained from the Riccati equation by writing
f ðtÞ ¼ f 0 ðtÞ þ ef 1 ðtÞ þ e2 f 2 ðtÞ þ e3 f 3 ðtÞ . . .
ð31Þ
in which e = x1/X. By substitution into Eq. (9), the values of f0(t) = f2(t) = f4(t) = 0 and
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114 3.5
γ (τp) = α(τp)
3.0
6
a
2.5
c
5
e 2.0
2.5
4
2.0
3
1.5
1.5
2
1.0
1.0
1
0.5
0
0.0
–1
2.0
0.5 0.0
b
d
f
|β(τp)|
1.5 1.0 0.5
0
1
2
3
4
0
1
2
3
0
4
1
2
3
4
Ω/ω1 Fig. 1. Euler angles (a, c, e) a(sp) = c(sp) and (b, d, f) b(sp) and for a square (a, b) 90°, (c, d) 270°, and (e, f) 60° x-pulse are shown as a function of the offset parameter X/x1. The limiting (red) near-resonance (X/x1 6 1 and (blue) off-resonance (X/x1 P 1) values also are depicted. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
30
a 20
20
a 10
0
α (τp)
α (τp)
10
–10
0
–20
–10
–30
b
–20
1.5
2.0
|β(τp)|
b 1.0
1.5
|β (τp)|
0.5
1.0
0.5
0.0 3
0.0
2
3
γ (τp)
1
2 0
1
γ (τp)
–1 –2 –3 –1.5
0 –1
c –1.0
–0.5
0.0
0.5
1.0
1.5
Ω/ω1 Fig. 2. Euler angles (a) a(sp), (b) b(sp), and (c) c(sp) for a EBURP-2 pulse are shown as a function of the ratio of resonance offset and B1 field, X/x1. (red) The limiting near-resonance linear dependence of a(sp) over the bandwidth ± 0.3X/x1 is shown. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
–2 –3 –1.5
c –1.0
–0.5
0.0
0.5
1.0
1.5
Ω/ω1 Fig. 3. Euler angles (a) a(sp), (b) b(sp), and (c) c(sp) for a Q5 pulse are shown as a function of the ratio of resonance offset and B1 field, X/x1.
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114
and the third-order term has been included in the expression for b for increased accuracy (the first-order result replaces xe by X). The propagator is
Table 1 z-Rotation scaling factors for shaped pulses.
1 2
Pulse
sa/sp
sc/sp
Square Sinc1 Gaussian2 E-BURP2 Q5
0.637 0.597 0.589 0.718 0
0.637 0.597 0.589 0 0
Defined as the center lobe of the sinc function. Truncated at an amplitude of 0.0471 x1.
2x1
eiXsp
X
X
eiXsp 3
Z sp
8x1
eiXt xðtÞ
Z
In the absence of the pulse, transverse magnetization of the off-resonance spin would acquire a phase Xsp; as shown by the first factor of Eq. (40), the phase acquired during the pulse is 2c(sp) so that the off-resonance phase error becomes:
t
0
eiXt xðt 0 Þdt
0
ð33Þ
2 dt
0
0
Thus, to second order in x1/X,
Z 2 1 sp iXt cos bðsp Þ ¼ 1 e xðtÞdt 2
ð34Þ
0
yielding the first-order estimate:
Z sp bðsp Þ ¼ eiXt xðtÞdt
ð35Þ
0
The z-rotation angle c(sp) is given by
cðsp Þ ¼ Arg½f ðsp Þ ¼ Xsp =2 þ Arg " Z
Z sp
eiXðtsp =2Þ xðtÞdt
0
sp
þ Arg 1 þ
eiXt xðtÞ
Z
0
t
0
eiXt xðt 0 Þdt
0
# 2 , Z sp dt 4 eiXt xðtÞdt 0
0
ð36Þ
As X ? 1, c(sp) ? Xsp/2; consequently, the second term in the above equation is negligible and
cðsp Þ Xsp =2 ¼ Im
"Z
sp
# Z t 2 Z s p 0 0 eiXt xðtÞ eiXt xðt0 Þdt dt 4 eiXt xðtÞdt 0
0
0
ð37Þ
The square pulse is analyzed first to illustrate the properties of the perturbation solutions. Close to resonance, Eq. (27) yields:
aðsp Þ ¼ cðsp Þ ¼ ðX=x1 Þ tanðw=2Þ e2
cos bðsp Þ cos w þ ð1 cos wÞ 2 cos2 w w sin w
ð38Þ
2
in which the nominal rotation angle w = x1sp. Thus, a square pulse with x-phase can be regarded near resonance as an x-rotation preceded and followed by a delay s = tan(x1sp/2)/x1 = sptan(w/2)/w. If x1sp = p/2 for a nominal 90° pulse, then s = 2sp/p as also can be shown by standard techniques [1]. Far from resonance, Eqs. (35) and (37) yield:
aðsp Þ ¼ cðsp Þ ¼ Xsp =2 þ bðsp Þ ¼
2x1
X
2
ð40Þ
ð32Þ
eiXt xðtÞdt
0
3
f 3 ðsp Þ ¼
Z sp
¼ exp½i2cðsp ÞIz 2x1 exp i sinðxe sp =2Þ Ix cosðcðsp ÞÞ Iy sinðcðsp ÞÞ
X
Solving these two equations yields
X
X
exp½i2cðsp ÞIz h x i 1 Ix sinðxe sp Þ Iy ð1 cosðxe sp ÞÞ exp i
df1 ðtÞ X ¼ iXf 1 ðtÞ þ xðtÞ dt 2x1 df3 ðtÞ X ¼ iXf 3 ðtÞ þ xðtÞf 21 ðtÞ dt 2x1
f 1 ðsp Þ ¼
2x1 U exp icðsp ÞIz exp i sinðxe sp =2ÞIx exp½icðsp ÞIz
x21 2
4X
½sin ðXsp =2Þ þ
Xsp sinðXsp Þ xe sp =2
x21 4X2
Xsp sinðXsp Þ
1=2
x21 4X2
2x1
X
sin
sinðXsp Þ
xe sp =2
ð39Þ
/NR ¼
x21 2
2X
Xsp sinðXsp Þ
ð41Þ
This expression approaches the well-known result /NR ¼ x21 sp =ð2XÞ of McCoy and Mueller [11] when Xsp 1, but is more accurate for intermediate cases. The second factor is a rotation by an angle around an axis in the transverse plane that is phase-shifted by an angle v = tan1[(cos(xesp) 1)/sin(xesp)] = xesp/2. In many applications [1], the amplitude of the x1 field is adjusted to yield the rotation angles x1sp on-resonance and 2np at an offset frequency X; consequently, b(sp) = 0. The expressions given in Eqs. (35) and (37), although more computationally demanding, are generally more accurate than the analogous expressions of McCoy and Mueller [11] and are related to expressions obtained from average Hamiltonian theory by Warren [3]. For illustration, the values of b(sp), and c(sp) for square x-pulses are shown in Fig. 1. The more useful applications of the above formalism are to rf pulses with time-varying parameters. EBURP-2 [12] and Q5 [13] pulses are widely used band-selective 90° excitation pulses. The results for a(sp), b(sp), and c(sp), calculated from the analytical forms of the pulse shapes by solution of Eqs. (14) for y(t), are shown in Figs. 2 and 3; identical results are obtained by discretizing the pulse shapes and applying Eqs. (19) and (20). The values of a(sp) and c(sp) are approximately linear functions of X for a bandwidth of ±0.3X/x1 for the EBURP-2 pulse and ±0.4X/x1 for the Q5 pulse. Thus, both pulses can be described within their bandwidths by the sequence sa–90°x–sc with sa = 0.718 sp and sc 0 for the EBURP-2 pulse and sa = sc 0 for the Q5 pulse, calculated using Eqs. (28) and (29). A numerical fit to the value of a(sp) near resonance in Fig. 2a yields an estimate of sa = 0.70. The value of sa for the EBURP-2 pulse agrees well with the fitted value of 0.67sp reported previously from numerical fitting of the phase of transverse magnetization following the pulse [8]. The calculated nearresonance z-rotation angles for the Q5 pulse are a(sp) = 0.013Xsp and c(sp) = 0.076Xsp; thus, as also illustrated by Fig. 3b and c, slightly better phase performance for excitation of Mz ? My is expected for the time-reversed Q5 pulse. Results for sa and sc are given in Table 1 for these and other common pulse shapes. A number of novel pulse shapes with exceptional properties have been determined by optimal control methods [14]. Fig. 4a shows the pulse shape developed by Li and coworkers for broadband excitation with minimal pulse power requirements [15]. The excitation profile is shown in Fig. 4b. The corresponding Euler rotation angles determined from Eq. (26) are shown in Fig. 4c–d. The results show that over the excitation bandwidth of ±20 kHz for x1 = 20 kHz, the pulse can be regarded as the sequence s–90°x, with s = 0.87sp. Reversing the pulse shape profile exchanges
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114 15
a
c 10
0.5
5
α (τp), γ (τp)
ωx(t)/ω1max, ωy(t)/ω1max
1.0
0
0 –5
–0.5 –10 –1.0
–15 0
0.2
0.4
0.6
0.8
1.0
t/τp 2.5
1.0
d 2.0
0.5
β (τp)
My
1.5 0
1.0
–0.5 0.5
b –1.0 –40
–20
0
20
0.0 –40
40
–20
Ω (kHz)
0
20
40
Ω (kHz)
Fig. 4. Minimum-energy broadband excitation pulse. (a) The (black) xx(t) and (red) xy(t) components of the pulse shape are shown. (b) The excitation profile for zmagnetization is shown as a function of the resonance offset X. (c) The Euler angles (black) a(sp) and (red) c(sp) and (d) b(sp) are shown as a function of the resonance offset X. The pulse had a length of 100 ls and x1/(2p) = 20 kHz. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
1.0
0.02
0.5 ^
β
0.01 0.0 –0.5
0.00
–1.0 –0.01 –1.5
^ ^ α/γ
–2.0 0.10
a
0.05
0.05
0.00
0.00
–0.05
–0.05
–0.10 –0. 3
c
–0.02 0.10
b –0. 2
–0. 1
0.0
0.1
0.2
0.3
–0.10 –0. 3
d –0. 2
–0. 1
0.0
0.1
0.2
0.3
Ω/ω1 ^ for (black) conventional two-pulse and (red) Yip and Zuiderweg [17] four-pulse phase alternating scheme. Fig. 5. CPMG pulse sequences. (a) The effective x-rotation angle b ^ and (d) (red) a ^ for the (black) conventional and (red) a ^ and (blue) c ^ for the Yip and Zuiderweg schemes. The (c) effective x-rotation angle b ^ (b) The effective z-rotation angles a ^ for the eight-pulse phase alternating scheme. All calculations used x1/2p = 6250 Hz, scp = 0.5 ms, in which scp is the spacing between p pulses. (For interpretation and (blue) c of the references to color in this figure legend, the reader is referred to the web version of this article.)
a(sp; X) = c(sp; X), c(sp) = a(sp; X), and b(sp; X) = b(sp; X); thus, the time-reversed pulse can be regarded as the sequence 90°x–s, allowing broadband transformations from y- to z-magnetization. The Euler angle representation allows convenient
representation of the properties of pulses with complex rf profiles for comparison with other pulse shapes, e.g. comparing Figs. 2–4. The formalism is also applicable to pulse trains consisting of pulse-interrupted-free-precession sequence blocks:
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s1 Pðx11 ; sp1 ; /1 Þ s2 Pðx12 ; sp2 ; /2 Þ s3
g ¼ tan1
in which sn is a time delay and P(x1n, spn, /n) is a square pulse applied with rf field x1n, length spn, and phase /n, is commonly encountered in NMR spectroscopy. Using Eqs. 3, 19 and 20 yields the propagator for this pulse sequence as:
U ¼ exp ½iðXs3 þ c2 þ cÞIz exp½ibIx exp ½iðXs1 þ a1 þ aÞIz
ð42Þ
in which an, bn, and cn are given in Eq. (18) for the nth pulse,
a ¼ tan1
2 sin b2 tanðd=2Þ sinðb1 þ b2 Þ þ sinðb1 b2 Þ tan2 ðd=2Þ 2
ð43Þ
and d = a2 + Xs2 + c1 = d0 /2 + /1. In the limit that s2 ? 0 (d ? a2 + c1), Eq. (43) yields the Euler angle composition rule [16], which defines the effective Euler angles from two successive rotations. If the two pulses are applied with values of x1 and sp (but potentially different phases), then a simpler result is obtained:
a ¼ c ¼ tan1
tanðd=2Þ cos b1
ð44Þ
The Carr–Purcell–Meiboom–Gill (CPMG) sequence, s–p/1–2s–p/2–s, with x1sp = p is an example of a pulse sequence with identical pulse lengths. Application of Eq. (44) gives: U/2 /1 ¼ exp ½ifg þ ð3/2 /1 Þ=2gIz exp ½ibIx exp ½ifg þ ð/2 3/1 Þ=2gIz ð45Þ
in which
y-magnetization
sin d cos d tan2 ðb1 =2Þ
1
ð46Þ
The conventional two-pulse CPMG pulse sequence element has the propagator Uxx with /1 = /2 = 0. Yip and Zuiderweg [17] proposed a four-pulse CPMG sequence, s–px–2s–px–2s–py–2s–py–s. A propagator for this sequence is given by U ¼ Uyy Uxx , in which:
Uxx ¼ exp ½ig Iz exp ibþ Ix exp ½ig Iz
Uyy ¼ exp igþ Iz exp ½ib Ix exp igþ Iz
g ¼ tan1
sin d0 cos d0 tan2 ðb1 =2Þ
ð47Þ
ð48Þ
h i 2 b ¼ cos1 1 sin b1 ð1 cos d0 Þ
This propagator can be recast by the same approach embodied in Eqs. (42) and (43) to give:
h i ^ x exp ½ia ^Iz exp ibI ^ Iz Uyy Uxx ¼ exp ½ic
ð49Þ
in which
"
a^ ¼ g þ tan1
h i 2 b ¼ cos1 1 2 sin b1 cos2 ðd=2Þ
a
and
b ¼ cos1 ½cosðb1 þ b2 Þ cos2 ðd=2Þ þ cosðb1 b2 Þ sin ðd=2Þ 2 sin b1 tanðd=2Þ c ¼ tan1 sinðb1 þ b2 Þ sinðb1 b2 Þ tan2 ðd=2Þ
# 2 sin b tan g þ sinðbþ b Þ sinðbþ þ b Þ tan2 g
sin2 g cos2 g ^ ¼ cos1 ½1 2 sin2 b 2 sin2 Db b " # 2 sin bþ tan g c^ ¼ gþ þ tan1 sinðbþ b Þ sinðbþ þ b Þ tan2 g
ð50Þ
¼ ðb þ b Þ=2; Db ¼ ðb b Þ=2, and g ¼ ðg þ gþ Þ=2. Illustrative b þ þ calculations of the rotation angles for the two CPMG variants are shown in Fig. 5a and b. The major difference between the sequences ^ c ^, with an absolute is that for the Yip and Zuiderweg scheme, a difference of 60.5° for 0.3 6 X/x1 6 0.3 for the parameters used
c
e
d
f
0.8 0.6 0.4 0.2 0
x-magnetization
b
1
0.8 0.6 0.4 0.2 0 –1 –0.8 –0.6 –0.4 –0.2
0
Ω/ω1
0.2 0.4 0.6 0.8
1
–0.4
–0.2
0
Ω/ω1
0.2
0.4
–0.4
–0.2
0
0.2
0.4
Ω/ω1
Fig. 6. Refocused transverse magnetization as a function of the ratio of resonance offset and B1 field, X/x1. Plots show refocused (b, d, f) x-magnetization or (a, c, e) ymagnetization following CPMG pulse trains beginning with initial (b, d, f) x-magnetization or (a, c, e) y-magnetization. Panels (a) and (b) show the refocused magnetization after one cycle of CPMG pulse train. Panels (c, d, e, f) show the refocused magnetization after n cycles, averaged over n = 1–100. For panels (e) and (f), 10% rf inhomogeneity (FWHM) was included in the calculation. Each cycle consists of 8 pulses for all four schemes. The color coding is (blue) conventional x-phase, (black) Yip and Zuiderweg phase alternating scheme, (red) proposed eight-pulse phase alternating scheme, and (green) XY-8 scheme. All calculations used x1/2p = 2500 Hz, scp = 1.25 ms, in which scp is the spacing between p pulses. The calculations were performed using the software package SIMPSON [25]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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for Fig. 5. Thus, the propagator for the complete CPMG train, consisting of n repetitions of the four-step cycle simplifies to
Uyy Uxx
n
h i ^ x exp ½ia ^Iz exp inbI ^ Iz exp ½ic
ð51Þ
and, unlike the conventional sequence, the effects of the z-rotations are not cumulative over the n cycles. Eq. (49) indicates that the eight-pulse phase alternating scheme formed by concatenating the Yip and Zuiderweg four-pulse scheme and its phase-inverted counterpart, yielding the phases x, x, y, y, x, x, y, y, has a propagator described by:
U ¼ Uyy Uxx Uyy Uxx ¼ Uyy Uxx Uyy Uxx h i h i ^ x exp ½ia ^ x exp ½ia ^Iz exp ibI ^ Iz exp ½ic ^Iz exp ibI ^ Iz ¼ exp ½ic h i h i ^ x exp ibI ^ x exp ½ia ^Iz exp ibI ^ Iz exp ½ic ^Iz exp ½ia ^ Iz E exp ½ic ð52Þ
1.0
1.0
0.8
0.8
x/y-magnetization
x-magnetization
in which E is the identity operator. The calculations shown in Fig. 5c demonstrate the effective x-rotation is reduced by two orders of magnitude in the eight-pulse scheme compared to the four-pulse scheme shown in Fig. 5a and nearly an order of magnitude smaller than the XY-8 scheme [18] (not shown). The residual x-rotation is 61° over the range 0.3 6 X/x1 6 0.3 (and 61.7° for a variation in x1 of ±10%); consequently the two z-rotations shown in Fig. 5d nearly cancel to yield a propagator proportional to the identity operator. Thus, the proposed eight-pulse phase alternating scheme should refocus transverse magnetization with arbitrary initial phase more accurately than XY-8 (the conventional or Yip and Zuiderweg
schemes do not efficiently refocus magnetization with initial yphase). Fig. 6a and b show the numerically calculated magnetization after a single cycle of each refocusing scheme as a function of resonance offsets. For fair comparison, each cycle consists of eight pulses for all schemes. The performance of the different refocusing schemes also depends on the extent to which phase errors accumulate as more than one cycle of the CPMG pulse train is applied. In particular, the residual x-phase rf fields in the conventional and Yip and Zuiderweg schemes dominate small non-commuting terms in the Hamiltonian; in contrast, sequences proportional to the identity operator, like the proposed sequence or XY-8, do not have this property and can be expected to be more sensitive to accumulated effects. The magnitudes of refocused magnetization have been calculated following 1–100 cycles of all four refocusing schemes. Fig. 6c and d show the average refocused transverse magnetization. The Yip and Zuiderweg scheme, as predicted from Eq. (51) has the largest bandwidth (0.35|X/x1|). The proposed eight-pulse scheme has a bandwidth ((0.16|X/x1|) substantially larger than the XY-8 scheme (0.1|X/x1|) both in the absence and the presence of rf inhomogeneity (Fig. 6c–f). Only the proposed eight-pulse scheme and XY-8 refocus y-magnetization. Experimental validation of these predictions is shown in Fig. 7. As shown in Fig. 7a, both the Yip and Zuiderweg and the proposed CPMG sequences conserve >95% of both initial x- and y-magnetization at resonance offsets up to 0.5|X/x1| for a single cycle consisting of eight pulses (and a total time of 8 ms). The bandwidth of the Yip and Zuiderweg scheme is largely maintained as the number of cycles in the pulse train increases, while the bandwidth for the proposed sequence is reduced to 0.3|X/x1| for 10 cycles (a total time of 80 ms). As shown
0.6 0.4 0.2
0.6 0.4 0.2
a
b
0
0 0
0.2
0.4
0.6
0.8
0
0.2
0.4
Ω/ω1
0.6
0.8
Ω/ω1
x−magnetization
1 0.8 0.6 0.4 0.2
c
d
e
0 0
0.2
0.4
Ω/ω1
0.6
0
0.2
0.4
Ω/ω1
0.6
0
0.2
0.4
0.6
Ω/ω1
Fig. 7. CPMG performance demonstrated from normalized intensities for the 1H resonance of H2O in a doped water sample. (a) Refocused x-magnetization for the (black) proposed and (red) Yip and Zuiderweg phase alternating scheme for (circles) 8 ms, (squares) 40 ms, and (triangles) 80 ms pulse trains, corresponding to 1, 5, and 10 repetitions of an 8-pulse CPMG block, as described in Section 3. (b) Refocused (black) x-magnetization and (red) y-magnetization for the proposed eight-pulse phase alternating scheme for (circles) 8 ms, (squares) 40 ms, and (triangles) 80 ms pulse trains. (c–e) Refocused x-magnetization for (red, circle) the proposed eight-pulse scheme and (green, square) the XY-8 scheme for (c) 8 ms, (d) 40 ms, and (e) 80 ms pulse trains overlaid onto the x-magnetization from numerical simulations using (black) the proposed scheme and (blue) the XY-8 scheme. 10% rf inhomogeneity (FWHM) was included in the calculations. The magnetizations in panels (c–e) were normalized to the values at zero resonance offset for clarity. The relative performance of the proposed and XY-8 schemes for y-magnetization are similar to the performance for x-magnetization and is not shown. The Yip and Zuiderweg scheme does not refocus orthogonal y-phase magnetization and results are not shown for this sequence. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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1.5
a
d
b
e
c
f
Tr {IxHeff}τc
1 0.5 0 −0.5 −1
Tr {IyHeff}τc
1.5 1 0.5 0 −0.5 −1
Tr {IzHeff}τc
1.5 1 0.5 0 −0.5 −1 −0.6
−0.4
−0.2
0
0.2
0.4
0.6
−0.6
−0.4
−0.2
Ω/ω1
0
0.2
0.4
0.6
Ω/ω1
Fig. 8. The x-, y- and z-components of effective Hamiltonians from EEHT calculations for (a, b, c) the XY-8 and (d, e, f) proposed 8-pulse schemes, multiplied by the cycle time, sc. The product reflects the residual x-, y- and z-rotations each cycle. The calculations were performed at scp/s180 values of (red) 0, (green) 1, (blue) 10 and (magenta) 100. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
in Fig. 7b, the proposed CPMG sequence is equally efficient in refocusing x- or y-magnetization. The bandwidth of the proposed sequence, consistent with the calculations shown in Fig. 6 is larger than that for the XY-8 sequence, as shown in Fig. 7c–e for increasing numbers of cycles. Additional insights into the relative performance of the XY-8 and proposed schemes over many cycles can be gained from calculation of effective Hamiltonian. These calculations were performed using exact effective Hamiltonian theory (EEHT) [19,20] and expanded to fifth order with respect to resonance offset. For the XY-8 scheme,
"
Heff
# 0:5rcp þ 0:25 4 0:393r cp þ 0:785r 2cp 5 ¼ e e x1 I x rcp þ 1 rcp þ 1 " # 0:5rcp þ 0:25 4 0:393r cp þ 0:785r 2cp 5 þ e þ e x1 I y rcp þ 1 rcp þ 1 r cp þ 0:5 5 þ e x1 Iz þ Oðe6 Þ r cp þ 1
ð53Þ
For the proposed eight-pulse scheme,
Heff ¼
rcp þ 0:5 5 e x1 Iz þ Oðe6 Þ rcp þ 1
ð54Þ
in which e = X/x1, rcp = scp/s180, scp is the spacing between pulses, and s180 is the length of the 180° pulse. The EEHT results show that the proposed eight-pulse scheme gives superior performance over XY-8, regardless of the number cycles applied, because the z-components of the effective Hamiltonians are identical in the fifth order expansion, whereas the new scheme compensates x- and y-components to higher order than the XY-8 scheme. Although simple and more intuitive, representation of effective Hamiltonian by power series is less accurate when the value of rcp is large. Fig. 8 shows
the exact effective Hamiltonian, without truncation to powers, as a function of resonance offsets at different rcp values. When rcp is large, the effective Hamiltonians oscillate at high frequency outside the effective bandwidth and therefore cannot be well represented by power series expansion. However, the numerical representations of effective Hamiltonian obtained from its exact expression are consistent with the numerical results shown in Fig. 6. The performance of the proposed eight-pulse scheme is not significantly altered by expanding into a 16-step scheme RRRR in which R ¼ Uyy Uxx and the bar indicates phase inversion. Thus, the improved performance of the proposed scheme is not simply a consequence of imposing a supercycle, as utilized in decoupling schemes [1]. The above considerations indicate that the Yip and Zuiderweg phase alternating scheme has a superior bandwidth than the other schemes if initial x-magnetization (i.e. magnetization with the same phase as the first 180° pulse) is to be refocused, but that the proposed scheme has the best performance if both x-magnetization and y-magnetization (orthogonal to the first pulse) must be retained. The Yip and Zuiderweg scheme is the choice for most CPMG relaxation dispersion experiments [21] and the proposed scheme for applications such as CPMG-INEPT polarization transfers [22,23] and a subset of CPMG relaxation dispersion experiments, in which the starting magnetization is of arbitrary phase [24]. 5. Conclusion The Euler-angle rotation operator factorization of the propagator for time-varying rf pulses and pulse-interrupted-free-precession trains provides insights into performance that are not as easily discerned by average Hamiltonian theory or by numerical calculation of the evolution of magnetization through the pulse. The Euler-angle representation is particularly useful in determining whether a shaped pulse or pulse train can be represented over
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some bandwidth by a sequence s1–R/(b)–s2, allowing phase evolution during the pulse to be compensated by adjusting time periods prior or subsequent to the pulse. Explicit solutions for the Euler rotation angles have been derived using perturbation theory nearand off-resonance; the resulting expressions are useful especially for characterizing phase evolution of magnetization during shaped pulses. The Euler-angle approach also is powerful in analyzing repetitive pulse trains, such as the CPMG sequence and has been used to identify an eight-step phase alternating scheme that performs better than the XY-8 scheme in refocusing magnetization orthogonal to the phase of the first pulse in the CPMG train. Acknowledgments Support from the National Institutes of Health Grants GM059273 (A.G.P.) and GM063855 (M.R.) are gratefully acknowledged. We thank Justin Ruths and Jr-Shin Li for providing the pulse shape file for the broadband excitation pulse described in [15]. References [1] J. Cavanagh, W.J. Fairbrother, A.G. Palmer, M. Rance, N.J. Skelton, Protein NMR Spectroscopy: Principles and Practice, second ed., Academic Press, San Diego, CA, 2007. [2] U. Haeberlen, J.S. Waugh, Coherent averaging effects in magnetic resonance, Phys. Rev. 175 (1968) 453–467. [3] W.S. Warren, Effects of arbitrary laser or NMR pulse shapes on populationinversion and coherence, J. Chem. Phys. 81 (1984) 5437–5448. [4] J. Zhou, C. Ye, B.C. Sanctuary, Rotation operator approach to spin dynamics and the Euler geometric equations, J. Chem. Phys. 101 (1994) 6424–6429. [5] D.J. Siminovitch, Rotations in NMR: Part II. Applications of the Euler–Rodrigues parameters, Concepts Magn. Reson. 9 (1997) 211–225. [6] D.J. Siminovitch, Rotations in NMR: Part I. Euler–Rodrigues parameters and quaternions, Concepts Magn. Reson. 9 (1997) 149–171. [7] R.M. Gregory, A.D. Bain, The effects of finite rectangular pulses in NMR: phase and intensity distortions for a spin-1/2, Concepts Magn. Reson. 34A (2009) 305–314. [8] E. Lescop, T. Kern, B. Brutscher, Guidelines for the use of band-selective radiofrequency pulses in hetero-nuclear NMR: example of longitudinalrelaxation-enhanced BEST-type 1H–15N correlation experiments, J. Magn. Reson. 203 (2010) 190–198.
[9] N.I. Gershenzon, T.E. Skinner, B. Brutscher, N. Khaneja, M. Nimbalkar, B. Luy, S.J. Glaser, Linear phase slope in pulse design: application to coherence transfer, J. Magn. Reson. 192 (2008) 235–243. [10] D.J. Siminovitch, Comment on ‘‘Rotation operator approach to spin dynamics and the Euler geometric equations’’ [J. Chem. Phys. 101, 6424 (1994)], J. Chem. Phys. 103 (1995) 2766–2768. [11] M.A. McCoy, L. Mueller, Nonresonant effect of frequency-selective pulses, J. Magn. Reson. 99 (1992) 18–36. [12] H. Geen, R. Freeman, Band-selective radiofrequency pulses, J. Magn. Reson. 93 (1991) 93–141. [13] L. Emsley, G. Bodenhausen, Optimization of shaped selective pulses for NMR using a quaternion description of their overall propagators, J. Magn. Reson. 97 (1992) 135–148. [14] M. Nimbalkar, B. Luy, T.E. Skinner, J.L. Neves, N.I. Gershenzon, K. Kobzar, W. Bermel, S.J. Glaser, The Fantastic Four: a plug ‘n’ play set of optimal control pulses for enhancing NMR spectroscopy, J. Magn. Reson. 228 (2013) 16–31. [15] J.S. Li, J. Ruths, T.-Y. Yu, H. Arthanari, G. Wagner, Optimal pulse design in quantum control: a unified computational method, Proc. Natl. Acad. Sci. USA 108 (2011) 1879–1884. [16] D.A. Varshalovich, A.N. Moskalev, V.K. Khersonskiı˘, Quantum Theory of Angular Momentum: Irreducible Tensors, Spherical Harmonics, Vector Coupling Coefficients, 3nj Symbols, World Scientific Pub., Singapore; Teaneck, NJ, USA, 1988. [17] G.N.B. Yip, E.R.P. Zuiderweg, A phase cycle scheme that significantly suppresses offset-dependent artifacts in the R2-CPMG 15N relaxation experiment, J. Magn. Reson. 171 (2004) 25–36. [18] T. Gullion, D.B. Baker, M.S. Conradi, New, compensated Carr–Purcell sequences, J. Magn. Reson. 89 (1990) 479–484. [19] T.S. Untidt, N.C. Nielsen, Closed solution to the Baker–Campbell–Hausdorff problem: exact effective Hamiltonian theory for analysis of nuclear-magneticresonance experiments, Phys. Rev. E 65 (2002) 021108. [20] D. Siminovitch, T. Untidt, N.C. Nielsen, Exact effective Hamiltonian theory. II. Polynomial expansion of matrix functions and entangled unitary exponential operators, J. Chem. Phys. 120 (2004) 51–66. [21] D.M. Korzhnev, L.E. Kay, Probing invisible, low-populated states of protein molecules by relaxation dispersion NMR spectroscopy: an application to protein folding, Acc. Chem. Res. 41 (2008) 442–451. [22] L. Mueller, P. Legault, A. Pardi, Improved RNA structure determination by detection of NOE contacts to exchange-broadened amino protons, J. Am. Chem. Soc. 117 (1995) 11043–11048. [23] F.A.A. Mulder, C.A.E.M. Spronk, M. Slijper, R. Kaptein, R. Boelens, Improved HSQC experiments for the observation of exchange broadened signals, J. Biomol. NMR 8 (1996) 223–228. [24] V.Y. Orekhov, D.M. Korzhnev, L.E. Kay, Double- and zero-quantum NMR relaxation dispersion experiments sampling millisecond time scale dynamics in proteins, J. Am. Chem. Soc. 126 (2004) 1886–1891. [25] M. Bak, J.T. Rasmussen, N.C. Nielsen, SIMPSON: a general simulation program for solid-state NMR spectroscopy, J. Magn. Reson. 147 (2000) 296–330.
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Journal of Magnetic Resonance journal homepage: www.elsevier.com/locate/jmr
Rotation operator propagators for time-varying radiofrequency pulses in NMR spectroscopy: Applications to shaped pulses and pulse trains Ying Li a,1, Mark Rance b, Arthur G. Palmer III a,⇑ a
Department of Biochemistry and Molecular Biophysics, Columbia University, 630 West 168th Street, New York, NY 10032, United States Department of Molecular Genetics, Biochemistry and Microbiology, University of Cincinnati, 231 Albert Sabin Way, Medical Sciences Building, Cincinnati, OH 45267-0524, United States b
a r t i c l e
i n f o
Article history: Received 22 June 2014 Revised 31 August 2014 Available online 22 September 2014 Keywords: Rotation operator Euler angle Perturbation expansion Shaped pulse CPMG
a b s t r a c t The propagator for trains of radiofrequency pulses can be directly integrated numerically or approximated by average Hamiltonian approaches. The former provides high accuracy and the latter, in favorable cases, convenient analytical formula. The Euler-angle rotation operator factorization of the propagator provides insights into performance that are not as easily discerned from either of these conventional techniques. This approach is useful in determining whether a shaped pulse can be represented over some bandwidth by a sequence s1–R/(b)–s2, in which R/(b) is a rotation by an angle b around an axis with phase / in the transverse plane and s1 and s2 are time delays, allowing phase evolution during the pulse to be compensated by adjusting time periods prior or subsequent to the pulse. Perturbation theory establishes explicit formulas for s1 and s2 as proportional to the average transverse magnetization generated during the shaped pulse. The Euler-angle representation of the propagator also is useful in iterative reduction of pulse-interrupted-free precession schemes. Application to Carr–Purcell–Meiboom–Gill sequences identifies an eight-pulse phase alternating scheme that generates a propagator nearly equal to the identity operator. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Numerous techniques in NMR spectroscopy utilize radiofrequency (rf) pulses in which the carrier frequency, amplitude, and/or phase are time-dependent. Examples include shaped pulses, shifted laminar pulses, adiabatic pulses, and pulse-interruptedfree-precession sequences [1]. For an isolated spin, in the absence of scalar couplings and ignoring relaxation, evolution of the density operator during such a pulse scheme is described by
rðtÞ ¼ U r ð0ÞU1
ð1Þ
in which r(t) is the density operator, t is the length of the pulse, and U is the propagator given by
Z t U ¼ T exp i xx ðt0 ÞIx þ xy ðt0 ÞIy þ Xðt0 ÞIz dt0
ð2Þ
0
T is the Dyson time-ordering operator, xx(t) and xy(t) are the timedependent rf amplitudes with x- and y-phase, respectively, and X(t) ⇑ Corresponding author. Fax: +1 212 305 7932. E-mail address: [email protected] (A.G. Palmer III). Present address: Department of Chemistry, University of Louisville, 2320 South Brook Street, Louisville, KY 40208, United States. 1
http://dx.doi.org/10.1016/j.jmr.2014.09.001 1090-7807/Ó 2014 Elsevier Inc. All rights reserved.
is the time-dependent resonance offset of the nuclear spin under consideration. The two most common approaches for calculating the evolution of the density operator under the propagator are average Hamiltonian theory (AHT) [2,3], which has the advantage of yielding analytically tractable results in many instances, and direct numerical integration, which has the advantage of high accuracy. In the latter case, the integral is expressed as a summation using a time step Dt sufficiently short that the exponential of the sum can be expressed as the product of individual rotations during each time step [1]. The rotation operator formalism of Sanctuary and coworkers [4] and Siminovitch [5,6] provides an alternative approach and is the subject of the present work. In this method, the propagator given in Eq. (2) is expressed as:
U ¼ exp ½icðtÞIz exp ½ibðtÞIx exp ½iaðtÞIz
ð3Þ
Thus, evolution of the density operator is calculated as the product of three individual rotation operations by the Euler angles a(t), b(t), and c(t). As this approach makes clear, any time-varying rf pulse train can be represented as an x-rotation flanked by two (not necessarily identical) z-rotations. The initial z-rotation is inconsequential for an excitation pulse applied to equilibrium z-magnetization, so that any excitation pulse is described by an x-rotation followed by
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a z-rotation. Inversion of the initial equilibrium density operator is described simply by cos[b(t)]. The present work employs the rotation operator formalism to characterize shaped and other time-dependent rf pulse schemes commonly employed in modern NMR spectroscopy of biological macromolecules. The off-resonance effects of a square 90° pulse are known to be approximated by a sequence s–90°x–s, in which s = 2sp/p and sp is the on-resonance pulse length [1,7]. Recent work has used numerical methods to approximate the off-resonance effects of shaped pulses by a pulse rotation preceded and/or followed by a time delay [8]. As Eq. (3) indicates, such relationships arise whenever a(t) and/or c(t) are proportional to the resonance offset of spins within the bandwidth of the shaped pulse and such dependence can be required as part of the optimization of shaped pulses [9].
1
bðtÞ ¼ cos
" # 1 jf ðtÞj2 1 þ jf ðtÞj2
ð10Þ
cðtÞ ¼ Arg½f ðtÞ and by construction, a(0) = c(0), b(0) = 0, and f(0) = 0. The Riccati equation also can be converted to a second-order linear homogeneous equation by making the substitution:
d ln yðtÞ 1 ¼ x ðtÞf ðtÞ dt 2
ð11Þ
to yield:
2
d yðtÞ dt
2
d ln x ðtÞ dyðtÞ 1 þ iXðtÞ þ jxðtÞj2 y ¼ 0 dt dt 4
2. Theory
If x(t) is written as |x(t)|exp[i/(t)], then:
The functions a(t), b(t), and c(t) are found by differentiating Eqs. (2) and (3) with respect to time. The derivative of Eq. (2) is simply:
d yðtÞ
dU ¼ i xx ðtÞIx þ xy ðtÞIy þ XðtÞIz U dt
2
dt
2
dU dcðtÞ dbðtÞ Iz U þ ¼ i exp ½icðtÞIz Ix exp ½ibðtÞIx exp ½iaðtÞIz dt dt dt daðtÞ exp ½icðtÞIz exp ½ibðtÞIx Iz exp ½iaðtÞIz þ dt daðtÞ dcðtÞ dbðtÞ Iz þ Ix cos½cðtÞ þ Iy sin½cðtÞ þ ðIz cos½bðtÞ ¼ i dt dt dt ð5Þ Iy sin½bðtÞcos½cðtÞ þ Ix sin½bðtÞsin½cðtÞ U
ð13Þ and the term in curly braces can be regarded as a time-dependent resonance offset of the affected spin. A phase-shifted pulse is treated by subtracting the phase shift / from a(t) and adding / to c(t). Once Eqs. (12) or (13) are solved, either analytically or numerically, to obtain b(t) and c(t), a(t) is obtained by direct integration of Eq. (7a). The resulting values for t equal to the pulse length are used in Eqs. (1) and (3). For clarity in the following, the pulse length will be denoted sp, so that a(sp), b(sp), and c(sp) are the quantities of interest. For amplitude-modulated pulses, d/(t)/dt = 0 and x+(t) = x (t) = x(t), and Eq. (13) becomes:
2
Equating the terms in Eqs. (4) and (5) that are proportional to the same operator Ix, Iy, or Iz yields a system of equations:
ð6Þ
dbðtÞ ¼ xx ðtÞ cos½cðtÞ þ xy ðtÞ sin½cðtÞ dt
ð7bÞ
ð7cÞ
Eqs. (7b) and (7c) are a pair of coupled differential equations for b(t) and c(t). As first demonstrated by Zhou et al. [4] in the context of NMR, defining:
bðtÞ exp½icðtÞ 2
ð8Þ
enables Eqs. (7b) and (7c) to be combined to give the Riccati equation:
2 df ðtÞ 1
1
¼ xx ðtÞ ixy ðtÞ f ðtÞ þ iXðtÞf ðtÞ þ xx ðtÞ þ ixy ðtÞ dt 2 2 1 1 2 ¼ x ðtÞf ðtÞ þ iXðtÞf ðtÞ þ xþ ðtÞ 2 2 where x±(t) = xx(t) ± ixy(t). From the definition in Eq. (8),
d ln xðtÞ dyðtÞ 1 2 þ iX þ x ðtÞy ¼ 0 dt dt 4
ð14Þ
2
d yðtÞ 2
iX
dyðtÞ 1 2 þ x1 y ¼ 0 dt 4
ð15Þ
yðtÞ ¼ eiXt=2 fcosðxe t=2Þ iðX=xe Þ sinðxe t=2Þg ð7aÞ
f ðtÞ ¼ tan
2
with solution:
daðtÞ ¼ xx ðtÞ sin½cðtÞ xy ðtÞ cos½cðtÞ = sin½bðtÞ dt
dcðtÞ ¼ XðtÞ xx ðtÞ sin½cðtÞ xy ðtÞ cos½cðtÞ = tan½bðtÞ dt
dt
dt
These equations can be rearranged to give:
d yðtÞ
For a square pulse with xx(t) = x1, xy(t) = 0, and X(t) = X. Eq. (14) reduces to:
xx ðtÞ ¼
d ln jxðtÞj d/ðtÞ dyðtÞ 1 þ i XðtÞ þ jxðtÞj2 y ¼ 0 dt dt dt 4
ð4Þ
The derivative of Eq. (3) is
dbðtÞ daðtÞ cos½cðtÞ þ sin½bðtÞ sin½cðtÞ dt dt dbðtÞ daðtÞ xy ðtÞ ¼ sin½cðtÞ sin½bðtÞ cos½cðtÞ dt dt dcðtÞ daðtÞ XðtÞ ¼ þ cos½bðtÞ dt dt
ð12Þ
ð16Þ
Substitution into Eq. (11) gives:
f ðtÞ ¼ sin h tanðxe t=2Þ
1 þ i cos h tanðxe t=2Þ 1 þ cos2 h tan2 ðxe t=2Þ
ð17Þ
in which h = tan1(x1/X) is the tilt angle and xe ¼ X2 þ x21 the effective field. Using Eqs. (7a) and (10) yields:
1=2
is
aðsp Þ ¼ tan1 cos h tanðxe sp =2Þ / h
i 2 bðsp Þ ¼ cos1 cos2 h þ sin h cosðxe sp Þ
cðsp Þ ¼ tan1 cos h tanðxe sp =2Þ þ /
ð18Þ
The pulse phase / has been added in Eq. (18) to give the most general result. Eqs. (17) and (18) have been reported previously [4]. The propagator for a square pulse is then: "
U¼
cosðxe sp =2Þ icos hsin i/ ie sin hsin xe sp =2
xe sp =2 iei/ sin hsin xe sp =2 cosðxe sp =2Þ þ i cos hsin xe sp =2
#
ð19Þ
ð9Þ
Eqs. (12)–(14) assume that the rf field is described by an explicit function. In many cases, the pulse shape is defined by a list of N paired values giving the amplitude and phase (or xx and xy
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114
components) of the x1 field applied for a time interval Dt, with sp = NDt. The propagator U of Eq. (1) can be determined by iterative application of Eq. (19) for each step in the pulse shape. The resulting propagator is expressed as [10]
2
3 i½aðsp Þcðsp Þ=2 i½aðsp Þþcðsp Þ=2 e cos½bð s ie sin½bð s p Þ=2 p Þ=2 5 U¼4 i aðs Þcðsp Þ=2 ie ½ p sin½bðs Þ=2 ei½aðsp Þþcðsp Þ=2 cos½bðs Þ=2 p
p
ð20Þ and the parameters a(sp), b(sp), and c(sp) are obtained directly from this matrix.
The NMR spectra of H2O in a doped water sample (0.1 mg/ml GdCl3, 0.1% 13CH3OH, 1% H2O, 99% D2O; Cambridge Isotope Laboratories, CDLM-5010) were recorded using a pulse sequence 90/°– CPMGk–90/°–Grad–90x°–acquire, in which the first two 90° pulses have / = x or y phase, k is the number of CPMG blocks, and Grad is a z-gradient pulse to dephase transverse magnetization. Each CPMG block consisted of 8 repetitions of the element scp/2–180°–scp/2; the 180° pulse separation scp = 1 ms. The phases of the 180° pulses were set as described in Section 4. The carrier frequency for the first two 90° pulses and the CPMG blocks was varied between experiments from 0 to 10 kHz in 250 Hz steps. The last 90° pulse was applied on-resonance with the H2O signal. Pulses were applied with x1/2p = 10 kHz. The first and last delays in the CPMG sequences were shortened by 2p/s90, in which s90 is the 90° pulse length, to account for phase evolution during the first and second 90° pulses. Each experiment was recorded using 8 scans, a recycle delay of 8 s, a spectral width of 25 ppm, and 8192 complex data points. Experiments were conducted at T = 300 K on a Bruker 500 MHz DRX NMR spectrometer with a triple-resonance tripleaxis gradient probe. 4. Results and discussion The Euler-angle decomposition provides a representation of the propagator that is particularly convenient for NMR spectroscopy, in which z-rotations assume critical prominence. The Euler-angle decomposition is used in the following to establish general properties of amplitude-modulated shaped pulses and repetitive pulse trains, such as the Carr–Purcell–Meiboom–Gill scheme. In particular, the results demonstrate when z-rotations induced by various pulse schemes can be compensated by adjustment of time delays flanking the pulse sequence or minimized by phase alternating schemes. When Eq. (14) cannot be solved analytically, perturbation theory can be used to determine the limiting properties of a particular rf sequence. Near resonance, the solution to Eq. (14) is assumed to have the form:
yðtÞ ¼ y0 ðtÞ þ ey1 ðtÞ þ e2 y2 ðtÞ þ
ð21Þ
in which e = X/x1 and x1 is the maximum strength of the rf field. Substituting for y(t) and equating terms of the same order in e transforms Eq. (14) into a system of equations:
d2 yn ðtÞ dt 2
d ln xðtÞ dy0 ðtÞ dt dt
þ x ðtÞy0 ¼ 0
d ln xðtÞ dyn ðtÞ dt dt
ðtÞ þ 14 x2 ðtÞyn ¼ ix1 dyn1 dt
1 4
2
ð22Þ ðn P 1Þ
The first equation is simply the result for an on-resonance pulse and has the solution y0(t) = cos[d(t)/2], in which:
dðtÞ ¼
Z
t 0
xðt Þdt 0
0
dun ðtÞ dv n ðtÞ yþ ðtÞ þ y ðtÞ ¼ 0 dt dt dun ðtÞ dyþ ðtÞ dv n ðtÞ dy ðtÞ dy ðtÞ þ ¼ ix1 n1 dt dt dt dt dt
ð23Þ
ð24Þ
Substituting for y(t) in Eq. (11) and solving for f(sp) to second-order in e yields:
ie e2 dðtÞ Sðsp Þ S2 ðsp Þ tan 2 sin dðsp Þ 2dðsp Þ ð25Þ þSðsp ÞCðsp Þ þ Gðsp Þ
f ðsp Þ tan
3. Experimental methods
d2 y0 ðtÞ dt 2
is the instantaneous accumulated rotation angle. The solutions of Eq. (22) for n P 1 have the form yn(t) = un(t)y+(t) + vn(t)y(t), in which y±(t) = exp[±id(t)] and un(t) and vn(t) satisfy the equations:
dðsp Þ 2
1þ
in which,
Sðsp Þ ¼ x1 Cðsp Þ ¼ x1
Z sp
sin½dðtÞdt
Z0 sp
cos½dðtÞdt
ð26Þ
0
Gðsp Þ ¼ x21
Z sp Z 0
t0
0
sin½dðt 0 Þ dðtÞdt dt
0
S(sp) is the integrated y-magnetization over the duration of the rf pulse and d(sp) is the nominal rotation angle of the pulse. Notably, the integrals in Eqs (25) and (26) depend only on the pulse shape and not on the resonance offset. The second-order expansion gives:
tan cðsp Þ eSðsp Þ= sin dðsp Þ e2
cos½bðsp Þ cos dðsp Þ þ Sðsp ÞCðsp Þ sin dðsp Þ þ Gðsp Þ sin dðsp Þ 2 Sðsp Þ cos dðsp Þ ð27Þ and the next correction to tan[c(sp)] is of order e3. If a(sp) and c(sp) are proportional to X over some bandwidth near resonance for a particular rf pulse, then the pulse can be treated as a sequence sa–R/[b(sp)]–sc, in which R/(b) is a rotation by an angle b around an axis with phase / in the transverse plane, sa = a(sp)/X and sc = c(sp)/X. Using tan[c(sp)] c(sp) near resonance yields,
sc
1 sin dðsp Þ
Z sp
sin½dðtÞdt
ð28Þ
0
When the average y-magnetization is zero, then c(sp) = 0 and no phase correction is needed following the pulse. An analogous expression for sa is obtained by time-reversing the pulse:
sa
1 sin ~dðsp Þ
Z sp
sin½~dðtÞdt
ð29Þ
0
in which
~dðtÞ ¼
Z sp
xðt0 Þdt0
ð30Þ
sp t
The above results suggest that all amplitude-modulated p/2 pulses will have a(sp) and c(sp) proportional to X over at least some bandwidth |X/x1| 1 and allow direct calculation of sa and sc as integrals over the rf field amplitude function. Far from resonance, a perturbation solution for f(t) is directly obtained from the Riccati equation by writing
f ðtÞ ¼ f 0 ðtÞ þ ef 1 ðtÞ þ e2 f 2 ðtÞ þ e3 f 3 ðtÞ . . .
ð31Þ
in which e = x1/X. By substitution into Eq. (9), the values of f0(t) = f2(t) = f4(t) = 0 and
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114 3.5
γ (τp) = α(τp)
3.0
6
a
2.5
c
5
e 2.0
2.5
4
2.0
3
1.5
1.5
2
1.0
1.0
1
0.5
0
0.0
–1
2.0
0.5 0.0
b
d
f
|β(τp)|
1.5 1.0 0.5
0
1
2
3
4
0
1
2
3
0
4
1
2
3
4
Ω/ω1 Fig. 1. Euler angles (a, c, e) a(sp) = c(sp) and (b, d, f) b(sp) and for a square (a, b) 90°, (c, d) 270°, and (e, f) 60° x-pulse are shown as a function of the offset parameter X/x1. The limiting (red) near-resonance (X/x1 6 1 and (blue) off-resonance (X/x1 P 1) values also are depicted. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
30
a 20
20
a 10
0
α (τp)
α (τp)
10
–10
0
–20
–10
–30
b
–20
1.5
2.0
|β(τp)|
b 1.0
1.5
|β (τp)|
0.5
1.0
0.5
0.0 3
0.0
2
3
γ (τp)
1
2 0
1
γ (τp)
–1 –2 –3 –1.5
0 –1
c –1.0
–0.5
0.0
0.5
1.0
1.5
Ω/ω1 Fig. 2. Euler angles (a) a(sp), (b) b(sp), and (c) c(sp) for a EBURP-2 pulse are shown as a function of the ratio of resonance offset and B1 field, X/x1. (red) The limiting near-resonance linear dependence of a(sp) over the bandwidth ± 0.3X/x1 is shown. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
–2 –3 –1.5
c –1.0
–0.5
0.0
0.5
1.0
1.5
Ω/ω1 Fig. 3. Euler angles (a) a(sp), (b) b(sp), and (c) c(sp) for a Q5 pulse are shown as a function of the ratio of resonance offset and B1 field, X/x1.
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114
and the third-order term has been included in the expression for b for increased accuracy (the first-order result replaces xe by X). The propagator is
Table 1 z-Rotation scaling factors for shaped pulses.
1 2
Pulse
sa/sp
sc/sp
Square Sinc1 Gaussian2 E-BURP2 Q5
0.637 0.597 0.589 0.718 0
0.637 0.597 0.589 0 0
Defined as the center lobe of the sinc function. Truncated at an amplitude of 0.0471 x1.
2x1
eiXsp
X
X
eiXsp 3
Z sp
8x1
eiXt xðtÞ
Z
In the absence of the pulse, transverse magnetization of the off-resonance spin would acquire a phase Xsp; as shown by the first factor of Eq. (40), the phase acquired during the pulse is 2c(sp) so that the off-resonance phase error becomes:
t
0
eiXt xðt 0 Þdt
0
ð33Þ
2 dt
0
0
Thus, to second order in x1/X,
Z 2 1 sp iXt cos bðsp Þ ¼ 1 e xðtÞdt 2
ð34Þ
0
yielding the first-order estimate:
Z sp bðsp Þ ¼ eiXt xðtÞdt
ð35Þ
0
The z-rotation angle c(sp) is given by
cðsp Þ ¼ Arg½f ðsp Þ ¼ Xsp =2 þ Arg " Z
Z sp
eiXðtsp =2Þ xðtÞdt
0
sp
þ Arg 1 þ
eiXt xðtÞ
Z
0
t
0
eiXt xðt 0 Þdt
0
# 2 , Z sp dt 4 eiXt xðtÞdt 0
0
ð36Þ
As X ? 1, c(sp) ? Xsp/2; consequently, the second term in the above equation is negligible and
cðsp Þ Xsp =2 ¼ Im
"Z
sp
# Z t 2 Z s p 0 0 eiXt xðtÞ eiXt xðt0 Þdt dt 4 eiXt xðtÞdt 0
0
0
ð37Þ
The square pulse is analyzed first to illustrate the properties of the perturbation solutions. Close to resonance, Eq. (27) yields:
aðsp Þ ¼ cðsp Þ ¼ ðX=x1 Þ tanðw=2Þ e2
cos bðsp Þ cos w þ ð1 cos wÞ 2 cos2 w w sin w
ð38Þ
2
in which the nominal rotation angle w = x1sp. Thus, a square pulse with x-phase can be regarded near resonance as an x-rotation preceded and followed by a delay s = tan(x1sp/2)/x1 = sptan(w/2)/w. If x1sp = p/2 for a nominal 90° pulse, then s = 2sp/p as also can be shown by standard techniques [1]. Far from resonance, Eqs. (35) and (37) yield:
aðsp Þ ¼ cðsp Þ ¼ Xsp =2 þ bðsp Þ ¼
2x1
X
2
ð40Þ
ð32Þ
eiXt xðtÞdt
0
3
f 3 ðsp Þ ¼
Z sp
¼ exp½i2cðsp ÞIz 2x1 exp i sinðxe sp =2Þ Ix cosðcðsp ÞÞ Iy sinðcðsp ÞÞ
X
Solving these two equations yields
X
X
exp½i2cðsp ÞIz h x i 1 Ix sinðxe sp Þ Iy ð1 cosðxe sp ÞÞ exp i
df1 ðtÞ X ¼ iXf 1 ðtÞ þ xðtÞ dt 2x1 df3 ðtÞ X ¼ iXf 3 ðtÞ þ xðtÞf 21 ðtÞ dt 2x1
f 1 ðsp Þ ¼
2x1 U exp icðsp ÞIz exp i sinðxe sp =2ÞIx exp½icðsp ÞIz
x21 2
4X
½sin ðXsp =2Þ þ
Xsp sinðXsp Þ xe sp =2
x21 4X2
Xsp sinðXsp Þ
1=2
x21 4X2
2x1
X
sin
sinðXsp Þ
xe sp =2
ð39Þ
/NR ¼
x21 2
2X
Xsp sinðXsp Þ
ð41Þ
This expression approaches the well-known result /NR ¼ x21 sp =ð2XÞ of McCoy and Mueller [11] when Xsp 1, but is more accurate for intermediate cases. The second factor is a rotation by an angle around an axis in the transverse plane that is phase-shifted by an angle v = tan1[(cos(xesp) 1)/sin(xesp)] = xesp/2. In many applications [1], the amplitude of the x1 field is adjusted to yield the rotation angles x1sp on-resonance and 2np at an offset frequency X; consequently, b(sp) = 0. The expressions given in Eqs. (35) and (37), although more computationally demanding, are generally more accurate than the analogous expressions of McCoy and Mueller [11] and are related to expressions obtained from average Hamiltonian theory by Warren [3]. For illustration, the values of b(sp), and c(sp) for square x-pulses are shown in Fig. 1. The more useful applications of the above formalism are to rf pulses with time-varying parameters. EBURP-2 [12] and Q5 [13] pulses are widely used band-selective 90° excitation pulses. The results for a(sp), b(sp), and c(sp), calculated from the analytical forms of the pulse shapes by solution of Eqs. (14) for y(t), are shown in Figs. 2 and 3; identical results are obtained by discretizing the pulse shapes and applying Eqs. (19) and (20). The values of a(sp) and c(sp) are approximately linear functions of X for a bandwidth of ±0.3X/x1 for the EBURP-2 pulse and ±0.4X/x1 for the Q5 pulse. Thus, both pulses can be described within their bandwidths by the sequence sa–90°x–sc with sa = 0.718 sp and sc 0 for the EBURP-2 pulse and sa = sc 0 for the Q5 pulse, calculated using Eqs. (28) and (29). A numerical fit to the value of a(sp) near resonance in Fig. 2a yields an estimate of sa = 0.70. The value of sa for the EBURP-2 pulse agrees well with the fitted value of 0.67sp reported previously from numerical fitting of the phase of transverse magnetization following the pulse [8]. The calculated nearresonance z-rotation angles for the Q5 pulse are a(sp) = 0.013Xsp and c(sp) = 0.076Xsp; thus, as also illustrated by Fig. 3b and c, slightly better phase performance for excitation of Mz ? My is expected for the time-reversed Q5 pulse. Results for sa and sc are given in Table 1 for these and other common pulse shapes. A number of novel pulse shapes with exceptional properties have been determined by optimal control methods [14]. Fig. 4a shows the pulse shape developed by Li and coworkers for broadband excitation with minimal pulse power requirements [15]. The excitation profile is shown in Fig. 4b. The corresponding Euler rotation angles determined from Eq. (26) are shown in Fig. 4c–d. The results show that over the excitation bandwidth of ±20 kHz for x1 = 20 kHz, the pulse can be regarded as the sequence s–90°x, with s = 0.87sp. Reversing the pulse shape profile exchanges
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114 15
a
c 10
0.5
5
α (τp), γ (τp)
ωx(t)/ω1max, ωy(t)/ω1max
1.0
0
0 –5
–0.5 –10 –1.0
–15 0
0.2
0.4
0.6
0.8
1.0
t/τp 2.5
1.0
d 2.0
0.5
β (τp)
My
1.5 0
1.0
–0.5 0.5
b –1.0 –40
–20
0
20
0.0 –40
40
–20
Ω (kHz)
0
20
40
Ω (kHz)
Fig. 4. Minimum-energy broadband excitation pulse. (a) The (black) xx(t) and (red) xy(t) components of the pulse shape are shown. (b) The excitation profile for zmagnetization is shown as a function of the resonance offset X. (c) The Euler angles (black) a(sp) and (red) c(sp) and (d) b(sp) are shown as a function of the resonance offset X. The pulse had a length of 100 ls and x1/(2p) = 20 kHz. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
1.0
0.02
0.5 ^
β
0.01 0.0 –0.5
0.00
–1.0 –0.01 –1.5
^ ^ α/γ
–2.0 0.10
a
0.05
0.05
0.00
0.00
–0.05
–0.05
–0.10 –0. 3
c
–0.02 0.10
b –0. 2
–0. 1
0.0
0.1
0.2
0.3
–0.10 –0. 3
d –0. 2
–0. 1
0.0
0.1
0.2
0.3
Ω/ω1 ^ for (black) conventional two-pulse and (red) Yip and Zuiderweg [17] four-pulse phase alternating scheme. Fig. 5. CPMG pulse sequences. (a) The effective x-rotation angle b ^ and (d) (red) a ^ for the (black) conventional and (red) a ^ and (blue) c ^ for the Yip and Zuiderweg schemes. The (c) effective x-rotation angle b ^ (b) The effective z-rotation angles a ^ for the eight-pulse phase alternating scheme. All calculations used x1/2p = 6250 Hz, scp = 0.5 ms, in which scp is the spacing between p pulses. (For interpretation and (blue) c of the references to color in this figure legend, the reader is referred to the web version of this article.)
a(sp; X) = c(sp; X), c(sp) = a(sp; X), and b(sp; X) = b(sp; X); thus, the time-reversed pulse can be regarded as the sequence 90°x–s, allowing broadband transformations from y- to z-magnetization. The Euler angle representation allows convenient
representation of the properties of pulses with complex rf profiles for comparison with other pulse shapes, e.g. comparing Figs. 2–4. The formalism is also applicable to pulse trains consisting of pulse-interrupted-free-precession sequence blocks:
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Y. Li et al. / Journal of Magnetic Resonance 248 (2014) 105–114
s1 Pðx11 ; sp1 ; /1 Þ s2 Pðx12 ; sp2 ; /2 Þ s3
g ¼ tan1
in which sn is a time delay and P(x1n, spn, /n) is a square pulse applied with rf field x1n, length spn, and phase /n, is commonly encountered in NMR spectroscopy. Using Eqs. 3, 19 and 20 yields the propagator for this pulse sequence as:
U ¼ exp ½iðXs3 þ c2 þ cÞIz exp½ibIx exp ½iðXs1 þ a1 þ aÞIz
ð42Þ
in which an, bn, and cn are given in Eq. (18) for the nth pulse,
a ¼ tan1
2 sin b2 tanðd=2Þ sinðb1 þ b2 Þ þ sinðb1 b2 Þ tan2 ðd=2Þ 2
ð43Þ
and d = a2 + Xs2 + c1 = d0 /2 + /1. In the limit that s2 ? 0 (d ? a2 + c1), Eq. (43) yields the Euler angle composition rule [16], which defines the effective Euler angles from two successive rotations. If the two pulses are applied with values of x1 and sp (but potentially different phases), then a simpler result is obtained:
a ¼ c ¼ tan1
tanðd=2Þ cos b1
ð44Þ
The Carr–Purcell–Meiboom–Gill (CPMG) sequence, s–p/1–2s–p/2–s, with x1sp = p is an example of a pulse sequence with identical pulse lengths. Application of Eq. (44) gives: U/2 /1 ¼ exp ½ifg þ ð3/2 /1 Þ=2gIz exp ½ibIx exp ½ifg þ ð/2 3/1 Þ=2gIz ð45Þ
in which
y-magnetization
sin d cos d tan2 ðb1 =2Þ
1
ð46Þ
The conventional two-pulse CPMG pulse sequence element has the propagator Uxx with /1 = /2 = 0. Yip and Zuiderweg [17] proposed a four-pulse CPMG sequence, s–px–2s–px–2s–py–2s–py–s. A propagator for this sequence is given by U ¼ Uyy Uxx , in which:
Uxx ¼ exp ½ig Iz exp ibþ Ix exp ½ig Iz
Uyy ¼ exp igþ Iz exp ½ib Ix exp igþ Iz
g ¼ tan1
sin d0 cos d0 tan2 ðb1 =2Þ
ð47Þ
ð48Þ
h i 2 b ¼ cos1 1 sin b1 ð1 cos d0 Þ
This propagator can be recast by the same approach embodied in Eqs. (42) and (43) to give:
h i ^ x exp ½ia ^Iz exp ibI ^ Iz Uyy Uxx ¼ exp ½ic
ð49Þ
in which
"
a^ ¼ g þ tan1
h i 2 b ¼ cos1 1 2 sin b1 cos2 ðd=2Þ
a
and
b ¼ cos1 ½cosðb1 þ b2 Þ cos2 ðd=2Þ þ cosðb1 b2 Þ sin ðd=2Þ 2 sin b1 tanðd=2Þ c ¼ tan1 sinðb1 þ b2 Þ sinðb1 b2 Þ tan2 ðd=2Þ
# 2 sin b tan g þ sinðbþ b Þ sinðbþ þ b Þ tan2 g
sin2 g cos2 g ^ ¼ cos1 ½1 2 sin2 b 2 sin2 Db b " # 2 sin bþ tan g c^ ¼ gþ þ tan1 sinðbþ b Þ sinðbþ þ b Þ tan2 g
ð50Þ
¼ ðb þ b Þ=2; Db ¼ ðb b Þ=2, and g ¼ ðg þ gþ Þ=2. Illustrative b þ þ calculations of the rotation angles for the two CPMG variants are shown in Fig. 5a and b. The major difference between the sequences ^ c ^, with an absolute is that for the Yip and Zuiderweg scheme, a difference of 60.5° for 0.3 6 X/x1 6 0.3 for the parameters used
c
e
d
f
0.8 0.6 0.4 0.2 0
x-magnetization
b
1
0.8 0.6 0.4 0.2 0 –1 –0.8 –0.6 –0.4 –0.2
0
Ω/ω1
0.2 0.4 0.6 0.8
1
–0.4
–0.2
0
Ω/ω1
0.2
0.4
–0.4
–0.2
0
0.2
0.4
Ω/ω1
Fig. 6. Refocused transverse magnetization as a function of the ratio of resonance offset and B1 field, X/x1. Plots show refocused (b, d, f) x-magnetization or (a, c, e) ymagnetization following CPMG pulse trains beginning with initial (b, d, f) x-magnetization or (a, c, e) y-magnetization. Panels (a) and (b) show the refocused magnetization after one cycle of CPMG pulse train. Panels (c, d, e, f) show the refocused magnetization after n cycles, averaged over n = 1–100. For panels (e) and (f), 10% rf inhomogeneity (FWHM) was included in the calculation. Each cycle consists of 8 pulses for all four schemes. The color coding is (blue) conventional x-phase, (black) Yip and Zuiderweg phase alternating scheme, (red) proposed eight-pulse phase alternating scheme, and (green) XY-8 scheme. All calculations used x1/2p = 2500 Hz, scp = 1.25 ms, in which scp is the spacing between p pulses. The calculations were performed using the software package SIMPSON [25]. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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for Fig. 5. Thus, the propagator for the complete CPMG train, consisting of n repetitions of the four-step cycle simplifies to
Uyy Uxx
n
h i ^ x exp ½ia ^Iz exp inbI ^ Iz exp ½ic
ð51Þ
and, unlike the conventional sequence, the effects of the z-rotations are not cumulative over the n cycles. Eq. (49) indicates that the eight-pulse phase alternating scheme formed by concatenating the Yip and Zuiderweg four-pulse scheme and its phase-inverted counterpart, yielding the phases x, x, y, y, x, x, y, y, has a propagator described by:
U ¼ Uyy Uxx Uyy Uxx ¼ Uyy Uxx Uyy Uxx h i h i ^ x exp ½ia ^ x exp ½ia ^Iz exp ibI ^ Iz exp ½ic ^Iz exp ibI ^ Iz ¼ exp ½ic h i h i ^ x exp ibI ^ x exp ½ia ^Iz exp ibI ^ Iz exp ½ic ^Iz exp ½ia ^ Iz E exp ½ic ð52Þ
1.0
1.0
0.8
0.8
x/y-magnetization
x-magnetization
in which E is the identity operator. The calculations shown in Fig. 5c demonstrate the effective x-rotation is reduced by two orders of magnitude in the eight-pulse scheme compared to the four-pulse scheme shown in Fig. 5a and nearly an order of magnitude smaller than the XY-8 scheme [18] (not shown). The residual x-rotation is 61° over the range 0.3 6 X/x1 6 0.3 (and 61.7° for a variation in x1 of ±10%); consequently the two z-rotations shown in Fig. 5d nearly cancel to yield a propagator proportional to the identity operator. Thus, the proposed eight-pulse phase alternating scheme should refocus transverse magnetization with arbitrary initial phase more accurately than XY-8 (the conventional or Yip and Zuiderweg
schemes do not efficiently refocus magnetization with initial yphase). Fig. 6a and b show the numerically calculated magnetization after a single cycle of each refocusing scheme as a function of resonance offsets. For fair comparison, each cycle consists of eight pulses for all schemes. The performance of the different refocusing schemes also depends on the extent to which phase errors accumulate as more than one cycle of the CPMG pulse train is applied. In particular, the residual x-phase rf fields in the conventional and Yip and Zuiderweg schemes dominate small non-commuting terms in the Hamiltonian; in contrast, sequences proportional to the identity operator, like the proposed sequence or XY-8, do not have this property and can be expected to be more sensitive to accumulated effects. The magnitudes of refocused magnetization have been calculated following 1–100 cycles of all four refocusing schemes. Fig. 6c and d show the average refocused transverse magnetization. The Yip and Zuiderweg scheme, as predicted from Eq. (51) has the largest bandwidth (0.35|X/x1|). The proposed eight-pulse scheme has a bandwidth ((0.16|X/x1|) substantially larger than the XY-8 scheme (0.1|X/x1|) both in the absence and the presence of rf inhomogeneity (Fig. 6c–f). Only the proposed eight-pulse scheme and XY-8 refocus y-magnetization. Experimental validation of these predictions is shown in Fig. 7. As shown in Fig. 7a, both the Yip and Zuiderweg and the proposed CPMG sequences conserve >95% of both initial x- and y-magnetization at resonance offsets up to 0.5|X/x1| for a single cycle consisting of eight pulses (and a total time of 8 ms). The bandwidth of the Yip and Zuiderweg scheme is largely maintained as the number of cycles in the pulse train increases, while the bandwidth for the proposed sequence is reduced to 0.3|X/x1| for 10 cycles (a total time of 80 ms). As shown
0.6 0.4 0.2
0.6 0.4 0.2
a
b
0
0 0
0.2
0.4
0.6
0.8
0
0.2
0.4
Ω/ω1
0.6
0.8
Ω/ω1
x−magnetization
1 0.8 0.6 0.4 0.2
c
d
e
0 0
0.2
0.4
Ω/ω1
0.6
0
0.2
0.4
Ω/ω1
0.6
0
0.2
0.4
0.6
Ω/ω1
Fig. 7. CPMG performance demonstrated from normalized intensities for the 1H resonance of H2O in a doped water sample. (a) Refocused x-magnetization for the (black) proposed and (red) Yip and Zuiderweg phase alternating scheme for (circles) 8 ms, (squares) 40 ms, and (triangles) 80 ms pulse trains, corresponding to 1, 5, and 10 repetitions of an 8-pulse CPMG block, as described in Section 3. (b) Refocused (black) x-magnetization and (red) y-magnetization for the proposed eight-pulse phase alternating scheme for (circles) 8 ms, (squares) 40 ms, and (triangles) 80 ms pulse trains. (c–e) Refocused x-magnetization for (red, circle) the proposed eight-pulse scheme and (green, square) the XY-8 scheme for (c) 8 ms, (d) 40 ms, and (e) 80 ms pulse trains overlaid onto the x-magnetization from numerical simulations using (black) the proposed scheme and (blue) the XY-8 scheme. 10% rf inhomogeneity (FWHM) was included in the calculations. The magnetizations in panels (c–e) were normalized to the values at zero resonance offset for clarity. The relative performance of the proposed and XY-8 schemes for y-magnetization are similar to the performance for x-magnetization and is not shown. The Yip and Zuiderweg scheme does not refocus orthogonal y-phase magnetization and results are not shown for this sequence. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
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1.5
a
d
b
e
c
f
Tr {IxHeff}τc
1 0.5 0 −0.5 −1
Tr {IyHeff}τc
1.5 1 0.5 0 −0.5 −1
Tr {IzHeff}τc
1.5 1 0.5 0 −0.5 −1 −0.6
−0.4
−0.2
0
0.2
0.4
0.6
−0.6
−0.4
−0.2
Ω/ω1
0
0.2
0.4
0.6
Ω/ω1
Fig. 8. The x-, y- and z-components of effective Hamiltonians from EEHT calculations for (a, b, c) the XY-8 and (d, e, f) proposed 8-pulse schemes, multiplied by the cycle time, sc. The product reflects the residual x-, y- and z-rotations each cycle. The calculations were performed at scp/s180 values of (red) 0, (green) 1, (blue) 10 and (magenta) 100. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
in Fig. 7b, the proposed CPMG sequence is equally efficient in refocusing x- or y-magnetization. The bandwidth of the proposed sequence, consistent with the calculations shown in Fig. 6 is larger than that for the XY-8 sequence, as shown in Fig. 7c–e for increasing numbers of cycles. Additional insights into the relative performance of the XY-8 and proposed schemes over many cycles can be gained from calculation of effective Hamiltonian. These calculations were performed using exact effective Hamiltonian theory (EEHT) [19,20] and expanded to fifth order with respect to resonance offset. For the XY-8 scheme,
"
Heff
# 0:5rcp þ 0:25 4 0:393r cp þ 0:785r 2cp 5 ¼ e e x1 I x rcp þ 1 rcp þ 1 " # 0:5rcp þ 0:25 4 0:393r cp þ 0:785r 2cp 5 þ e þ e x1 I y rcp þ 1 rcp þ 1 r cp þ 0:5 5 þ e x1 Iz þ Oðe6 Þ r cp þ 1
ð53Þ
For the proposed eight-pulse scheme,
Heff ¼
rcp þ 0:5 5 e x1 Iz þ Oðe6 Þ rcp þ 1
ð54Þ
in which e = X/x1, rcp = scp/s180, scp is the spacing between pulses, and s180 is the length of the 180° pulse. The EEHT results show that the proposed eight-pulse scheme gives superior performance over XY-8, regardless of the number cycles applied, because the z-components of the effective Hamiltonians are identical in the fifth order expansion, whereas the new scheme compensates x- and y-components to higher order than the XY-8 scheme. Although simple and more intuitive, representation of effective Hamiltonian by power series is less accurate when the value of rcp is large. Fig. 8 shows
the exact effective Hamiltonian, without truncation to powers, as a function of resonance offsets at different rcp values. When rcp is large, the effective Hamiltonians oscillate at high frequency outside the effective bandwidth and therefore cannot be well represented by power series expansion. However, the numerical representations of effective Hamiltonian obtained from its exact expression are consistent with the numerical results shown in Fig. 6. The performance of the proposed eight-pulse scheme is not significantly altered by expanding into a 16-step scheme RRRR in which R ¼ Uyy Uxx and the bar indicates phase inversion. Thus, the improved performance of the proposed scheme is not simply a consequence of imposing a supercycle, as utilized in decoupling schemes [1]. The above considerations indicate that the Yip and Zuiderweg phase alternating scheme has a superior bandwidth than the other schemes if initial x-magnetization (i.e. magnetization with the same phase as the first 180° pulse) is to be refocused, but that the proposed scheme has the best performance if both x-magnetization and y-magnetization (orthogonal to the first pulse) must be retained. The Yip and Zuiderweg scheme is the choice for most CPMG relaxation dispersion experiments [21] and the proposed scheme for applications such as CPMG-INEPT polarization transfers [22,23] and a subset of CPMG relaxation dispersion experiments, in which the starting magnetization is of arbitrary phase [24]. 5. Conclusion The Euler-angle rotation operator factorization of the propagator for time-varying rf pulses and pulse-interrupted-free-precession trains provides insights into performance that are not as easily discerned by average Hamiltonian theory or by numerical calculation of the evolution of magnetization through the pulse. The Euler-angle representation is particularly useful in determining whether a shaped pulse or pulse train can be represented over
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some bandwidth by a sequence s1–R/(b)–s2, allowing phase evolution during the pulse to be compensated by adjusting time periods prior or subsequent to the pulse. Explicit solutions for the Euler rotation angles have been derived using perturbation theory nearand off-resonance; the resulting expressions are useful especially for characterizing phase evolution of magnetization during shaped pulses. The Euler-angle approach also is powerful in analyzing repetitive pulse trains, such as the CPMG sequence and has been used to identify an eight-step phase alternating scheme that performs better than the XY-8 scheme in refocusing magnetization orthogonal to the phase of the first pulse in the CPMG train. Acknowledgments Support from the National Institutes of Health Grants GM059273 (A.G.P.) and GM063855 (M.R.) are gratefully acknowledged. We thank Justin Ruths and Jr-Shin Li for providing the pulse shape file for the broadband excitation pulse described in [15]. References [1] J. Cavanagh, W.J. Fairbrother, A.G. Palmer, M. Rance, N.J. Skelton, Protein NMR Spectroscopy: Principles and Practice, second ed., Academic Press, San Diego, CA, 2007. [2] U. Haeberlen, J.S. Waugh, Coherent averaging effects in magnetic resonance, Phys. Rev. 175 (1968) 453–467. [3] W.S. Warren, Effects of arbitrary laser or NMR pulse shapes on populationinversion and coherence, J. Chem. Phys. 81 (1984) 5437–5448. [4] J. Zhou, C. Ye, B.C. Sanctuary, Rotation operator approach to spin dynamics and the Euler geometric equations, J. Chem. Phys. 101 (1994) 6424–6429. [5] D.J. Siminovitch, Rotations in NMR: Part II. Applications of the Euler–Rodrigues parameters, Concepts Magn. Reson. 9 (1997) 211–225. [6] D.J. Siminovitch, Rotations in NMR: Part I. Euler–Rodrigues parameters and quaternions, Concepts Magn. Reson. 9 (1997) 149–171. [7] R.M. Gregory, A.D. Bain, The effects of finite rectangular pulses in NMR: phase and intensity distortions for a spin-1/2, Concepts Magn. Reson. 34A (2009) 305–314. [8] E. Lescop, T. Kern, B. Brutscher, Guidelines for the use of band-selective radiofrequency pulses in hetero-nuclear NMR: example of longitudinalrelaxation-enhanced BEST-type 1H–15N correlation experiments, J. Magn. Reson. 203 (2010) 190–198.
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