Shaped pulse excitation in multi-quantum magic-angle spinning spectroscopy of half-integer quadrupole spin systems

16May 1997

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 270 (1997) 81-86

Shaped pulse excitation in multi-quantum magic-angle spinning spectroscopy of half-integer quadrupole spin systems Shangwu Ding, Charles A. McDowell

*

Department of Chemistry, University of British Columbia, 2036 Main Mall, Vancouver, British Columbia, Canada V6T IZ1

Received22 January 1997; in final form 24 March 1997

Abstract

Coherence excitation by shaped pulses in the multi-quantum magic-angle spinning spectroscopy of half-integer quadrupole spin systems was explored. A theoretical analysis shows there exists a class of shaped pulses that are superior to the rectangular pulses normally used in MQMAS studies. Significant sensitivity enhancement can be achieved without loss of resolution and quantification. This is verified by numerical simulations based on decaying triangular and exponential shaped pulses. 23Na multi-quantum magic-angle spinning spectra of polycrystalline anhydrous di-sodium hydrogen phosphate (Na2HPO4) are shown to be well-explained by the theoretical analysis, t~) 1997 Elsevier Science B.V.

1. I n t r o d u c t i o n

Significant progress in acquiring high-resolution solid-state spectra free of second-order quadrupole interactions of quadrupole spin systems was achieved recently by Frydman and Harwood [ 1 ]. The most practical advantage of this technique, called multiquantum magic-angle spinning (MQMAS) spectroscopy which correlates the multi-quantum coherence (MQC) and single quantum coherence, is that it does not need the rotor spinning axis to be flipped as is the case for dynamic angle spinning NMR [2-4], or spinning of the sample around two axes simultaneously (double rotation NMR [5-7] ) during the experiment. A number of publications following the original work, which used the three-pulse scheme [ 1 ], showed that the commonly used two-pulse scheme can be employed better to excite and convert the multi-quantum coherence [8-13]. Several groups of * Correspondingauthor; E-mail: [email protected]

research workers have reported experimental methods to yield pure absorptive lineshapes, these include appropriate phase cycling [8-13], Z-filtering [ 14-16] and synchronized data acquisition [ 16,17]. Extensive investigations into the dependencies of the excitation efficiency on crystallite axes orientations, quadrupole coupling constant (QCC), rotor spinning speed, RF off-set, RF field amplitude and pulse widths were carried out recently [8-13], expanding the earlier studies of MQC in quadrupole systems [ 18-22]. The basic conclusions are that stronger RF fields with smaller off-sets, and higher rotor spinning speeds, generally produce better MQMAS spectra. The MQMAS spectra are highly resolved but the spectra are generally not quantitative. A scheme which chooses the pulse widths according to the rotor spinning speed was proposed recently in an effort to obtain sitequantitative MQMAS spectra [ 23 ]. It was found that much better quantification can be achieved by use of this technique. All the above works use rectangular pulses for ex-

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82

S. Ding, C.A. McDoweU/Chemical Physics Letters 270 (1997) 81-86

citation and conversion of MQC. For spin-½ systems, it is well known that shaped pulses, besides other advantages, can be used to generate better excitation efficiencies [24]. Shaped pulses are usually considered as selective. Therefore, at first sight, the use of shaped pulses in MQMAS may seem inadequate because that could render the quantitative problem more severe. In this work, we present preliminary results on an investigation of shaped pulses applied to MQMAS spectroscopy. Our theoretical and experimental results show that the selectivity of shaped pulses does not aggravate the quantification problem, but instead yields the benefits of higher sensitivity, maintained resolution and less demanding requirements on the experimental conditions, can be achieved by use of certain types of shaped pulses. Spin -3 systems are considered in this work, but it is likely that the conclusions may apply, mutatus mutandis, to higher-spin systems.

2. Experimental 23Na MQMAS spectra were recorded with both rectangular and decaying-triangular pulse excitations. All experimental spectra were acquired by using a Bruker MSL-200 spectrometer with the proton resonance frequency tuned to 200.13 MHz. A Doty DAS probe was used for the measurements at a rotor spinning rate of 5.5 kHz. The two-pulse sequence used in our experiments is shown in Fig. 1. The maximum RF field amplitude of the 23Na channel was about 100 kHz and the proton decoupling power was about 60 kHz. The phase cycling scheme used in this work is the same as in Ref. [9] and one cycle consists of 24 scans. The TPPI method was employed to obtain quadrature detection in the first dimension. The 2D spectra were skewed by a shear transformation so that the projections to the first dimension correspond to the isotropic spectra. To compare the results with different excitation pulse shapes, and for rectangular pulse excitation, the RF field amplitude of 23Na channel was set at about 60 kHz. The shaped pulses used in our experiments were amplitude modulated, realized by varying the RF field amplitude. Up to 12 steps were used for the excitation time of about 12/xs. 128 increments were used for the first dimension with a spectral width of 25 kHz. The relaxation delay was 10 s. To test the resolution, sensitivity and quantification of the

I

DEC

:

lq

Fig. 1. The two-pulse scheme of MQMAS of half-integer quadrupole spin systems. The first pulse is rectangular (solid line) or shaped pulse (dashed line). This sequence is the same as the one in Ref. [9] except that the first pulse is amplitude modulated.

MQMAS spectra with shaped pulses excitation, the compound used should have at least two inequivalent sites, therefore, in our experiment di-sodium hydrogen phosphate Na2HPO4 was used which has three magnetically inequivalent 23Na sites [25]. As shown below, all three sites can be clearly resolved and the difference between the rectangular and triangular pulse can be readily demonstrated.

3. Theoretical The problem of obtaining the optimal excitation of certain orders of MQC's in quadrupole systems can be generally expressed as: given the quadrupole interaction Hamiltonian and initial density matrix, it is necessary to find the proper expression of the RF Hamiltonian so that a certain order of MQC's is maximized. The initial state of the system is usually at thermodynamic equilibrium and the density matrix can be written approximately as p ( 0 ) = Iz, where Iz is the zcomponent of angular momentum operator of spin I. The total Hamiltonian can be expressed in the rotating frame as H=02q[] 2 - ½1(1+

1)]

+ 021(t)l¢

(1)

where o21 is the amplitude of the RF field which is generally time-dependent when shaped pulses are used. ~b

S. Ding, C.A. McDowell/Chemical Physics Letters 270 (1997) 81-86

is the phase of the RF field. The quadrupole frequency Wq can be written as +2

0)q = v/'~h ~ O2(0)rt, m,n=--2

O,O)OZnm(Ot,~,9")Pfn" (2)

The Euler angles ( a , fl, 9') correspond to the transformation from the principal axis system (PAS) of the quadrupole interaction tensor to the rotor axis frame. 0 is the spinning axis direction relative to the direction of the applied magnetic field and 0)r is the rotor spinning rate, D the Wigner transformation matrix, and p2Qnthe principal elements of electric field gradient (EFG) tensor

83

can be satisfied so long as both the quadrupole coupling constant (QCC) and 0)1 are much larger than the rotor spinning rate. In Eq. (6), Ha(t) is the instant eigenmatrix of the Hamiltonian at time t and U71H(t)Ut = Ha(t). With Eq. (6), a much simpler formulation of Eq. (3) can be obtained as

p(t) = U71 exp -i

dt'Hd(t ') Uo 0

x p(O)Uo lexp i

dt~Ha(/) G. o

(7)

j

For spin-~ systems, we have

eqQ 20-

Ha(t) = U71H(t)Ut = "( E1

= V 2 4 1 ( 2 I - - 1)h'

pQ22 = pQ2--2 = 1 ?.p,,,Q.

(3)

The density matrix at any time t ( > 0 ) is

p( t) = U( t, O)-l p(O)U( t, O),

-d-[ Ut << Ha(t)

E4

E1 = 0.50)1 + D_, E2 = 0.50)i - D_,

(5)

E3 = -0.50)1 + D+, E4 = -0.5o)1 - D+

Here T is the time-ordering operator. Obviously it is extremely difficult, if not impossible, to find the general solution of Eq. (4) based on Eqs. (1) and (5). Numerical calculation can be always carried out if the form of the Hamiltonian is known, but this, generally, cannot provide the insights sought about the types of pulses which are best employed for optimal MQC excitation. An analytical solution, even an approximate one, may provide useful clues to help identify the optimal pulse excitation protocols. In the following, we present an approximate analytical solution of Eq. (4) by retaining only the most important terms. Some useful conclusions can be drawn regarding the choices of the most desirable pulse shapes. In most cases, in practice, the RF field does not change rapidly, therefore, the Hamiltonian ( 1 ) can usually be taken to be a sufficiently smooth function of time. Furthermore, the condition dUt

\

(8)

where (4)

where the evolution operator is written as

U(t,O) =Texp ( - i fo dt'H(t') ) .

E3

(6)

(9)

and D+ = [0)q24- 0)10)q + 0)~]1/2

(10)

and 1

Ut = --~

cos 0_ sin0_ sin0_ - c o s 0 _ sin0_ - c o s 0 _ cos0_ sin0_

cos 0+ sin0+ sin0+ - c o s 0 + | - s i n 0 + cos0+ ] ' -cos0+ -sin0+] (11)

where tan 20+ = v~0)1/(20)q

4- Wl ).

(12)

Therefore, the density matrix elements can be generally expressed as 4

Pin(t)=

Z Utji Utmn Uojk UOmlPkl ( 0 ) e --iEmt , j,k,l,m=l

(13)

84

S. Ding, C.A. McDowell/Chemical Physics Letters 270 (1997) 81-86

where the unitary property of Ut is used and Ej,, = Ei - En,. Substituting the expressions for U0 (which is an identity matrix in our case) and p ( 0 ) , Eq. (13) can be simplified as 4 Pin(t) =

E UtjiUtjmUOjkUOmkPkk(O)e-iEj't" j,k,m=l

(14) Because each term of above equation contains an oscillating factor e - m j ' t , the most important terms are those with small values of Ejm. From Eqs. (9) and 910) it can be seen that the terms with oscillating factors e 4-iEIEt, e "4-iE34t, e "4-iE23t and e -I-iE14t are the smallest. We can neglect these "high-frequency terms". The zero-frequency terms of Eq. (14) can be calculated as p~Zf) (t) = (p01 COS2 0 0 4- p02 sin 2 0 ° 4- p03 cos 2 0 ° 4- 1904 sin 2 0 ° ) ( UmUt,n + Ut4iUt4n )

4- (pO 1 sin 2 0 0_ + p02 cos2 0 0_ + p033sin2 00 4- p04 COS2 00+) (Ut2iUt2n nt- Ut3iUt3n)

(15)

where the superscript 0 stands for the initial time (0). It is readily seen that the above equation contains no contribution to the 3Q coherence P l 4 ( t ) . The zero-frequency nutation peaks result from the (zf) (zf) (zf) , . ,O12 , 1034 ' Pii k t = 1 , 2 , 3 , 4 ) terms. Therefore, we conclude that the most important contributions to the 3Q coherence in spin -3 systems come from the "low-frequency" terms with time factors e :t:ie23t and e +iEut. After some algebra the 3Q coherence with "low-frequency" oscillations can be written as p 14 0 f )kt-t , . , = ½(sin20 o _ sin200 ) X [sin0+ s i n O _ ( e iE'3t -- e - i E u t ) 4- c o s 0+ c o s 0 _ ( e iE13t -- e -iE24t) ].

(16)

In the case of a rectangular pulse, i.e. w l ( t ) = O.)rf (constant), the 3Q coherence becomes (If) (- t ) :- i ( s i n PI4

20 ° - sin 20 ° )

x [sin0+ s i n O _ e i(D--D+)t - sin 0+ sin O _ e i(D- --D+)t ] sin wrrt.

(17)

In practice, the condition wrf << Wq is met, hence D+ - D _ '~ taft, El3 "" 0, E24 ~'~ 2rOrf, so Eq. (17) can be written as

p(lf),,.,, 14 k t ) = ½(sin20 o -- sin200-- )

x [sin 0+ sin 0_ ( 1 - e m~t) - cos 0+ cos 0_ ( 1 - e -iE24t) ].

(18)

Therefore, the 3Q coherence nutates at a frequency of 2Ogrf. Because 0 ° = 0+ for a rectangular pulse excitation, the best 3QC excitation can be achieved by setting ,-~ 2Wrftp = (2k 4- 1)~r where tp is the pulse width and k is an integer. This analytical result confirms the correctness of the conclusions drawn from numerical simulations [ 13,23 ]. If a shaped pulse is used, the problem then becomes (If) ~ .~ how to maximize Pl4 tr) by optimizing the shape of Wrf(t). A universal solution cannot be found but some important conclusions can be drawn from Eq. (18): (i) because of the coefficient (sin 20 ° - sin 200- ), the initial RF field must be as large as possible to maxi0f) mize Pl4 (t); (ii) the value of RF field at time t must deviate as far as possible from ~Orftp = 2klr (where k is an integer). Considering that a practical transmitter has a definite RF power output, the general pulse shape should be of a decaying type, i.e., the initial RF field should be the strongest. The exact excitation profile can be obtained by numerical simulation. As shown in the following sections, the above conclusions can be verified by simulations and experimental results.

4. N u m e r i c a l s i m u l a t i o n s a n d e x p e r i m e n t a l results

The excitation profiles of various coherence orders of spin -3 systems under irradiation with different types of pulses were calculated starting from the exact expression of the evolution operator Eq. (5). Shown in Fig. 2 are the normalized intensities o f 1Q, 2Q and 3Q coherences immediately after irradiation with three different types of shaped pulses (rectangular, decaying triangular and decaying exponential). The typical quadrupole parameters for spin -3 systems are used (our simulations with different parameters gave similar results). As can be seen from this figure, the double quantum coherence, generally, is rather weak regardless of the pulse shape, while the 3Q coherence can be as strong as, or even stronger than, the single quantum coherence. It is noteworthy that the intensities of 1Q and 2Q coherences depend on the excitation pulse shape less than that of 3Q coher-

S. Ding, CA. McDowell/Chemical Physics Letters 270 (1997) 81-86

3

. . . .

I

/I"~ ....

"~"~

. . . .

I

. . . .

85

I

3Q -

-

~ ~

m m .~ .7..Z .Z Z. 7.7L

Z'. 2-. 27..~ .7: .v..7:.7..

77.7~ ~"

"'l''''l''''l

.

''

'1'''

2 ~ O

2Q~3 /

\

r,

i ~ / \ O0

1

2

t 3

71"

X,~,,I,,,,I,, 0 5 10 F1

Jr,,,, 15

O

Fig. 2. The IQ (bottom), 2Q (middle), and 3Q (top) theoretical coherence excitation efficiencies of a spin -3 quadrupole system with different pulse shapes: The solid lines are for a rectangular pulse; the dashed lines for decaying triangular pulse and the dotted lines for decaying exponential pulse. The intensities of 2Q coherences are multiplied by 3 and then shifted up by 1 unit. The intensities of 3Q coherences are shifted by 2 units. For the rectangular pulse, the RF field to I = 50 kHz; for the triangular pulse tol (t) = o~lo( 1 - t / t p ) where tp is the pulse width, to10 = 100 kHz; for the exponential pulse Oil (t) = toloe -2t/tl' with tOl0 = 100 kHz. The RF offset is 4 kHz. The quadrupole coupling constant and asymmetry parameter are 2.4 MHz and 0.8, respectively. The rotor spinning speed was 5 kHz.

ences. Moreover, it can also be seen that all orders of coherences excited by shaped pulses have almost the same intensities at the initial stage of excitation. More importantly, it is readily seen that the results obtained from the analytical solution in the preceding section are confirmed: decaying pulse shapes are better than the rectangular pulse. The sensitivity advantage of certain types of shaped pulses over the rectangular (and other shaped pulses such as the growing type) pulse can be as large as 50%. Of course, here we must stress that the sensitivity is compared at the same effective RF power. If the maximum RF power is used as the criterion of comparison, then the best excitation is always obtained by using a rectangular pulse. This is easy to understand because for the same period of excitation, a rectangular pulse consumes the most actual power. Thus the shaped pulse can be used for the situation when the maximum power can be

t~

o

0

5

FI

I0

15

Fig. 3. The 23Na 3 Q - I Q correlation spectra of di-sodium hydrogen phosphate with rectangular pulse excitation (bottom) and decaying triangular pulse excitation (top), respectively. The other experimental parameters are the same as for the spectra in the earlier figures and are given in the text. The spectra are skewed by shear transformation. Seven equal levels were used in each 2D spectrum.

very high but the pulse width is limited. The sensitivity enhancement by the use of shaped pulses clearly has been demonstrated experimentally. The polycrystalline sample of di-sodium hydrogen phosphate used has three different magnetically inequivalent sites.

86

s. Ding, C.A. McDowell/Chemical Physics Letters 270 (1997) 81-86

The M Q M A S experiment was carried out following the standard procedure [8-13] except that we used a shaped pulse for one spectrum. The shaped pulse used is a decaying triangular pulse generated by gradually reducing the R F power o f the 23Na channel. The first pulse (for M Q C excitation) is about 1 2 / z s and the second pulse is about 4 / z s . Because the second pulse is short, a rectangular pulse was used. The 23Na MQM A S spectra recorded with rectangular and decaying triangular excitation pulses are shown in Fig. 3. To contrast the excitation efficiencies o f different pulses, the same experimental parameters (except the pulse shape) were used for both spectra. The sensitivity enhancement resulting from the use of a shaped pulse is obvious from the projections along both dimensions. Comparing the two spectra in Fig. 3 which are skewed by a shear transformation, clearly there is no loss o f resolution evident for the top spectrum in Fig. 3. Another interesting phenomenon shown in Fig. 3 is that the spectrum acquired by use o f the shaped pulse excitation, within experimental errors, has the same site quantification as that resulting from a rectangular pulse excitation though the quadrupole coupling constants o f the three inequivalent 23Na sites o f d i - s o d i u m hydrogen phosphate differ, significantly. This can be explained as follows: though in general the shaped pulses have more selectivity, the decaying triangular pulse used in our experiment has the same pulse length as the rectangular pulse, thus the excitation width is the same as the rectangular pulse. Moreover, the m a x i m u m pulse power o f the triangular pulse is larger than that o f the rectangular pulse, so it is possible, at least theoretically, the shaped pulse may have better quantification.

5. Conclusions In this work, we first derived an approximate analytical expression for the excitation o f M Q C intensities o f spin -3 systems. Some general principles for designing shaped pulses for the efficient M Q C excitation o f quadrupole spin systems were obtained, which were further verified by numerical simulations and experimental spectra o f a typical sodium compound. We believe the conclusions can be applied to higher-spin systems. A more general investigation is being undertaken and the results will be presented elsewhere.

Acknowledgements We are grateful to the Natural Sciences and Engineering Research Council o f Canada for research grants to C.A.McD.

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