- Email: [email protected]
Third-order effect in solid-state NMR of quadrupolar nuclei
Chemical Physics Letters 367 (2003) 163–169 www.elsevier.com/locate/cplett
Third-order effect in solid-state NMR of quadrupolar nuclei Zhehong Gan a b
a,*
, Parthasarathy Srinivasan b, John R. Quine b, Stefan Steuernagel c, Benno Knott c
National High Magnetic Field Laboratory (NHMFL), Tallahassee, FL 32310, USA Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA c Bruker Analytik GmbH, D-76287 Rheinstetten, Germany Received 17 May 2002
Abstract The theory and experimental observation of the third-order effect in solid-state NMR of quadrupolar nuclei are presented. The third-order effect consists of spherical harmonic terms up to rank l ¼ 6 and shifts NMR frequencies between two spin states that are not symmetric such as satellite transitions. Two-dimensional satellite transition magicangle spinning experiment averages both the first and the second-order quadrupolar interactions making the quantitative measurement of the third-order effect possible. The third-order quadrupolar effect in andalusite has been measured at 11.7 T and its powder patterns are fitted with numerical simulations. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction For nuclei with spin S > 1=2, nuclear quadrupolar coupling is often the dominant spin interaction that governs solid-state NMR spectra. At magnetic fields with Larmor frequency higher than quadrupolar coupling frequency, the effect of quadrupolar interaction can be treated as a perturbation. The first-order effect can cause a large shift to satellite NMR transitions, but it vanishes for central NMR transition in the case of half-integer quadrupolar nuclei [1]. The second-order effect is one order smaller than the leading first-order term;
*
Corresponding author. Fax: 1-850-644-1366. E-mail address: [email protected] (Z. Gan).
however, it contains the l ¼ 4 rank anisotropic terms that still remain under magic-angle spinning (MAS). A complete average of the anisotropic second-order effect requires double rotation (DOR) [2,3], dynamic-angle spinning (DAS) [4,5], multiplequantum magic-angle spinning (MQMAS) [6,7] or satellite transition magic-angle spinning (STMAS) [8,9]. Quadrupolar effects higher than the secondorder are usually assumed negligible at high fields and the average of the second-order terms yields so-called isotropic NMR spectra. Although the third-order quadrupolar effect has been mentioned in low-field NMR experiments in the past [10–12], a direct observation and its effect to high-resolution NMR spectra have not yet been reported and discussed. In this Letter, we present the first experimental observation of the third-order quadrupolar
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 6 8 1 - 0
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Z. Gan et al. / Chemical Physics Letters 367 (2003) 163–169
effect at high magnetic fields. The high-order effect is observed through satellite transitions for which the third-order effect is not zero. With careful magic angle calibration and rotor synchronization, twodimensional STMAS experiment averages both the first- and the second-order quadrupolar effect. The resulting powder spectra show mostly the third-order effect. The spectra of andalusite show a surprisingly large third-order effect about 2.5 kHz for an aluminum site with mq about 2.3 MHz at 11.7 T.
2. Theory We consider the Hamiltonian of a spin S > 1=2 including Zeeman and quadrupolar interactions H ¼ x0 Sz þ xq
2 X
quadrupolar coupling Hamiltonian directly give the first-order quadrupolar effect DEmð1Þ ¼
m¼2
3Sz2 S 2 pffiffiffi ; 6 ðS Sz þ Sz S Þ ; ¼ 2 S2 ¼ ; S ¼ Sx iSy : 2
T2;0 ¼ T2;1 T2;2
ð2Þ
2 X
q2;m0 D2m0 ;m ða; b; cÞ;
m0 ¼2
q2;0
rffiffiffi 3 ; ¼ 2
ð3Þ q2;1 ¼ 0;
q2;2
Vm;mþDm VmþDm;mþDm0 VmþDm0 ;m DmDm0 x20
X Vm;mþDm VmþDm;m Vm;m Dm6¼0
ðDmx0 Þ
2
;
ð6Þ
where V is the quadrupolar coupling. Here we focus on the third-order term DEmð3Þ x3q Q2;2 Q2;2 Q2;0 pb DEmð3Þ ¼ 2 Q2;1 Q2;1 Q2;0 pa þ 4 x0 þ ½Q2;1 Q2;1 Q2;2 þ Q2;1 Q2;1 Q2;2 pc ; ð7Þ pffiffiffi 6 ½20m4 þ 7m2 ð12m2 þ 1ÞSðS þ 1Þ ; 4 pffiffiffi 6 ½5m4 þ 7m2 ð6m2 þ 2ÞSðS þ 1Þ þ S 2 ðS þ 1Þ2 ; pb ¼ 4 1 pc ¼ ½30m4 þ 15m2 24m2 þ 3 SðS þ 1Þ þ 2S 2 ðS þ 1Þ2 : 4 pa ¼
The spatial tensor components for the quadrupolar coupling can be obtained by a rotation from the principal axis frame to the laboratory frame, Q2;m ¼
X Dm;Dm0 6¼0
ð1Þ
At high magnetic field B0 , the Larmor frequency x0 ¼ cB0 is large compared to the quadrupolar coupling frequency xq ¼ e2 qzz Q=ð2Sð2S 1ÞhÞ. The quadrupolar coupling is expressed in irreducible second-rank tensors:
ð4Þ
where m ¼ S; S þ 1; . . . ; S describes the 2S þ 1 spin states. The m2 dependence in DEmð1Þ implies zero first-order effect to the central and multiplequantum transitions that are between two symmetric spin states m () m. The second- and the third-order quadrupolar effects can be derived by applying high-order perturbation theory [13] X jVm;mþDm j2 DEmð2Þ ¼ ; ð5Þ Dmx0 Dm6¼0 DEmð3Þ ¼
ð1Þm Q2;m T2;m :
½SðS þ 1Þ 3m2 pffiffiffi Q2;0 xq ; 6
g ¼ ; 2
where D2m0 ;m ða; b; cÞ are Wigner matrix elements and g ¼ ðqyy qxx Þ=qzz is the asymmetry factor of the electric field gradient (EFG) tensor. The effect of quadrupolar interaction to NMR transition frequencies can be derived using perturbation theory. The diagonal elements of the
ð8Þ Similar to the first-order effect, the third-order effect is also an even function of m and the difference is zero between symmetric spin states. This explains the reason why the third-order effect has been safely neglected even at moderate magnetic fields because most NMR spectra of half-integer quadrupolar nuclei have been detected through central transitions or indirectly through mq =2 () mq =2 MQ transitions. The third-order effect contains products of three spatial tensor components Q2;m . A product of two components can be expressed in terms of Clebsch–
Z. Gan et al. / Chemical Physics Letters 367 (2003) 163–169
Gordon coefficients cðl1 l2 l; m1 m2 Þ and tensor components Wl;m : X Ql1 m1 Ql2 m2 ¼ cðl1 l2 l; m1 m2 ÞWl;m1 þm2 ; X
Wl;m ¼
Dxð3Þ ¼
w0 ¼ ½12SðS þ 1Þ 30m2s 12 ms ;
ð9Þ
w2 ¼ ½3SðS þ 1Þ 15m2s 15=4 ms ;
m1 þm2 ¼m
w4 ¼ ½102SðS þ 1Þ 230m2s 269=2 ms ;
The product of three tensor components can be expanded consecutively in two steps leading to the expansion of the third-order quadrupolar effect DEmð3Þ ¼
w6 ¼ ½43SðS þ 1Þ=4 þ ð125=4Þm2s þ 235=16 ms : ð13Þ
l x3q X X ðal pa þ bl pb þ cl pc Þ x20 l¼0;2;4;6 m¼l
cl;m Dlm;0 ða; b; cÞ:
Fig. 1 shows transition frequencies obtained by exact numerical calculation with the full Hamiltonian minus from the first-, the second- and the third-order quadrupolar effect. The reduction of residual frequency with increasing order of quadrupolar effect demonstrates the convergence of perturbation theory noting that the vertical axes of the three plots are scaled down by mq =mL . The comparison between numerical results and perturbation theory also serves as an error check for the derivation of the expansion coefficients in Table 1.
ð10Þ
It should be noted that the tensor components are non-zero only for even l due to the relations l þl þl cðl1 l2 l; m1 m2 Þ ¼ ð1Þ 1 2 cðl2 l1 l; m2 m1 Þ and q2;m ¼ q2;m . It is also interesting to note that the expansion coefficients among the three parts in Eq. (7) are related by constant ratios. Table 1 lists explicitly the expansion coefficients cl;m and the ratios al ; bl ; cl . For rotating samples, various expansion coefficients are scaled by Pl ðcos hS Þ accordingly with their rank l leading to the third-order effect to m1 $ m2 transition frequency m 2 3 X x q Dxð3Þ ¼ 2 ðal pa þ bl pb þ cl pc Þ x0 l¼0;2;4;6
3. Experimental All experiments were performed on a 500 MHz Bruker Avance spectrometer with a 2.5 mm MAS probe. The magic-angle adjustment device was improved for fine angle increment of the spinning axis with respect to the magnetic field. Two-dimensional STMAS spectra were acquired with 30 kHz spinning frequency using a shifted-echo pulse sequence [14]. The diagonal peaks in STMAS spectra were reduced by presaturating the central transition peaks. Numerical simulations were performed using
m1
Pl ðcos hS Þ
l X
cl;m Dlm;0 ða; b; cÞ:
l X x3q X w P ðcos h Þ cl;m Dlm;0 ða; b; cÞ; l l M x20 l¼0;2;4;6 m¼l
ð12Þ
l¼l1 l2 ;...;l1 þl2
cðl1 l2 l; m1 m2 ÞQl1 ;m1 Ql2 ;m2 :
165
ð11Þ
m¼l
For single-quantum ms 1=2 () ms þ 1=2 satellite transitions, the third-order shift can be simplified as
Table 1 Expansion coefficients cl;m and ratios al : bl : cl for the third-order quadrupolar effect cl;m l¼0 2 4 6
m¼0 9 ð1 g2 Þ 70
9 g2 1þ 14 3
27 4 1 g2 385 9
54 g2 1þ 77 6
2 pffiffiffi
3 6g g2 1þ 28 3 pffiffiffiffiffi
9 10g g2 1þ 1540 3 pffiffiffiffiffiffiffiffi
18 105g g2 1þ 385 36
4
pffiffiffiffiffi 3 70 2 g 770 pffiffiffiffiffi 9 14 2 g 154
6
pffiffiffiffiffiffiffiffi 231 3 g 154
al
bl
cl
pffiffiffi 1= 6
pffiffiffi 2= 6
)2
pffiffiffi 1= 6
pffiffiffi 1= 6
0
pffiffiffi 1= 6
pffiffiffi 8= 6
)7
pffiffiffi 1= 6
1 pffiffiffi 4 6
1/3
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Z. Gan et al. / Chemical Physics Letters 367 (2003) 163–169
Fig. 1. Residual frequencies of the first satellite (solid lines) and the central (dotted lines) transitions of a spin S ¼ 5=2 after subtracting (a) the first, (b) the second and (c) and the third-order quadrupolar effect sequentially from the frequencies obtained by numerical diagonalization of the full Hamiltonian. The frequencies were calculated over a 16 16 grid (0 < h 6 p=2; 0 < u 6 p=2) for the polar angles of the magnetic field in the EFG principle axis frame. Quadrupolar coupling parameters of the larger mq site in andalusite qr ¼ e2 qzz Q=h ¼ 15:3 MHz and g ¼ 0:13 were used in the calculation. The frequencies are plotted against 16 16 ¼ 256 polar angles and the vertical frequency axes are scaled down from (a) to (b) to (c) by mq =mL ¼ 0:0177 with the quadrupolar coupling constant of andalusite and 27 Al Larmor frequency at 11.7 T.
MATLAB software (MathWorks, MA, USA). The isotropic chemical shift, the quadrupolar coupling constant and asymmetry factor are 11.9 ppm, 15.3 MHz, 0.13; and 35.2 ppm, 5.6 MHz and 0.76, respectively, for the two aluminum sites in andalusite.
4. Results and discussions Single-quantum satellite transitions are between two non-symmetric spin states and hence are shifted by quadrupolar effects of various orders. In order to quantitatively measure the very small high-order effect, the much larger first- and second-order effects must be averaged or at least reduced to the same magnitude as the third-order term. The two-dimensional STMAS experiment averages the first-order interaction with sample spinning precisely at the magic-angle and rotorsynchronized two-dimensional pulse sequence. The correlation between satellite and central transitions refocuses the second-order effect. The averaging of both the first- and second-order quadrupolar effects makes the observation of the third-order quadrupolar effect possible. Fig. 2 shows two-dimensional STMAS spectra of andalusite. The sample has two aluminum sites with quadrupolar coupling frequencies mq about 0.8 and 2.3 MHz. Because STMAS experiments are extremely sensitive to magic-angle setting, a
series of spectra were acquired by incrementing the spinning axis near the magic-angle. Two features can be found from these spectra. First, the powder patterns of the larger mq site would not collapse into a sharp horizontal ridge as expected by STMAS experiment through magic-angle calibration. The low but visible spectral intensities between )140 and )200 ppm split. Narrowing in this region through angle adjustment broadens the peak intensities from )20 to )120 ppm. Second, a comparison between the large and small mq sites shows that sharpening of the main peak intensities of the two sites occurs at different angle settings: (a) for the large mq site and (d) for the small mq site. These features are the results of the interplay between the third-order quadrupolar effect and the residual first-order term. Small deviations of the spinning axis from the magic-angle introduce a small residual first-order quadrupolar interaction. Varying the angle changes the scaling factor that partially cancels the third-order effect. The cancellation narrows the powder pattern at some regions as in (a); however a complete cancellation for total collapse of the powder pattern is not possible between two terms of different ranks. Because the third-order effect (/ m3q =m20 ) decreases rapidly with mq , the splitting of the small mq site mainly comes from the residual first-order interaction. The spectrum in (d) shows the sharpest line shape for the small mq site indicating that it is the closest to the
Z. Gan et al. / Chemical Physics Letters 367 (2003) 163–169
167
Fig. 2. Two-dimensional STMAS 27 Al spectra (left) and numerical simulations (right) of andalusite. The series of STMAS spectra were acquired with angle increment 0.00275° of spinning axis near magic angle (hS hM ) and the simulations were calculated including the high-order quadrupolar effect as described in the text.
magic-angle (0.001° as estimated by simulations). At this angle setting, the large mq site shows mainly the third-order effect that amounts about 20 ppm (or 2.5 kHz) in the STMAS spectrum at 11.7 T. The experimental results and numerical simulations in Fig. 2 show good agreement after the
third-order quadrupolar effect is taken into account. The four powder patterns are fitted by varying only a single parameter: the starting angle for the spinning axis. The angle increment 0.00275° is known for the angle adjustment device. Quadrupolar coupling and chemical shift para-
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Z. Gan et al. / Chemical Physics Letters 367 (2003) 163–169
meters are all known for andalusite. There may be other small effects that could affect the powder patterns. The cross terms between chemical shift anisotropy, dipolar coupling and quadrupolar coupling are expected to be small for this sample. The geometric or BerryÕs phase could also shift the resonance frequencies under sample spinning [15–18]. This shift adds to the transition frequency calculated from the eigenvalues of the Hamiltonian. The quadrupolar coupling tilts the quantization axis from the magnetic field roughly by an angle mq =mL . The tilted axis sweeps through a tra2 jectory enclosing a solid angle ðmq =mL Þ for a spinning sample. Repeated accumulation of the solid angle contributes to a shift of mr ðmq =mL Þ2 from the geometric phase to the resonance frequencies. For typical quadrupolar coupling and spinning frequencies, this geometric phase effect is about two orders of magnitude ( mq =mr ) smaller than the third-order quadrupolar effect. There could be another possibility that we have not yet considered. Nuclear hexadecapole interaction is expected from nuclear theory for S > 3=2 spins [1,19,20]. The existence of such spin interaction has not yet verified by experiments. The high-rank (l ¼ 4) property of the hexadecapole coupling means that the interaction, if it exists, would remain under MAS. The hexadecapole Hamiltonian is also an even function of Sz and would therefore shift satellite transition frequencies in the same way as the third-order effect. The nearly perfect fit of the STMAS spectra after taking the third-order effect into account indicates that the mentioned effects are small or at least below the level of detection for this specific sample and nuclei.
5. Conclusions It has been shown that the third-order quadrupolar effect does not shift the commonly observed central and multiple-quantum transitions in half-integer quadrupolar nuclei but does affect transitions between non-symmetric spin states such as satellite transitions. Theory and numerical simulations of the third-order effect has explained the unexpected features that have been observed in two-dimensional STMAS spectra of andalusite.
The third-order effect may add complications to the STMAS experiment in the aspect of spectral resolution. On the other hand, the non-symmetric nature of satellite transitions and the ability to refocus both the first and the second-order quadrupolar effects make the STMAS experiment capable of detecting small spin interactions that vanish between symmetric spin states. Examples of such interaction include hexadecapole coupling and the cross terms between quadrupolar coupling and chemical shift [21]. In this aspect, full understanding of the third-order quadrupolar effect is important.
Acknowledgements The authors thank Professor A. Samoson of National Institute of Chemical and Biophysics of Estonia and Professor F. Taulelle of Strasbourg University for discussions. The work is supported by the National High Magnetic Field Laboratory through National Science Foundation Cooperative Agreement DMR0084173 and by the State of Florida. J.R.Q. acknowledges support from National Science Foundation Grant DMB 9986036.
References [1] A. Abragam, Principles of Nuclear Magnetism, Clarendon Press, Oxford, 1961. [2] A. Samoson, E. Lippmaa, A. Pines, Mol. Phys. 65 (1988) 1013. [3] B.F. Chmelka, K.T. Mueller, A. Pines, J. Stebbins, Y. Wu, J.W. Zwanziger, Nature 339 (1989) 42. [4] A. Llor, J. Virlet, Chem. Phys. Lett. 152 (1988) 248. [5] K.T. Mueller, B.Q. Sun, G.C. Chingas, J.W. Zwanziger, T. Terao, A. Pines, J. Magn. Reson. 86 (1990) 470. [6] L. Frydman, J.S. Harwood, J. Am. Chem. Soc. 117 (1995) 5367. [7] A. Medek, J.S. Harwood, L. Frydman, J. Am. Chem. Soc. 117 (1995) 12779. [8] Z. Gan, J. Am. Chem. Soc. 122 (2000) 3242. [9] Z. Gan, J. Chem. Phys. 114 (2001) 10845. [10] L. Brown, D. Williams, J. Chem. Phys. 24 (1956) 751. [11] F. Wolf, D. Kline, H.S. Story, J. Chem. Phys. 53 (1970) 3538. [12] G.M. Volkoff, Can. J. Phys. 31 (1953) 820. [13] J.J. Sakurai, Modern Quantum Mechanics, AddisonWesley, New York, 1994.
Z. Gan et al. / Chemical Physics Letters 367 (2003) 163–169 [14] D. Massiot, B. Touzo, D. Trumeau, J.P. Coutures, J. Virlet, P. Florian, P.J. Grandinetti, Solid State Nucl. Magn. Reson. 6 (1996) 73. [15] M.V. Berry, Proc. R. Soc. Lond. A 392 (1984) 45. [16] R. Tycko, Phys. Rev. Lett. 58 (1987) 2281. [17] D. Suter, G.C. Chingas, R.A. Harris, A. Pines, Mol. Phys. 61 (1987) 1327.
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[18] N.R. Skrynnikov, B.C. Sancturay, Mol. Phys. 83 (1994) 1133. [19] E.B. Doering, J.S. Waugh, J. Chem. Phys. 85 (1986) 1753. [20] M.Y. Liao, G.S. Harbison, J. Chem. Phys. 100 (1994) 1895. [21] S. Wi, L. Frydman, J. Chem. Phys. 116 (2002) 1151.
Third-order effect in solid-state NMR of quadrupolar nuclei Zhehong Gan a b
a,*
, Parthasarathy Srinivasan b, John R. Quine b, Stefan Steuernagel c, Benno Knott c
National High Magnetic Field Laboratory (NHMFL), Tallahassee, FL 32310, USA Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA c Bruker Analytik GmbH, D-76287 Rheinstetten, Germany Received 17 May 2002
Abstract The theory and experimental observation of the third-order effect in solid-state NMR of quadrupolar nuclei are presented. The third-order effect consists of spherical harmonic terms up to rank l ¼ 6 and shifts NMR frequencies between two spin states that are not symmetric such as satellite transitions. Two-dimensional satellite transition magicangle spinning experiment averages both the first and the second-order quadrupolar interactions making the quantitative measurement of the third-order effect possible. The third-order quadrupolar effect in andalusite has been measured at 11.7 T and its powder patterns are fitted with numerical simulations. Ó 2002 Elsevier Science B.V. All rights reserved.
1. Introduction For nuclei with spin S > 1=2, nuclear quadrupolar coupling is often the dominant spin interaction that governs solid-state NMR spectra. At magnetic fields with Larmor frequency higher than quadrupolar coupling frequency, the effect of quadrupolar interaction can be treated as a perturbation. The first-order effect can cause a large shift to satellite NMR transitions, but it vanishes for central NMR transition in the case of half-integer quadrupolar nuclei [1]. The second-order effect is one order smaller than the leading first-order term;
*
Corresponding author. Fax: 1-850-644-1366. E-mail address: [email protected] (Z. Gan).
however, it contains the l ¼ 4 rank anisotropic terms that still remain under magic-angle spinning (MAS). A complete average of the anisotropic second-order effect requires double rotation (DOR) [2,3], dynamic-angle spinning (DAS) [4,5], multiplequantum magic-angle spinning (MQMAS) [6,7] or satellite transition magic-angle spinning (STMAS) [8,9]. Quadrupolar effects higher than the secondorder are usually assumed negligible at high fields and the average of the second-order terms yields so-called isotropic NMR spectra. Although the third-order quadrupolar effect has been mentioned in low-field NMR experiments in the past [10–12], a direct observation and its effect to high-resolution NMR spectra have not yet been reported and discussed. In this Letter, we present the first experimental observation of the third-order quadrupolar
0009-2614/02/$ - see front matter Ó 2002 Elsevier Science B.V. All rights reserved. PII: S 0 0 0 9 - 2 6 1 4 ( 0 2 ) 0 1 6 8 1 - 0
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Z. Gan et al. / Chemical Physics Letters 367 (2003) 163–169
effect at high magnetic fields. The high-order effect is observed through satellite transitions for which the third-order effect is not zero. With careful magic angle calibration and rotor synchronization, twodimensional STMAS experiment averages both the first- and the second-order quadrupolar effect. The resulting powder spectra show mostly the third-order effect. The spectra of andalusite show a surprisingly large third-order effect about 2.5 kHz for an aluminum site with mq about 2.3 MHz at 11.7 T.
2. Theory We consider the Hamiltonian of a spin S > 1=2 including Zeeman and quadrupolar interactions H ¼ x0 Sz þ xq
2 X
quadrupolar coupling Hamiltonian directly give the first-order quadrupolar effect DEmð1Þ ¼
m¼2
3Sz2 S 2 pffiffiffi ; 6 ðS Sz þ Sz S Þ ; ¼ 2 S2 ¼ ; S ¼ Sx iSy : 2
T2;0 ¼ T2;1 T2;2
ð2Þ
2 X
q2;m0 D2m0 ;m ða; b; cÞ;
m0 ¼2
q2;0
rffiffiffi 3 ; ¼ 2
ð3Þ q2;1 ¼ 0;
q2;2
Vm;mþDm VmþDm;mþDm0 VmþDm0 ;m DmDm0 x20
X Vm;mþDm VmþDm;m Vm;m Dm6¼0
ðDmx0 Þ
2
;
ð6Þ
where V is the quadrupolar coupling. Here we focus on the third-order term DEmð3Þ x3q Q2;2 Q2;2 Q2;0 pb DEmð3Þ ¼ 2 Q2;1 Q2;1 Q2;0 pa þ 4 x0 þ ½Q2;1 Q2;1 Q2;2 þ Q2;1 Q2;1 Q2;2 pc ; ð7Þ pffiffiffi 6 ½20m4 þ 7m2 ð12m2 þ 1ÞSðS þ 1Þ ; 4 pffiffiffi 6 ½5m4 þ 7m2 ð6m2 þ 2ÞSðS þ 1Þ þ S 2 ðS þ 1Þ2 ; pb ¼ 4 1 pc ¼ ½30m4 þ 15m2 24m2 þ 3 SðS þ 1Þ þ 2S 2 ðS þ 1Þ2 : 4 pa ¼
The spatial tensor components for the quadrupolar coupling can be obtained by a rotation from the principal axis frame to the laboratory frame, Q2;m ¼
X Dm;Dm0 6¼0
ð1Þ
At high magnetic field B0 , the Larmor frequency x0 ¼ cB0 is large compared to the quadrupolar coupling frequency xq ¼ e2 qzz Q=ð2Sð2S 1ÞhÞ. The quadrupolar coupling is expressed in irreducible second-rank tensors:
ð4Þ
where m ¼ S; S þ 1; . . . ; S describes the 2S þ 1 spin states. The m2 dependence in DEmð1Þ implies zero first-order effect to the central and multiplequantum transitions that are between two symmetric spin states m () m. The second- and the third-order quadrupolar effects can be derived by applying high-order perturbation theory [13] X jVm;mþDm j2 DEmð2Þ ¼ ; ð5Þ Dmx0 Dm6¼0 DEmð3Þ ¼
ð1Þm Q2;m T2;m :
½SðS þ 1Þ 3m2 pffiffiffi Q2;0 xq ; 6
g ¼ ; 2
where D2m0 ;m ða; b; cÞ are Wigner matrix elements and g ¼ ðqyy qxx Þ=qzz is the asymmetry factor of the electric field gradient (EFG) tensor. The effect of quadrupolar interaction to NMR transition frequencies can be derived using perturbation theory. The diagonal elements of the
ð8Þ Similar to the first-order effect, the third-order effect is also an even function of m and the difference is zero between symmetric spin states. This explains the reason why the third-order effect has been safely neglected even at moderate magnetic fields because most NMR spectra of half-integer quadrupolar nuclei have been detected through central transitions or indirectly through mq =2 () mq =2 MQ transitions. The third-order effect contains products of three spatial tensor components Q2;m . A product of two components can be expressed in terms of Clebsch–
Z. Gan et al. / Chemical Physics Letters 367 (2003) 163–169
Gordon coefficients cðl1 l2 l; m1 m2 Þ and tensor components Wl;m : X Ql1 m1 Ql2 m2 ¼ cðl1 l2 l; m1 m2 ÞWl;m1 þm2 ; X
Wl;m ¼
Dxð3Þ ¼
w0 ¼ ½12SðS þ 1Þ 30m2s 12 ms ;
ð9Þ
w2 ¼ ½3SðS þ 1Þ 15m2s 15=4 ms ;
m1 þm2 ¼m
w4 ¼ ½102SðS þ 1Þ 230m2s 269=2 ms ;
The product of three tensor components can be expanded consecutively in two steps leading to the expansion of the third-order quadrupolar effect DEmð3Þ ¼
w6 ¼ ½43SðS þ 1Þ=4 þ ð125=4Þm2s þ 235=16 ms : ð13Þ
l x3q X X ðal pa þ bl pb þ cl pc Þ x20 l¼0;2;4;6 m¼l
cl;m Dlm;0 ða; b; cÞ:
Fig. 1 shows transition frequencies obtained by exact numerical calculation with the full Hamiltonian minus from the first-, the second- and the third-order quadrupolar effect. The reduction of residual frequency with increasing order of quadrupolar effect demonstrates the convergence of perturbation theory noting that the vertical axes of the three plots are scaled down by mq =mL . The comparison between numerical results and perturbation theory also serves as an error check for the derivation of the expansion coefficients in Table 1.
ð10Þ
It should be noted that the tensor components are non-zero only for even l due to the relations l þl þl cðl1 l2 l; m1 m2 Þ ¼ ð1Þ 1 2 cðl2 l1 l; m2 m1 Þ and q2;m ¼ q2;m . It is also interesting to note that the expansion coefficients among the three parts in Eq. (7) are related by constant ratios. Table 1 lists explicitly the expansion coefficients cl;m and the ratios al ; bl ; cl . For rotating samples, various expansion coefficients are scaled by Pl ðcos hS Þ accordingly with their rank l leading to the third-order effect to m1 $ m2 transition frequency m 2 3 X x q Dxð3Þ ¼ 2 ðal pa þ bl pb þ cl pc Þ x0 l¼0;2;4;6
3. Experimental All experiments were performed on a 500 MHz Bruker Avance spectrometer with a 2.5 mm MAS probe. The magic-angle adjustment device was improved for fine angle increment of the spinning axis with respect to the magnetic field. Two-dimensional STMAS spectra were acquired with 30 kHz spinning frequency using a shifted-echo pulse sequence [14]. The diagonal peaks in STMAS spectra were reduced by presaturating the central transition peaks. Numerical simulations were performed using
m1
Pl ðcos hS Þ
l X
cl;m Dlm;0 ða; b; cÞ:
l X x3q X w P ðcos h Þ cl;m Dlm;0 ða; b; cÞ; l l M x20 l¼0;2;4;6 m¼l
ð12Þ
l¼l1 l2 ;...;l1 þl2
cðl1 l2 l; m1 m2 ÞQl1 ;m1 Ql2 ;m2 :
165
ð11Þ
m¼l
For single-quantum ms 1=2 () ms þ 1=2 satellite transitions, the third-order shift can be simplified as
Table 1 Expansion coefficients cl;m and ratios al : bl : cl for the third-order quadrupolar effect cl;m l¼0 2 4 6
m¼0 9 ð1 g2 Þ 70
9 g2 1þ 14 3
27 4 1 g2 385 9
54 g2 1þ 77 6
2 pffiffiffi
3 6g g2 1þ 28 3 pffiffiffiffiffi
9 10g g2 1þ 1540 3 pffiffiffiffiffiffiffiffi
18 105g g2 1þ 385 36
4
pffiffiffiffiffi 3 70 2 g 770 pffiffiffiffiffi 9 14 2 g 154
6
pffiffiffiffiffiffiffiffi 231 3 g 154
al
bl
cl
pffiffiffi 1= 6
pffiffiffi 2= 6
)2
pffiffiffi 1= 6
pffiffiffi 1= 6
0
pffiffiffi 1= 6
pffiffiffi 8= 6
)7
pffiffiffi 1= 6
1 pffiffiffi 4 6
1/3
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Fig. 1. Residual frequencies of the first satellite (solid lines) and the central (dotted lines) transitions of a spin S ¼ 5=2 after subtracting (a) the first, (b) the second and (c) and the third-order quadrupolar effect sequentially from the frequencies obtained by numerical diagonalization of the full Hamiltonian. The frequencies were calculated over a 16 16 grid (0 < h 6 p=2; 0 < u 6 p=2) for the polar angles of the magnetic field in the EFG principle axis frame. Quadrupolar coupling parameters of the larger mq site in andalusite qr ¼ e2 qzz Q=h ¼ 15:3 MHz and g ¼ 0:13 were used in the calculation. The frequencies are plotted against 16 16 ¼ 256 polar angles and the vertical frequency axes are scaled down from (a) to (b) to (c) by mq =mL ¼ 0:0177 with the quadrupolar coupling constant of andalusite and 27 Al Larmor frequency at 11.7 T.
MATLAB software (MathWorks, MA, USA). The isotropic chemical shift, the quadrupolar coupling constant and asymmetry factor are 11.9 ppm, 15.3 MHz, 0.13; and 35.2 ppm, 5.6 MHz and 0.76, respectively, for the two aluminum sites in andalusite.
4. Results and discussions Single-quantum satellite transitions are between two non-symmetric spin states and hence are shifted by quadrupolar effects of various orders. In order to quantitatively measure the very small high-order effect, the much larger first- and second-order effects must be averaged or at least reduced to the same magnitude as the third-order term. The two-dimensional STMAS experiment averages the first-order interaction with sample spinning precisely at the magic-angle and rotorsynchronized two-dimensional pulse sequence. The correlation between satellite and central transitions refocuses the second-order effect. The averaging of both the first- and second-order quadrupolar effects makes the observation of the third-order quadrupolar effect possible. Fig. 2 shows two-dimensional STMAS spectra of andalusite. The sample has two aluminum sites with quadrupolar coupling frequencies mq about 0.8 and 2.3 MHz. Because STMAS experiments are extremely sensitive to magic-angle setting, a
series of spectra were acquired by incrementing the spinning axis near the magic-angle. Two features can be found from these spectra. First, the powder patterns of the larger mq site would not collapse into a sharp horizontal ridge as expected by STMAS experiment through magic-angle calibration. The low but visible spectral intensities between )140 and )200 ppm split. Narrowing in this region through angle adjustment broadens the peak intensities from )20 to )120 ppm. Second, a comparison between the large and small mq sites shows that sharpening of the main peak intensities of the two sites occurs at different angle settings: (a) for the large mq site and (d) for the small mq site. These features are the results of the interplay between the third-order quadrupolar effect and the residual first-order term. Small deviations of the spinning axis from the magic-angle introduce a small residual first-order quadrupolar interaction. Varying the angle changes the scaling factor that partially cancels the third-order effect. The cancellation narrows the powder pattern at some regions as in (a); however a complete cancellation for total collapse of the powder pattern is not possible between two terms of different ranks. Because the third-order effect (/ m3q =m20 ) decreases rapidly with mq , the splitting of the small mq site mainly comes from the residual first-order interaction. The spectrum in (d) shows the sharpest line shape for the small mq site indicating that it is the closest to the
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Fig. 2. Two-dimensional STMAS 27 Al spectra (left) and numerical simulations (right) of andalusite. The series of STMAS spectra were acquired with angle increment 0.00275° of spinning axis near magic angle (hS hM ) and the simulations were calculated including the high-order quadrupolar effect as described in the text.
magic-angle (0.001° as estimated by simulations). At this angle setting, the large mq site shows mainly the third-order effect that amounts about 20 ppm (or 2.5 kHz) in the STMAS spectrum at 11.7 T. The experimental results and numerical simulations in Fig. 2 show good agreement after the
third-order quadrupolar effect is taken into account. The four powder patterns are fitted by varying only a single parameter: the starting angle for the spinning axis. The angle increment 0.00275° is known for the angle adjustment device. Quadrupolar coupling and chemical shift para-
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meters are all known for andalusite. There may be other small effects that could affect the powder patterns. The cross terms between chemical shift anisotropy, dipolar coupling and quadrupolar coupling are expected to be small for this sample. The geometric or BerryÕs phase could also shift the resonance frequencies under sample spinning [15–18]. This shift adds to the transition frequency calculated from the eigenvalues of the Hamiltonian. The quadrupolar coupling tilts the quantization axis from the magnetic field roughly by an angle mq =mL . The tilted axis sweeps through a tra2 jectory enclosing a solid angle ðmq =mL Þ for a spinning sample. Repeated accumulation of the solid angle contributes to a shift of mr ðmq =mL Þ2 from the geometric phase to the resonance frequencies. For typical quadrupolar coupling and spinning frequencies, this geometric phase effect is about two orders of magnitude ( mq =mr ) smaller than the third-order quadrupolar effect. There could be another possibility that we have not yet considered. Nuclear hexadecapole interaction is expected from nuclear theory for S > 3=2 spins [1,19,20]. The existence of such spin interaction has not yet verified by experiments. The high-rank (l ¼ 4) property of the hexadecapole coupling means that the interaction, if it exists, would remain under MAS. The hexadecapole Hamiltonian is also an even function of Sz and would therefore shift satellite transition frequencies in the same way as the third-order effect. The nearly perfect fit of the STMAS spectra after taking the third-order effect into account indicates that the mentioned effects are small or at least below the level of detection for this specific sample and nuclei.
5. Conclusions It has been shown that the third-order quadrupolar effect does not shift the commonly observed central and multiple-quantum transitions in half-integer quadrupolar nuclei but does affect transitions between non-symmetric spin states such as satellite transitions. Theory and numerical simulations of the third-order effect has explained the unexpected features that have been observed in two-dimensional STMAS spectra of andalusite.
The third-order effect may add complications to the STMAS experiment in the aspect of spectral resolution. On the other hand, the non-symmetric nature of satellite transitions and the ability to refocus both the first and the second-order quadrupolar effects make the STMAS experiment capable of detecting small spin interactions that vanish between symmetric spin states. Examples of such interaction include hexadecapole coupling and the cross terms between quadrupolar coupling and chemical shift [21]. In this aspect, full understanding of the third-order quadrupolar effect is important.
Acknowledgements The authors thank Professor A. Samoson of National Institute of Chemical and Biophysics of Estonia and Professor F. Taulelle of Strasbourg University for discussions. The work is supported by the National High Magnetic Field Laboratory through National Science Foundation Cooperative Agreement DMR0084173 and by the State of Florida. J.R.Q. acknowledges support from National Science Foundation Grant DMB 9986036.
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