Analysis of the Quadrupolar Coupling Effect on the Line Intensities Using Sin

Analysis of the Quadrupolar Coupling Effect on the Line Intensities Using Sin

e-Crystal Nutation NMR in a -AbO

Crystals

Ae Ja Woo,^* Duk-Young Hanj andSoHyun Cho

Departmentof Science Education,

、

With ID-nutation NMR for a spinI=5/2 system,therelative line intensities of thecentralandthe inner- and outer-satellite transitions are calculated as functionsof quadrupolar couplingoqand rf pulse strength af Ex­ perimentallymeasured line intensitiesincludingbothcentralandsatellites are used toextract thevalues of^oq

and Of fromnonlinearleast-squares fits. The method is illustrated in a-ALO3 crystals (ruby andcorundum) withthesingle-crystal 27Alnutation NMR spectra.As a result,thenew feature that therf pulse strength shows reduced effect on thesatellite transitionlines according tothe quadrupolar couplingisdiscussedbyusing a fic­ titiousspin-1/2 operator.

Introduction

Solid-state 27Al (I = 5/2) NMR spectroscopy has been widely usedto investigate the inorganic solids such as zeo­

lites, glasses, ceramics, and clay minerals. The 27Al NMR quadrupolar coupling, which describes the principal ele­ ments of the electric fieldgradient tensor, depends onthe local symmetry around the nucleus and gives direct struc­ tural information. For a powdered material, MAS (magic an이e spinning),1,2 VAS (variable an이e spinning),3 DAS (dynamic angle spinning),4,5 DOR (double rotation),6,7 and MQ (Multiple Quantum) MAS8,9 NMR have been applied for determining quadrupolar coupling constante2qQ/h and asymmetry parameter n by acquiring high-resolution spec­

tra. Sin 이 e-crystal NMR10,11 also provides the most direct approachtodeterminethe quadrupolarcouplingparameters.

In these NMR techniques, theeffect of quadrupolar interac­

tion duringrfpulseexcitationiscompletelyignored assum­ ing itis much weaker than the excitationpulse, which is not easyto be thecaseinrealexperiments. On theotherhand,in the nutation NMR technique, the spin dynamics nutating aroundrfmagnetic field underthe influence of the stronger quadrupolarinteractionduringpulsing can be observed.

For half-integer quadrupolar spin systems, 1D- and 2D­ nutation NMR areeffectivetechniquestodeterminethequa- drupolarcoupling parameters andtoresolvethe overlapping powder patterns with dissimilar quadrupolar interactions, respectively.For a spinI= 5/2 system,extensiveworkshave been done theoretically and also experimentally. A. P. M.

Kentgen, W. S. Veeman, andP. P.Man havegiven theoreti­

cal descriptions of 1D- and2D-nutationNMR.12-15 The 27Al 1D-nutation NMR hasbeen used to determinethequadrupo­ larcoupling in a corundum (a-AbOa) crystal.12Applications of the 27Al 2D-nutation NMR on the structural studies of inorganic solids16-18 present successful results. All of these worksfocusedonly on theanalysis ofthe central line inten­ sities.

Herein, we aretryingto determinethe 27Al NMR quadru- polar coupling in a-Al2O³ crystals from the relative line

intensities of the central and the inner- and outer-satellite transitions inthe 1D-nutation NMR spectra. In this course, we expect two distinctive features: (1) The modulation pat­

tern ofcentral transition is affected by the satellite transi­

tions. (2) The rf pulsestrengthover thesatellitetransitionsis reduced compared to that overthecentraltransition.

Theory

Scheme Ishows a graphical descriptionforthex-nutation NMR pulse sequence.

耳f+ *

I

For a spinI= 5/2system, theHamiltonianduring rf pulse excitation in therotatingframeisrepresentedby

H^on = Hrf+ H^Q ) = - O^rfIx+ 1-Oq ( 31-1 (I+1 ))

oQ = 이*(3cos2。- 1+nsin2^cos20) (1) 80 h

where H^fand H^Q) representthe coupling of the spin with the rf magnetic fieldandthe firstorderquadrupolar interac­

tion,respectively. 0and 0are Euler angles neededtoorient the principal axis of electric field gradient tensor with respect to theappliedmagneticfield.

The spin dynamics excitedby a x-nutation pulse can be described by the density matrix. The evolution of density matrix during t¹ is obtained by solving the Liouville-von Neumann equationasfollows;

P( 11) ij= {exp(-Hn11 )p( 0) exp(Hn11)} ^ij

={Texp(- iT+^心 Tt 1) T+p( 0) Texp(iT+ H°n Tt 1) T+} 〃

=» TiTiexp{-id 111)(T+p(0) T) k 1} (2)

k, 1

where Qk 1 = (Ek — E¹), represents nine nutation frequen­ ciesasshowninScheme II, whichonlydependon theeigen­ values.

E6

e5

E4

e3 E2 Ei

f

k

y

）、

f f ^、

n

Unitary matrix (T) diagonalizing Hamiltonian Hon can be found by solving the eigenvalue equation and its secular equation like

Hn中i =冗i中i and Det(% — XI) = 0. (3) The equilibriumdensitymatrixp(0) corresponds to Izand T+p(0)T isexpressedby

T+p( 0)T =

0 0 0 S11 S¹² S13

0 0 0 S²¹ S22 S²³ 0 0 0 S31 S32 S33

5¹¹ S 21 S 31 0 0 0

512 S22 S32 0 0 0 5¹³ S23 S³³ 0 0 0

(4)

And, the line intensity of the central (S(t¹)34) and inner- (S(t1)23) andouter- (S(t1)12) satellitetransitionsduring t1 are representedas

S(1¹), = [p( 1^{1 山}(I+) j

Thus,S(1¹)34 = 3p(1¹)43 = 2i 文 文 SZ+Zj-sin(*】)

i = 1 j = 1

S(11) 23 = W2p( 11) 32 = 马力力 i = 1 j = 1

Sij {i (Yⁱ⁺Zj-- Yj - Zi+)}sin 以 〃t 】) S(1¹)12 = J5p( 1¹)21 =T^主文

i = 1 j = 1

S, {i(X + Y^j--Xj- Yⁱ⁺)} sin *^】).(5) where S^j= 5X+X^- + 3 ^珞 Y^- + Z⁺Z^- and 扁=^儿•+—*•-. The termsXi+,Xj-, Yi+, Yj-, Zi+, andZj- aredescribed indetailsin

thereference.19 Eqn. (5) shows that eachline intensityisthe sumofnine sinecurvesofdifferentamplitudesandfrequen­ cies. And therelativeline intensitiesofthecentraland satel­

litetransitiondependon thevaluesof气and «f. ExperimentalSection

27Al nutation NMR spectra were acquired on a Bruker MSL200 solid-state NMR spectrometer operating at 52.2 MHz. a-AbO³ crystals(rubyand corundum) were mounted on a high-powerwidelineprobe andorientedby a goniome­ ter. Thex-nutationpulse sequencewas applied.The rf pulse length t¹ was incrementedinsteps of 1卩s from 0卩s. The rf pulse strengthes 아^f were adjusted to 25 kHz and 63 kHz which correspond to 90o pulse lengthes of 10您 and 4 您， respectively, for corundum andruby. An aqueoussolution of Al(H2O)63+ was usedtodetermine90o pulse length. A relax­

ationdelay of 20 s and a dead timeof 8 卩s wereused. Each spectrum was obtained by summing 10 free induction decays. Exponentialline broadening of 800 Hz andnozero filling were applied. The relative line intensities of central andsatellitetransitionsweremeasuredfrom a Fourier-trans­ formedandmanually-phasedspectrum.

ResultsandDiscussion

Shown in Figure 1(a) is the 27Al single-crystal nutation NMRspectra for acorundum crystal at a fixed orientation with respect to the applied magnetic field. Signal-to-noise ratio (S/N〜50) of the central transition linefor t¹ = 1 卩s is enough to measure the line intensity with high accuracy.

Because of thelarge spectral width (2.5 MHz), onlythe cen­

tral transition line was properly phased in a spectrum. All spectra weredrawnon theabsoluteintensityscale. The rela­

tive line intensities of inner- and outer-satellite transitions were obtained by averaging each pair ofthe satellites. In Figure 1(b), we show the experimental symmetrized stick spectra recreatedfrom the relativelineintensitiesof the 27Al nutation NMR spectra. From the line separation between inner-satellitetransitions,theoreticallyfour times of quadru-

Figure 1. (a) Experimental 27Al nutation NMR spectra of a corundum crystal at a fixed orientation for rf pulse length t1= 1 to 6

“s. (b) Experimental symmeterized stick spectra recreated from the relative line intensities of the 27Al nutation NMR spectra.

으

uo

드

은 —-

(D

루 -

(D Qr

Figure 2. Results from nonlinear least-squares analysis of the relative line intensities of the central

transitions measured from Figure

andinner- and outer-satellite Experimental data are represented as circles (^•) and diamonds (^♦ ) and squares (^■) for the central and the inner- and outer-satellite transitions, respectively. The solid lines ( __ ) were calculated with the best fitted parameters. The dashed lines (---) were obtained without the effect of the reduced rf pulse strength over the satellite transitions.

polar coupling (=4^q), 27Al NMR quadrupolar coupling with axialsymmetryin a corundumcrystalwereexperimen­ tallydeterminedtobe ~85 kHz.

Figure 2showsthenonlinearleast-squares fitstothe rela­

tive line intensities ofthe central and the inner- and outer­ satellite transitions measured from Figure 1. Experimental data are represented as circles (^•) and diamonds (^♦) and squares(■ )for the centralandtheinner-andouter-satellite transitions, respectively. The best calculated fitsare repre­

sented as the solid lines ( __ ) andthe calculations aredone by using Eqn. (5). The Levenberg-Marquardt nonlinear least-squares algorithm20 was used. The important fitted parameters are (DqandQ^f. In order to correct the systematic experimental errors such as rf pulse rising time, probe fre­ quency response effect, and finite rf pulse effect, imperfec­ tions in t1 and D^f are also considered. Finally, variable contribution factors a and /3are also included in Eqn. (7), whichhave aneffectonthe line intensities ofthe inner-and outer-satellitetransitions, respectively.Fittedparameters for thecorundumcrystalare listed in Table 1.

Figure 3 shows the 27Al nutation NMRspectra for a ruby crystalatthree orientations (I,II, and III) withrespecttothe appliedmagnetic fieldat a constant rf fieldstrength.Figures 3(a), 3(b), and 3(c) correspond tothe orientations I, II, and III. These orientations were purposely chosen to show the spectra of the different quadrupolar interaction. 27Al NMR quadrupolar couplings with axial symmetry were experi­

mentally determined to be ~25, ~67, and ~83 kHz, respec­ tively, for three orientations. Figure 4 represents the nonlinear least-squares fits to theexperimental line intensi­ ties ofthe central andthe inner-satellite transitions.Experi­ mental data are represented as circles (^•) and diamonds

Table 1. Results of Nonlinear Least-Squares Fits to the Sin이e- Crystal 27Al Nutation NMR Spectra of a-AhO3Crystals

oq (kHz) orf (kHz) a P

Corundum 81.2(6) 67.2(5) 0.89(7) 0.84(2)

Ruby Ia 26.7(7) 19.6(6) 0.71(9) 0.39(2)

II” 62.9(2) 19.6(6) 0.84(2) 0.35(4) IIIc 77.6(2) 19.6(6) 0.86(7) 0.46(4) acorresponds to Figures 3(a), 4(a), and 5(a). ”corresponds to Figures 3(b), 4(b), and 5(b). ccorresponds to Figures 3(c), 4(c), and 5(c).

Figure 3. Experimental 27Al nutation NMR spectra of a ruby crystal at three orientations for rf pulse length t1= 1 to 10 “s. Three orientations I, II, and III correspond to figure (a), (b), and (c), respectively.

(^♦) and squares (^■) for the central and the inner- and outer-satellite transitions, respectively. The best calculated fits arerepresentedasthesolid lines ( __ ). Fittedparameters for a rubycrystalarealsolistedin Table 1. Thevaluesofdq and (O^rf extracted from nonlinear least-squares fits are in goodagreement with those determinedfromthe experimen­ taldata.

There is an interesting feature investigated in this work:

the line intensitiesof the satellite transitions relative to the central transition are affected by the reduced rf pulse strength. In case the effect of the reduced rf pulse strength over the satellite transitions is not considered, the relative lineintensitiesoftheinner-andouter-satellitetransitions are representedasthedashed lines(---) showninFigures 2 and 4. It could be explained in terms ofthe fictitious spin-1/2 operator. Hamiltonianforthe rf pulse excitation O^fIx canbe expressed like

orfIx = orf(M5『a/12 + 3l^X4+a2T2 後+财V) (7) Theadditionalfactorsa and/3shouldbeconsideredbecause the rf pulsestrengthovertheinner-andouter-satellite transi­

tions can bereducedinreality. Ideally,thevalues ofa andp areequalto 1 in thenutation NMR theory.21,22 The values of aandpextracted fromthenonlinearleast-squaresfitfora- Al²O³ crystals are listed in Table 1. Moreover, it is quite

Figure 4. Results from nonlinear least-squares analysis of the relative line intensities of the central and inner-satellite transitions measured from Figure 3. Experimental data are represented as circles (^•) and diamonds (^♦) and squares (^■) for the central and the inner- and outer-satellite transitions, respectively.. The solid lines (—) are calculated with the best fitted parameters. The dashed lines (---) were obtained without the effect of the reduced rf pulse strength over the satellite transitions.

interestingtoshow therelationshipbetween thevaluesof (Oq anda and&. However,at this moment,itisdifficulttoquan­ titativelyexplainthetrends ofthe factors a and/3correlated with thevaluesofoqand O^f.

Figure 5 shows the simulated single-crystal 27Al nutation spectrafort1 =0to 31 卩s, whichare calculatedbyusingthe same setofthe fittedparametersasextractedfromnonlinear least-squares fittings.As shown in Figure 5,thenutation of the quadrupolar nuclei are represented as functions of the quadrupolar couplingandthe rf pulse strength. The ampli­ tude and frequency modulation clearly shows that there exists themutual relationbetween the centraland the satel­ litetransitions.

Conclusions

27Al NMR quadrupolar couplings in a-AbO3 crystals (corundum andruby)were determined fromthe analysis of the single-crystal 27Al nutation NMR spectra. The experi­ mentallymeasuredrelative line intensities ofthe centraland

Figure 5. Simulated ID-nutation spectra for a spin I = 5/2 system, which are calculated using the same values of the fitted parameters for a ruby crystal at three orientations as listed in Table 1.

the inner- and outer-satellite transitions were used for the nonlinearleast-squaresfitting. In thecourse of thiswork,we found that the rf pulse strengthaccordingtothe quadrupolar coupling has anreduced influence on the line intensitiesof the satellitetransitions. Asa result, the analysis ofthe satel­ litetransitionsis necessary forthe correct understanding of thecentraltransition.

Acknowledgment. This work was supported by the Korea Science andEngineering Foundation (971-0305-036­

1). The Korea Basic Science Institute is acknowledged for the useof the BrukerMSL200 solid-state NMR spectrome­ ter.

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