Off-resonance effects on 2D NMR nutation spectra of I = 32 quadrupolar nuclei in static samples

SOLID STATE Nuclear Magnetic Resonance Solid State Nuclear Magnetic Resonance 5 (1995) 227-232

Off-resonance

effects on 2D NMR nutation spectra of I = 3/2 quadrupolar nuclei in static samples Youlin Xia, Feng Deng, Chaohui Ye

*

Laboratory of Magnetic Resonance and Atomic and Molecular Physics, Wuhan Institute of Physics, The Chinese Academy of Sciences, P.O. Box 71010, Wuhan 430071, People’s Republic of China

Received 17 January 1995; 7 April 1995

Abstract The off-resonance effects on 2D NMR nutation of I = 3/2 quadrupolar nuclei are demonstrated with perturbation theory and numerical calculation in static samples. The off-resonant (Aa) rf field (w,) enlarges a nutation frequency and consequently increases the measurement range of nuclear quadrupolar interaction parameters. When W, > WY, and arctg(w,/Aw) = k54.7” (magic angle), the satellite lines (produced by coherence transfers) in a nutation spectrum are superimposed with the line of central transition, and hence the nutation spectrum is simplified and its sensitivity is enhanced. The nuclear quadrupolar interaction parameters of 23Na nuclei in NaR molecular sieve are obtained using 2D NMR nutation. Keywords:

Molecular sieve; Quadrupolar

nuclei; Static sample nutation

1. Introduction

2D NMR nutation experiment [l-5] is a powerful tool measuring nuclear quadrupolar interaction parameters. Because the rf field is applied during its evolution time of 2D NMR nutation experiment, the chemical shift interaction causes a second-order influence in comparison with the strong rf field, and can usually be neglected. The nutation experiment not only eliminates the chemical shift interaction, but can also enhance resolution. In certain cases, the resolution of nutation spectra can even be better than that of DAS and DOR 161.

* Corresponding 0926-2040/95/$09.50

author 0 1995

SSDZ 0926-2040(95)01185-4

I = 3/2

The strength of the rf field must be increased in order to enlarge the measurement range of nuclear quadrupolar interaction parameters. Besides, it is sometimes necessary to obtain the nutation spectra under various rf field strengths. However, there are some experimental difficulties in enlarging or varying the strength of the rf field. Kentgens [7] has suggested an off-resonance method to meet the two needs mentioned above. This off-resonance nutation experiment has advantages: firstly, the axial peaks, which usually appear in frequency-stepped adiabatic half-passage experiments [8,9], can be removed; secondly, higher sensitivity can be obtained. However, experimental nutation spectra are generally rather complicated when a few non-equivalent sites are encountered in the experiment, so spectral simulations have to be performed.

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Y. Xia et al. /Solid State Nuclear Magnetic Resonance 5 (199.5) 227-232

228

The off-resonance effects of 2D NMR nutation of Z = 3/2 quadrupolar nuclei are treated with perturbation theory and numerical calculation in this paper. The nuclear quadrupolar interaction parameters wzax and 77 can therefore be extracted by the spectral simulation. The experimental results for 23Na nuclei in a typical sample of NaNO, agree with theoretical calculations. For a powdered sample of spin Z = 1, Wu and Sanctury [lo] pointed out that the effective nuclear quadrupolar interaction is zero when do = fw,/fi. We will show in this paper that the same conclusion may be applied to the 2D NMR nutation experiment of the nucleus of spin Z= i.e. 3/2. When wi > WY, and Aw = +ml/&, arctg(w,/Aw) = k54.7” (magic angle), the satellite lines (produced by coherence transfers) are superimposed with the line of central transition, and hence the nutation spectrum can greatly be simplified with its sensitivity correspondingly enhanced. The nuclear quadrupolar interaction parameters of 23Na nucleus in Nail molecular sieve are measured using 2D NMR nutation.

2. Basic theory

where 6 = arctg(w,/Aw), and transform the Hamiltonian in Eq. (1) into a tilted frame (I:, z;, Z:, z:, ZL) H, = U;HlJ, -z; + L2(zi2 - 173)

=W, i

+&[$(‘;+ZL) +(z:+z’)z;] z:‘+ z,2)

+ L( 2fi

i

where

o, = /w: + (do)*! , a = (3 cos2 19 - 1) b = &kin 26)wQ/2w,, c = 6(sin2 WQ/2%, is obvious that if 0, nuclear spins would nutate purely with frequency w, about Zi in tilted rotating frame. Therefore, we can see that the influence of nuclear quadrupolar interaction in the nutation experiment causes the nutating frequency to vary. Eq. (3) cannot normally be diagonalized. We note that when o, > oEa” (strong effective rf field), the nuclear quadrupolar interaction HQ may serve as a perturbation term. The total Hamiltonian can then be written as &Q/h,.

It

uQ

=

The pulse sequence used in our nutation experiments is shown in Fig. 1. The rf field with its strength wi and frequency offset Aw is applied during time t,. However, the rf frequency offset is removed during detection time t,. During time t,, the Hamiltonian in the rotating frame is written as

where H, = -w,I\. Then, we try to diagonalize Eq. (3) using perturbation theory. Let us suppose the wave functions (YPn) be accurate to the second-order approximation. We obtain

H = -oiZSsx + Awl, + we,‘;

*n = G*,

- Z*/3)

(1)

where WQ=W;jax

eF=

3 cos2 p - 1

(

2

+ tsin*

+ IY)Z~]

(5)

p cos 2y 1

4Z(2Z - 1)h

U, = exp[i(r

(4)

where ?Pr = ( I3/2), I l/2), I -l/2), I -3/2)Y are the eigenstates of Ho in a tilted rotating

3 e2qQ

and other parameters ings. Let

H=H,+H,

have their ordinary mean-

1

kfQ-

t1

t2

n

(2)

b

Fig. 1. The pulse sequence used for 2D NMR nutation experiment.

229

Y Xia et al. /Solid State Nuclear Magnetic Resonance 5 (1995) 227-232

frame, and ’ means transposed matrix. With d = 1 - (c2 + 4b2)/8 and b(1 + 2a)

d -b(l+ -c(l

u, =

2a)

d

+ a)/2

- 3bc/2 -c(l

bc/6

- a)/2

- bc/6

c(1+ a)/2

c(1 - a)/2

3bc/2

-b(l-

d b( 1 - 2~)

P,(O) =

2~) d

The eigenenergies lated in this case

can be analytically

E, = E3/* = oe[ -3/2

+ a - (2b2 + c’)

calcu-

/2 + u( c* + 4b2)/2] -a

- (2b2 - c”)

/2 + a( c* + 4b2)/2] E4=E-3,2=

w,[3/2

+ a + (2b2 + c’)

/2 - a( c2 + 4b2)/2]

(6)

So, through the transformations of ZJ, and Up, the Hamiltonian in Eq. (1) has been diagonalized when o, > wza” I for the second-order perturbation approximation (energies are accurate to the third-order perturbation approximation). It may be seen from Eq. (3) and Eq. (6) that in the case of o, > w;lax, when arctg(wi/Aw) = f 54.7” (magic angle), i.e. do = -fwl/ a, a = 0, the first-order effective nuclear quadrupolar interaction is zero, then E, = E3,2 = w, [ -3/2

- (2b2 + c2)/2]

E, = E,,/, = w, [-l/2

+ (2b2 - c*)/2]

E, =E-,,,

+ (2b2 + c2)/2]

(8)

= exp( -i&t,)po(O)

exp(%tJ (9)

where wij = Ei - Ej. It is obvious that the pulse during t, has not only created single-quantum coherence, but also double and triple-quantum coherence. Because of anisotropic broadening from the nuclear quadrupolar interaction, no satellite lines can be observed in the experiment. But the central transitions are free from the influence of the nuclear quadrupolar interaction in the first order. Consequently we can observe only the central transition signals. Therefore, the NMR signal in orthogonal phase-sensitive detection mode can be written as M(t,)

=tr[~(t,)(L+i1,)~~] = tr[ pD(tl)(ZX + il*)E] =

thm%l

= i $ 1Pijsj,

exP( -iwijtl)

(10)

where s, = (I, + iZY)Z = U~U,+(Z, + iZy)23UTUD

= oe[ l/2 - (2b2 - cz)/2]

E, = E_3,2 = w,[3/2

= (Pi;)

= ( pij exp( -iwijt,))

4b2)/2]

E, = El,2 = w, [-l/2-u+(2b2-c2)

= w&/2

wJu,+PwwD

where pij is the matrix element of the ith row and the jth column in matrix p,(O). The density matrix at any time t, is p&J

/2 - a(~‘+

E, = E_,,,

spectrum. If only the first-order perturbation approximation is considered, we have Ed-E3 = E,-E, = E,-E, = w, because a = 0, and lines of three transitions are superimposed. So there will only be one line of spectrum, and the satellite lines (produced by the transitions from 1 to 2, and 3 to 4) seem to disappeared. The density matrix under thermal equilibrium is p(O) = I,, so

= (‘ii) (7)

The remaining second-order perturbation approximation has much less influence on the resultant

Sij is the matrix element of the ith row and the jth column in matrix S,. There are a total of sixteen terms in Eq. (10); among them four are

Y. Xia et al. /Solid State Nuclear Magnetic Resonance 5 (1995) 227-232

230

the signal in orthogonal written as

zero-quantum coherence terms, six single-quantum coherence terms, four double-quantum coherence terms and, finally, two triple-quantum coherence terms. In the second dimension, only the central transition can be directly detected. However, the satellite lines and the multi-quantum transitions in time t, can also be indirectly observed in the detection time because the relevant coherence transfers happens during t,. For a polycrystalline or powdered sample, the powdered average [ll] must be used for Eq. (10). Most of the time we have o, < WY, owing to the limited values of oi and Aw (limited by the Q value of the probe). The perturbation approximation is no longer valid, and we must diagonalize Eq. (1) with a numerical calculation. That is, through a unitary transformation U, let VHU

= H,

M(t,)

= tr[ &)(I,

detection

mode can be

+ $)“]

= P23(fl)

(12)

where P( ti> = exp( -iHti)~(O)

exp(%)

= U exp( -iH,t,)U+p(O)U

exp(iH,t,)U+

&I) is the element of the second row and the third column in matrix p(t,).

3. Experimental All the nutation spectra were recorded on a Bruker MSL 400 spectrometer, its static magnet is 9.7 T, the 23Na Larmor frequency is 105.8 MHz. Static spectra were recorded with powdered samples. The rf field strength wi was 93110 kHz. Polycrystalline NaNO, and Na0 molec-

(11)

then H becomes a diagonalized matrix H,, its diagonal elements are eigenenergies. Therefore,

Q I

I

-2SOKHz

0

A

I

-3OOKHz

0

+300KHz

0

+300KHz

1

I

-3OOKHz

+250KHz

I

L

-250KHr

0

+25OKHz

0

+300KHz

0

+300KHz

1

I

-3OOKHz

I

I

-3OOKHz

Fig. 2. The experimental (left) and simulating (right) spectra of 2D NMR nutation under various rf frequency offset. The frequency offset of (a), (b), and (c) are 0, 50, and 100 kHz; with corresponds to effective RF field w, = 110, 121, and 149 kHz, respectively. Lorentz broadening of simulating spectra is 2.5 kHz, rf field strength wr = 110 kHz. The sample is polycrystalline “NaNO,; 23Na nucleus is observed; wEax = 84 kHz.

231

Y Xia et al. /Solid State Nuclear Magnetic Resonance 5 (1995) 227-232

see from Fig. 2 that the simulation spectra on the right are in agreement with the experimental spectra on the left. We can also see from Fig. 2 that the off-resonant nutation spectra are simpler than those of on-resonant nutation spectra. This fact is beneficial to nutation spectral analysis when non-equivalent sites exist, and the nuclear quadrupolar interaction parameters can be obtained accurately. Furthermore, the nutation spectra with various rf frequency offsets can be compared with each other; it is not necessary to change the rf field strength during the experiment courses. Fig. 3a shows the on-resonant (do = 0) experimental (left) and simulating (right) spectra for 23Na nuclei in Na0 molecular sieve, where the rf field strength w1 = 93 kHz. It may be seen from the experimental spectrum of nutation that 23Na nuclei in NaR molecular sieve have two nonequivalent sites. Among them, the ones with a larger nuclear quadrupolar interaction have resonant lines at sites f 2w,, i.e. & 186 kHz, but the ones with smaller nuclear quadrupolar interaction, have nutation spectrum which looks like those shown in Fig. 2a. The nuclear quadrupolar interaction parameters can be obtained by simulation, they are C, = e2qQ = 0.4 MHz, 77= 0.15. The simulated spectra are shown on the righthand side of Fig. 3a. Because of the inhomogeneity of the rf field, there are some differences

ular sieve were studied. The nuclear quadrupolar interaction parameters of 23Na nuclei in the polycrystalline NaNO, are C, = e2qQ = 0.337 MHz 77= 0. The 23Na nucleus in the Na0 molecular sieve has two non-equivalent sites; their nuclear quadrupolar interactions and chemical shifts are different. The nuclear quadrupolar interaction parameters of one site have been measured by VASS [12]
4. Results and discussion Fig. 2 shows the NaNO, experimental spectra (left) and the simulation spectra (right) of 2D NMR nutation under various rf frequency offsets. The rf frequency offsets in Figs. 2a, 2b, and 2c, are 0,50, and 100 kHz, respectively. The accumulated scans are 4, 8, and 64, respectively. We can

b -5DOKHz

0

+SOOKHz

0

+500KHz

i

I

-5OOKHz

-5OOKHz

0

+500KHz

0

+500KHz

I

-5OOKHz

/

Fig. 3. The experimental (left) and simulating (right) spectra of 2D NMR nutation for 23Na nucleus in 23NaR molecular sieve sample. o1 = 93 kHz. (ajAw = 0; (b)Aw = wl/ \/z = 65 kHz. In (b) W, = 113 kHz.

232

Y. Xia et al. /Solid State Nuclear Magnetic Resonance 5 (1995) 227-232

between the experimental and the simulated spectra. The part of the 2D NMR nutation spectrum of 23Na nuclei in the NaO molecular sieve along F, (w2 is set at the second non-equivalent site) is shown on the left-hand side of Fig. 3b. Its simulated spectrum is shown on the right-hand side of Fig. 3b. In Fig. 3, wi = 93 kHz, Aw = wl/ fi = 65 kHz. It is obvious that the smaller nuclear quadrupolar interaction is suppressed in the first order, and the nutation spectrum is therefore simplified. This helps in the analysis of the 2D NMR nutation spectrum of some samples with a few non-equivalent sites.

Acknowledgment Financial support by the National Natural Sciences Foundation of China is gratefully acknowledged.

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