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Analytical expression for thespin-5/2 line intensitiesPascal P. Man aa Laboratoire de Chimie des Surfaces, CNRS URA 1428 ,Université Pierre et Marie Curie , 4 Place Jussieu, Tour 55,75252, Paris Cedex 05, FrancePublished online: 22 Aug 2006.

To cite this article: Pascal P. Man (1993) Analytical expression for the spin-5/2 lineintensities, Molecular Physics: An International Journal at the Interface Between Chemistryand Physics, 78:2, 307-318, DOI: 10.1080/00268979300100251

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MOLECULAR PHYSICS, 1993, VOL. 78, NO. 2, 307-318

Analytical expression for the spin-5/2 line intensities

By PASCAL P. MAN

Laboratoire de Chimie des Surfaces, CNRS URA 1428, Universit6 Pierre et Marie Curie, 4 Place Jussieu, Tour 55, 75252 Paris Cedex 05, France

(Received 14 April 1992; accepted 7 July 1992)

The density matrix of a spin I = 5/2 excited by a radiofrequency pulse is calculated. The interaction involved during the excitation of the spin system is first order quadrupolar. Consequently, the results are valid for any ratio of the quadrupolar coupling, tgQ to the pulse amplitude CORF. The behaviour of the central and the two satellite line intensities versus the pulse length is discussed. The aluminium nuclei (27AI) in a single crystal of corundum (AI20~) are used to illustrate some results.

1. Introduction

Over the last ten years, tremendous work in solid state NMR has dealt with quadrupolar nuclei possessing half-integer spins I. The simplest" case is the spin I = 3/2; the most common nuclei are 23Na, ~TRb, ~IB or 7Li, where extensive exper- imental as well as theoretical results are available. The various fields of investigation can be divided into two parts. The first focuses on the frequency domain response (line shape analysis) of the spin system when the sample is static or under mechanical rotation. The second is related to the time domain response of the spin system to radiofrequency (RF) pulse excitation: line intensity measurement [1-7], quadrupole nutation [1, 4, 5, 8, 9], rotary echo nutation [9, 10], spin-echo [1 1, 12], spin-lock [13, 14], cross-polarization [1 5], or multiquantum transitions [14, 16].

There is a great deal of interest in spin I = 5/2 systems, and 27A1 [17-19] and 170

[20, 21] in inorganic chemistry have led to both experimental and theoretical develop- ments. In this paper, we focus on the simplest NMR experiment, the measurement of spin-5/2 line intensities after a single RF pulse excitation. This is of interest because a series of line intensities obtained with increasing pulse length t allows the deter- mination of a~Q, the amplitude of the first order quadrupolar interaction (or the quadrupolar coupling). Fenzke et al. [22] have performed numerical calculations and have shown that the line intensity for a give RF pulse amplitude ogRv and t depends on co o. A sytematic study of half-integer spins, up to I = 9/2, undertaken by Samoson and Lippmaa [8], has given the principal trend. A more specific investigation devoted to a spin I = 5/2 system was carried out by Van der Mijden et al. [23]. However, in both cases, the final results concerning the line intensities are not given explicitly, and only the formal are available. In this study, we fill this gap and give the analytical expressions for the density matrix, as well as the central and the two satellite line intensities of a spin 5/2. Some results are illustrated with 27A1 in a single crystal of corundum, A120 3. This paper is the first stage towards the study of a spin-echo sequence which recovers broad NMR lines lost in the dead time of the receiver. This happens very often in zeolite characterization with 27A1 NMR, where some AI atoms are not observable [24, 25] with a single RF pulse excitation.

0026-8976]93 $10.00 �9 1993 Taylor & Francis Ltd

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308 P .P . Man

) + O RFI,

I I

t i m e

Figure 1. The Hamiltonians associated with the one RF pulse excitation.

2. Theory

The Hamiltonians throughout the paper are defined in angular frequency units. Neglecting relaxation phenomena and second order quadrupolar effects, the dynamics of a spin I = 5/2 system, excited by a - x pulse (figure 1), is described by the density matrix p(t) expressed in the rotating frame associated with the central transition:

p(t) = exp(-- iJcf (a) t)p(0) exp(Def (a) t), (1)

where

~0Q

p(0) = I~, (2a)

~1 ) = 1 a~Q(3# -- I(I + 1)), (2b)

3e 2 qQ 8I(2-i -- ])1t [3 cos z fl - 1 + r/sin 2 t8 cos 2~], (2c)

~,ZP (") = a~RV + Jcg~l), (2d)

O~RF = (.0RF/x, (2e)

jfrl) is the first order quadrupolar interaction. ~ and fl represent the Euler angles describing the orientation of the strong static magnetic field in the principal axis system of the electric field gradient (EFG) tensor and r/the asymmetry parameter. The matrix representation Mr (table 1) of jg(a), expressed in the eigenstates of L, is not diagonal. The matrices of eigenvalues 12 and eigenvectors T of g(a) are related by

= T-tM~T, (3)

the symbol - 1 in the superscript meaning the inverse of the matrix. Equation (1) can be rewritten as:

p(t) = T e x p ( - iOt)T-~ p(O)T exp(if2t)T -~ . (4)

The major problem is to determine ~ and T. A method was proposed by Van der Mijden et al. [23], who have shown that T is the product of two matrices:

T = AoB. (5) The matrix A 0 [23] is reported in table 1. The transformation AtoM~Ao produces a matrix in the form of two 3 x 3 diagonal symmetric blocks. The superscript letter t means the transpose of the matrix. We propose another way of obtaining a similar

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Spin-5~2 line intensities

Table 1. The 6 x 6 matrices involved in this work.

309

M~ =

M~ =

,40

10toO/3 `/509RF/2

`/5tORF/2 -- 2tOQ/3

0 ,/2~OR~

0 0

0 0

0 0

10o0/3 `/5CORF/2

`/50)RF/2 -- 2~%/3

0 `/2OgRF

0 0

O 0

o 0

`/2 245 `/10

2`/5 -2`/2 2 `/10 2 - 3 , / 2

,/10 2 - 3 , / 2

2,/5 -242 2 ,/2 2`/5 `/10

1 0 0 0

0 1 0 0

0 0 1 1

0 0 --1 1

0 --1 0 0

--1 0 0 0

0 0

420)RF 0

-- 80)Q ]3 3tORF/2

3CORF/2 -- 8C00/3

0 ,/2r

0 0

0

,/2tORF

--(8tOQ/3 + 3tOaF/2)

0

0

0

410 2`/5 2 - 2`/2

- 3 ` / 2 2

3`/2 - 2

- 2 2 , / 2

--`/10 --2`/5

0 1

1 0

0 0

0 0

1 0

0 1

0 0

0 0

0 0

`/2toRt 0

- 2r `/5r

`/509R~/2 10OOQ/3

0

0

0

- (8t%/3 - 3tORy/2)

`/2tORF

0

,/2 245

, / lo - , / 1 0

-2`/5 - , / 2

0 0

0 0

0 0

`/2eoRr 0

-- 2to0/3 `/5tORF/2

`/5tOR~/2 10t%/3

result, but the matrix A o is replaced by a simpler form, A~ (table 1). Samoson and Lippmaa [8] have proposed the same kind o f matrix.

First, equat ions (2b) and (2e) are written with the fictitious spin-l /2 operators [16, 26] associated with a spin-5/2

.~Ql) = ~ CO ~ (5(I).2 _ is.6) + 4(i:.3 _ i:.5)}, (6)

5,6 2x/2(I~ '3 + I~ ) + 3Ix3"}. ~ = , o ~ { , / S ( t ~ ',~ + t~ ) + '.~ (7)

In this formalism, the eigenstates o f It, Im > (figure 2), are redefined as: li > = II - m + 1 > , so i = 1 . . . . , 21 + 1 [16, 26]. ~6~) is quadrat ic in Is, so it remains unchanged if the eigenvalue m of I~ is replaced by - m . As a result, the eigenstates I I > and 16 > , 12 > and 15 > , 13 > and 14 > are degenerate. A series of rotat ions within each pair o f these eigenstates will partially diagonalize ~ a ) . This

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310 P .P . Man

I 16> I

/ t' ~10 o ~ / 3

1-512>

I i 15> 1-3/2>- I/

~,- 2 ~ / 3

U

I / 14> I-1/2> \ ~ ~-..80)Q/3 t

I~ [3> 11/2>\ ,~ ~ . 8 0 ~ / 3

i~' T- 2 0 ~ / 3 I ]2> 13/2> l /,, I / ~ 10 O)0/3

'll> 15/2>

Figure 2.

Hz = 0 Hz r 0 Hz + H~ a)

The energy levels, their shifts and the two forms of eigenstates of a spin-5/2 (Hz means Zeeman interaction).

occurs with the following rotation operator,

~r = exp - - i ~ (13: + I~ '5 + i~,6) , (8)

whose matrix representation is A~. It is easy to check that this kind of matrix is suitable for spins I = 7/2 and 9/2 [8]. The modified expression of ~o(,) is

= d + j :a )~d

=~_ O3~Q (1) -~- ~ + ~ R F , ~

d~Q(l) q - CORF{--3Iz 3'4 q- 45(]x L2 q- Ix 5"6) q- 242(Ix 2'3 =k 14"5)}. (9)

The superscript + means the adjoint of the operator. The matrix representation/142, of rig is given in table 1. We do not proceed to further diagonalize ~ with the fictitious spin-l/2 operator formalism, and the standard method for diagonalizing a 3 x 3 symmetric matrix is used [23]. The notation used throughout this paper is: the subscripts + and - are related to the two 3 x 3 submatrices of a 6 x 6 matrix. The six eigenvalues of M2, expressed in angular frequency units, are

O)RF + 2 COS m~_ -- + 2 COS COl+ -- 2 ' 2 '

) (ORF 2 /S+ ( n 4)+ tORy 2 COS CO2+ -- 2 k/-3 -c~ 3 ' c~ = " - ~ -- 3 '

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Spin-5~2 line intensities 311

0)3+ - - (~ 2 cos ~ + , o93_ = - - 2 - 2 cos + ,

(10)

with

84 s_+ -- --~ to~ -t- 4tOoO)av + 4to~v, (1 la)

X i + -= X i.._..~ ~

- Qi+_ '

with i = 1, 2 or 3, and

tOQ (160o9~ + 36COQt~RF -- 144(O~F), ( l ib) q+- = 2-"ff

3 q _ J ~ + 3 q + J 3 ( l lc) c o s ~ + = 2s+ , c o s ~ _ = Z~_ s-7

The components of the normalized eigenvectors associated with the six eigenvalues coi_+ are:

Yi+ = Yi+_ Zi+ - zi+ _ Q,_+, _ Q,_+, (12)

45~aF 42~aF Xi+ ~ X i _

2ai+ ' hi_ '

yi+ = 1, Yi- = 1,

42~RF 45~gF Zi+ ~ Z i_

b i+ ' 2ai_ '

10 8 3

5 ~ F 2 ~ , ) ',2 = 1 + b-U_+ ] .

(13)

The matrix of eigenvectors B of M 2, those of eigenvalues .f2, the transformation matrix T~, and J associated with I= are

(14)

The 3 x 3 submatrices N• fl_+ and J_+ are defined in table 2. Equation (4) can be written as

p(t) = T~ exp(- iOt)Ti- ' JT~ exp(iOt)T~ -l . (15)

The matrix multiplications in (15) were performed using Mathematica version 2.0 operating on a 386SXC 20 MHz microcomputer equipped with a coprocessor. The computation took 15min. Then, the factorization of each element po.(t) (table 3) of the density matrix took 3 min. Knowledge of the density matrix p(t) allows the determination of the line intensity P~(t), and the relative line intensity FiJ(t), related by

F~J(t) = I 'J(t) (16a) Tr[L2] '

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312 P.P. Man

Table 2. The 3 x 3 submatrices involved in this work.

. [ x ~ + x~+ x3+) (xl_ x~_ N+ = | Y~+ Y2+ Y3+ N_ = Yx- Y2-

\ Z1 + Z2 + Z3 + Z I _ Z 2 _ (o0 0 603+ 0

J+ = 3 J_ = - 3

0 0

X3-

Z3_

0

0

093 _

Tr[I~] = � 8 9 1 ) ( 2 I + 1).

The relative intensity o f the central line F3'4(t) is

F3,4(/) = ~ Tr[p(t)313y '41 3 3

---- 3"~ E E Zi+ Xj- Kij sin c%t, i=l j=~

with

(16b)

(17)

/~0 = 5x,+zj_ + 3~+g_ + z ~ + ~ . (18)

The nine angular frequencies ~% are formed f rom the difference o f components in ~ +

Table 3. Components pij(t) of the density matrix p(t) (equation (15)). For clarity, the symbol �88 Z~=~ E3=1 K~j in front of each term is missing. For example p21(t) = �88 Z3=1 Z3=1 Kij[(X,+Yj_ + Y/+Zj_) cosc%t + I(Xi+Yj_ - Yi+Zj_) sinoifl] is written as below. i = x/-Z"(.

Pu(t) = 2X~+Zj_

P21(t) = p*2(t) =

P22(t) = 2Yi+ ~_

P31(t) = p*3(t) =

P32(t) = p*3(t) =

P33(/) ~-" 2Zi+Xj_

/941 (t) = p*4(t) =

P42(t) = p~4(t) =

P43(t) = p~'4(t) =

P51(t) = p*5(t) =

P52(t) = p*s(t) =

ps4(t) = p*5(t) =

p61(t) = p*6(t) =

P63(t) = p*6(t) =

p65(t) = p~'6(t) =

COS ogijt

(X/+~_ + Y~+Zj_)cosc%t + I(X~+~_ - Yi+~-)sinco~jt

COS ~Oij t

(X~+Xj_ + Z i + ~ _ ) c o s t o J + I(X,+Xj_ - Zi+Zj_)sincojj t

(Y,+Xj_ + Zi+ Yj_) cosc%t + I(Y~+Xj_ - Zi+ Yj_) sinc%t

COS f.Oq l

(Xi+ Xj_ - Zi+ Zj_ ) cosoif l + I(X,+~_ + Z~+ Zj_ ) sinoijt

(Yi+Xj_ - Zi+ Yj_) cosc%t + I(Y,+Xj_ + Zi+ Y)_) sinc%t

2IZi+Xj_ sine%t, p~(t) = --P33(t)

(Xi+Yj_ -- Yi+Zj_) cosc%t + I(X,+Yj_ + Y/+Zj_) sinc%t

2IY~+~_ sine%t, P53(0 = p~5(t) = -p*2(t)

- -p*2( t ) , Pss( t ) = --P22(t)

2IXi+Zj_ s i n o J , p62(t) = p*6(t) = -p* l ( t )

-p*l ( t ) , p6~(t) = p~(t) = - p ~ ( t )

- -p~ l ( t ) , P 6 6 ( t ) = - - P l l ( t )

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Spin-5/2 line intensities 313

A

,r

|

9

6

3

0

-3

-6

/ "x_ ".. " \ / " .X, ' x ""~ / ' x x . ~ N~,

'~N. " ' . / / X . oo

,%

I I I t I I I I I

0 1 2 3 4 5 6 7 8 9 10

R.F. pulse length t (its) Figure 3, Relative intensity of the central line -FS'~( t ) , equation (17), versus the RF pulse

length t, for several values o f coQ: - - - , 0 kHz; . . . . ,20 kHz; - - - , 50 kHz; . . . . . . . , 500kHz. 09RF/2rC -~ 50kHz.

and ~

to~ = toi+ - toj-. (19)

The relative intensity of an inner line F2'3(t) is

F2.3(t) = 2 TrLo(t)~/8i~.3]

- 48 ~ ~ (Y~+ Xj_ - Z,+ Y~._ )K o sin to,jr. (20) 70 i=l j=l

Finally, the relative intensity of an outer line FL2(t) is

F,.2(t) = 2 Tr[p(t)x/5i~.2]

~/5 ~ ~ (Xi+YJ- Yi+Zj_)Kijsintoijt. (21) 70 if t i=l

The total relative line intensity of a spin 5[2 is given by

F(t) = FS'4(t) + 2F2'S(t) + 2Fn'2(t). (22)

The three relative line intensities F3"4(/), F2"s(t) and Fl2(t) are a sum of nine sine curves of different amplitudes and frequencies. Figures 3-5 represent these functions versus the pulse length for a typical RF amplitude toRF/2n = 50 kHz; toQ is taken as a parameter. For the central transition (figure 3), the maximum of the relative line intensity FS'4(t) decreases, as well as the associated pulse length, when toQ increases from a small value to a large one, but both reach a limiting value which is one third of those when toQ ~ 0:

Fs'4(/) = -- ~ sin toRF t, for o) O '~ O)RF, (23a)

= _ s sin 3toRvt, when r >> toRF. (23b)

In other words, when too >> tORE, the magnetization precesses around the RF magnetic field three time faster than in the opposite case (too '~ tORe)" There is also a loss of relative line intensity by a factor o f three. This is due to the fact that part of the

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4 / : - - , . . \ 2 - " " ' "~' " '~ ' ,~ , " \ ~ , . _,o - " = - ,,,.

/ " ' ' . " ~ ' . " \ - ' - " ' . . ~Nx-

0 / " " - ' I - - - " ' - ~ "~ ' ' , ' < . " - " ~ " - " " " . ~ ' - - I

=4 ' : ', : I I I I I I

0 1 2 3 4 5 6 7 8 9 10

R.F. pulse length t (~ts)

Figure 4. Relative intensity of an inner line --FZ'3(t), equation (20), versus the RF pulse length t, for several values of ~oQ :- ,0 kHz; . . . . ,30 kHz; . . . . ,50 kHz; . . . . ,100 kHz;

, 500kHz. ~ORF/2n = 50kHz.

magnetization remains in the z axis. There appears also a linear region, defined by t < 0.5 Ixs, where the relative line intensity is proportional to t and independent of ~%. Indeed, if t is short enough, so that sin 3~RF t ~ 3~ORF t, then equations (23a) and (23b) become identical. This linear region is therefore available for the distribution of o)Q, which can occur in a powdered sample. This excitation condition [5, 27] must be applied in order to quantify spin populations in powdered compounds. For the other transitions, the maximum magnitude of F2'3(t) and Fl'2(t) as well as the associated pulse length decrease continually towards zero when o)Q increases. There is no linear region where the relative line intensities are independent of o)Q, as in the case of the central transition.

The behaviour of equations (17), (20) and (21) are better analysed in the frequency domain (nutation spectra). For simplicity, we focus mainly on the relative intensity

|

5

4

3

2

1

0

-1

- . . . , - - - , . . . ~ , - -, .,.- --...._-,~-

~ : ~ 1 " - ~ i I

0 1 2 3 4 5 6 7 8 9 10

R.F. pulse length t (its) Figure 5. Relative intensity of an outer line --Fl'2(t), equation (21), versus the RF pulse

length t for several values of o)Q: - - - , 0 kHz; ,20 kHz; - - - , 50 kHz; . . . . . . . , 100kHz. OgRF/2n = 50kHz.

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Spin-5/2 line intensities 315

1.2

0.8

0.4

-0.4

22 33 i ' " =- =UY

13 "''"" 32

1 10 100 1000

(.0o/27g (kHz)

Figure 6. The nine amplitudes Z~+ ~_ K 0 (equation (17)) versus log~0(c%[2~) for 0)RF[2~ =- 50 kHz. The paired numbers are related to the two subscripts/j of Z~+ ~_ Ko.

of the central line. So its Fourier transform is a set of nine lines located at <no whose amplitudes are the terms Z~+Xj_ K~. These nine amplitudes versus log~0(toQ/2r0 are represented in figure 6, which shows that mainly five of them are important for tOQ/O)R r < 0,2, seven for 0.2 < ~oQ/a)R~ < 2, and one for 2 < too/togv, The tO~/tO~F < 0.2 case was well analysed by Van der Mijden et al. [23],

"r v

0.1

0,01

0.001

-d . . . . . . . . . ~ e g "r 0 ~ :[ ~ , i r oo

. .') #" ~oO~ ~***.******.4)* .~0~ O

~ s ~ x l x

\ III

X

Ig

I f

z llg

1 10 1QQ 1000

Figure 7,

(,0Q/2/t (kHz)

A log-log plot of the a)0]21r (equation (l 9)) versus 0)o]27t for 0)ar]2~ = 50 kHz: I , O)ll ; O) O)12 ; A, (Oi3 ; (~, (.O21 ; r], 0)22: O, (./323; A, 0)31; •, (/)32; (x) , 0)33.

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7 ~s 14 ~s

I I I 300 0 -300

kHz

8 kts

Figure 8. 27A1 NMR spectra of a single crystal of A120 3 for increasing RF pulse length t.

Figure 7 represents logl0(coo/2r 0 versus log10(COQ/21t ) for CORF/2~Z = 50 kHz. For COQ]oaRv < 0.2, five lines are located around CORF. As the value of c% increases, some lines shift towards higher positions, and some others towards lower positions. For 2 < C%/C%F, mainly a single line remains and is located at 3COR F. Our results are in perfect agreement with those of Samoson and Lippmaa [8].

3. Experimental The sample was a single crystal of corundum, A1203. 27A1 N M R spectra were

obtained on a Bruker MSL-400 multinuclear spectrometer operating at 104.2 MHz.

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Spin-5/2 line intensities 317

:5

2

~U. 0 |

-1 0 :~ 4 6 8 1() 12 14

R.F. pulse length t (l~s)

Figure 9. Experimental 27A! central line intensities from figure 8 (e) and calculated central line intensities with -F3"4(t), equation (17).

The high power static probehead was equipped with the standard 5 mm diameter horizontal solenoid coil. The amplitude of the pulse, determined using aluminium nitrate solution, was toRv[2n = 42 kHz, corresponding to a re/2 pulse of 5.8 laS. Each spectrum was obtained with a recycle delay of 5 s, 48 scans, a sampling dwell time of 0.5 Ixs and a dead time of 6#s.

27A1 spectra on the absolute intensity scale, obtained with increasing RF pulse length, are presented in figure 8. The experimental value of coo ]2n is 60 kHz. Only the central line in each spectrum was phased properly. Due to the 6 las acquisition delay that yields additional dephasing, no attempt was taken to phase the inner and outer lines. In figure 9, the curve corresponds to a fit of the experimental central line intensities (full circle) with (17). The fitting parameters used are the same as the experimental ones, except for toQ/2n which was 65kHz instead of 60 kHz.

4. Conclusions

The analytical expressions of the density matrix and the line intensities of a spin 5/2 were obtained using conventional matrix algebra. This study is mainly an extension of the previous one on a spin I = 3/2 system. The results provide the first stage in the study of the spin-echo sequence which is required for recovering broad lines lost in the dead time of the receiver.

We thank Technic de Bouregas, Lincoln, MA 01773-0421, USA for the NMR simulation program Antiope. In particular, we wish to thank Dr. P. Tougne for valuable discussions, Dr. L. Oger for technical help and Dr. M. A. Hepp for a critical reading of the manuscript.

References [1] SAMOSON, A., AND LIPPMAA, E., 1983, Phys. Rev. B, 28, 6567.

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