MOLECULAR PHYSICS,1993, VOL. 78, NO. 2, 307-318
Analytical expression for the spin-5/2 line intensities
By PASCAL P. M A N
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 N M R 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 experimental 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 developments. In this paper, we focus on the simplest N M R experiment, the measurement of
spin-5/2 line intensities after a single R F pulse excitation. This is of interest because
a series of line intensities obtained with increasing pulse length t allows the determination 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 coo. 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 N M R lines lost in the dead time of the receiver. This
happens very often in zeolite characterization with 27A1NMR, where some AI atoms
are not observable [24, 25] with a single RF pulse excitation.
0026-8976]93 $10.00 9 1993Taylor & Francis Ltd
308
P . P . Man
) + O RFI,
I
I
time
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)
p(0) = I~,
(2a)
where
~1)
~0Q
= 1 a~Q(3# -- I(I + 1)),
3e2qQ
8I(2-i -- ])1t [3 cos z fl - 1 + r/sin 2 t8 cos 2~],
~,ZP(") =
a~RV + Jcg~l),
O~RF =
(.0RF/x,
(2b)
(2c)
(2d)
(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
309
Spin-5~2 line intensities
Table 1. The 6 x 6 matrices involved in this work.
10toO/3
`/509RF/2
0
0
0
0
`/5tORF/2
-- 2tOQ/3
0
3tORF/2
0
0
0
0
-- 8C00/3
`/2toRt
,/2r
- 2r
0
,/2~OR~
420)RF
-- 80)Q]3
0
0
3CORF/2
0
0
0
0
0
0
M~ =
0
0
0
0
,/2tORF
0
0
0
O
0
0
`/2tORF
o
0
0
0
245
`/10
410
2`/5
2`/5 -2`/2
2
2
- 2`/2
- 3`/2
`/2
,40
10OOQ/3
0
0
`/2OgRF --(8tOQ/3 + 3tOaF/2)
0
0
- (8t%/3 - 3tORy/2) `/2eoRr
0
0
M~ =
`/509R~/2
0
10o0/3 `/5CORF/2
`/50)RF/2 -- 2~%/3
0
`/5r
`/10
2
-3,/2
,/10
2
-3,/2
2,/5 -242
,/2
2
2`/5
`/10
2
,/lo
- 2
-,/10
- 2
2,/2
-2`/5
-,/2
1
0
0
0
0
1
0
1
0
0
1
0
0
0
1
1 0
0
0
0
1
0 0
0
--1
0
0
1
0
--1
0
0
0
0
1
--1
--2`/5
0
-- 2to0/3 `/5tORF/2
`/5tOR~/2 10t%/3
,/2
245
3`/2
--`/10
0
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, equations (2b) and (2e) are written with the fictitious spin-l/2 operators [16,
26] associated with a spin-5/2
~ CO~ (5(I).2 _ is.6) + 4(i:.3 _ i:.5)},
(6)
, o ~ { , / S ( t ~ ',~ + t~5,6) + 2x/2(I~'3 + I~'.~ ) + 3Ix3"}.
(7)
.~Ql) =
~
=
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 quadratic in Is, so it
remains unchanged if the eigenvalue m o f 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
rotations within each pair o f these eigenstates will partially diagonalize ~ a ) . This
310
P . P . Man
/
16>
t'
1-512>
Ii
I/
U
15>
1-3/2>-
I/
14>
I-1/2> \ ~
~-..80)Q/3
I~
[3>
11/2>\ ,~
~.8 0~/3
]2>
13/2>
I
~10 o ~ / 3
I
t
i~'
I
l
I
~,- 2 ~ / 3
T- 2 0 ~ / 3
/,,
/
'll>
Hz = 0
~ 10 O)0/3
15/2>
Hz r 0
Hz + H~ a)
Figure 2. 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{--3Iz3'4 q- 45(]x L2 q- Ix5"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
COl+ --
CO2+ --
O)RF
2 + 2
(ORF
2
COS
2 /S+
k/-3 -c~
(n
'
m~_ --
)
4)+
3 '
2
c~
+ 2
COS
tORy 2
= " - ~ --
'
COS
3
'
Spin-5~2 line intensities
0)3+
-
-
(~
2
cos ~ +
,
311
o93_ = - - 2 - 2
cos
+
,
(10)
with
84
s_+ -- --~ to~ -t- 4tOoO)av + 4to~v,
tOQ (160o9~ + 36COQt~RF -- 144(O~F),
2-"ff
q+- =
cos~+
(1 la)
3q_J~+
=
2s+
,
cos~_
3q+J
=
(lib)
3
Z~_
s-7
(llc)
The components of the normalized eigenvectors associated with the six eigenvalues
coi_+ are:
Xi +
-=
Yi+_ = Q,_+,
Yi+_
X i.._..~
~
-
Zi+_ -
zi+
Q,_+,
(12)
Qi+_ '
with i = 1, 2 or 3, and
Xi+
45~aF
~
2ai+
yi+ =
42~aF
Xi_
1,
Yi- =
42~RF
Zi+
~
Zi_
bi+
hi_
'
45~gF
2ai_
'
'
'
10
1,
8
5~F
=
1
3
2 ~ , ) ',2
+ 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•
written as
p(t)
=
fl_+ and J_+ are defined in table 2. Equation (4) can be
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)
Tr[L2] '
(16a)
312
P . P . Man
Table 2. The 3 x 3 submatrices involved in this work.
.[ x ~ +
x~+
x3+)
N+ = | Y~+
Y2+
Y3+
\ Z1 +
Z2 +
Z3 +
0
J+ =
(xl_x~_
X3-
N_ =
Yx-
Y2-
(o0
ZI _
603+
Z 2_
0
3
J_ =
-
Z3_
0
0
093 _
3
0
0
Tr[I~]
=
89
1)(2I+
1).
(16b)
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,
(17)
i=l j=~
with
/~0 = 5x,+zj_ + 3 ~ + g _ + z ~ + ~ .
(18)
The nine angular frequencies ~% are formed f r o m the difference o f c o m p o n e n t s in ~ +
Table 3. Components pij(t) of the density matrix p(t) (equation (15)). For clarity, the symbol
88Z~=~ 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_ COS ogijt
P21(t) = p*2(t) = (X/+~_ + Y~+Zj_)cosc%t + I(X~+~_ - Yi+~-)sinco~jt
P22(t) = 2Yi+ ~_ COS ~Oijt
P31(t) = p*3(t) = (X~+Xj_ + Z i + ~ _ ) c o s t o J + I(X,+Xj_ - Zi+Zj_)sincojjt
P32(t) = p*3(t) = (Y,+Xj_ + Zi+ Yj_) cosc%t + I(Y~+Xj_ - Zi+ Yj_) sinc%t
P33(/) ~-"
2Zi+Xj_
/941(t) =
p*4(t) = (Xi+ Xj_ - Zi+ Zj_ ) cosoifl + I(X,+~_ + Z~+ Zj_ ) sinoijt
COS f.Oql
P42(t) = p~4(t) = (Yi+Xj_ - Zi+ Yj_) cosc%t + I(Y,+Xj_ + Zi+ Y)_) sinc%t
P43(t) = p~'4(t) = 2IZi+Xj_ sine%t,
p~(t)
=
--P33(t)
P51(t) = p*5(t) = (Xi+Yj_ -- Yi+Zj_) cosc%t + I(X,+Yj_ + Y/+Zj_) sinc%t
P52(t) = p*s(t) = 2IY~+~_ sine%t,
ps4(t) = p*5(t) = --p*2(t),
Pss(t) =
p61(t) = p*6(t) = 2IXi+Zj_ s i n o J ,
P63(t) =
p*6(t) = -p*l(t),
p65(t) = p~'6(t) =
--p~l(t),
P53(0 = p~5(t) = -p*2(t)
--P22(t)
p62(t) = p*6(t) = -p*l(t)
p6~(t) = p~(t) = - p ~ ( t )
P66(t)=--Pll(t)
Spin-5/2 line intensities
313
9
6
3
A
/
,r
"x_
'x
0
|
"..
"\/"
""~
'~N.
"'.
//
/'x x
.X,
X . oo
.~
N~,
,%
-3
-6
0
I
I
I
t
I
I
I
I
I
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; . . . . , 2 0 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~._)Ko sin
70 i=l j=l
Finally, the relative intensity of an outer line
F,.2(t)
=
to,jr.
(20)
FL2(t) is
2 Tr[p(t)x/5i~.2]
~/5 ~ ~ (Xi+YJ70 ift i=l
Yi+Zj_)Kijsintoijt.
(21)
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 o f nine sine
curves o f 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 toRFt,
for o)O
_ s sin 3toRvt,
when r
'~ O)RF,
>> toRF.
(23a)
(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
314
P.P. Man
/:--,..
4
2
- ""
/
0 /
' "~' "'~',~,
"''.
-'I
""
=4
0
"\~,.
"~'.
- - - "'-~
\
_,o -" =- ,,,.
"\-'-
"'..
"~'','<."-" ~ "-"
"".~'--
~Nx-
I
'
:
',
:
I
I
I
I
I
I
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~ORFt, 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 m a x i m u m 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
0
1
--,... ~,-
-, .,.-
--...._-,~-
~
:
~
1
"-
~
i
I
2
3
4
5
6
7
8
9
R.F. pulse length t (its)
of an outer line --Fl'2(t), equation (21),
Figure 5. Relative intensity
length t for several values of o)Q: - - - , 0 kHz;
100kHz. OgRF/2n = 50kHz.
10
versus the RF pulse
,20 kHz; - - -, 50 kHz; . . . . . . . ,
Spin-5/2 line intensities
315
1.2
22
=-=UY
33
0.8
0.4
-0.4
i'"
13 "''"" 32
1
10
100
1000
(.0o/27g (kHz)
Figure 6. The nine amplitudes Z~+ ~_ K0 (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)Rr < 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],
-d
:[
.
.
.
.
~***.******.4)*
.
.
.
.
.~eg
"r
~,ir
0~
oo
. .') #" ~oO~
.~0~
O
0.1
"r
v
~
s~xl
x
0,01
X
Ig
If
z
\
llg
III
0.001
1
10
1QQ
1000
(,0Q/2/t (kHz)
Figure 7, 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.
316
P . P . Man
7 ~s
14 ~s
8 kts
I
I
300
0
kHz
I
-300
Figure 8. 27A1 NMR spectra of a single crystal of A120 3 for increasing RF pulse length t.
Figure 7 represents logl0(coo/2r0 versus log10(COQ/21t) for CORF/2~Z = 50 kHz. For
0.2, five lines are located around CORF.As the value o f 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 3CORF. Our results are in
perfect agreement with those of Samoson and Lippmaa [8].
COQ]oaRv<
3. Experimental
The sample was a single crystal o f corundum, A1203. 27A1 N M R spectra were
obtained on a Bruker MSL-400 multinuclear spectrometer operating at 104.2 MHz.
Spin-5/2 line intensities
317
2
:5
~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 N M R
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
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318
P . P . Man
PANDEY,L., TOWTA, S., AND HUGHES,D. G., 1986, J. chem. Phys., 85, 6923.
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