Irreducible tensor operators and weak, selective, and nonselective, RF fields

JOURNAL

OF MAGNETIC

RESONANCE

Irreducible

83,79-96 ( 1989)

Tensor Operators and Weak, Selective, and Nonselective, RF Fields

G.J. BOWDEN, J. KHACHAN,AND

J.P.D.

MARTIN

School ofPhysics, University ofNew South Wales, P.O. Box I, Kensington, New South Wales 2033, Australia Received July 1, 1988; revised October 7, I988 A general method, based on Newton’s method of successiveapproximations, and irreducible tensor operators, is used to examine the effect of weak, selective, and nonselective, RF fields, in multiple-quantum NMR and NQR experiments. Results are given for an I = 1 ensemble, subject to a chemical shift, an axially symmetric quadrupole interaction, and a soft RF field directed along the x’ axis in the rotating frame. Combined hard and soft RF pulse sequences are briefly examined, and the “Fan0 sum rule” is used to show that in certain cases 5 0 % of the original multipolar NMR order oscillates at the double-quantum frequency of the order 2wo, with the remainder at the slower frequency of \/zw, . The treatment is related to (i) the earlier work of S. Vega and A. Pines (J. Chem. Phys. 66,5624 ( 1977)) and A. Wokaun and R. R. Ernst (.I. Chem. Phys. 67, 1752 ( 1977)) and (ii) the “single-shot” Raman MQNMR experiment of C. S. Yannoni, R. D. Kendrick, and P. K. W a n g (Phys. Rev. Lett. 58, 345 ( 1987)). In addition, it is shown that the results presented in this paper can be used to describe NQR experiments using circularly polarized RF fields. o 1989 Academic PBS, IK.

In the description of multiple-quantum NMR experiments, irreducible tensors possess many advantages over fictitious spin-f operators, products of Cartesian operators, etc. ( l-3). Tensor operators form an orthonom-ial set, with well-documented rotational properties (4, 5). Recently, it has been suggested that such operators are only useful in describing transformations under hard RF pulses (6)) because they form a basis set for the rotation group. By way of contrast, however, we believe that irreducible tensor operators reduce the necessary algebra for any spin manipulation or evolution, to a bare minimum. In addition, the use of symmetric and antisymmettic contributions of irreducible tensors allows considerable insight to be gained, through the clear identification of both the rank and the order of the multipolar states, constants of the motion, harmonics of the motion, etc., created during the course of multipulse NMR experiments (2, 3). Surprisingly, tensor operators have rarely been used to describe the evolution of a given nuclear ensemble in the presence of weak, selective, RF fields. However, the reason for this omission is not hard to find. For either hard RF pulses or, say, evolution under the action of a quadrupole interaction, simple expressions for both the nuclear eigenvalues and eigenfunctions are easily found. However, even for an I = 1 spin system, evolving under a chemical shift, an axially symmetric quadrupole interaction, and an RF field directed along the x’ axis in the rotating frame, analytical expressions for eigenvalues and eigenvectors are extremely cumbersome. The prob79

0022-2364189 $3.00 Copyright 0 1989 by Academic Press, Inc. All rights of reproduction in any form -ed.

80

BOWDEN,

KHACHAN,

AND

MARTIN

lem reduces itself to a determination of the roots of a cubic equation. This has been addressed by ( 7)) and more recently by (8)) who have given an exact, if somewhat unwieldy solution, for all values of chemical shift AU, RF field strength wI , and the quadrupolar parameter WQ. Only the case of AU = 0 proved to be amenable to solution, and the reader is referred to (8) for a complete solution of this particular problem. In this paper, however, we tackle the problem from another point of view. In the case of selective irradiation, the RF field strength w, is necessarily weak. Consequently, the eigenvalues of the Hamiltonian are dictated primarily by the diagonal components of the chemical shift and the quadrupolar interaction. This suggests, therefore, that it should be possible to use perturbation theory to determine the eigenvalues and eigenvectors, in the presence of the RF field u1 . However, some care is necessary. In the case of selective excitation (AU = fuQ), two of the unperturbed eigenvalues are degenerate, so nondegenerate perturbation theory cannot be used. Instead, we use Newton’s method of successive approximations to determine the eigenvalues, and subsequently the eigenvectors, in some cases to order w:. Given the eigenvalues and eigenvectors, it is then a relatively straightforward matter to determine the time dependence of the eight irreducible tensor operators available to the Z = 1 spin ensemble. It should also be acknowledged, of course, that many aspects of this problem have already been examined by earlier workers. For example, a discussion of selective excitation, in terms of irreducible tensor operators, has already been given by (9) for RF fields very much greater than the quadrupole interaction ( w1 $ uQ) . In addition, the problem of weak selective excitation w 1 < uQ has also been examined ( 10, Z 1) , using (i) fictitious spin-$ operators and (ii) the “simple line” approximation of Abragam ( 12). In this paper, it is shown that the latter leads to a simplified set of time-dependent expressions, for the evolution of the eight irreducible tensor operators ?a( a) available to the Z = 1 spin system. SPIN I = 1 ENSEMBLE. AXIALLY INTERACTION WITH

SYMMETRIC QUADRUPOLE CHEMICAL SHIFT

Consider an Z = 1 nuclear ensemble evolving in the rotating frame under the action of(i) a chemical shift, (ii) an axially symmetric quadrupole interaction, and (iii) a weak RF field directed along the X’axis, x=2?c)+A? = h{ AwT:, f (2/3) “2~QTa =

- qTf(a)}

111

h{ EAo?A + (2/3)1’2wQ?i - fi,,?t(a)),

where (i) x0 9 p and (ii) Tz( s, a) (?i( s, a)) are nonunit (unit) tensor operators, respectively (2). In matrix form AU+ wQ/3 s?/h

=

[

(4lfi 0

0

wlfi

-2‘dQ/ti

o,/ti

WlllJz -Lh+U,/3

Our task therefore is to diagonalize [ 21, in the limit of small wI .

1 *

[21

TENSOR OPERATORS FOR WEAK RF FIELDS

81

Using standard techniques ( 7)) the characteristic equation of [ 2 ] can be written in the deceptively simple form x3 - ax-p

= 0,

131

a = w: + Au2 + 43

[41

fl= (wo/3)[2Aw2 - w: - 2w;/9].

151

where ( i ) and (ii) As noted earlier, while it is possible to obtain the energy eigenvalues of [ 21 in closed form (8)) the resulting expressions are unwieldy and difficult to use in practice. Instead, we use Newton’s method of successive approximations to obtain analytical expressions for the roots of % / h, in an iterative fashion. This approach holds for all I, both on and off resonance, and therefore provides a valuable alternative to the problem of solving cubic, quartic, etc., equations. ON RESONANCE Aw = uQ

In this case, it is easily shown, using algebraic computer routines such as MACSYMA, that the eigenvalues are given by

x 1 - 40, ; w: 3

4wo

0: 256w6 3tiw: 55tiw: + 0: 128~6 + 16384~; 51206

,,++p&

--3fiw:

W2

h,=-?-z-&+128w$

55fiw: WY 16384w;+512w&

[61

to order w 7. Comparison with numerical diagonalization shows that the eigenvalues of [ 61 are accurate to (5 parts in 10 lo provided w r / wo < 1 / 10. In practice, of course, the selective RF field is very weak, and so it is often a good approximation to terminate [ 61 at order w :. We are now in a position to examine the harmonic “beating” frequencies, induced by the RF field. Following (8)) the harmonic frequencies are given by 2

h,-h2=Zwo-$+$$ h2-

X3=

tiW1, 2

X,-X3=2wQ+~+~

171

to order WT. Note that while two of the frequencies are very high, -2wo, the third frequency, is dictated solely by the size of the RF field fiw 1. The latter, of course, is responsible for the generation of sizable transverse magnetizations, even when the RF field strength is very weak.

82

BOWDEN,

Following

(8) the eigenvectors

KHACHAN,

AND

MARTIN

of [ 21 can be written in the form

l~n)=a,Il)+bnlO)+cnl If the eigenvalues

a,,=-tiu.fn

- 1).

181

A, are known, then it is easily shown that

y-X,-Au

,

b,=$-[(y-h,)l-Au’],

I

1

, [9]

where the normalization

constant TV, is given by

DOI Given the eigenvalues of [ 61 therefore, it is a relatively straightforward that the eigenvectors for Au =

matter to show

wQ

2

respectively, to order w :. In this regard, it should perhaps be noted that in order to derive the eigenvectors to order w 2, , it is necessary to use eigenvalues correct to w :. Once the eigenvalues and eigenvectors of %’ have been established the time dependence of the nuclear density matrix can be obtained via the usual prescription (2, 8))

1121 where (i) the time-dependent

Fano coefficient

( 23) is given by

~,“
multipolar

- WI

[I31

state, respectively,

TENSOR

OPERATORS

FOR WEAK

RF FIELDS

83

The complete set of results is summarized in Table 1. Given the initial density matrix p( 0 ) , Table 1 can be used in conjunction with [ 12 ] and [ 13 ] to compute p( t ) at a later time t. Note that (i) the time dependences of the eight irreducible tensor operators ii( (Y) have been expressed in terms of unit, as opposed to nonunit, irreducible tensor operators, for reasons which will soon become apparent, and (ii) for simplicity, the calculations have been carried through only to order wl. However, the information given in [ 61 and [ 111 is sufficient to determine the evolution of the ?,“( a) to w :, and further if required. It should also be mentioned that the time-dependent expressions presented in Table 1 possess a high degree of symmetry. For example, the time dependence of the multipolar operators could be reexpressed in the form of a single matrix equation T(t) = M(t)T, where (i) T(t) and T are the column

1141

matrices

- u+(t)i;u(t) u+(t>if(s)u(t) U+(t)it(a)U(t) u+(t)iigJ(t) T(t) = u+(t)i:(s)U(t) U+(t)i:(a)U(t) u+(t)i;(s)u(t) _ U+(&(aPJ(O

[I51

respectively, and (ii) M(t) is the time-dependent evolution matrix, which can be obtained from Table 1. Further, it is easily established, from an examination of [ 131, that M(t)+ = M(f).

1161

So the information presented in Table 1 could be reexpressed in an abbreviated form. However, for convenience in application, we have chosen to present the time dependence of each of the eight irreducible tensor operators available to the spin I = 1 ensemble. Finally, it should be remarked that the results of Table 1 can only be used in the limit t < T, and T2, since relaxation effects have been explicitly ignored. FAN0

SUM

RULE

Before embarking on a discussion of the results presented in Table 1, it is worth noting a simple “sum rule,” which can be used to (i) partially check the accuracy of the evolutions given in Table 1, and (ii) establish how much of the “nuclear order” is oscillating at a given frequency. Let us imagine that the initial density matrix has been magnetically polarized along the z axis in a pure TA multipolar state. Under the influence of the weak selective RF field, ?A will evolve into the entire multipolar set available to the spin- 1 nucleus, as detailed in Table 1. However, since X is nondissipative (relaxation effects have been ignored), the original nuclear order, of strength unity, must be conserved. Consequently, the sum of the squares of the Fano statistical tensors must equal unity.

TABLE 1 The Evolution of the Unit Tensors i;( s, a), for Z = 1 Spin Systems, under the Action of the Hamiltonian 8

= h { [Awfifb

+ (2/3)1’2~Q?ia] -

w,f%i(a)l,

4wcand(ii)Aw=

+wc

Where(i)w, -ii;(s)

[ ++:(a):

[

-i:(a):

:(I

-p

-1-sin(V5w,r)+~{sin(2w&)+sin(2w$)} 2V5

1

+ i8G

u&t) + cos(2u$)

-~+~cos(fiqr)+cos(2

-Lsin(tiqt) [ 2ti

+ ii:(s)

{ sin(2o&t) + sin(2wQt))

- cos(Vk,r))+i(t

[( 1 - cos(f2’w,t))]

I

- cos(2o&t))+~(l

- cos(2wQt))

I

[ + i:(s)x U(t)+il(s)U(t)

4fi

[cos(2wdt) - cos(2w$)] Isin(fiw,l) [ 2k5

= -iid

+it(s) [ iii(a)

+ iii

- ii:(a)x

+ sin(2wpt)} - 2

-i:(s)

- %

+os(\rZw,t)

-i

{ sin(2w$)

U(t)+if(a)U(t)

16ti

[

$cos(fiu,t)+J-

= +i:i

i {cos(kil~w,t)

2v5

iif

Xsin(VYw,t) I 8fi

++;(a)

$[2+ [

- ia?

-

ii:(s)

[

- xsin(k5iw,t) Slci

1

- sin(2w$)}

1

-~{cos(2w~l)+cos(2w~t)}

I

cos(2w$)} I

+$ (sin(2wGt) + sin(2wot)}

-3x {sin(2w$) 16fi

+ 3) -;

{cos(2w$)

+ cos(2w$)}

+ $ { sin(2 w$) + sin(2w$)} 84

- sin(2wGt))

I

1

{cos(2w$)+cos(2w$)}]-~{cos(2w~t)-cos(2w$)} 16\/2

i { cos(fiw,t)

1 1

I

{cos(2w~t)-cos(2wQt)}

- 5) + {cos(24t)+

- cos(2w$)}

[ -

- sin(2w$)}

{ sin(2w$)

nn(2oQt)~-~{sin(2wbr)+sin(2wQt)]

&{sin(2w$+ i:(a)

-A{cos(2w$) 16fi

+ $ { sin(2wGt) + sin(2wGt)) +x

- ii:(s) 1

- cos(2w$)}

1

+ sin(2w$)}

(cos(2w~t) + cos(2w$)}

- sin(2w$)}]

1

16\/2 { sin(2w$)

[

-ii:(a)L3xsin(15w,t) 8fi

+ {sin(2w$)

{cos(2w$)

Xsin(tio,t)+${sin(2w$)+sin(2w$)}-X [ 8\/5

3 sin(fiw,t) [ 2fi

[sin(Vk,t)

4V5

+ f { sin(2w$) u;t, + cos(2w$)}

;cos(v2w,t)+~{cos(2

1

1 [

$(3+cos(tiw,t))

U(t)+i&(l)=i~

1

+ ?{ sin(2w$) 16fi

- sin(2wpt))

I

TABLE l-Continued -i:(a)

+ [2 - {cos(2&jt)

[

+i:(s,

+ cos(2w$)}]

~{COS(2o~t)-cos(2w$)} [ 2v5

+;

= +id$[l

{cos(2w:t)

- cos(2qt))

1 -~{cos(20~t)+cos(20,t)} [

- ii:(a) U(t)+iflJ(t)

--&

1 { sin(2wGt) - sin(2w$)} [ 2vi

-cos(VTw,t)]

+ iii(s)

-d

1 11 1 1

{ sin(2wbt) + sin(2wot))

f3. -sm(V?w,t)-%(sin(2o&t)+sin(2w;t)j [ 2fi

- i;(a)%

-

VT.

ii:(s)

-sm(L,t)

+%{sin(2+)

$ { 3 cos(V?w,t) + 1) -i [

v3.x

+ i:(s)-

i;(s)

4ti

[cos(2&jt)

;cos(vL,t)

-i

{cos(2w$)

-i

{cos(2w$)

+ cos(2w$)}

1/5x .

+ iif - iif

Xsin(filSwlt) 1 se

[VT -sin(fiqt) 2v?

+ i:(s) [

-i

+ COS(2U$)} -A

;cos(L,t)

{ sin(2w&t) + sin(2wot))

l5lsx .

+8

{ sin(2w$)

+$ (cos(2w;t)

+ sin(2w$)}

*sin(filZw,t) [ 8fi

- ii:(s)

1 { sin(2o&t) - sin(2wQt)) + f { sin(2w$) [ 2b5

U(t)+i:(a)U(t)

[

-d { sin(2wGt) + sin(2w$)}

$cos(fiw,t)

= -i”f

[ iii(s)

-i:(a)

*sin(hw,t)+$ [ Sti [

:[2-

1

-x

{cos(2w$)

- cos(2&$)}

{ sin(2o$)

16fi

i { 7 - 3 cos(L,t)}

-*

16\/2

- sin(2wGt)}

{ sin(2wGt) - sin(2oGt))

+ sin(2wQt)}

-Jj {cos(2&jt)

(sin(2wGt) +sin(2wGt)}

+x

16fi

+ COS(2W$)} 85

- ff { cos(2w&t) + cos(2w$)}

1

{sin(2w$)-sin(2wGt))

16ti

1

1 1

1

+ cos(2o$)}

-L{cos(2w&t)-cos(2wQt)}

+ 1) - {cos(2w$)

1 1

1

- -!2~ { cos(2w$) - cos(2w&}

{cos(2w~t)+cos(2w$)}]

; { 3cos(b5qt)

16fi

- sin(2w$)}]

+ COS(2U$)} +* 16~ { cos(2w$) - cos(2o$,}

+ ii:(a)

- i:(a)

- {sin(2w$)

{ sin(2o&t) + sin(2w$)}

[

[I + 3 cos(V?~,t)]

1

+ ii:(a)- 4~ [sm(fiw,t)

- cos(2w$)]

J- sin(tiw,t) [ 2\/2

= +iiA

U(t)+ij(s)U(t)

-

+ iai

1

+ sin(2w$)}

[ 2G

-if(a)?

-

1

i { cos(V?w,t) + 3) - { cos(2wht) + cos(2wot)) [

1

1

1

TABLE l-Continued + ii:(s)

-i

3xsin(V3w,t)

[ Sfi

+ i:(a)

$ [2 + {cos(2o$)

{ sin(2wGt) + sin(2oQt)) + cos(2w$)}]

+5x

+i:(s) [

-&cos(2w~t)-cos(2w$}

-a[

- ii:(a)

u(t)+i:(s)u(t) - iii(s) +?!(a) +i&

4fi

1 {sin(2w$) [ 2V5

[cos(2w$) - sin(2w$)}

+ i:(s)

{cos(2w~t)+cos(2Wyt)}]]

- sin(2w$)}

-f{sin(2w$) +f

+t

+i

r

+i

{cos(~c.o~~)- cos(2+)}

; {cos(2&@)

+ COS(2U$)}

-2

- ii:(a)

11

1 -~{cos(2o~t)+cos(2~~t)}

{ sin(2wbt) - sin(2wQt))

8\/2

(cos(2wbt)

+ cos(2wQt) - 2)

(cos(2w3)

- cos(2wQt))

Jj { sin(2w;t) + sin(2w$)}

= -ii~“[sin(ti’w,t) 4\/2

1 { sin(2o(;t) - sin(2w$)} [ 2V5

+ i:(s)

- 12 cos(vL,t)

-I

1

-x

+L

2V5

{cos(2w$)

- COS(2W$)} + 9 {cos(Zw~t)

i { sin(2wGt) + sin(2wQt)) -x

{ sin(2wbt) - sin(2wQt))

+ {cos(2w$)

+ sin(2&jt)}

+ cos(2w$)}

[ Note. Thistable holds for evolution times t where 3/8(w:/q)t

+ w,/fi,

2w, = 2WQ- o,/\lz,x

1

1

]

{ sin(2w$)

+ i:(a)

- sin(2wpt))

1

{ sin(2w&t) + sin(2o$)}

-!- { sin(2wGt) - sin(2wQt)) +t [ 2fi 8\/2

{ sin(2w$)

Sfi

-~{cos(2w~t)+cos(2w~t)}

- { sin(2wGt) - sin(2w$)}

[

1

- sin(2w$)}]

2fi

+ ii:\IS;;[sin(V?wlt) 4V5

+ ii:(a)

+ (sin(2w;t)

~cos(\lZw,t)+~{cos(2w&t)-cos(2w~t)} [

- iii(a)

1

- sin(2wQt))

[

+i;(s)

1

I

[

U(t)+i:(a)U(t)

{ sin(2w(;t) + sin(2wGt)}

- cos(2&5t)}]

1 { sin(2wGt) - sin(2w$)} [ 2V5 [ 2V5

1

- cos(2~Qt))

[

{cos(2w$)

1

- cos(2w$)]

‘{cos(~u$)-cos(2+)}

- ii:(s)

1-i

J- { sin(2w;t) [ 2V5

[ 2V5

4V7

+ i:(a)

= +i;”

{ cos(24t)

16\/2

[

-5x { sin(2wGt) - sin(2wot)) 16k5

= (W,/WQ). 86

4 I.

-x

1

1 1

8fi

U(f)

+ cos(2&$)}

{ cos(2o;lt) = exp[ -izt/h],

- COS(~U$)}

I

2~6 = 2wQ

TENSOR

OPERATORS

FOR WEAK

87

RF FIELDS

For example, from Table 1 the sum of the squares of the Fano coefficients for the evolution of a pure ih state is found to be ~{3+cos(fifd,t)}

'+2 I

[

=

&sin(\iIqt)]

ti +[q{l-Co~~V2~~~))l2=l

iI71

[

to order wl, in amplitude. Note that this simple check can only be applied if the unit irreducible tensors are employed. ON RESONANCE

Au = -wQ

So far we have considered only the specific resonance Aw = I- 1) to IO). For the other satellite resonance AU = - w IO) to 11) , it can be shown that the eigenvalues can be written in the form of [6], with the eigenvectors of [ 81, but with the interchange a,, S c,,, respectively. Note that since we have chosen to list the eigenvalues of the (AU = resonance in the same order as those of (AU = the beating frequencies (X, - X2), etc., are identical. The new eigenvectors, however, can be used to show that the matrix elements for the Ahw= resonance are simply related to those of Ao = In particular, the matrix elements of the ii (a), ii, f:(s), and i:(s) tensors are all identical, for both resonances, while those of ?,$, T f (s), ?:(a), and ?:(a) change sign. Thus the evolution of the irreducible tensor operators for the AU = case can be easily obtained from Table 1, by inserting appropriate sign changes. wQ,

Q,

WQ)

+wQ),

-uQ

+uQ.

aQ

DISCUSSION.

SELECTIVE

EXCITATION

Aw = uQ

As a first example consider the usual situation where the initial density matrix is proportional to Iz or ?A. From Table 1, we see that under selective excitation Au = only half of the original i Aorder can be converted into other multipolar states. For example, if we ignore terms of amplitude (X = w1/ ) , WQ,

aQ

U(t)+i&J(t)

= ih

+(3 +cos(VSw,t))

+it,[(l

ti

1

1 [ - iif

Lsin(\/2w,t) 2v-5

1

. 1181

-co~(\lZw~t))]+ii:(s)

Thus the original ?A order is partially converted into other multipolar states, at a frequency of fi wl. This conclusion was first reached by Vega and Pines (IO), who examined this problem using fictitious spin-4 operators. Note that if we choose Ew,t = ?r, [I91

which exhibits the minimum (maximum) coefficient of ?A (?z), respectively. Alternatively, ifwe set VTw,t = 7r12, id~2id-ii’(s)-/5i’+‘i’(s) 4 2\/2'

4

O

2w

.

[201

Thus the original ?A order is only partially converted into detectable ? i (s) y’-axis

88

BOWDEN,

KHACHAN,

AND

MARTIN

NMR signal. For hard RF pulses, of course, ?A order can be entirely converted into pure ? f (s, a) states, using pulses of the appropriate length. Explicitly i; + i~cos(w,t)

- iif(s)sin(w,t),

[211

where w1 is now 9 . Second, consider the case of a hard RF pulse followed by soft selective irradiation. The hard RF pulse is first used to convert the initial i; order into y’-axis magnetization. Soft irradiation at Aw = is subsequently applied along the x’ axis. For Table 1,wefind aQ

tdQ

C(s) +

-ii: [

+ + { cos(24t)

-

L

iii(s)

-L

[

iif

~cos(VTw~t)

- iit

$ { sin(2Q)

I

VT [ 2E

- ii:(a)[$

+

cos(2qt))

+

-if(S)

{ sin(2w$) 2vT

{

~COS(fiw,t)-i

+ sin(2wot))

sin(2w$)

+ sin(2Gjt))

[

1 [

-fjin(Ew,t)

+ iif

1 [

--I-sin(fiw,t) 2v5

(CoS(2fd~t)

1

+ coS(2@~t)}

1

1

A

- sin(2wot))

I

1

+ i;(a)

L { cos(24t) - cos(2oot)) , [221 [ 2v5 where, for simplicity, we have dropped terms whose amplitudes are of the order ( w1/ Consequently, NMR signals are induced along all three principal axes. Note that while the z’-axis signal is dominated by the slow harmonic beating frequency fiw,, the x’- and y’-axis signals also oscillate at the higher frequencies 2~6, 2wo defined in Table 1. Using the Fano sum, it is easily shown that 50% of the original ? f (s) order is transformed into multipolar components evolving at the double-quantum frequencies 2~6 and 206, with the remainder evolving at the slower frequency of ew , . Note also that if we choose a time 7 such that uQ).

COS(2W~T)= 1,

COS(2W,T) = -1

~231 then reasonable conversion of the ? f (s) order into the double-quantum state ?s( a) can be achieved. Explicitly, it(s)

+

-ii; [

Lsin(V5W,T) 2fi -;

+iif [

1 [ 1 [ +

ii:(s)

sin( I5,,7-)

~cos(fiwls)

-T:(s)

1

~cos(tiw,s)

1 I5 +i:(a)I.

[241

However, this method is not efficient as the soft, nonselective, Aw = 0, RF-pulse method detailed by [ 8, lo], where almost complete transformation of ?A into ?$( a) can be achieved.

89

TENSOR OPERATORS FOR WEAK RF FIELDS

Finally, it should be acknowledged that in deriving the results shown in Table 1, we have neglected terms of the order - 3 / 8 ( w ?/ wQ) , in the beating frequencies of [ 7 1. Thus strictly speaking, Table 1 only holds for evolution times t where 3w: t / 8~o < 1, with the exception of the evolution frequency ( h2 - X,) = fiw , , which is accurate to at least w :. However, more precise beating frequencies, to order w f if required, can be obtained using Eq. [ 6 1. NEARLY

ON RESONANCE Ao = wQ - 6w

It is also of interest to examine the case where the resonance condition Aw = is nearly satisfied. In this situation we write

+wQ

where 6w $ wQ . Using the simple-line approximation (see Appendix), it is easily shown that the original beating frequency of Ew, is modified to become (2~: + 6w2) ’12. However, using Newton’s method of successive approximations, we find

x2= x3

=

21Q 27

I ~+W+6w2Y2 (24

I 60

2

w: I

2

~WQ

+ 8w2)“2

2

--_

aww: 8W~(2Wf + 6W2 ) l/2 6WW:

w:

~WQ

8W,(2W:

+ 6w2 ) 112’

1261

So the beating frequency in question, to second order in w I and 6w, is given by h2 - x3 = (2w: + 6w2)1’2 +

aww: 4wQ(2w: + 6w2 ) l/2’

Equations [ 12 1, [ 13 1, and [ 261 can now be used as a starting point for (i) the examination of the near-resonance case and (ii) adiabatic passage(see, for example, ( 12)). OFF RESONANCE Aw f fwQ OR 0

In this case, all the nuclear eigenvalues are nondegenerate. Using MACSYMA we find 2(Aw +

w:( AGJ- WQ) + wf(Aw - 3‘0,) - 8Aw(Aw + wQ)3 16Aw(Aw + wQ)’

WQ)

W~WQ(

Aw2

+

Wb)

+

“&Q(24

2( Aw2 + ~6)~ 2( Aw -

WQ)

w:( Aw + + 8Aw( Aw -

WQ) WQ)3

+ 5Aw2w$ + Aw4) (Aw2-WC)’

w:(Aw+ 3‘00) - 16Aw( Aw - wQ)5

1281

90

BOWDEN,

KHACHAN,

AND

MARTIN

to WT. The corresponding eigenvectors are given by a, = l-

w:

b, =

4( Aw + wQ)’ ’

w: fi( A::.

” = 4Aw(Aw + wo)

wo) ’

a2=-\IZ(A:l+wa)’

b = 1 _ w:(Aw2 + W$> 2( Aw2 - ~6)~ ’ 2

w: a3 = 4Aw( Aw - wo) ’

b3 = - fi(A:I

” = fi(A:l

w: 4( Aw - wQ)2

c3=1-

wo) ’

wo)

v91 to order w : . The time dependences of the eight multipolar states available to the I = 1 spin system, correct to first order in w, , are summarized in Table 2. However, the information given in [ 281 and [ 291 is sufficient to determine the evolution of the multipolar states it< s, a) to higher order, if required. In the use of [ 281, [29], and Table 2, two words of caution are necessary. In the first place, they cannot be used in the limits Aw = 0 and Aw = +wo. In establishing [ 281 and [ 291, nondegenerate perturbation theory has been used. Second, it can be shown that for long evolution times t, the results presented in Table 2 must break down, due to the neglect of terms proportional to WT. From [ 281, we find that the multipolar evolutions given in Table 2 only hold to w :, provided w:Aw 2(Aw2 - WC)

w:( Aw f 2wo) t’l’

(Aw2-w3

1.

t-g

[301

However, once again, we note that [ 28 ] could be used to generate much more precise beating frequencies, to order w 7, if required. DISCUSSION.

NONSELECTIVE

EXCITATION

Aw # +uQ OR 0

From an examination of the results presented in Table 2, several conclusions can be drawn. For example, if the initial density matrix is p( 0) = ?A, a weak RF field, off resonance, will leave the ?A multipolar state largely unchanged, to order w:, while generating NMR signals of strength w i /( Aw f along both the x’ and the y’ axes. From Table 2 we find wQ)

sin( Aw -

wQ)t

sin( Aw +

+

(Au-WQ)

- i+i(a)w,

1

wQ)t

(Aw+wQ)

( 1 - cos( Aw + ( 1 - cos( Aw + (Aw - WQ) (Aw+wQ) [ sin( Aw + sin( Aw WQ)

+;i:cs,,, [

(Au-WQ)

1^ 2

+-T:(a)wl

wQ)t

t)

(Aw

(1 - cos(Aw (Aw - WQ)

+

wQ)t)

t)

1

1

wQ)t

-

WQ)

WQ) _

(1

This is in marked contrast to the “on-resonance” ( Aw = conversion of ?A into other multipolar states can occur.

-

cos(Aw + (Aw+wQ)

fwQ)

‘dQ)t)

1.

[311

case where substantial

91

TENSOR OPERATORS FOR WEAK RF FIELDS TABLE 2

The Evolution of the Unit Tensors ?;(s, a), for I = 1 Spin Systems, under the Action of the Hamiltonian &” = h { [Aw\lzid +(2/3)‘%cT~]

ti:Awt w:( AC,I+ 2w )t 41,2(Ao2-w~)4~ (Aw2pw6~

where(i)wldwQand(ii)

sin(Ao - wQ)t (Au - WQ)

u(t)+i@ (t)= i:, +(s)w, -iil(a)w, [

i -2

+ sin(Aw + ‘dQ)t (Au + wQ)

( 1 - cos( Aw - wQ)t) + ( I - cos( Ao + wO)t (Au-WQ) (Au + WQ) sin(Aw - w )t sin(Aw + wQ)f (AwewQT -

+ph

(Aw+wQ)

u(t)+i~(s)u(t) =-+o, sin( Aw 2

3i -

sin(Aw - WQ)? [ (AU-W,)

-iii(s)[cos(Aw

wQ)t

(A~++Q)

sin(Aw + wQ)t (Aw+uQ)

U(t)+Ti(a)U(t)

VT - 1 i&d,

cos( Aw [

+iT:(s)[sin(Aw + ; i:(s)w,

cos( Aw - oo)t - cos( Aw + oc)t + ~WQCOS( 2Awt) (Aw + oQ) ( Aw* - w&) (Aw-wQ) + ( 1 - cos( Aw + uQ)t

wQ)t)

(Au +

WQ)

2WQ - cos( Au + wc)t -(Aw’+) (Au + WQ)

-w,)t-sin(A&~++~)t] cos( Aw - wQ)t (Aw+wQ)

_ cos( Aw +

WQ)

-ii:(a)[cos(Aw -~~)t-cos(Aw wQ)t

(Aw-wQ)

+

2Awt) ( Aw* - ~6)

~WQCOS(

U(t)+T;lJ(t)

(Aw-wQ)

sin(Aw + wo)t (A~+wQ)

1 +uQ)t]

1

wQ)t-

(A~+wQ)

sin(Aw - wc)t

1

1

sin(Aw VTi = i; + -T;(s)wl 2

wQ)t]

- sin(Aw + wc)t+ 2w&n(2Awf) ( Aw* - ~6) I (Am - WQ)

(Au -

wQ)t

1

wQ)t] +iT:(a)[sin(Aw-wQ)t-sin(Aw+

( 1 - cos( Aw -

(Aw-uQ)

1

1

l -2 +TT2(a)w’ I = --T&I, 2

‘dQ)t

(Aw+wQ)

-wc)t-cos(Aw+ sin(Aw -

- ; i:(s),,

-

1

(1 - cos(Aw - wQ)t) _ (1 - cos(Aw + wc)t) (A~-wQ) (Au + wQ)

+ sin( Aw +

wQ)t

(Aw-~Q)

1

1

+ ii:(a),,

+y+

- w,fiTi(a)},

I

sin(A\w + wc)t + 2wosin(2Awt) ( Aw’ - w;) (A~-wQ)

1

92

BOWDEN, KHACHAN,

AND MARTIN

TABLE 2-Continued - cos(Aw + wQ)t (Au + UQ) sin(Aw -

+ sin(Aw +

wQ)t

2OQ - (A& -u$)

I

(Aw+~Q)

cos(Aw-wQ)t+cos(Aw+oQ)t (A~+wQ) (A@ - uQ)

+ eii(a)w, 2

u(t)+i:(s)u(t)= +Li&o, 2 -;i;(s)[cos(A

w-

[

-

sin(Aw -

i2

+ -T:(s)w,

[

+ cos(Aw +

wQ)t

+ sin(Aw + wQ)t (A~-wQ)

wQ)t

(Au+uQ)

- ii:(a)w, I 2

U(t)+Tj(a)U(t)

= -T&

(1 - cos(Aw (Au - WQ) [

q,)t)

-

wQ)t

- sin(Aw +

wQ)t]

-

uQ)t

+ sin(Aw +

wQ)t]

1 -i?:(a)[sin(Aw

wQ)t]

2Ao - (Au’ - ~6)

1

wQ)t

(Aw+wQ)

+ sin(Aw + wo)t (Am + WQ)

wQ)t

(Au-wQ)

+if:(s)[cos(Aw

-

(A~-wQ)

+

- cos(Aw + wQ)t] + jfi(a)[sin(Aw

wQ)t

sin(Ao -

I- -YT&

wQ)t sin(Aw

sin(Aw -

1

wQ)t

- 2Aw sin(2Awt)

(Au’-

1

~6)

cos( Ao - wQ)t [ (Au + WQ)

+ cos( Aw + wQ)t (Au - WQ)

_ (1 - cos(Aw + (Am + UQ)

wo)t)

2Aw cos( 2Aut) (Au* - u$

1

+~~f(s)[sin(Aw-wQ)t-sin(Aw+uQ)t]-~~~(a)[cos(Aw-~Q)t-cos(A~+wQ)t] +%gw, 2

cos(Aw [

oQ)t

+

cos(Aw +

wQ)t

(Aw+wQ)

(Au-~Q)

2Aw - ( Au2 - w&)

1

-~~~(s)[sin(Aw-wQ)t+sin(Aw+wQ)t]+~~~(a)[cos(Aw-wQ)t+cos(Ao+wQ)t] - ; i:(s),,

cos(Aw-wQ)t+cos(Aw+wQ)f (Au - @Q) [ (A~+wQ) +iS(ah

U(t)+T:(s)U(t)

i2

= --T;(s)w,

sin(Aw -

wQ)t

(A~+wQ)

- i i:(a)q

wQ)t

cos(Aw - wQ)t (Au + UQ)

+ sin(Aw +

wo)t

(A~-wQ)

+ cos(Aw + (A~-wQ)

q,)t

- 2Awsin(2Awt) (Au’-w&)

[

- sin(Aw + (Au-wQ)

(A~+wQ)

1

sin(Aw -w )t+sin(Aw +wo)t (Aw+wQC; (Au - WQ) wQ)t

+ 2wQsin(2Ad) (Ad - w&)

cos( Ao - wQ)t _ cos( Aw + wQ)t + ZwQcos(2Awt) ( Au2 - w&) (Au - WQ) (A~+wQ)

[ +fi:(s)u, [

+iii(a)w,

sin(Aw [

- 2Awcos(2Awt) (Au’ - u;)

1

1

- 2Awsin(2Awt) (Ad - ~6) I _ 2Aw cos(2Awt) (Ad - u&,

1

+ f:(s)cos(2Awt)

- if:(a)sin(2Aut)

1

93

TENSOR OPERATORS FOR WEAK RF FIELDS TABLE Z-Continued U(t)+i:(a)U(t)

=fif(s)u,

cos((Aw - w,$ cos(Ao + wo)t (Aw+uQ) - (hw-uQ) sin(Ao +

wQ)t

+ 2wocos(2Awt) (Au’ - w&)

1

+ 2wosin(2Awt) (Au* - w&) I

cos( Au - wa) t + cos( Aw + wo)t - 2Aw cos( 2Awt) (Au2 - w&, I - 2Aw sin(2Awt) (Au* - a$)

1 - i?:(s)sin(2Awt)

+ $(a)cos(2Awt)

Note. U(t) = exp[-izt/h].

Table 2 can also be used to suggest an alternative method of creating the doublequantum state ?z( a), using a m ixture of hard and soft RF pulses. In the first place a hard RF pulse is used to produce a -?;(a) ( aZx) multipolar state. Second, a soft nonselective RF pulse is applied along the x’ axis for a time t such that cos(Ao + wQ)t = -1.

cos(Aw - wo)t = 1,

~321

At the end of this period the initial ?i (a) is converted almost entirely, to order w:, into a if< a) multipolar state. Finally, a hard ?r/2 RF pulse applied about the x’ axis will produce the desired double-quantum coherent state i;(a). Table 2 can also be used to discuss the YKW M Q N M R pulse sequence for Z = 1 spin systems, in which multiple-quantum coherence is both created and detected, in a single-shot experiment ( 14,8). In the YKW experiment, two hard RF pulses, separated by an appropriate evolution time, are used to create the double-quantum multipolar state ?:(a). Subsequently, a weak off-resonant RF field is applied along the X’ axis, while simultaneously NMR signal is acquired along the y’ axis. From Table 2 we see that the presence of the weak RF field will give rise to a y’-axis NMR signal %(t)=--w

1 2

sin(Aw -wQ)t (Aw+WQ)

-

sin(Aw+wo)t (A@-WQ)

Im (&2-4)

1 *

t331

[ Thus the Fourier transform of the NMR signal will be characterized by two singlequantum peaks at (AU f and a double-quantum signal at 2Aw. Normally, timeconsuming 2D methods are used to detect M Q N M R signals (6). However, while the pulse sequence of ( 14) can be used to reduce the time taken to detect multiple-quantum coherence, it must be acknowledged that the double-quantum signal is necessarily weak. wQ)

CIRCULARLY

POLARIZED

NQR

Table 1 can also be used to detail nuclear quadrupole experiments using circularly polarized RF fields. In the laboratory frame, the Hamiltonian in question can be written in the form

94

BOWDEN,

KHACHAN,

AND MARTIN

A!?= ti{[$]‘%oi3-V?w,[Tt(a)cos(wt)--iif(s)sin(wt)]}.

1341

On transferring to the rotating frame on resonance ( w = wo), we find A?= h{wo~~:,+[3]“2Woi8-~W,if(a)}.

[351 Thus [ 35 ] is formally equivalent to [ 1 ] with Aw replaced by wQ . Since this is identical with the on-resonance case AU = wQ, we can therefore use Table 1 to describe NQR experiments. For example, if we start with the nuclear density matrix in the quadrupolar state p( 0) = ?g, then following the application of the weak RF field p(t) = ++A7

[

1 - cos(tiw,t)

X {sin(2w$)

I

+ iii(s)

lb

zsin(llZwrt)

[

1

-i’(a)& I

+sin(2w$)}

- {COS(2w~t)+COS(2w$)}

-8

\/5X

12 {cos(fiwlt)

8

+ 3)

[

+i+

+3cos(fiw,t)]

3 v3 psin(V?w,t)

- ii:(s)

tiX

+8

{ sin(2w$)

+ sin(2wot))

[ 2fi

-yf(a) fixl

4 { 3 COS(lEWJ) + l} - ; { cos(2w$)

4

1

+ coWw$)}

[

V3X

+ i;(S)-[cos(2w$)

4E

I

- cos(2w$)] V3X

+z?$(a)--&sin(fiw,t)-

{sin(2w$)-sin(2w$)}],

[36]

Consequently, if we set fiw,t = 7r/2, much of the initial Tz order where x = wl/wQ. is converted into a detectable y’-axis NMR signal (amplitude ( fi/ 2fi). Such conclusions, of course, were reached long ago by ( 15-17)) using the so-called “quadrupolar interaction picture.” However, it should be noted that Eq. [ 361 contains far more information than that provided by the interaction picture. In the latter, high-frequency components are explicitly ignored, in order to allow simple expressions for the time dependence of the density matrix to be obtained. No such approximations have been made in arriving at [ 36 1. CONCLUSION

In this paper, the problem of determining the evolution of a nuclear ensemble perturbed by weak RF fields has been addressed. In particular, evolution tables for the eight irreducible tensor operators i,“( cy), for both Aw = fwQ and Aw # fwQ, 0, have been presented and discussed. In addition, the tables have been used to illustrate (i) a useful Fano sum-rule, (ii) a new method of preparing the double-quantumcoherent state ?$ (a), using a mixture of hard and selective RF pulses, and (iii) a new method of describing NQR experiments. In addition, the results have been related to

TENSOR OPERATORS FOR WEAK RF FIELDS

95

the recently reported Raman experiment of Yannoni, Kendrick, and Wang (14), who have used a m ixture of hard and soft RF pulses to both create and detect multiplequantum coherence in a single-shot experiment. It should be stressed however that the techniques developed in this paper are not specifically restricted to I = 1 ensembles. Newton’s method of successiveapproximations can be used to obtain eigenvalues, correct to wl, w:, or more for any spin. Once these have been found, eigenvectors can be determined, and hence the time evolutions of the individual multipolar states under the action of weak RF fields can be determined. APPENDIX.

THE WOKAUN-ERNST

OR SIMPLE-LINE

APPROXIMATION

Although fictitious spin-4 operators were used by Wokaun and Ernst (II), it is a relatively easy matter to recast their treatment in terms of irreducible tensor operators. Consider the on-resonance case AGJ= +wo . The Hamiltonian [ 2 ] reduces to % ‘/h =

% I3 w,/fi

Wl/E -2w,/3

0

0 w*llJz -2w,/3

1 .

[AlI

W lfi [ Thus the energy difference between the two unperturbed energy levels is 2wo. If wQ is large therefore, it is a reasonable approximation to omit the RF m ixing term linking the 11) and 10) energy levels. This is equivalent to the simple-line approximation of Abragam ( 12) [see also the discussion following [ 261 of ( 1 I)]. In this approximation, the energy levels are given by [A21 which are almost the same as those of [ 6 1, except for differences of the order w :I wQ . At first sight therefore, one m ight expect little difference between the two approximations. However, differences to order w I emerge in the wavefunctions. Using [ A21 we find

IhJ=4111)+~,10)+cnl-l),

[A31

where the coefficients are now given by a1 = 1,

b, = 0,

a2 = 0,

b,=--&

Cl = 0

Note that [ A3 ] can be obtained from [ 111 simply by setting wr = 0. Thus the evolutions of the irreducible tensors, in the Wokaun and Ernst approximation, can be obtained from Table 1 by setting x ( =wI / wQ) = 0, and 2aQ f wI / fi = 2wQ. This leads to a considerable simplification in the time development of the eight multipolar

96

BOWDEN,

KHACHAN,

AND

MARTIN

states ?i( s, a) available to the I = 1 spin ensemble. For example, in the simple-line approximation, the coefficients of t;(a), ?:(a), i:(s), and i:(a) which appear in the evolution of the ?A (see Table 1) are now all zero. In addition, the high-frequency 20, components of i f (s) and i:(s) multipolar states are not reproduced. Moreover, as aQ + 0, substantial deviations from the simple-line approximation will occur. ACKNOWLEDGMENTS The authors acknowledge

some usehtl discussions with F. Separovic and C. S. Yannoni. REFERENCES

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Il.

12. 13. 14. 15. 16. 17.

B. SANCTUARY, T. K. HALSTEAD, AND P. A. OSMENT, Mol. Phys. 49,753 ( 1983). J. BOWDEN AND W. D. HUTCHISON, J. Magn. Reson. 67,403 ( 1986). J. BOWDEN, W. D. HIJTCHISON, AND J. KHACHAN, J. Mugn. Reson. 67,415 ( 1986). A. BUCKMASTER, Can. .I Phys. 40,167O ( 1962). A. BUCKMASTER, R. CHAI-TETERJEE, ANDY. H. SHING, Phys. Status Solidi 13,9 ( 1972). R. ERNST, G. BODEHAUSEN, AND A. WOKAUN, “Principles of Nuclear Magnetic Resonance in One and Two Dimensions,” Clarendon, Oxford, 1987. G. M. MUHA, J. Chem. Phys. 73,4 139 ( 1980). G. J. BOWDEN, W. D. HUTCHISON, AND F. SEPAROVIC, J. Mugn. Reson. 79,413 ( 1988). B. B. SANCTUARY, Mol. Phys. 49,785 (1983). S. VEGAANDA. PINES, J. Chem. Phys. 66,5624 ( 1977). A. WOKAUNANDR. R. ERNST, J. Chem. Phys. 67,1752 (1977). A, ABRAGAM, “The Principles of Nuclear Magnetism,” Clarendon, 196 1. U. FANO, Rev. Mod. Phys. 29,76 (1957). C. S. YANNONI, R. D. KENDRICK, AND P. K. WANG, Phys. Rev. Lett. 58,345 ( 1987). M. BLOOM, E. L. HAHN, ANDB. HERZOG, Phys. Rev. 97,1699 (1955). T. P. DAS AND E. L. HAHN, “Solid State Physics, Nuclear Quadrupole Resonance Spectroscopy,” Suppl. 1, Academic Press, New York, 1958. C. P. SLICHTER, “Principles of Magnetic Resonance, with Examples from Solid State Physics,” Harper & Row, New York, 1963. B. G. G. H. H. R.