Design of composite pulses for nuclear quadrupole resonance spectroscopy

Journal of Molecular Structure, 192 (1989) 333-344 Elsevier Science Publishers B.V., Amsterdam - Printed

333 in The Netherlands

DESIGN OF COMPOSITE PULSES FOR NUCLEAR QUADRUPOLE RESONANCE SPECTROSCOPY

A. RAMAMOORTHY

and P.T. NARASIMHAN*

Department of Chemistry, Indian Institute of Technology, Kanpur 208 016 (India) (Received

16 March 1988)

ABSTRACT Composite pulses that compensate the inhomogeneities of electric field gradient (efg) and radio frequency (rf) field strength for nuclear quadrupole resonance (NQR) spectroscopy are presented. The theoretical procedure for constructing these composite pulses employs the Magnus expansion technique in a manner similar to NMR average Hamiltonian theory. Composite pulse sequences, for which the zeroth-order term in the Magnus expansion vanishes, have been obtained using the fictitious spin-l/2 operator formalism for spin I= 1 case. Some typical NMR composite pulses have also been examined in the NQR context. The efficiency of these pulses for broadband excitation in NQR of single crystals is analyzed by simulation of the appropriate responses. We propose the sequence (go),-(300),, as a composite n/2 pulse for compensating efg inhomogeneity and (go),- (8& (9O)0 as a composite inversion rr pulse which, for a large range of 8, compensates both efg and rf field inhomogeneities in pure NQR of single crystals containing physically equivalent spin I= 1 nuclei. The results obtained are independent of the asymmetry parameter of the electric field gradient in the absence of a Zeeman field.

INTRODUCTION

A common problem in magnetic resonance is the excitation of broadband resonance frequencies by a resonant radio frequency (rf ) pulse. This problem has largely been overcome by using composite pulses [ 1 ] in the area of NMR and coherent optics. A “composite pulse” is a sequence of phase-shiftedrf pulses, without any time delay between them, to excite the spin populations over a larger range of some experimental parameters compared to a single rf pulse. The individual pulses are generally rectangular pulses with constant phase. Thus, in NMR we have composite pulses that are suitable for population inversion ( TCpulse) and for creating transverse magnetization (n/2 pulse) in the case of broadband excitation as well as narrow-band excitation. In nuclear quadrupole resonance (NQR) spectroscopy one has generally broad lines and this is mainly because of the electric field gradient (efg) inhomogeneity, and *Author to whom correspondence

0022-2860/89/$03.50

should be addresssed.

0 1989 Elsevier Science Publishers

B.V.

334

dipole-dipole interactions. Dislocations, strains, presence of impurities, torsional motions of units in the crystals and fluctuations in the temperature cause a random distribution of efg at the nuclear site. Hence there will be a distribution (Awe) of quadrupole frequencies around the central frequency (oo). The excitation of such broad spectral lines demands high rf power. If one uses relatively low rf power the signal intensity will be poor. It is here that the use of composite pulses in NQR would be advantageous. Although the problem in NMR is usually one of exciting a spectrum which consists of large groups of multiplet lines, in NQR the spectral lines are often separated very widely. Multiplet structures are rarely met with in NQR. However, such situations do arise even in NQR and the need for excitation of multiplets is felt occasionally. In any case, the line widths in NQR being much larger than those met with in high resolution NMR, design of composite pulses for broadband excitation in NQR seems worthwhile. To our knowledge, no attempt has so far been made in the NQR literature for the design of composite pulses. We have, therefore, undertaken a programme of design of composite pulses for NQR spectroscopy. In the present work we confine our attention to the problem of broadband excitation using composite pulses that compensate for efg and rf field inhomogeneities for the spin I= 1 case in single crystals. Unlike in NMR, composite pulse sequences with a large number of rf pulses cannot be used in NQR because of the shorter spin-spin relaxation time (Z’.,), which is of the order of milliseconds. NMR currently employs composite pulses starting with a minimum of two pulses going up to several, typically nine pulses. Since the flexibility available with a two-pulse sequence is limited, the present work mainly considers three-pulse sequences of the type (8, ) e1- ( O2) e - (0, ) e3 with flip angle (0,) and rf phase ($,) for the nth rf pulse in the sequence. There are various methods available for construction of composite pulses in NMR such as the geometrical approach [ 21, recursive expansion [ 3, 41, reversed nutation pulses [ 51, Magnus expansion approach [ 61, fixed-point analysis [7] and numerical approach [B]. Following the trajectory of a magnetization vector starting from some given initial condition in three-dimensional space is convenient with the geometrical approach in NMR for the spin I= l/2 case. However, such a visualization is not possible in NQR because of the invalidity of the simple Bloch vector model for spin 12 1 cases. The numerical approach involves the search for the optimum pulse sequence in which, in general, 2N variables consisting of N flip angles and N phases are involved for an N-pulse sequence. The computer time associated with such a multi-dimensional search could be extremely large. In the present work we have employed the Magnus expansion [g-11 ] approach in the spirit of the work of Tycko [6] on composite pulses in NMR. The Magnus expansion approach reduces the problem of finding composite pulses to that of solving a set of equations to get the flip angles and phases of rf pulses in the sequence. It treats the composite pulse sequence entirely rather than a pulse-by-pulse analysis. For the representation of the relevant spin operators involved in the Ham-

335

iltonian in the present problem we have found it convenient to make use of the fictitious spin-l/2 operator formalism [ 121. The next section presents the general theoretical procedure for the design of composite pulse sequences in pure NQR for spin 1~1. We consider the single crystal case in the absence of a Zeeman field and assume that all the resonant nuclei are at physically equivalent sites. We also confine ourselves here to “zeroth-order” composite pulse sequences for which the zeroth-order term of the Magnus expansion either vanishes or commutes with the thermal equilibrium density matrix. Subsequently we present details of the procedure for designing zeroth-order composite pulses that compensate for efg inhomogeneity, followed by consideration of composite pulses for rf field inhomogeneity compensation. The performance evaluation of these composite pulses is then presented and discussed. GENERAL

THEORY

Our goal is to find a spin system under those trum and thus achieve the Magnus expansion theory [lo]. We briefly The total Hamiltonian is given by X=

pulse sequence which suppresses the evolution of the interactions which cause broadening of the NQR specthe broadband excitation. This can be realized using [ 61 in the same manner as in the average Hamiltonian outline the general theoretical procedure here. of the system in the quadrupole principal axis system

z$o+X$+v

(I)

where %o is the quadrupole Hamiltonian, & is the rf Hamiltonian and V is the Hamiltonian which causes broadening of the NQR spectrum. In the quadrupole interaction frame (QIF) [ 131 eqn. (1) can be written as %=

u,

%U,’

= *,+o

(2)

where U, = exp[ -i3(e,t]

(3)

In QIF, the evoution by the Dyson evolution U(t)

= Texp-_

s

of the spin system under the Hamiltonian operator U (t), as

dt’2(t’)

2 is given

(4)

0

where T is the Dyson time-ordering operator. Equation (4) gives the evolution of the spin system simultaneously under 2, and 0. However, we are interested in studying the evolution of the spin system under the Hamiltonian V separately. This can be done by transforming into the rf interaction representation. In the rf interaction frame, the complete Hamiltonian is given by

336

2= U,,(t)-’ aJ&)

= O(t)

(5)

where U,,(t)

= T exp-i

5

(6)

dt’*,(t’)

0

Now, the system evolves only under 3 (t) and the evolution

operator

is given

by t

U,(t)

= Texp-i

I

dt’q(t’)

(7)

0

In QIF, eqn. (4) can be written U(t)

= U,,(t)&

(3)

(t)

0 ( t) is time dependent, Q(t)

= V’o’+V”‘+

as

and using the Magnus expansion

we may write

...

(9)

where V(O) and V(l) are respectively the Magnus expansion.

the zeroth-order

and first-order

terms in

(10) If we could make U,(t) equal to the identity operator then the system would evolve in the QIF as if there were no interactions causing the broadening of the frequency spectrum (see eqns. (7) and (8) ) . Uniform excitation of spin populations over a broadband is then achieved. An mth-order composite pulse sequence for this purpose is then one for which Vci’ =O; 0 6 i< m. An m-thorder composite pulse can be designed as follows. Consider a sequence of N rf pulses, which can be completely defined by N flip angles (13,) and N rf phases (#,). By selecting @I= 0 as the reference phase, only 2N- 1 variables are sufficient to define the sequence. These variables can be optimized by satisfying the following conditions (eqns. (ll), (13) and (14) ). The mth-order composite pulse should satisfy one or both of the following conditions: V’“’ = 0

(lla)

or

[Vi’,p(0)]

(lib)

= 0

Here p (0) is the reduced thermal

equilibrium

density matrix and is given by

337

p(o) =

(12)

.fQ

We have spin inversion

with a composite

n pulse if

Tr[I,,,U,,(t)p(O)U,,(t)-ll

= - (lz)maximum

and maximum

decay with a composite

free induction

Tr[I,,U,,(t)p(O)U,,(t)-‘l

(13) n/2 pulse if

= (ls)maximum

(14)

The performance of the composite n/2 pulse can be judged by evaluating signal amplitude W,, which can be written as

w, = Tr[I~z~U(~)p(0)U(~)-‘l Tr [ I’XZ1

the

(15)

and similarly the performance of the composite inversion n pulse can be judged by evaluating the extent of inversion W,, which can be expressed as w

=

2

-Tr[I~~~U(~)p(O)U(~)-‘l Tr t 1’~s 1

U (t ) is the evolution pulse. DESIGN

operator

OF ZEROTH-ORDER

(16)

(see eqn. (4) ) for the appropriate

COMPOSITE

composite

PULSES FOR COMPENSATING

EFG

INHOMOGENEITY

The quadrupole Hamiltonian moment eQ is given by [ 141 e2qQ ,*o = 41(21-l)

-&fQ,

for the nucleus with spin I and quadrupole

[312,-12+~(12X-12Y)]

where e 2qQ is the quadrupole eter defined as

(17)

coupling constant

and q is the asymmetry

param-

(18) V,, are the components of efg and V,, = eq with 1V,, 1< 1V,, 1< 1V,, I. Using the fictitious spin-l/2 operator fOrmdiSm [121 for spin I= 1, be written as [ 151 Xo = ~[I&&,,] The fictitious

= 0, I& + o&*; spin-112 operators

are written

p = x,

Y, z

8Q

can

(19)

in general as IP1 where p has

338

three components X, Y and 2 and for each p there is a subspace characterized by 1= 1,2,3. These operators obey the commutation relations

[I&d,21 = $I

(or) cyclic permutation

of 1,2,3

(29a)

Ip4in eqn. (19) is given by Ix4 = Iy3- Iz3;X, Y, 2 taken in cyclic order and

(29b)

&&Ill

(2Oc)

= 0

for 1 = 1,2, 3.

(23)

(24)

(25)

(26) cox, oy and wz are the three pure quadrupole resonance frequencies for a spin I= 1 nucleus. We consider one of these three transitions, for example, the transition with frequency ~0~ and let COG=oQ; ok = cob. The quadrupole Hamiltonian can now be written in the form %Q

=

mQIX3+&&

(27)

The rf Hamiltonian &for an rf field of frequency (rad s-l ) and phase @(t) is given by c%$ = -2w,

aQ

with strength

cos[wQt+$b(t)]I,

co1= yH,

(28)

Here it is assumed that the x axis of the quadrupole principal axis system is aligned with the rf coil axis. Using the fictitious spin-l/2 operator formalism eqn. (28) becomes c%$ = -40,

(29)

cos[wQt+@(t)]&l

In QIF, after truncating

the high frequency

terms, we obtain

339

&

co@(t) -Ixz sin@(t)]

= -204[Ixl

(30)

The efg inhomogeneity arises from a spread Sq of the field gradient. Consequently, there is a distribution in @Q and ob. The Hamiltonian corresponding to the efg inhomogeneity can be written as (31) AoQ is the width of the NQR frequency spectrum. In QIF 0

1 =

(32)

Vl

,o,(t)

Using eqn. (5 ) operator is U,(t). pessed as

can be calculated for a composite pulse whose evolution For the three-pulse case using eqn. (lo), V!O) can be ex-

(33) where t = r1+ z2+ r,, is the duration of the pulse sequence and z, is the width of the nth pulse in the sequence. a,, b, and cl are functions of flip anlges (&.,=ByH,r,) and phases (&,) of rf pulses in the sequence. Since, the expression for a,, b, and cl are lengthy they are not reproduced here. However, in order that V(O) satisfy the conditions given in eqns. (ll), we must optimize 8, and $,, such that a,, b, and c, vanish identically or the resulting V(O) commute with p(0) of eqn. (13). Using this approach we obtained the composite inversion z pulse sequence (90) o- (270) 9o(90)o. This pulse sequence is also a composite R pulse in the case of NMR for compensating resonance offset [ 161. In view of the similarity in performance noticed for this pulse sequence in both NQR and NMR we decided to examine the efficiency of several other composite pulses that are employed in NMR for use in the NQR context. One of the composite n pulses proposed for NMR is the ( 90)o- (225) 18o-(315 ). pulse sequence. This sequence works well in NQR also for compensating efg inhomogeneity (see section on performance evaluation). The (go),-(go),, pulse sequence has been proposed as a composite n/2 pulse for compensating rf field inhomogeneity in NMR. We found that the performance of this pulse sequence was not as good in NQR. We therefore decided to vary 13,and found that the (90)o- (300)90 sequence gives better performance as a composite x/2 pulse for compensating efg inhomogeneity. Using this analogy with NMR pulses we also obtained another composite 7c/2 pulse, namely (385)o- (320) lso-(25 )o. A summary of the zeroth-order term in the Magnus expansion associated with these pulse sequences is presented in Table 1. Detailed evaluation of the performances of these pulse sequences with respect to broadband excitation are given below.

340 TABLE I Zeroth-order Magnus expansion term for different NQR composite pulse sequence that compensate efg inhomogeneity in the spin I= 1 case Composite pulse

Zeroth-order Magnus expansion term, Vi”

A. Composite x/2 pulses (i) @O)o-(300)90 (ii) (385)o-(320),00-(25)o

Ao, [ - 0.00261,, + 0.00261, ]

B. Composite x pulses (i) (90)o-270)~o-(90)o (i) (90)o-(225)1so-(315)0

0 Ao$[ -0.05331,+0.31051x,]

Aw,[ -0.07341X1 -0.01971,,+0.14691,,]

DESIGN OF ZEROTH-ORDER COMPOSITE PULSES FOR COMPENSATING RF FIELD INHOMOGENEITY

The Hamiltonian V 2 = -4A0,

corresponding

is given by

cos[coat+~(t)]Ixl

where Aw, is the inhomogeneity In QIF, eqn. (34) becomes 8,

to the rf field inhomogeneity

(34) in the rf field strength.

= -2Aco,[cos@(t)Ix1-sin@(t)Ixz]

Using eqn. (5), 6, (t) can be calculated for a composite pulse consisting three rf pulses. Using eqn. (lo), Vi’) can be expressed as

(35) of

u2, b2 and cz are functions of 0, and &. Using the conditions given in eqn. ( 11) , we arrived at three pulse sequences shown in Table 2. An evaluation of these pulse sequences in the context of broadband excitation is given in the next section. TABLE 2 Zeroth-order Magnus expansion term for different NQR composite pulse sequences that compensate rf field inhomogeneity in the spin I= 1 case Composite x pulse

Zeroth-order Magnus expansion term, V$”

(i) (90)o-(180)~o-@O)o

Am, 1x3

(ii) (180)o-(180)120-(180)o (ii) (90)0-(360)1zo-(90)0

0 1.1547 Ac&s

341 EVALUATION

OF PERFORMANCE

The performance

OF COMPOSITE

of the different

1 and 2 can be judged by evaluating

I

-0.8

,

I

-0.4

I

0

*

0.4

I

PULSES

composite pulse sequences given in Tables WI and W, given by eqns. (15) and (16))

IIIII,I

,

0.8

-1.4

Aw,/Wl-

I,,,,,,

-1.0

-0.6

0.2

-0.2 AL&/W,

0.6

1.0

1.4

-

Fig. 1. The signal magnitude as a function of the electric field gradient inhomogeneity for a single n/2 pulse (-m--m-), the xeroth-order composite pulses (90),-(300)w (-O---O-) and (385),(320),,,-(25), (-A-A-). Fig. 2. The extent of the inversion of spin population, as a function of the electric field gradient inhomogeneity for a single apulse (-m---m-), the zeroth-order composite pulses (go),,- (270),(go),, (-A-A-) and (90)0-(225)180-(315), (-O--O-).

0.6~ -1.6

-1 2

-0.8

-0.4

0 Aw,lw,

o-4

V-8

1.2

16

-

Fig. 3. The extent of the inversion of spin populations, as a function of the inhomogeneity of the rf field strength for a x pulse (-O-a--), the zeroth-order composite pulses (go),- ( 180)s0(9O)0 (-A-A-) and (180),-(180)120-(180)00r (90)0-(360)lz0-(90)0 (-m--m-).

342

respectively. In these equations as U(t)

U(t)

for a three-pulse

sequence can be written

= exp ( $4 1x3 ) exp ( iPr,, ) exp ( - & IX3 ) exp ( - i&s

(-i~~IIX3)exp(i~,Ix,)exp(iPIx2)exp(-ie,I,,)exp(

) exp

-i/31xz)exp

(37)

(-i~~I1~3)exp(iPIx,)exp(-ie,I~~)exp(-_iPlxz) where 8n = z,( 2c0, sinp+

Aoo co@)

(37a)

with p = tan-l

(o,/Aoo)

for efg inhomogeneity 8n = 27,(o), +A@)

(37b) and for rf field inhomogeneity (37c)

with p=90°. Computer simulations of W, and W, as a function of efg and rf field inhomogeneities are shown in Figs. l-3 for spin I= 1. For comparison, the broadband excitation by a single n/2 or a single n pulse, as the case may be, is depicted in these figures. DISCUSSION

Computer simulations presented in the previous section show that the composite pulses designed using the Magnus expansion approach to compensate the inhomogeneities of efg and rf field perform significantly better than a single rf pulse. From the form of V(O)given in Tables 1 and 2, one would expect that a pulse sequence with V (O) of smaller size should perform better than a pulse sequence with V (O)of larger size. But in our case, performance of (90)o(25).with (300)90 with larger VI')is better than that of ( 385)o-(320)180smaller Vi’). Similarly for the (90)o-( 225),so- ( 315). sequence with larger Vi’), the performance is better than that of the ( 90)o- ( 270)90- ( 9O)o sequence with a smaller Vi’). The ( 90)o- (360) iso- ( 9O)o sequence with larger VA')shows better performance than the (90)o- ( 180)90- (9O)o sequence with smaller V&O’. These results clearly reveal the importance of the higher order terms in the Magnus expansion. Surprisingly, results of (90)_o-(360)1Zo-(90)o and (180)o-(180),20-(180)o are exactly the same for large ranges of Aal. The pulse sequence (go),(8)9o(9O)oworks as an inversion n pulse for compensating both efg and rf field inhomogeneities. It gives better performance for compensating efg inhomoge-

343

neity when 8 is anywhere between 150’ and 300’ and for compensating rf field inhomogeneity when 0 is anywhere between 90” and 270”. When 13=90” results are better for positive values of Aw, and 8=270” for negative values of A#,. The results obtained here for the NQR case are independent of the asymmetry parameter of efg and hence these composite pulses should have a wide validity. It is also interesting to note that while some of the pulse sequences employed in NMR for compensating resonance offset and rf field inhomogeneity work well in the context of NQR with respect to efg and rf field inhomogeneities, respectively, one cannot assume a priori that all NMR sequences behave in a similar fashion in the NQR case as well. This has been borne out by our results in the case of 7c/2 pulses as well as z pulses of NMR. The zerothorder term in the Magnus expansion in the NQR context does not vanish for some of these sequences. Even the commutation relation given by eqn. (llb) is not satisfied by some of the NMR pulse sequences when used in the NQR case. It was pointed out that in eqn. (33) and also in eqn. (36) the V(O) term has to be zero for the chosen values of the parameters defining the pulse sequence. In particular the values of uk, bk, ck (k = 1 or 2 ) have to go to zero individually, in order to satisfy eqns. (11). Here we have employed an approach based on an examination of the qualitative behaviour of these trignometric functions and thus arrived at the values of 8, and c&. A better approach would be to make a computer search for the best values of these parameters such that V(O) vanishes. Such an approach combines the advantages of the Magnus expansion procedure with that of the numerical procedure but at the same time will not be as time consuming as a complete numerical search procedure [ 81. Interestingly, none of the zeroth-order composite 7c/2 pulses which compensate for rf field inhomogeneity in the NMR case gives better performance than a single n/2 pulse in NQR. In this context design of higher order composite pulses may be interesting, but the Magnus expansion-based theoretical design of composite pulses becomes extremely complicated when one wishes to incorporate the higher order terms. It is known from NMR that larger numbers of pulses are involved when one incorporates the higher order terms. In view of the shorter spin-spin relaxation times in NQR we feel that it is appropriate to limit oneself to zeroth-order terms in the Magnus expansion and consequently to a fewer number of pulses in the sequence. CONCLUSIONS

We have demonstrated that the Magnus expansion approach provides a useful route for designing composite pulses for pure NQR. This approach is less time-consuming than the pure numerical procedure. The composite inversion 7cpulse (90)o-( 0),,-(go), compensates both efg and rf field inhomogeneities for large ranges of 8. The pulse sequence reported in this paper for compensat-

344

ing efg inhomogeneity are also capable of compensating resonance offset effects. NQR of powder specimens involves a spread in the flip angle of an applied rf pulse due to the random orientation of the crystallites. Composite pulses can be expected to yield a better signal-to-noise ratio in NQR of powders on account of their ability to ensure uniform excitation of a large number of crystallites. Design of composite pulses for higher spin systems using the procedure presented in this paper, as well as sequences applicable to powder specimens, will be published elsewhere.

REFERENCES 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

(a) M.H. Levitt, Prog. Nucl. Magn. Reson. Spectrosc., 18 (1986) 61, and references cited therein. (b) A.J. Shaka and A. Pines, J. Magn. Reson., 71 (1987) 495. (c) B.C. Sanctuary and H.B.R. Cole, J. Magn. Reson., 71 (1987) 106. M.H. Levitt and R. Freeman, J. Magn. Reson., 33 (1979) 473. M.H. Levitt and R.R. Ernst, J. Magn. Reson., 55 (1983) 247. A.J. Shaka and R. Freeman, J. Magn. Reson., 59 (1984) 169. A.J. Shaka and R. Freeman, J. Magn. Reson., 55 (1983) 487. R. Tycko, Phys. Rev. Lett., 51 (1983) 775. R. Tycko, A, Pines and J. Guckenheimer, J. Chem. Phys., 83 (1985) 2775. D.J. Lurie, J. Magn. Reson., 70 (1986) 11. W. Magnus, Commun. Pure Appl. Math., 7 (1954) 649. U. Haeberlen, High Resolution NMR in Solids: Selective Averaging, Academic Press, New York, 1976. M. Mehring, Principles of High Resolution NMR in Solids, 2nd edn., Springer, New York, 1983. S. Vega and A. Pines, J. Chem. Phys., 66 (1977) 5624. M. Goldman, Spin Temperature and Nuclear Magnetic Resonance in Solids, Oxford University Press, Oxford, 1970, Chapt. 1. T.P. Das and E.L. Hahn, Nuclear Quadrupole Resonance Spectroscopy, Solid State Phys., Suppl. 1, Academic Press, New York, 1958. R.S. Cantor and J.S. Waugh, J. Chem. Phys., 73 (1980) 1054. M.H. Levitt and R. Freeman, J. Magn. Reson., 43 (1981) 65.