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The WAHUHA-4 multipulsesequence in nitrogen-14 N.Q.R.V.L. Ermakov a & D.Ya. Osokin aa Kazan Physico-Technical Institute of the U.S.S.R. Academyof Sciences , Sibirski trakt 10/7, Kazan, 420029, U.S.S.R.Published online: 22 Aug 2006.

To cite this article: V.L. Ermakov & D.Ya. Osokin (1984) The WAHUHA-4 multipulse sequencein nitrogen-14 N.Q.R., Molecular Physics: An International Journal at the Interface BetweenChemistry and Physics, 53:6, 1335-1353

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MOLECULAR PHYSICS, 1984, VOL. 53, No . 6, 1335-1353

The WAHUHA-4 mult ipulse sequence in nitrogen-14 N.Q.R.

b y V. L. E R M A K O V and D. Ya. O S O K I N

K a z a n P h y s i c o - T e c h n i c a l I n s t i t u t e of the U . S . S . R . A c a d e m y of Sciences , S ib i r sk i t r ak t 10/7, K a z a n 420029, U . S . S . R .

(Received 29 February 1984 ; accepted 19 August 1984)

The influence of the W A H U H A - 4 multipulse sequence on a system of dipolar coupled spin-1 nuclei each subject to a local electric field gradient in zero external magnetic field is studied. To interpret the experimental results the two equivalent spin-1 model is used, leading to results which reproduce the experimentally observed quasi steady-state of magnetization. Using a fictitious spin-l /2 formalism a theory is developed which provides a conve- nient description of the evolution of the density matrix of the model system during the pulsed N.Q.R. experiments. The experimental curves of spin echo envelopes obtained for a single crystal of NaNO 2 are given and discussed.

1. INTRODUCTION

Sp in re laxa t ion rates have b e c o m e a m a j o r source of i n fo rma t ion for the s tudy of m o l e c u l a r d y n a m i c s in solids. Re laxa t ion rates are sensi t ive to bo th s low and fast m o t i o n a l processes and p e r m i t one to inves t iga te the a n i so t ropy of mo lecu l a r mo t ion . In m a n y cases it is poss ib le to ob ta in the des i r ed i n fo rma t ion f rom s tudies of q u a d r u p o l a r re laxa t ion which is pa r t i cu l a r ly s imple to i n t e rp re t in t e rms of mo t iona l pa rame te r s . In p r ev ious pape r s [ 1 - 5 ] it has been shown that the p h a s e - a l t e r n a t e d m u l t i p u l s e sequence ( P A M S ) , P~x-(r-P~ x-2z-P~x-z)n, leads to a so-ca l led quasi s t eady-s t a t e in a q u a d r u p o l a r spin-1 sys tem in a t ime of the o rde r of T 2 . As e x p e r i m e n t s show, this s tate re laxes t owards e q u i l i b r i u m in a t ime of the o rde r of T 1. I t is seen tha t m u l t i p u l s e sequences can c ons ide r a b ly shor ten the t ime for re laxa t ion m e a s u r e m e n t s . T h e resul ts ob t a ined for P A M S [5] make it poss ib le to ob ta in all t h ree t r ans i t ion p robab i l i t i e s cha rac te r i z ing the t ensor of mo lecu l a r m o t i o n us ing the o n e - f r e q u e n c y i r rad ia t ion . In this connec - t ion a s t u d y of the processes of c rea t ion and re laxa t ion of these s tates for o the r m u l t i p u l s e sequences shou ld be of in teres t . In this p a p e r we d iscuss the inf luence of the W A H U H A - 4 pu lse sequence on a q u a d r u p o l a r sp in-1 sys tem us ing a two- sp in model . T h i s s imple m o d e l will be shown to conta in the necessa ry phys i - cal e l emen t s to qua l i t a t ive ly r e p r o d u c e our e x p e r i m e n t a l obse rva t ions . I t is the p u r p o s e of this p a p e r (i) to p rov ide a conven ien t theore t ica l de sc r ip t i on of m u l t i - pu lse e x p e r i m e n t s for the t w o - s p i n sys tem, (ii) to ascer ta in to wha t ex ten t this s imple m o d e l accounts for the e x p e r i m e n t a l b e h a v i o u r of real sol ids and (iii) to give a p r e l i m i n a r y r epo r t of some typ ica l e x p e r i m e n t a l examples .

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1336 V . L . Ermakov and D. Ya. Osokin

2. THE MODEL

The system to be studied is a single crystal of NaNO2, containing the ~4N spin-1 nuclei, in zero external magnetic field. Since we are discussing a system of many spins coupled by dipolar interactions, we cannot hope to follow the evolu- tion of the system exactly. To simplify the problem some characteristic features of nitrogen-14 N.Q.R. may be used.

The nitrogen nuclei occupy equivalent positions in the crystal. The linewidth in the absence of dynamic effects comes from two major sources : (i) a distribution in the quadrupolar interaction constants-- inhomogeneous broadening, and (ii) broadening due to homonuclear and heteronuclear dipolar interactions (DI). The DI between nitrogen nuclei yields linewidths up to a few hundred hertz, while inhomogeneous broadening results in a linewidth of a few kilohertz. Therefore, for a given nitrogen nucleus only part of the other nitrogen nuclei are ' like' spins, which yield the homonuclear dipolar broadening. The others have other quadru- polar transition frequencies, i.e. there is no overlap between the quadrupole res- onances. Taking into account these facts we may consider, as a first approximation, an ensemble of isolated groups of completely equivalent coupled spins, each group having a given deviation of electric field gradients (EFG) from that of the ideal crystal.

Then it is necessary to take into account the homonuclear DI within a group. The main purpose of the paper is to provide a qualitative understanding of the behaviour of the signal (spin echo envelope) in multipulse N.Q.R. experiments for times longer than T 2 . Assuming the validity of the general concepts of spin- temperature theory in the rf-interaction frame (in this section all the terms such as ' s ignal ' , ' equa t ion ' etc. refer to this frame) the task may be broken into two steps. In the first one needs a steady state solution of the equation of motion of the many spin system in the absence of spin-lattice relaxation. This state of the spin system is attained when the transients have decayed by modified (averaged) spin spin interactions in a time of the order of T z . The next step, taking any model for the T~ process, is to obtain the rate equation describing the evolution of the spin system to equilibrium with the lattice. The separation into two steps is justified by the two different time scales for these processes in the multipulse N.Q.R. experiments [1-3, 5]. A quasi steady-state is rapidly established within a t i~e of the order of the spin spin relaxation time T2; the envelope of the echo signal, in this case, takes the form of damped oscillations. This is followed by relaxation of this state to a state of equilibrium with the lattice in a time of order T 1. Since T2 ~ T1 the spin system is effectively disconnected from the lattice on the time scale of the modified spin-spin interactions and the approach outlined above is valid.

As pointed out above, the inhomogeneous broadening is dominant in the experiments considered and, for a given group of equivalent spins, it is equivalent to the presence of a common resonance offset. A resemblance, therefore, might be expected between N.Q.R. experiments and off-resonance multipulse N.M.R. experiments. However, in N.Q.R. one has a significantly different kind of spin system and it is important, first of all, to develop a microscopic model to provide the basis for a physical understanding of the more precise and formal further developments of nitrogen-14 multipulse N.Q.R.

The approach adopted in the first step is to solve the two spin problem and then to average the solution over the distribution of the interaction constants in

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W A H U H A multipulse sequence in N . Q . R . 1337

order to in t roduce damping. Since d ipole-d ipole coupl ings are interactions of the t w o - b o d y type it is natural to expect that the steady state solution obta ined for the equat ion of mot ion of the two-spin system provides an adequate not ion con- cerning the main propert ies of the steady state solution in the case of a m a n y spin system. Of course, some of the characterist ics thus obta ined may be specific to the two-sp in system and not a general feature of an arbi t rary N- sp in system. In order to ascertain to what extent the propert ies of real solids are de te rmined by the model system the experimental data mus t be used.

T h u s the model , which has already been used for a similar purpose by Cantor and W a u g h [6], is a two, equivalent, spin-1 system. T h e cont r ibut ions to the total hamil tonian are: (i) the quadrupola r hamil tonians for each spin Hol , (ii) the resonance offset hamil tonians Hag , produced by the E F G deviations f rom the average values defined by Hog, (iii) the D I between these nuclei Ha, and (iv) the interact ions with the rf field Hrf i . I t is assumed that the fol lowing condi t ions are satisfied :

IHQI >> [Hrfl >)" tc 1 >> IHal ~ [Hal.

Here tr is the cycle t ime of the mul t ipulse sequence. T h r o u g h all the paper the hamil tonians are expressed in f requency units, that is we set h = 1. T h e condi t ion I Hrfl >> I Hal ensures that, a l though the spins possess a dis t r ibut ion in resonance frequencies, they behave identically dur ing the rf pulses, that is, the rotat ion angle is the same for all the spins in the sample.

3. OPERATOR FORMALISM

We consider the two-spin sys tem in the representat ion where the quadrupole hamil tonian for each sp in- l , HQ = K [ 3 I 2 -- 12 + q(I 2 - - /2 )] , has diagonal form. T h e eigenstates and eigenvalues of HQ are

I + ) = ([ + 1 ) + I - -1 ) ) /x /2 ; E+ = U(1 + q);

10) = 10); E o = - - 2 K ;

I - - ) = (] + 1 ) - - [ - -1 ) ) /x /2 ; E_ = K(1 -- q).

Here Ix , Iy and I z are nuclear spin componen t s in the principal axis sys tem of the E F G tensor ; 1 + 1 ) , 10) and [ - 1 ) are the eigenstates of the opera tor Iz with eigenvalues + 1, 0 and - 1 respectively, K = e2qQ/4 is the nuclear quadrupole coupl ing constant and t/ is the a symmet ry parameter of the E F G tensor. In accordance with the formal ism developed by Vega and Pines [7] for the single spin-1 sys tem we define for every pair of levels + 0 , 0 - - and -- + three fictitious sp in - l /2 operators. We presume familiari ty with [7] and do not discuss all the propert ies of these operators. For the convenience of the reader the definitions and commuta t ion relations of the operators needed in the fol lowing calculations are collected in tables 1 and 2. T o facilitate the calculations in addi t ion to the nine operators in t roduced in [-7] their three linear combina t ions are used in the present paper, which have the special notat ions and are s imply the uni t operators for every pair of levels. In the notat ion adopted the lower index of S-opera to rs cor responds to that of Pauli matr ices and the upper one denotes the pairs of energy levels. For notat ional convenience let us write p, q and r instead of + 0 ,

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1 3 3 8 V . L . E r m a k o v a n d D . Y a . O s o k i n

O9

O9 r O

~ ~

~ O9

~

O9 ~

2

II II II

v ~ v . v~ ~ ~ . - o ~ - ~ o ~ o .~

i i i . ~

r

II II II x

I

~leq ~ltq ~lt~

II II II

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Table 2.

W A H U H A mul t ipulse sequence in N . Q . R . 1339

The commutation, anticommutation and trace relations among the fictitious spin- l /2 operators used in the paper.

[s~, s1]

[s~, s~] +

[s~, s1]

[s~, s~] +

[s~, s1]

[ s f , s1] +

[s f , s~]

[s~, N] +

[s~, s~]*

= - [ s ~ , s ~ ] = 1. , -~zSy ,

= - [ s , ~ , s ; ] + 1 , ~ + = : s , , [ & , , s , ] = [ s , ~ , ~ ] + = �89

= [SPx, S r'] = I S p , Sqe] = - - [ S P x , Sre -] = -~lSyl" p ,

= - [ s ~ , s ~ ] + = [ s ~ , s g ] + = [ s ~ , x l ] + -- ~&~ ~,

= [ s ~ , s'~] = [ s ; , s g ] = - [ s ; , s ; ] = - v " & , ~

= - [spy, s ~ ] + = [spy, s g ] + = [ s~ , , S'e] + = ~S,1,,

= [aPe, N ] = E a r , N ] = [ S ~ , S~] = [ S ~ , &~] = [ S , ' , S ~ ] + = 0,

= [ s ~ , s ~ ] + = - [ s f , s ~ ] + = - [&,, sg] + = � 8 9 - 2 s ; ) ,

= [S~, S~] + = S~, [SPa, S~] + = SP., [SP., S~] = i S f ,

[ • , S ] = 0 , [Se, S] + = 2 S , T r ( S e S ) = T r ( S ) ;

the linear relations:

s ~ = �89 - s ~ + s ~ ) = �89 - s r - s ~ ) = s E - s ~ - s ~ = � 8 9 + s ~ - s ; ) ,

s f i = ~ ( S e 3 S ~ e - - S ~ ) = I , _ _ ~ _ ~, - ~ ( - - S E + 3 S ~ - - S ~ ) = - - S q e + S ~ - - S~ S~

x f + s g + s i = 0 , x ~ + s g + x ; = & ;

the trace relations:

T r ( S E ) = 3 , T r ( S ~ ) = 1, T r ( S a p ) = 0 , Tr [(SE) 2 ] = 3 , Tr [(S.P) 2 ] = y r [(S~) 2 ] = � 8 9

Tr (S. p S ~ ) = T r (S. p S ~ ) = T r (S. pS q ) = T r ( S ~ S q ) = T r (S~Sy q ) = 0 ,

P q P q P q Tr (S~ S,) = Tr (S~Sq~) = - T r (S~ S=) = - T r (S~ S~) = �88

here a, b, c = x, y, z and S denotes any operators of the set; these relations hold also after cyclic permutations of p, q, r and/or a, b, c.

0 - - and -- + respec t ive ly . T h e uni t 3 x 3 m a t r i x is d e n o t e d b y SE, whi le the uni t 9 x 9 one b y E .

E x p r e s s e d in t e rms of S - o p e r a t o r s the s ing le - sp in q u a d r u p o l e hami t t on i an H O takes the fo rm

HQ = copXf + 1 ( % _ co,)(S~ - S~) = % S f + ( % - co,)(S~ - ~SE), (1)

where

cop = E+ -- E0 , coq = E 0 -- E _ , co, = E_ -- E + ,

are the t r ans i t ion f requenc ies be tween q u a d r u p o l a r ene rgy levels. T h e o the r two equ iva len t fo rms of HQ are o b t a i n e d b y cycl ic p e r m u t a t i o n of p, q, r. F o r a s ingle spin-1 sys t em the set of the n ine f ict i t ious s p i n - l / 2 ope ra to r s is comple t e and is ve ry he lp fu l for the de sc r ip t i on of pu re nuc lea r q u a d r u p o l e resonance in solids. T h e dens i ty ma t r i x of two coup led spin-1 nuc le i is r e p r e s e n t e d by a 9 x 9 ma t r i x and thus is, in general , a l inear c o m b i n a t i o n of 81 i n d e p e n d e n t opera to r s . Since the choice of the basis set is largely a m a t t e r of conven ience it is des i rab le to take m a x i m u m advan tage of s y m m e t r y and m a g n e t i c equiva lence . T h e use of a set c o m p r i s i n g K r o n e c k e r p r o d u c t s of s ing le - sp in ope ra to r s [8] seems to be the mos t adap t ab l e a p p r o a c h for this pu rpose . I t was shown by Packer and W r i g h t [8] tha t the use of a s ing le - sp in o p e r a t o r basis set is bo th s imple to v isual ize in phys ica l

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1340 V . L . E r m a k o v and D. Ya. Osok in

Table 3. The twin operators comprising the linear combinations of the products of the two single-particle S-operators. They are introduced in the paper as a basis for the expansion of the density matrix of the two equivalent spin-1 system.

p p

K~ = S~ Sr +

g x =

L y =

L z =

Le

Mx

M,.

M~

M e p p N~ = S~ S~ ,

s~ s~ , p p Sy S~ ,

p p Sy Sy +

�89 -

� 8 9 -

P ~ - ~ P K ~ = ~ P ~ ~, K p = S p SP~ -[- S p S p , gPz = S z S z S y S y , S x S x -]- S e S e

P P P P q P P P P P g q = S z S x + S x S z , K z = S P x S P x - S z S P z , K q e = S y S y -t- S e S e ,

= p p p p p __ S P S P x I ~ e = S z S z - { - S e S e , ICy S ~ S y + S y S ~ , IC~= SPySy , P p P P

SP~ S~ + 3 S~ S~;

2S~e)S~ + S~(S~ -- 2S~) + S~ S~ + S~ S~ + S~, S~, + S; S~], r �9 q r - - �9 q

q q r r r r �89 -- 2S~)S~ + S f ( S e - 2S~) - Sq~ S q - S r S r + S x S x + Sy Sy],

1_ q q q q �9 r p p 1 p = 2 ( S x S x -t- S y S y + SrxSr x + SySry) -- 2 S e S e + g(SESTe + SPeSe);

= �89 2S~)S~ + S~(SE 2S~) ,, . . . . . ,~

q �9 r q q �9 r q

q q q q r r r r = 1[(s~ - 2s~)s f + s~(sE - 2Se ~) + Sx S~ + S, S , Sx S~ - S , S,],

1 q q q �9 r r r p p 1 p = 2 ( S x S x + S y S q y + S ~ S x + S y S y ) - 2 S e S e + g ( S E S e + S P e S E ) ;

N : = Sq~ Sq~ + S~ S~ + S~ S~ + S~y S~ = L e ~ M e ,

N3 1 q = = = ~ [ S E ( s z s'~) + (s~ - s~)s~] - ~ E + (sE s~ + S~e X~) --~E + s + Me + 4N1 ;

E = S e S e is the unit 9 x 9 matrix.

t e rms and easi ly used to expla in and p r ed i c t the o u t c o m e of m a n y pu lse sequences . He re we p re sen t the set c o m p r i s i n g p r o d u c t s of the two s ing le -pa r t i c l e f ic t i t ious s p i n - l / 2 ope ra to r s wh ich are a subse t of the total set of 81 ope ra to r s , bu t which gives a comple t e and conven i en t de sc r ip t i on of the state of the two equ iva- lent spin-1 sys tem d u r i n g s i ng l e - f r equency pu lse e xpe r ime n t s . T h e set used in the p a p e r is g iven in table 3. T h e ca r r i e r f r equency of the pu lses is a s sume d to be the t r ans i t ion f r equency cop. A s imi la r a p p r o a c h has been used also by Can to r and W a u g h [6] . H o w e v e r , it was a d e q u a t e for the i r pu rposes , b u t is less genera l than the one p r e s e n t e d here. Since the ope ra to r s used in this p a p e r c o r r e s p o n d to the sys tem of two equ iva len t sp ins they are s y m m e t r i c wi th respec t to par t i c le in te r - change. T o emphas i ze this fact we will use the t e r m ' twin ope ra to r s '.

T h e s ing le -pa r t i c l e ope ra to r s of d i f fe ren t sp ins c o m m u t e , so tha t the o r d e r of occu r rence of the d i f ferent ope ra to r s in a p r o d u c t A 1 B 2 is in p r inc ip le a rb i t r a ry . But in o r d e r not to o v e r b u r d e n the no ta t ions we omi t the labels of the i nd iv idua l sp ins and o r d e r the p r o d u c t so as to avoid los ing t rack of the i r ident i t ies . W i t h

P q P q this no ta t ion , for example , S x l S y 2 = S x S y . In def in ing a s ing le - sp in o p e r a t o r in a t w o - s p i n sys t em it is u n d e r s t o o d that , for ins tance , the s y m b o l SxPl impl ies the K r o n e c k e r p r o d u c t wi th the un i t o p e r a t o r SE2 for o the r spin. In the no ta t ion a d o p t e d one ob ta ins tha t S~I = S ~ I S E 2 = S ~ S E. T h e set of twin ope ra to r s con- sists of th ree m u t u a l l y c o m m u t a t i v e and a n t i c o m m u t a t i v e subse ts K , L and M . T h e mos t i m p o r t a n t p r o p e r t y of the twin ope ra to r s is tha t in the s i ng l e - f r equency N . Q . R . e x p e r i m e n t s the m o d e l sys t em will evolve wi th no t r ans i t ions be tween these subse t s and will consis t of i n d e p e n d e n t ro ta t ions in each space. T h i s fact is a c o n s e q u e n c e of the s y m m e t r y of the cons ide r e d sys t em and may, in p r inc ip le ,

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W A H U H A multipulse sequence in N . Q . R . 1341

be proved with group theory. It is, certainly, a general feature of any system consisting of two coupled equivalent three-level subsystems. It is not the object of the paper to present a rigorous mathematical treatment of the problem. Since all the relations among the twin operators are calculated and obey simple rules, the use of the operators does not present any difficulties.

Now we wish to discuss some characteristic features of each set K , L and M. The K-operators are completely equivalent to fictitious spin-l /2 operators (S- operators) in the sense that all relations among S-operators hold also for K- operators with the same indices. For example, from [S~, S~] = 1. r -- ~zSy and [S t , S~] + = S~e- �89 it follows that [K~, Ky q] = - � 89 and [K~, K~] + =/C~ - � 8 9 and so on. That is why we do not present the relations among K-

operators ; they are obtained by replacing symbol S by K in table 2. The physical reason for this analogy between S and K-operators can be

visualized as follows. The K-operators consist of the S-operators which corre- spond to p-transitions only. It means that the evolution of the K-component of the density matrix is governed by the single quantum transitions between levels p = +0. Therefore the K-operators correspond to two coupled fictitious spins- 1/2. But two coupled equivalent spins-I/2 give a three level system in which the fictitious spin-l /2 operators, that is K-operators, again may be introduced. The L and M-operators correspond to the part of density matrix the evolution of which is governed by the double quantum transition between the same levels, passing through an intermediate third level. The operators of both sets obey the usual spin-l /2 commutation and anticommutation relations, L e and M e being the unit operators for each set. Since the transformation properties of the L and M- operators are identical and differ from those of the K-operators, let us use the terms LM-space and LM-reservoir to stress the physical equivalence of the L and M sets.

The three N-operators, which were not mentioned above, commute mutually and also with all K, L and M operators, so the terms in the initial density matrix corresponding to these operators are time independent and can be neglected in

P P the hamiltonian. Of them N 1 = S e S e is the only independent operator, while N z and N 3 are linear combinations of others. The special notations for them are used only for brevity.

Taking into account the linear relations among the K-operators one obtains 9 linear independent operators of the K set (including the unit operator KE) ; then 4 of L, 4 of M, N 1 and the unit 9 x 9 matrix E completes the set of the twin operators. It should be noted that K~ but not E, serves as the unit operator/for ' /

the K set. To calculate the bilinear commutators and anticommutators the following

identities have been used:

[ A , B z , CID2] = �89 C,][B2, D2]+ + [A,, C,]+[B2, DE]),'{ (2) [A,B~, C , D d + �89 C~]+[B~, D~] + + [A,, C,][B=, D~]),J

t "

so one needs only the single spin commutation and anticommutation relations of table 2. The relations of this table and expressions (2) are quite suitable for computer programming. By taking advantage of this fact a special program for analytic computations was prepared and the tedious calculations of all bilinear commutators and anticommutators of table 4 and of table 2 (for K-operators) were carried out on a EMG-666 type computer.

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Table 4. The relations among the twin operators. These ones among the K-operators are not presented because they are equivalent to the relations among S-operators and are obtainable from table 2 by replacing symbol S by K. The relations among M-operators are identical to the ones for L-operators and are not presented also.

[ g , L ] = [ g , L ] * = [ g , M ] = [ K , M ] + = [L , M ] = [L , M ] * = 0,

[N, K] = IN, L] = [N, M-] = IN1, N21 = IN1, N3] = [N2, N3] = 0,

T r (KL) = Tr (KM) = Tr (LM) = 0,

[L., Lb] = iL c , [L., Lb] + = 0, [L a, L.] + = [L e, Lr + = L., [La, L~]* = L. ,

T r ( L a ) = 0 , T r ( L ~ ) = I , Tr (L. Lb)=O, Tr [(La) 2] =�89

here a, b, c = x, y, z or cyclic permutations; symbols K, L, M and N denote any operator of the K, L, M or N sets respectively.

4. H AMILTONIANS

T h e total h a m i h o n i a n of the two-sp in system inc lud ing all the in teract ions

m e n t i o n e d above is

H = HQ1 + HQZ + Hrf 1 + Hrf 2 + Hox + H~2 + Hd.

In the in terac t ion representa t ion defined by the total quad rupo le h a m i h o n i a n

HQ1 ~- HQ2 one obtains , taking into account tha t /~a i = Hal,

/~ = /~rfl "[- /~rf2 ~1- Hal -1- Ha 2 -k- H i .

Here H~ denotes the secular par t of H d. Accord ing to the average hami l ton ian theory [9], the evolut ion of the spin system du r ing a cyclic sequence of s t rong rf pulses can be character ized by r emov ing the rf t e rm f rom the hami l ton ian and

replacing the other te rms by average express ions:

H = / ~ a l +/Qa2 + Hd-

Now we need to calculate all te rms in this expression.

4.1. The rf hamiltonian

T h e hami l ton i an of the in terac t ion of a spin wi th a rf field may be expressed

in te rms of S-opera tors as follows :

Hrf = - - pB 1 = - y B l ( s i n 0 cos tpI x + sin 0 s in ~0Iy + cos 0I=) cos (cot + 4>)

-~ - 2 7 B 1 ( s i n 0 cos q0S p + sin 0 sin (pS q + cos OS~) cos (cot + 49)

-- - 2 cos (cot + 49) ~ a m Sm. r e = p , q, r

Here 7 is the gyromagnet ic ratio, B t is the ampl i tude of the rf field, co and 49 are its f r equency and init ial phase, 0 and q0 are the polar and az imutha l angles of the field vector in the E F G pr incipal axes system. Us ing for each S~ the proper form of HQ defined by (1), one obta ins in the in terac t ion represen ta t ion :

/Qrf = exp (iHQ t)Hrf exp ( - iHQ t)

= -- ~ am{cos E(com - co)t - 49]Sx m -t- cos E(co,n -t- co)t -t- 4913 7 r e = p , q, �9

- - sin [(corn -- co)t -- 49]S 7 - sin [(com + co)t + 49]S~"}.

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W h e n the f requency of rf field coincides with the t ransi t ion f r equency ~op one has, omi t t ing rap id ly oscil lat ing t e rms :

/~rf ---- --apSPx, if qb = 0; / - I r f = apSPx, if ~) = u;

7~ / t r f = - - a p S e , if q ~ = ~ ; / t r f = a p S ( , if q b = - 2 ;

As it is easy to show by analogy, that in -phase (q~ = 0) and quadra tu re (qb = 7c/2) reference vol tages for the phase-sens i t ive detector , yield ou tpu t signals p ropor t iona l to the t e rms with S~ and S~, respect ively, in the express ion for the densi ty mat r ix of the single spin sys tem.

T o express the opera tors of rf pulses act ing on the two-sp in sys tem in t e rms of the twin opera tors we use the fol lowing identi t ies :

SES~ + S~SE = 2K~ + L x + Mx,] SE S~ + S~ SE 2K~ + Ly + My. ; (3)

Neglec t ing the dipolar and offset t e rms dur ing the pulses one can easily integrate the equa t ion of mo t ion of the densi ty ma t r ix in the ro ta t ing f r ame to obtain jus t after the pulse of dura t ion t w

p+ = P p - P + ,

where P = exp [-- i ( /~rr 1 + /JrfZ)tw] and plus denotes ' a f t e r pulse ', m inus denotes ' b e f o r e pu l s e ' . T h e n the opera tors co r r e spond ing to the pulses in the x(qb = 0) and y(qb = 7:/2) direct ions of the ro ta t ing f rame are defined by the express ions :

/ ~ = exp [i~9(2K~ + L x + Mx)] ,~

/~y exp [i~9(2Kq~ + Ly + My)]. J (4)

He re ~ = apt w is the rotat ion angle. T o calculate the average hami l ton ians and the initial dens i ty ma t r ix we use the

fol lowing p r o p e r t y of exponent ia l opera tors . I f [A, B] = zkC and [A, C] = --z~B then

exp ( - i $ A ) B exp ( + i~9A)= B cos (k$) + C sin (k~h),

exp ( - i O A ) C exp (+itpA) = C cos (kO) - B sin (kS).

4.2. The Dipolar hamiltonian

T h e hami l ton ian of the h o m o n u c l e a r D I of two like spins can be wri t ten using S -ope ra to r s as fol lows:

H d = 72r13[I l i2 - 3rl-22(Itr12)(I2 r12)]

= 4[S~S~npp + Sq~S~nqq + S~S~flrr + (S~S q + sqs~)npq

r r r p p r + (S~x sx + Sx s~)nqr + (Sx Sx + S~ s~)n,~], (5) where fl=p is the tensor of D I in the E F G pr incipal axes sys tem with diagonal e lements , which are actual ly necessary in the subsequen t calculat ions, def ined by

~"~pp = (1 - - 3 sin 2 0 COS 2 ~O)O)d; ~')qq = (1 -- 3 sin 2 0 sin 2 tp)coa;

~ ' ) r r = (1 - - 3 COS 2 0 ) ( / ) d ; (/)d = 7 2 r 1 2 3 .

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1344 V . L . Ermakov and D. Ya. Osokin

T h e other symbols in (5) have their usual meanings . As the details of ob ta in ing the secular part of H d are no t i mpor t a n t to the

p resen t discussion and can be found, for example, in [~6], only the result is given

here

' = P P Sy Sy) + Sy Sy) + 2 ~ r r ( S x S x + S y Sty). (6) H d 2~-,2pp(Sx S x + p p 2~-~qq(Sqx Sqx q_ q q r r r

In what follows we omit the second lower index in the diagonal e lements f ~ . Us ing the fact that T r (f)~a) = 0 one obta ins the two equiva len t forms of H~ in

te rms of the twin operators :

He, = ~ v ( K l ~ + K ~ - K ~ - - 4 N ~ - N2) + (f~q - ~ , . ) ( M z - L z ) (7 a)

= ~ p ( K E + Kqe + K q - - 4 N t - N2) + (['~q - - ~']r)(Mz - L~) . (7 b)

T h e advantage of each form will become clear later on. Cons ider a sl ightly modif ied W A H U H A - 4 pulse sequence, that is one with an

arb i t ra ry rota t ion angle of the pulses in the cycle :

/:~ (~ F~_x-z - /~y-2Z- /* , z - P ~ - z ) , .

T h e condi t ion of cyclicity P~xP~y/~_yI~ = 1 is satisfied, so one can use average hami l ton i an theory for a rb i t ra ry 0. In the present d iscuss ion we need the zero- order hami l ton ians only. Detai ls of the exact steps involved in the calculat ion of a zero-order average hami l ton i an are no t given, since this has been presented

several t imes. T h e result is wr i t ten down direct ly:

/_~(0) = �89 +/~H/~_x +/~P~ HP~xP~ j.

Here H denotes any hami l ton ian of in te rna l interact ions. It now remains for us, us ing this formula and the t r ans fo rmat ion proper t ies of the twin operators, to calculate the zero-order average hamiltonian/-Tga ~ R e m e m b e r i n g that the sets K, L and M mutua l l y c o m m u t e and taking H~ in the form (7 a), one obta ins :

P~H~/xo_:, = npEKE + K [ -- exp ( i 2 0 K ~ ) K P = exp ( - i 2 0 K ~ ) - 4 N t - N2]

+ (flq -- flr)[exp ( i O M x ) M = exp ( - i O M J

- exp ( i O L : , ) L ~ exp ( - i 0 L x ) ]

= f ~ p [ K E + KPe - - cos 2 0 K [ -- sin 2 0 K x P -- 4 N 1 -- N2]

+ (~o -- ~ , ) [cos 0(M= -- L=) + sin O ( M y - - Ly)].

T h e n , taking H~ in the form (7 b), one obta ins :

PO_xH'dt~y : n v ( K l { + Kqe + cos 20K~ -- sin 20K~ -- 4N1 -- N2)

+ (f~q -- f~,)[cos 0(M= -- Lz) + sin 0 (Mx -- Lx)]

: �89 - cos 2 0 ) K e + (3 cos 20 - 1)K[ - (1 + cos 2 0 ) K [

- 2 sin 2 0 K ~ - 8N~ - 2N2] + (~q -- ~ )

x [cos 0(M~ -- Lz) + sin @(M~ -- L~)].

I n the last equal i ty the l inear relat ions among K-ope ra to r s have been used to t r ans fo rm the expression into the form which is the most conven ien t one for the next step

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W A H U H A multipulse sequence in N . Q . R . 1345

/:~P*_y Ha/~yP~ :, = �89 - cos 2~p)K E + (3 cos 2~p - 1)K~

- cos 2~p(1 + cos 2~)K~ - sin 2~p(1 + cos 21p)K~

- - 2 sin 2~ cos qJK~ - 2 sin 2~ sin ~/,/Cy -- 8N1 - 2N2]

+ (~q -- f2r)[cos: th(M= - L~) + cos ~ sin g,(M,, - Ly)

+ sin ~(M~ - L~)].

F ina l ly one obta ins

Hd wn" = -~np[(7 -- cos 2~9)K~ + 3(1 + cos 2O)K~

- - (2 + 3 cos 2(p + cos 2 2O)K~ - sin 2~(3 + cos 2~)K~

- - 2 sin 2~ cos ~Ky q - 2 sin 2~ sin ~h/~y -- 24N1 - 6N2]

+ ~ (~q . - ~ ) [ ( 1 + cos tp + cos 2 (p)(M~ -- L=) + sin

x (1 + cos ~J)(My - Ly) + sin ~ ( M x - Lx)]. (8)

Here, and later on, we omit the p r ime and the superscr ip t (0) in /Q'm)for brevi ty. In the fol lowing discussion we need also the zero-order average hami l ton ian

for the P A M S , which is readily ob ta ined in an analogous way

/~PAMS = ~ ( H d l ' + /~HaP*_~) = f2p[IC~ + KPe -- �89 + cos 2O)K= p -- �89 sin 2 0 K f

-- 4 N 1 -- N2] + �89 -- ~1~)[(1 + cos ~J)(M= -- L~)

+ sin t~(My -- Ly)].

(9)

4.3. The resonance offset hamiltonian

T h e s ingle-sp in hami l ton ian of i nhomogeneous b roaden ing , or offset, is

defined by the fol lowing expression :

1 q __ r . Hg, i = ASPzi + ~B(Szi S=i), i = 1 2. (10)

In this expression A = Amp, B = (Ao.)q - A(Or), where Aa) is the f requency devi- ation f rom the average value. T h e resonance offset hami l ton i an of the two equiva- lent spin-1 system in te rms of the twin operators is as follows:

Ha = Hal + Ha2 = A(S~ SE + SE S~) + B[ (Sq - S~)SE + S~(Sq= - S~)]/3

= A(2/C~ + L= + Mz) + B N 3 . (11)

Now it is easy to obta in the zero-order average hami l ton ians

I~l wuu = �89 + cos ~ + cos 2 O)2K'~ + sin ~(1 + cos O)2Kq~ + sin O2K~

+ (1 + cos ~ + cos 2 O)(M= + L=) + sin ~(1 + cos O)(My + Ly)

+ sin ~b(M~ + L~,)] + B N 3 ,

/4a eAMs = �89 + cos ~9)(2/~ + L= + 21//=) + sin ~h(2Kq~ + L , + My)] + B N 3.

(12)

(13)

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4.4. Densi ty ma t r i x

In the h igh t e m p e r a t u r e a p p r o x i m a t i o n the e q u i l i b r i u m dens i ty ma t r i x of the t w o - s p i n sys t em is g iven by the express ion

Peq = ~[E - e(HQ1 + HQ2)], (14)

where HQ/ are the q u a d r u p o l e h a m i l t o n i a n s of the fo rm (1) and ~ = 1 / k T is the inverse t e m p e r a t u r e . T h e total q u a d r u p o l e h a m i l t o n i a n can be expres sed in t e rms of the twin ope ra to r s as fol lows :

HQ1 q- HQ2 = O ) p ( S E S P z q.- S P z S E ) -]- ((Oq - - o , ) r ) [ S E ( S q - Srz) -~- ( S q - S~z)SE]/3

= 09p(2/C~ + L~ + Mz ) + (09q -- ~or)N 3 .

T h e t e rms wi th E and N 3 are unaffec ted by any evo lu t ion of the sys tem, so we need on ly fol low the evo lu t ion of the r e d u c e d dens i ty ma t r i x a(t) , wh ich is def ined b y the fo l lowing equa l i ty :

p(t) = ~{E - a[ogva(t) + (e)q - e),)N3] }. (15)

Af te r the p r e p a r a t o r y pulse P~ the r e d u c e d dens i ty m a t r i x in the in te rac t ion r ep re sen t a t i on takes the fo l lowing f o r m :

(~0 = 0.0 = p~xO.eq(/~x ) +

= cos q)(2/~ x + L z + Mz ) + sin (p(2Kq~ + Ly + My). (16)

5. THEORETICAL RESULTS

N o w we p re sen t some exact so lu t ions of the L iouv i l l e equa t ion in the in te r - ac t ion represer~tat ion

= iEa, H] (17)

wi th the h a m i l t o n i a n H = /~WHn + /~wnn def ined b y (8) and (12) and wi th the ini t ia l dens i ty ma t r i x (16). S ince the sets K , L and M m u t u a l l y c o m m u t e , then one ob ta ins th ree i n d e p e n d e n t equa t ions of mot ion . T h e c o m p o n e n t s of h a m i l t o - n ians and dens i ty ma t r i x in each set (or. space) we will deno te b y lower indices K , L o r M .

5.1. Double quantum spaces

F o r the L space one has the fo l lowing h a m i l t o n i a n and in i t ia l dens i ty ma t r i x :

x [s in S L x + sin ~k(1 + cos $ ) L r + (1 + cos ~ + cos z ~p)Lz],

rr ~ = sin q~L r + cos rpL z .

F r o m the fo rm of H L we see tha t it genera tes ro ta t ion abou t the d i r ec t ion in L - s p a c e def ined b y the second factor in the above express ion . U s i n g the t race re la t ions a m o n g L - o p e r a t o r s one read i ly ob ta ins the so lu t ion in the fo l lowing fo rm :

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aL(t ) = a[ + (a ~ -- crq) cos COLt + C L sin ~OL t ;

col = �89 - ( n 0 - f ~ r ) l x / [ 2 ( 1 + c o s qj)2 + 1];

~r~ = T r (~o HL)HLFCr ( H 2) = flq nL ; (18)

flq = [(1 + cos ~ + cos 2 ~) cos q) + sin ~(1 + cos ~) sin (p]/

[2(1 + cos ~9) 2 + 1];

nL = sin OL~ + sin ~(1 + cos O)Ly + (1 + cos ~ + cos 2 O)L=.

He re co L is the f r equency of p recess ion and ~q is the s t eady - s t a t e so lu t ion , which is, in this case, s imply the p ro j ec t ion of the ini t ia l dens i ty ma t r i x on the axes def ined by the hami l ton ian . In the p a p e r we need not spec i fy the cons tan t C L. Rep l ac ing A -- (f2q -- ~r) by A + (flq - ~ ) in these express ions one ob ta ins the ana logous resul ts for the M - s p a c e .

I t is necessa ry to note that even in the absence of the offset t e rm (A = 0) the p resence of D I leads to a n o n - z e r o s t eady- s t a t e so lu t ion for the L M - s p a c e . I t is a consequence of the fact tha t HaL and HaL (and, consequen t ly , HaM and HaM ) have the same t r a n s f o r m a t i o n p rope r t i e s u n d e r pu lse ro ta t ions . N a m e l y , s ince the L and M sets obey s p i n - l / 2 c o m m u t a t i o n re la t ions and these par t s of the hami l to - n ians are l inear in the opera to r s , t hey t r a n s f o r m like f i r s t - r ank tensors . T h i s fact is the charac te r i s t i c fea ture of the sys t em cons ide red and leads to the poss ib i l i t y of e x p e r i m e n t a l ver i f ica t ion of the a p p r o x i m a t i o n s used. T h i s is d i scussed in detai l in w

5.2. Single quantum space

I t is r a the r d i f f icul t to ob ta in the genera l so lu t ion of the equa t ion of m o t i o n for the K - s p a c e wi th the h a m i l t o n i a n and ini t ia l dens i ty ma t r i x def ined by

HK = /~WKn. + /~wnn = ~f~v[( 7 _ cos 2~9)K~ + 3(1 + cos 20 )Ke p

-- (2 + 3 cos 2~ + cos 2 2~k)Kf -- sin 2~(3 + cos 2O)K~

-- 2 sin 20 cos 0 K y q -- 2 sin 20 sin 0/~y] (19)

+ �89 + cos ~9 + cos 2 ~ ) 2 / ~ x

+ sin 0(1 + cos 0 )2K~ + sin O2K~],

~r ~ = cos (p2/~x + sin g02K q .

W e p r e sen t the total so lu t ions for the fo l lowing two pa r t i cu l a r cases: (i) when one of the in te rac t ions is absen t and (ii) when the s t eady- s t a t e so lu t ion for the genera l case (~p r 0 and A 7 ~ 0) appl ies .

(a) f ~ p = 0 , A r

Since the th ree ope ra to r s ( 2 K v, 2K~, 2 / ~ ) have the same c o m m u t a t i o n rela- t ions as the L - o p e r a t o r s , tha t is, as the th ree c o m p o n e n t s of angu la r m o m e n t u m , and bo th the h a m i l t o n i a n and ini t ia l dens i ty m a t r i x cons is t of these ope ra to r s only , in th is case the so lu t ion has the same fo rm as for the L- space . Rep l ac ing A - (f~q - ~r) and (Lx, L r , L=) b y A and ( 2 K v, 2K~, 2/~x) respec t ive ly in (18), one ob ta ins the so lu t ion for th is case. I t is seen tha t the t r a n s f o r m a t i o n p r o p e r t i e s

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of H a are the same in all th ree spaces K , L and M : its c o m p o n e n t s t r a n s f o r m like a f i r s t - rank i r r educ ib l e t ensor u n d e r pu lse ro ta t ions .

(b) f 2 v r A = 0

T h i s is the more c o m p l i c a t e d case. T h e m e t h o d of solving such a p r o b l e m was d e v e l o p e d by us in [4] and, s ince the deta i l s of the ca lcu la t ions are no t the ob jec t of the paper , on ly the resu l t is g iven here :

3

aK( t ) = ~ , ( A i c o s co i t + B i sin co it) , i 1

eo I = 6*-f~px/[(cos 2~ + 17)(cos 2~ + 1)],

o)2, 3 = �89 + lf~p( cos 2~ + 1).

(20)

T h e so lu t ion has osc i l la t ing t e rms only , that is in the absence of the offset t e r m the D I t e r m gives no c o n t r i b u t i o n to the s t eady- s t a t e so lu t ion in the K - s p a c e . In this p a p e r we do not need the expl ic i t express ions for the cons tan t s A i and B i ,

which are ob ta ined by s t r a i gh t fo rwa rd ca lcu la t ions knowing co i .

(c) ~ p ~ 0 , A ~ 0

T h e fo rmal i sm, deve loped in [4] , also p e r m i t s one to ob ta in the s t eady- s t a t e

so lu t ion of the equa t ion (17) in the genera l case

where

Az[2 (cos @ + 1) 2 + 1]

flq = fl~ AZ[Z(cos ~ + 1) 2 + 1] + f~2 cos 2 @(cos 2 ~9 + 2 ) '

n r = (1 + cos ~ + cos 2 @)2/C" x + sin @(1 + cos ~9)2K~ + sin ~2K~ (21)

+ ( f ~ p / A ) [ l ( 3 + 4 cos 2~ + cos 2 2~)(K= q - K= p) - �88 sin 2 2 ~ / ~

- - �89 sin 2~(3 + cos 2~)K~ - sin 2~9 cos ~K~ -- sin 2~ sin ~/~y].

T h e so lu t ion m a y be easi ly checked by s t r a i gh t fo rwa rd ca lcula t ion of the c o m - m u t a t o r [ H K , aq]. T h e resul t is z e r o - - t h i s pa r t o f the ini t ia l dens i t y m a t r i x is not

affected by the hami l ton ian . I t has thus been d e m o n s t r a t e d tha t the so lu t ions o f the equa t ions of mo t ion

give eve ry dens i ty ma t r i x c o m p o n e n t 6K, a L and a m as a supe rpos i t i on of a s t eady-s t a t e so lu t ion and a s u m of osc i l la t ing te rms . A c c o r d i n g to the m o d e l a d o p t e d the m a n y - p a r t i c l e so lu t ion is now o b t a i n e d b y averag ing over the dis- t r i bu t i ons of in te rac t ion cons tan t A and ~ . Af t e r tha t the osc i l la t ing t e rms vanish (on the t ime of o r d e r of Tz) and the o b s e r v e d signal (spin echo envelope) will be whol ly def ined by a s t eady- s t a t e so lu t ion . W e shall hence fo r th d i s r ega rd the t r ans i en t t e r m s and focus our a t t en t ion on the solu t ion , which we call quasi s t eady- s t a t e (QSS) of magne t i z a t i on ( ' q u a s i ' is a d d e d to s t ress the t r ans i en t b e h a v i o u r of the t r ansve r se magne t i z a t i on d u r i n g a cycle of sequence) .

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W A H U H A mult ipulse sequence in N . Q . R . 1349

T o re la te the theore t ica l p r e d i c t i o n s to e x p e r i m e n t a l obse rva t ions we ob ta in the dens i ty ma t r i x of a single sp in b y t rac ing over the second sp in

o.qWHH T r 2 (0 "q + 0 "q + 6~) s

( ) = fl l 1 + A2[2(cos ~k + 1) 2 + 1] + n 2 cos 2 ~k(cos 2 O + 2)

x [ s i n ~ S ~ + s i n ~ h ( l + c o s ~ ) S ~ + ( l + c o s ~ + c o s 2~)S=p]. (22)

T h e o b s e r v e d signal is p r o p o r t i o n a l , as shown in w to the t e rms wi th S~ and S~. T h e r e f o r e the m o d e l p red ic t s tha t af ter a t ime of the o rde r of T 2 the signal , in genera l , is not equal to zero in the W A H U H A e x p e r i m e n t . T h e phys ica l sense of the resul t o b t a i n e d is man i f e s t ed in the fo l lowing ma nne r .

S ince in each space K, L and M

[H/( , a q] = [HL, cr q] = [HM, a q ] = 0, (23)

we have all the p re requ i s i t e s for spin locking of these c o m p o n e n t s of the ini t ial dens i ty m a t r i x (16). As was po in t ed out b y Pines and W a u g h [10], the cond i t ions (23) a r e the correc t c r i te r ia to e m p l o y for sp in lock ing ; the p resence of an average or mean field is ne i the r a suff ic ient nor a necessa ry cond i t i on as is some t imes e r roneous ly as sumed . As the c o m p o n e n t a q of the s ingle sp in dens i ty ma t r i x is spin locked, it will change on ly by sp in - la t t i ce re laxa t ion in the ro ta t ing f rame, which has been neg lec ted up to now. In such a way the inves t iga t ion of the l o n g - t i m e d e p e n d e n c e of the Q S S p rov ides a p r o m i s i n g new source of re laxa t ion i n fo rma t ion in N . Q . R . expe r imen t s .

I t is necessa ry to po in t out tha t the obse rva t ion of the Q S S is not a fea ture

specific to N . Q . R . T h e r e is a close ana logy be tw e e n the Q S S and the so-ca l led pedes ta l in the N . M . R . of f - resonance W A H U H A e x p e r i m e n t s [10, 11]. In bo th cases the pedes ta l , or Q S S , is caused by the t e rms of the to ta l h a m i l t o n i a n which t r a n s f o r m like a f i r s t - rank i r r educ ib l e tensor . In the N . M . R . case these are the offset and chemica l shif t t e rms only . In our case, in add i t ion , the secu la r pa r t of the h a m i l t o n i a n of h o m o n u c l e a r D I also has par t s which t r a n s f o r m like f i r s t - rank t ensors ; these are, as it can be seen f rom w prec i se ly H~L and H~l M. T h e e x p e r i m e n t a l ver i f ica t ion of this fact will be p r e s e n t e d in the next sect ion.

N o w we wish to d iscuss the t r a n s f o r m a t i o n p rope r t i e s of H~l K . I t can be seen d i rec t ly by subs t i t u t ion of tp = 90 ~ i n /~wun , def ined by (8), tha t

[~WKHH(I / / ---- 9 0 ~ = ~ n p K E .

T h e o p e r a t o r K E c o m m u t e s wi th all the o the r s and the re fo re has no inf luence on the ini t ia l dens i ty mat r ix . As a consequence in ~q, express ion (22), the t e r m d e p e n d i n g on ~p is we igh ted b y cos 2 ~ and van ishes at ~ = 90 ~ T h a t is at this angle the c o n t r i b u t i o n of the K - c o m p o n e n t of the dens i ty m a t r i x to the obse rved Q S S takes the m a x i m u m value i n d e p e n d e n t of f~p and A, as in the absence of D I , which is the case tha t has been d i scussed in w 5.2 (a) (tip = 0, A # 0). T h e resul t becomes c learer if we wr i te down the t rue h a m i l t o n i a n H'dK, def ined b y (7 a), in t e rms of the i r r educ ib l e tensors f o r m e d f rom the SP-ope ra to r s :

t 2 q HdK = ~"~p(K E + K~ -- K~) = P~p['~Kl~ + "~(Kz - K~P)]

4"l ~ p ~ p -~ P P P P = k")p[4N1 + 3k~'~ *"x S y S y + S z S z )

2 p p p 3SPS=p)]

= f2p[4N~ - (4/x/3) Too -- (2x/6/3)T2o ]. (24)

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1350 V . L . E r m a k o v and D. Ya. Osok in

H e r e we use the no ta t ion of M e h r i n g [12]. T h e last t e r m in (24), wh ich is a s e c o n d - r a n k tensor , vanishes in the W A H U H A e x p e r i m e n t at ~ = 90 ~ as is no t u n e x p e c t e d on the basis of the average h a m i l t o n i a n theory . T h e first two t e rms are invar ian t u n d e r all pulse ro ta t ions and c o m m u t e wi th all o the r twin opera to r s . Neg l ec t i ng these t e rms we m a y say tha t the K - c o m p o n e n t of H~ t r a ns fo rms like a secular pa r t of h o m o n u c l e a r D I in the N . M . R . case.

6. EXPERIMENTAL RESULTS

T h e e x p e r i m e n t s were ca r r i ed out wi th a cohe ren t pu l s ed N . Q . R . s p e c t r o m - eter on a N a N O 2 single c rys ta l at 77 K. A br i e f desc r ip t ion of the s p e c t r o m e t e r has been given e l sewhere I-1]. T h e s p i n - s p i n re laxa t ion t ime T 2 in the sample at the t r ans i t ion f r equency f+0 = 4 " 9 3 1 2 5 M H z is abou t 7 m s and T 1 is abou t l s . T h e 180 ~ pulse w i d t h was 0"05ms and the cycle t imes were 3 m s for the W A H U H A sequence and 2 ms for the P A M S .

F i r s t of all we check the ma in consequence of the t h e o r y - - t h e p resence of the two u n c o u p l e d reservoi rs in the m u l t i p u l s e N . Q . R . e xpe r ime n t s .

Cons ide r , f rom a genera l po in t of view, the P A M S e x p e r i m e n t wi th ~, = 180 ~ which is the case of the C a r r - P u r c e l l sequence wi th a l t e rna t ing phase . I t is well k n o w n tha t the sequence averages to zero all f i r s t - rank in te rac t ions . But the s e c o n d - r a n k s p i n - s p i n in te rac t ions take the i r o r ig ina l fo rms s ince they are invar i - ant u n d e r 180 ~ ro ta t ions . T h e m e a n rf-f ie ld of the app l i ed pu lse cycle is equal to zero in the P A M S expe r imen t s . The re fo re , in the ' r o t a t i n g ' f rame (us ing the t e r m i n o l o g y a d o p t e d in N . M . R . ) the re is no stat ic effective ' f i e l d ' wh ich may, u n d e r cer ta in cond i t ions p r o d u c e the so-ca l led second t r unc a t i on or averag ing of the r e m a i n i n g s e c o n d - r a n k in te rac t ions and, in such a way, give rise to a k ind of s p in - l ock ing effect [10]. One m u s t expec t tha t the ini t ia l dens i ty ma t r i x will be d e s t r o y e d by the nonze ro d ipo la r t e rm (or, to pu t it more exact ly , by the r e ma in - ing nonze ro pa r t of it) on the t ime of the o r d e r of T 2 . But the e x p e r i m e n t shows (figure 1) tha t the s ignal decays on a t ime of the o r d e r of T1, which indica tes tha t the ini t ia l dens i ty ma t r ix , or pa r t of it, is no t affected by this t e rm. I n t e r m s of the exis tence of two weakly coup led reservoi rs in m u l t i p u l s e e x p e r i m e n t s on real, m a n y - s p i n , sys tems this fact is r ead i ly exp l icab le as fol lows.

T h e genera l f o rmu la for the Q S S in the P A M S case is g iven in [5] so we need not to p re sen t it here and l imi t ourse lves to a shor t d i scuss ion in t e rms of the

f o r m a l i s m deve loped in the p re sen t paper . T h e t r a n s f o r m a t i o n p rope r t i e s of H~K and H a t are d i f ferent ; it has been

shown in the p rev ious sect ion tha t H~K t r ans fo rms like a s u m of the z e r o - r a n k and s e c o n d - r a n k tensors , whi le H a t t r ans fo rms , as is clear f rom w 5.2 (a), like a first- rank tensor . S u b s t i t u t i n g ~ = 180 ~ in (9) and (13) one ob ta ins the expec ted resul t

/_~PAMS , /~PAMS O" (@ = 180~ d K = U d K ; xXcSK = '

T h i s resul t means tha t the K - c o m p o n e n t of the ini t ia l dens i ty m a t r i x will be de s t royed by H~g in a t ime of the o r d e r of T 2 a s if the sequence were no t app l i ed at all. T h e r e f o r e the on ly c o n t r i b u t i o n to the obse rved n o n - z e r o Q S S af ter the t ime T 2 m a y be that of the L M - c o m p o n e n t of the dens i ty mat r ix . I nde e d , the

same subs t i t u t i on also gives tha t

/_~PAMS T'~PAMS = 0; (~ ---- 180~ d L , M ~" z~cSL, M

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W A H U H A multipulse sequence in N .Q .R . 1351

co

Clj

(2)

I i i

2 3 4

Figure 1. Nitrogen-14 N.Q.R. spin echo envelope in the phase-alternated multipulse sequence at the transition frequency f+0=4-93125MHz for the case q~=90 ~ (preparatory pulse) and ~ = 180 ~ (pulses of the sequence).

Thus the LM-component of the initial dens~y matrix is not affected by two major interactions and gives the observed QSS which will change by spin-lattice relaxation only.

No alternative physical description can be suggested at present which might account for the observation of the QSS in this PAMS experiment. Since the observation of two reservoirs or, in other words, quasi-constants of the motion, is not unexpected in the three-level system, the description offered appears to be essentially correct.

In order to examine the results for the W A H U H A sequence the time depen- dences of the spin echo intensity were measured for the two characteristic values of~.

It follows from (22) that at ~ = 90 ~ the dependence of the QSS on the pre- paratory pulse flip angle is defined by the following expression :

aqWnn(~0) = X/2 cos ((0 -- 45~ + S~ + S~); (~ = 90~ (25)

Up to now spin-lattice relaxation was neglected in the derivation of the QSS, while in general this effect is the main purpose of the experiments. The detailed discussion of the relaxation processes will be the subject of future publications, but the final state may be obtained in the framework of the present theory.

Under the influence of the T x process the initial QSS defined by (25) relaxes towards an equilibrium value a~ q. In the absence of a multipulse sequence it is known to be given by the Boltzmann distribution. When a sequence is applied the equilibrium value is assumed to be the Boltzmann density matrix transformed by the sequence and may be obtained from equation (25) by setting ~0 = 0, that is

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1352 V . L . E r m a k o v and D. Ya. Osok in

co

CD

C2~ -C2 Q) Clo

1 2 3

Figure 2. Spin echo envelopes in the W A H U H A sequence (flip angle of the pulses in the cycle ~ = 90 ~ at the same transition frequency for the different preparatory pulses: (0 = 45 ~ (1); in the absence of the pulse, q) = 0 ~ and at q~ = 90 ~ one obtains the same curve (2); (p = --45 ~ (3); q) = --90 ~ (4)and q0 = - 1 3 5 ~ (5).

a ~ q = asq(0). T h i s a s s u m p t i o n was checked e x p e r i m e n t a l l y in the p rev ious p a p e r [5].

U s i n g d i f ferent p r e p a r a t o r y pu lses we ob ta in the set of curves dep i c t ed in f igure 2. T h e ini t ia l and final values of the sp in echo enve lopes are desc r ibed by equa t ion (25) which conf i rm the theory .

By subs t i t u t ion of the second charac te r i s t i c value, ~ = 180 ~ in (8) and (12) one ob ta ins :

/_~KHH , ' . a W H H 1 , I~'WH H 2BN3, = 1 8 0 o ) . = H d K , "'dL, u = S H d L , U; = � 8 9 " (0

I t is seen tha t the c o m p o n e n t s of the to ta l average h a m i l t o n i a n have the same fo rm as in the absence of the sequence and some of t h e m are s imp ly scaled by the fac tor 1/3. T h e e x p e r i m e n t gives the fast decay of the o b s e r v e d signal on the t ime of the o r d e r of T 2 as if the sequence is not a p p l i e d at all, wh ich agrees wi th the theory .

T h u s we see tha t the e x p e r i m e n t s conf i rm the ma in p r ed i c t i ons of the devel - oped theory .

T o desc r ibe the evo lu t ion of the Q S S to the final s tate in the W A H U H A N . Q . R . e x p e r i m e n t s it is necessa ry to ob ta in , us ing the m e t h o d deve loped b y one of us in [5] , the rate equa t ion and solve it. T h i s will be done in fu tu re p u b - l icat ions.

7. CONCLUSIONS

T h e o p e r a t o r f o r m a l i s m d e v e l o p e d in the p a p e r offers a s imple , b u t r igorous , de sc r ip t i on of the b e h a v i o u r of the two equ iva len t spin-1 N . Q . R . sys tem d u r i n g

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W A H U H M multipulse sequence in N .Q .R . 1353

the mult ipulse experiments. It has been found that the set of the operators needed for the complete description of the state of the system is separated into three independent subsets K, L and M, two of them (L and M) being physically identical. The physical reality expressed by this mathematical s tatement is that one has to distinguish the two reservoirs K and L M , which have different physi- cal properties. Therefore the two-spin model predicts the observation of two physically distinguishable reservoirs in mult ipulse N.Q.R. experiments on systems comprising many coupled spins-1.

In order to examine this prediction the fact has been used that, in accordance with the formalism developed for the model system, the parts of the hamiltonian of the homonuclear dipolar interactions corresponding to these two reservoirs must have different t ransformation properties under pulse rotations. The P A M S exper iment has been per formed in which the proper ty of this sequence with 180 ~ pulses to reduce selectively all the first-rank interactions has been used. The result obtained suggests that the separation into two weakly coupled reservoirs is a general feature for many-spin systems subject to the influence of a mult ipulse sequence and is not specific to the model system only.

The results of the W A H U H A N.Q.R. experiments are in a good agreement with the predictions of the two-spin model and show the possibility of utilization of the W A H U H A sequence for relaxation measurements in N.Q.R.

We hope that the operator formalism developed for the two equivalent spin-1 model should serve as a guide to interpreting nitrogen-14 multipulse N.Q.R. experiments.

REFERENCES

[1] OSOKIN, D. YA., 1980, Phys. Stat. Sol. (b), 102, 681. [2] OSOK[N, D. YA., 1982, Phys. Star. Sol. (b), 109, K7. [3] OSOKrN, D. YA., 1982, J. molec. Struct., 83, 243. [4] ERMAKOV, V. L., and OSOKIN, D. YA., 1983, Phys. Stat. Sol. (b), 116, 239. [5] OSOKIN, D. YA., 1983, 2Vlolec. Phys., 48, 283. [6] CANTOR, R. S., and WAUGH, J. S., 1980,ff. chem. Phys., 73, 1054. [7] VECA, S., and PINES, A., 1977,ff. chem. Phys., 66, 5624. [8] PACKER, K. J., and WRICHT, K. M., 1983, Molec. Phys., 50, 797. [9] HAEBERLEN, U., 1976, High Resolution N.M.R. in Solids--Selective Averaging

(Academic Press). [10] PINES, A., and WAUGH, J. S., 1972,J. magn. Reson., 8, 354. [11] HAEBERLEN, U., ELLET, J. D., JR., and WAIJCH, J. S., 1971, J. chem. Phys., 55, 53. [12] MEHRING, M., 1976, High Resolution N.M.R. Spectroscopy in Solids, Appendix A

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