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Pulsed spin locking in pure nuclear quadrupole resonanceR. S. Cantor and J. S. Waugh Citation: J. Chem. Phys. 73, 1054 (1980); doi: 10.1063/1.440277 View online: http://dx.doi.org/10.1063/1.440277 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v73/i3 Published by the American Institute of Physics. Additional information on J. Chem. Phys.

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R. S. Cantor and J. S. Waugh: Pure nuclear quadrupole resonance 1055

that the quadrupole principal axis systems (QPAS) for

each of the two spins have the same orientation with re

spect to the lab. (This is true of the t4N spins within a

crystallite- in the sample used by MK : NaN02 at 77 K).

III. GENERAL DESCRIPTION OF THE METHOD

The time development of the model system will be

described by the evolution of a reduced density matrix

p. Since the system contains two coupled spin-l nuclei,

this density operator will be represented by a 9 x 9

Hermitian matrix, and thus will be, in general, a time

dependent linear combination of 81 operators, an un

pleasant prospect.

The method to be used relies on a major simplifica

tion resulting from the nature of this particular experi

ment. When the proper interaction representation is

chosen, and a special notation is used, then the evolu

tion of the system is describable by a (transformed) p

which is a linear combination of only si x operators, with

time-varying coefficients. The bulk of the calculation

will be concerned with this reduction. I f we represent

this transformed reduced density matrix p by a vectorin a six-dimensional space, the components of whichare the time dependent coef ficients of the six operators

comprising p, then it will be shown that both the effect

of a pulse and the effect of JeD (during the time the rf is

off) appear as rotations in this six-dimensional space.

In fact, i t can be simplified further, for i f we break up

this space into a direct product of two spaces, each of

three dimensions, then each of the two types of evolu

tions, pulse and JeD' is represented by independent

analogous rotations, one in each of the two spaces.

Having demonstrated this, the (T-p-T) evolution will be

evaluated in this vector representation of p, as a reduction from three successive rotations to one rotation

about some axis, of some precession angle, in each of

the two spaces. Then it will be Simple to determine the

effect of N such (T-P-T) sequences; the two axes remain

the same, and the two preceSSion angles ar e each multi

plied by N. Now, of these six operators, only two of. them, one from each se t of three, can generate any

m ~ n e t i z a t i o n . Thus the time dependence of the appro

priate linear combination of these two operators will be

proportional to the magnetization resulting from this

two-spin system. Remembering that the sample con

sists of many such systems of random angles of orienta

tion, appropriate integrals can then be performed to ob

tain the time dependence of the magnetization of the entire polycrystalline sample.

The evolution of the model system will be followed in

the interaction representation defined by the unitary

operator R =exp(i Jeot) in the following way. For any

operator A, we define an operator AU) such that A(t)

=RAR'. We have, fo r the equation of motion of the re

duced density matrix p (to be defined more precisely

later)

p(t) == i[p(t), Je(t») , (1)

with Je(t) =:JCQ+Jer f +Je D • Then it is easily shown that

~ ( t ) == i[p(t), feU) + iRtR]==

i[p(t), fert(t) +ieD(t)] . (2)

To obtain :lCr f and :feD' we need to express R == exp(iJeQt)

explicitly, which we choose to do in the quadrupolar

principal axis system (QPAS). We will then obtain JeD

and Jer f in that frame, and will calculate fer f and :lCD • We

then calculate p(O) as well. In so doing, we will adopt

the fictitious spin notation used by Vega and Pines6 and

Shattuck7 discussed in Appendix A.

IV. HAMILTONIAN OF THE SYSTEM

The quadrupole Hamiltonian for each of the two spins

(call them a and i3) expressed in the ~ o o r d i n a t e system

defined by the QP AS of each spin, is 8

Jeio= t w ~ [ 3 I ~ _II (Ii + 1) + 1]1 ( I ~ i + I ~ ~ ) l , i === a or {3 ,

(3)

Now, if we assume that the QPAS's of the two spins co

incide, then the coordinate systems will coincide. Also

== w ~ ' " wo; 1]'" == 1J1l= 1]. In fictitious spin notation6•

1

(see Appendix A), we obtain

Using the relation 1:••a+ I y • a+ I ...a=0, we can express

J e ~ in two other ways:

J C ~ = - iw Q{a(3 - 1 J » ) I ~ . 3 + WI + 1 ] ) J I ~ . 4 } ,J e ~ == ~ w o { ( i ( 3 + 1]) ] I ~ . 3 - WI - 1 ] » ) ~ . 4 } .

In general, then

J e ~ = = W : I ; . 3 + w ! ~ . 4 ' P==x,Y,z,

where

w ~ = [ « 3 + 1 J ) ] w Q ' w ~ = - ( < < l - 1 J ) ] w Q ' w ~ = =

- (t(3 -1J)JwQ,w ~ =

- [t(1+1J)]w

o ,a (2 ) b 2W. = - 31) WQ ' W = 3Wo .

(4)

(5)

(6)

(7)

Note that w;, w ~ , are the three pure quadrupole reso

nance frequencies. Now in each of the three ways of ex

pressing J e ~ , the two terms commute; [ I ! . 3 , I ~ . 4 ) =O.

Thus, we can factor R as a product of four terms:

R =exp(iJeQt) = exp[i(w:I:.3 + W!I:.4 + w;I:.3 + W;I!.4)t]

=exp(iw;tl:. a) exp(iw!tl:,4) exp(iw;tJ:. 3) exp(iw!tl:. 4). (8)

(This will be useful in calculating JC...f and JeD')

We need to express the angular momentum operators

in three different coordinate systems, corresponding to

the quadrupole, dipole, and lab (rf) principal axis systems. To avoid confusion we adopt the following nota

tion:

coordinate system notation

QPAS

DPAS

rfPAS x" , y", z";IxII) 1 I' " I ~ , To calculate both XD(t) and fc..t(t), we will need to know

how I!(p == x, y, or z) transforms going into the interac

tion representation. That is , we must evaluate expres

sions of the form

(9)

J. Chem. Phys • Vol. 73. No.3, 1 August 1980

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1056 R. S. Cantor and J. S. Waugh: Pure nuclear quadrupole resonance

Now it was shown earlier that JeQ can be written in three

ways having the form J C ~ =w; I!.3 + w!.c;" 4 ' Note that I! wt(t)=OM2:0[t-T(2k+1)] (all other pulses),k=O

== 2I!.1 (going into fictitious spin notation). Remember

ing that [ I ! , 3 , I ! ' 4 J = [ I ! . t , I ! , 4 J = [ I : ' i , I ~ , j J = 0 (see AppendixA) and that the operators {I!.t,I!,2,I!,S} have the same

commutation relations as the three components of angu

la r momentum V., J " J z}, then we obtain

1!,t(t)=RIp,tRt=I!,ICOS(W;t) -I!,zsin(w;t) . (10)

Now, the relative orientation of I , . and the QPAS de

pends on the crystalline orientation, whiCh is random.

In general, we can represent this relation by two angles

(e L' (h ) in the usual way (with y as the e=:o axis):I! . =: sine L sin¢LI! + cose L I ~ + sine L c o s ¢ L I ~ . (12)

A. rf Hamiltonian

We irradiate on one of the three quadrupole reso

nances; for example, w ~ , as was done in the MK ex

periment. Then

(lla)

(lIb)

Thus, in the QPAS frame, we get

J e ~ t =wt(t) c o s ( w ~ t + ¢)(sineL sin¢LI! +cose L I ~ (13)

where

¢ =: 0 (first pulse) ,

¢ = h (all other pulses) .

1 C r f = W l ( t ) C O S ( w ~ t ) I , .. (first pulse, t=O),

Jerf = - w1(t) s i n ( w ~ t ) l y " (all other pulses) ,

whereConverting to fictitious spin notation and USing Eq. (10),

we c an now obtain :iCr f :

: i C ~ f ( t ) = 2w t (t) c o s ( w ~ t + ¢)[sineL sin¢LU!, t c o s w ~ t -1;,z s i n w ~ t ) + c o s e L ( I ! , t C O s w ~ t - 1 ~ , 2 s i n w ~ t ) + s i n e L c o s ¢ L ( I ~ . t C o s w ~ t - 1 ~ , z s i n w ~ t ) ] , i= a or f3; (14)

- - a -:!C,.f = er f +Jert .

If we truncate in the usual way, i . e., ignoring terms which oscillate rapidly (or order wQ) , we obtain

: i C ~ f = : e M o c o S e L o ( t ) I ~ ' 1 (first pulse) (15a)

= eM cose Lei;; o( t - T(2k + 1 » ) I ~ ' 2 (all other pulses) . (15b)

B. Dipole Hamiltonian

Expressed in its own principal axis system, we have

J e D = w D ( 1 / v ' 6 ) ( 2 I ; I ~ - I ; . I ~ . - I : . I ~ . ) = w D T f . o , (16)

where Tr.o is an irreducible tensor operator; we use the convention of Haeberlen. 8 Here, wD=: - 2 yZn , l f r - ; ' ~ . As

suming, for generality, that the DPAS and QPAS are randomly oriented with respect to each other, then we can ex

press JeD in the QPAS by using the appropriate Wigner matrix for that transformation. Again adopting the notation

and conventions of Haeberlen, 8 we have .

Z~ J e D = 2: :nt_m T z.m(-I)mwD m=-Z

(17)

=,p;sinZe (T D e-2i I/! D+ TD e+2i I/! D_ v'Tsin20 (T D

e-/I/! D_ TD e iI/! D) + 1.(3 cos28 _1)TD

a D Z,Z Z,-2 8 D Z,1 2,-1 2 D Z,O

.ff . 2 2 (' " 8 .. 8) (3 coszeD -1) (2 a 8 a 8 .. 8)= llsmeDCOs ¢DI"Ix-I ,I , + 2N IzIz-I"I,,-IyI y

+ ,If [+ si n2e D s i n 2 ¢ D ( I : I ~ +I;m + sin2e Dcos¢D(I:I!+ I : I ~ ) + sin2e Ds i n ¢ D ( I : l ~ + I ; I ~ ) J . (18)

Changing notation, as before, we obtain

JeD =2w D[a" I: , 11:,1 + ayI;.l Ie.1 + a.I:, l I ~ , l + bU:, l I ~ . l + I; . 1I!. 1) + cU:.1 I ~ , 1 + I : , 1 I ~ , I ) + d ( I : . 1 I ~ . 1 + I : . 1 I ~ . t ) ] , (19)

where

a,,=v'f sin28Dcos2¢D -(1/,16)(3 COs 2eD -1 ) , ay = - v'fsin28Dcos2¢D - (1/,I6)(3cos

2e D -1 ) ,

az=(2/v'6)(3cos2eD-1) •

To calculate JCD(t), consider a bilinear operator of the general form 1:.11:.1; JeD is composed of a sum of nine such

terms. Now

J:.ti!.t(t)=exp{iJeot)I:. tI:.t exp( - iJeot) . (20)

Using Eqs. (8) and (10), and remembering that [ I ~ . h I ~ . 4 J = 0 , we obtain

i:. t i:, t(t) =exp(iw;tI:. 3)1;, t exp( - iw;tI:. 3) exp(iw:tI:, 3)1:,1exp( - w:tI:. 3)

= (1;,1 cos(w;t) -1;.2 sin(w:t) ][1:,1 cos(w:t) -1:,2 sin(w:t)1

J. Chern. Phys., Vol. 73, No.3, 1 August 1980

(21)





Now, unless p == q, this is purely oscillatory. If p == q,

then there i$ a nonzero static part, so we use the fol

lowing truncation as an approximation:

(22)

Thus, in this approximation, the only terms in :ito tosurvive ar e the first three in Eq. (19):

:itD==W [ax(I:.l 1 ~ . 1 + I : . 2 I ~ . 2 ) + a/I;. I I ~ . 1+1;.2 I ~ . 2 ) + a . ( I ~ . I I ~ , 1 + I : . 2 1 ~ . 2 » ) •

We choose to regroup terms for reasons that will be

come clear later:

where

L '" ( I : . l 1 ~ . 1 + I : . 2 1 ~ . 2 + I : . l 1 Z . 1 + 1 : . 2 1 ~ , 2 ) ' 1:==woTWa.+a x) ] ,

(-1:,11=.1 - 1 : , 2 I ~ . 2

+ I : . l 1 ~ . 1 + 1 : , 2 1 ~ ' 2 ) ' ~ = = W o T W a . - a x ) ] , N ~ 2 ( I Y . l I Y . l +IY.2IY,2) ' l 1 = = ~ W v T a y .

V. TIME DEVELOPMENT OF THE SYSTEM

(23)

I t has been shown that to good approximation, :iCv is

independent of time, and 3Crf depends on time only as

does the pulse envelope. Now when the rf is turned on,:!CD +:fc", ",3Crt • When it is off, :iCo +3Crf ==1eD • Thus, dur

ing the preparatory pulse we can easily integrate Eq.(2), the equation of motion of p, to obtain

p(O) == exp( - i8 01hl)P(t<0) exp(+ i8 0Iy.l) ,

where

(JO===(JMOCOSOL, ly.l==I:.1 + 1 ~ . 1 .

(24)

We have se t t == 0 as the time of the preparatory pulse.

In the same way, for the other pulses

(25)

where ,A denotes after pulse, <B denotes before pulse,and where

(J == 0MCOSOL, I y • 2 = I ~ . 2 + I ~ . 2 .

During the time (c..t) when the rf is off, we can integrate

the equation of motion just as easily:

iXto + c..t) == exp( -3CDc..t)p(to) exp(+ ilcDc..t) •

Setting c..t==T, i f we le t D==exp(-i3CDT), Po

=exp(-i8 01y • 1), and P = = e x p ( - i 8 I ~ . 2 ) ' then we can solvefor p(2kT) by calculating

p(2k7-)= ( D P D } ~ P o p ( t < O)P1{DPD)U

= DPD)"p(O)(DPD}tk (26)

We wrote :reD as a sum of three terms in Eq. (23).But [L,M]== [L,N]== [M,N]=O (see Appendix B). Thuswe can write D=DLDMDN , or any permutation, where

DL=exp(-if:L),

(27)

This will be useful, because i t will be seen that of these

three propagators, at least two will have no effect onany given term in p(t)-at any time.

A. Effect of preparatory pulse

Before the preparatory pulse, the system is assumed

be at equilibrium under the influence of the Hamilton!an JC =JC

Q+JC

D;:::JC Q• Using the high temperature ap

proximation, 9 we get

o( t < 0) "" {Tr[ exp( - JCQ/kT»}-l exp( - JCQ/kT)

"" [Tr(I»)-I(I-JCQ/kT) . (28)

The first term is unaffected by any evolution of thesystem, so we need only follow the evolution of the sec

ond term, proportional to the reduced density matrix p,

which we define by setting a=[Tr(1»)-1[1_(1/kT)p).

Thus, p(t< 0) ==JCQ • )CQ is invariant going into the inter

action representation. Thus,

p(t<O)=3CQ =JCQ = L ( W : I ! . 3 + w ~ I ! . 4 ) ' (29)i =0:,13

[From this point on, unless specified otherwise, al l cal

culations will be done in the interaction representation,

so the tilde (-) notation will be dropped. )

To calculate p(O), we use the following properties ofexponential operators. For three operators A, B, C;

scalar k, i f [A, B]=iC and [A, C]= - iB, then

e-ikAB e+

ikA:= B cosk + C sink,

e-ikAC e+ i kA := C cosk - B sink .

Also, of course, i f [A, B]== 0, then e- ikAB e+ikA =B.

Thus, we calculate

p(O) == Pop(t< 0)P1

(30)

(31)

where we have chosen the form of p(t< 0) with p == y,

[Eq. (29») and where Iy,i =1;,1 + I ~ , i ==/:. 116+ 1"'Ie. I (1'" is

the single-spin identity operator for spin <1; likewisefor (3).

We now define a few new operators:

Y i = = I ) 1 . i = I ; . i 1 a + 1 " ' I ~ . j , i=1,2 ,3 ,

U1 = = I : . i I ~ . 4 + I ~ . 4 I ~ . i ' i= 1,2,3 ,

WI == i( Yj + U i ), i == 1, 2, 3 ,

Vj==}(Y i -2U j ) , i==I,2,3

(so Y == Wi + Vj)' Thus, in this new notation,

p(O) = cosO o(Wa+ Va)

(32)

(33)

We choose these operators because they have the above

mentioned properties of operators with respect to theexponential operators D and P [see Eq. (30)], as weshall see, so their time development can easily be ex

pressed.

J. Chern. Phys • Vol. 73, No.3, 1 August 1980





B. Evolution after preparatory pulse

To begin, we calculate Dp(O)Dt. We will need to know

certain commutation relations, the method of calculationof which is outlined in Appendix B (as will be al l the

commutator calculations henceforth):

[lCD•Wa]= [:TeD' Va]:::: [:TeD, Iy,4]=0 ,

[L , W2]= [L , V2]= [M, W2]=[N, V2] =O.

Thus,

per) == Dp(O)Dt== cosBo(Wa+ V3) + W ~ l y , 4

- sinBo(DN W 2 D ~ + D M V 2 D ~ ) •

Consider the following commutation relations:

[N, W2J==iS 2 , 5 2 = 2 ( I : , I 1 ~ , a + 1 : , a 1 ~ , j ) ,

(34)

[M , V2]=::iR2 , R 2 = 1 : , I [ ~ , 2 + I : , 2 I ~ , 1 +1:,1[:,2 + [ : , 2 [ ~ , 1 ;also

[N, 52] == - iW2 , [M, R 2] =:: - iV 2 .

Thus, by Eq. (30),

DW2Dt:::: DN W 2 D ~ =W2cos1J +52 sin?] ,

DV2Dt=DM V 2 D 1 = v z c o s ~ + R 2 s i n ~ .

(35)

Now we know the effect of D on two other operators as

well:

DS2Dt=DNS2D1 =S2 COS?] - Wzsin?] ,

D R 2 D t = D M R 2 D 1 = R 2 C O S ~ - V 2 s i n ~ .

Thus,

p( r) == cosBo(Wa+ Va) + W ~ [ Y , 4 - sinBo(WzCOS?]

(36)

+ S2 sin1J + Vz c o s ~ +R2 s i n ~ ) . (37)

We started with a density matrix at t= 0 which was a

linear combination of five operators: {Wa• Va, 1Y,4 ' Wz,Vz}. After following it s evolution under the influence of

:TeD for a time r, we find the first three unaffected, thecoefficients of the last two changed, and the addit ion of

two more operators, {S2' R 2}, bringing the total numberof operators comprising p(r) to seven. Now we must

examine how each of these seven operators is affectedby a pulse, and how many new operators ar e generated.

Remembering that P:::: exp( - iB I , 2) :::: exp{ - iBY2), we needto know the commutators of Y2 with each of the seven

operators comprising per). Calculations reveal

[Y2, Wz]= [Y2• V2]= [Y 2,1y ,4]= 0 ,

[Y2,R 2]==iM,

[Y2,S2J=::i2Q2, Q Z = 2 ( I : , l l ~ , l - 1 : , 3 1 ~ , s ) , [Y 2, Wa] == iW I ,

[Y z, Va)==iV j •

In addition, we calculate

[Y2,M ]= - iR 2 , [Y2,Q2]==-i2S2 •

[Y 2• Wd==-iW a , [Y 2• Vd= - iV 3 ·

ThUS, by Eq. (30) we obtain the effect of a pulse on al l

seven operators:

PWzpt =W2 ,

PV2pt=V2 ,

P1 Y ,4Pt =ly,4 ,

PR 2Pt =R2 cosB +M sinB ,

P5 2Pt=52 cos2B + Q2 sin2B ,

PWapt:::: Wa cosB + Wt sinB ,

PVapt =Va cosB + Vj sinB .

(38)

We also know now how P affects four other operators:

PM pt =M cosB - R2 sinB ,

PQ2p t = Q2 cos2B - 52 sin2B ,

pwjpt == Wj cosB - Wa sine,

PVjpt= Vj cosB - Va sinB .

(39)

We have added four more operators {Wj. Vj , Q2' M} tobring the number of operators in our set to 11, a linear

combination of whic h constitu tes p( r - P) . We have also

determined how these four ar e affected by pulses. However, we now need to know how these additional four op-

erators ar e affected by the propagator D in order to cal-

culate p(r-P-r) ; we already know the effect of D on the

other seven.

We obtain the following commutation relations of thefour operators with the three (commuting) parts of :TeD'

{L,M,N}:

[L , wd=[M, Wj]=O,

[N, wd = - i5 j , 5 j =2 ( J : , 2 [ ~ , 3 + [ : , 3 1 ~ , 2 ) ,

[L , Vj)=[N, Vj)=O,

[M, Vt1=- iR j , R t = I : , j l ~ , j + I : . j l : . j

[L , Q2] = M, Q2)= [N, Q2)= 0 ,

[L ,M]=[M,M]==[N,M]=O.

Thus,

DW1Dt=D N W 1 D ~ = WI cos1J - SI sin7) ,

D V 1 D t = = D M V I D 1 = V j c o s ~ - R t s i n ~ , DQ2Dt ==Q2,

DMDt=M .

(40)

Also, we obtain the effect of D on two new operators:

DSjD1=DNS jD1=Sj cos7)+ Wjsin?] ,

DRjDt =D.vRjD1=R l c o s ~ + VI s i n ~ .(41)

We have added two new operators {Rt> SI} so the se t nowcontains 13. We know how all 13 ar e affected by the di -

pole evolution operator, and how al l but these two ar e

affected by a pulse . To obtain these last two bits of in -

formation, we calculate

[Y2, st1 == iSs , S 3 = 2 ( I : , I I ~ , 2 +1:,2 / :,1) ,

[Y 2,Rtl== 0 ,

[Y2,Ss):::: - iS t •

J. Chem. Phys., Vol. 73, No.3, 1 August 1980




R. S. Cantor and J. S. Waugh: Pure nuclear quadrupole resonance l059

Thus

PStpt=StcoSB +S3 sine ,

PRtPt=R t •(42)

Also, we have the effect of P on the new operator:

PSsPt=S3COse -Sjsine . (43)

Having generated one more operator {S3} we need to

know how it is affected by the three evolution operators

comprising D. Calculations give

Thus,

DS 3Dt=S3 (44)

and no new operators are generated.

The set of 14 operators is now complete in the follow

ing sense. I f we start with a reduced density matrix

which is any linear combination of these 14 operators,

and allow i t to evolve under an arbitrary sequence of the

propagators P and D, the result will always be some lin

ear combination of these 14.

We can now group these operators into five sets as

follows:

{Iy,4}; {V3, V1, R 1}; {W3, W1, S3, S1}; {W2, S2' Q2}; {V2' R 2, M} .

A glance at the results of the preceding calculations re

veals that with respect to the propagators P and D (con

sider them as operations) acting on the 14 operators

(consider them as elements), these subsets are indepen

dent. Thus, although p{O) is actually a linear combina

tion of five of the 14 elements (each in a different sub

set), we can treat the evolution of each separately, in

that an element of one of these five sets can never gen

erate any amount of an element in any of the other four

sets.

In addition, since we can only measure magnetization

in an experiment, we would like to know only about those

terms in P which ca n result in magnetization. Calling

this subset of the 14 operators {m}, then if A is in {m},Tr{AM y) * . Now, the magnetization operator (at the

resonant frequency in phase with the pulses after the

first pulse) expressed in the (static) lab frame, is

My=Iy.. s i n w ~ t (in units of yn). (45)

In the interaction representation,

M, = (- s i n w ~ t ) elJCot(sineL sin<pL1x + coseLly + sinBL cos <PLI.) e-IJCo t

= - 2 s i n w ~ t { s i n e L s i n < p L [ c o s { w ~ t ) I x , 1 - s i n ( w ~ t ) I x , 2 ] + cosB L [ c o s ( w ~ t ) I y , 1 - s i n ( w ~ t ) I Y , 2 ] + sine L cos<p L [ C O S ( w ~ t ) I . , 1 - s i n ( w ~ t ) I . , 2 ] } ~ cose L1y, 2 (46)

(truncating the rapidly OSCillating terms). Thus, we

need to know which operators (A) of the 14 obey Tr{AIy ,2)

*0.

Of the 14, only two have this property: W2 and V2•

ThUS, we need follow the evolution only of that part of P

containing the two independent sets {W2, S2' Q2} and {V2,

R2, M}. Analyzing the previous calculations, we see that

these two sets of operators behave analogously with re

spect to each of the "operations" P and D. This is sum

Jllarized in Table I. If we replace W2 by V2, S2 by R 2,

Q2 by M, 1) b y ~ , and 2e bye, we see that the second se t

behaves in the same way as the first.

To understand the effect of the entire pulse sequence

on that independent part of P which results in magnetization, call i t Pm, which is composed of these six opera

tors, we develop a new representation for Pm' We rep

resent Pm as two three-component vectors, each in it s

own space. In one space the axes are represented by

the three operators {W2,S2,Q2}; in the other, by {V 2,

R 2, M}, in the following sense. If Pm=aW 2 + bS2 + cQ 2+ dV2+ eR2 + fM , where {a, • . . ,j } are (time-dependent)

scalar coeffiCients, then the vectors ar e defined by the

coefficients: (a, b, c) and (d, e,f) . Let

Pm! == aW 2 + bS 2 + CQ2' Pm2 '" dV 2 + eR 2 + M • (47)

We examine just the first vector Pm!> since all that fol

lows will be applicable to the second as well. Identify

W2 with y, S2 with X, and Q2 with z. Then the dipole

evolution (D) appears as a rotation of angle (-1)) around

the Q2=z axis. A pulse (P) appears as a rotation of

angle ( -2e) around the W2=y axis. Thus, the evolution

of Pmt corresponding to (T - P - T) can be viewed, in this

space, as a sequence of three rotations; first a rotation

of angle (-11) around Z, then (-29) about y, then (-1))

about z. But this is Simply the Euler-angle definition of

one rotation of coordinates about some axis in 3 -space

with a =1), (3=2e, 1'=1); (a , (3,y) are defined using the

Haeberlen8 conventions. This three step process can

therefore be represented by defining the axis of the one

overall rotation, the initial vector [the representation of

Pmt(t= O)}, and the precession angle. We accomplish

this in the follOwing way, using Pauli algebra notation. to

I f we le t r be the vector form of Pmt, then we repre

sent it by the operator R= r · O ' = r x a x + r y a , + r ~ a . , where

ax, ay, a. ar e the Pauli spin-i matrices. Now, we ca n

TABLE 1. Effect of propagators on Pm'

Component of P Effect of P Effect of D

{W 2W 2 W2cosT) + 52 sinT)

Pm! S2 52 cos20 + Q2 sin20 82 cosT) - W 2 sinT)

Q2 Q2cos20 - 8 2 sin20 Q2

r V 2 V2 c o s ~ + R2 s i n ~

P.,,2 R2 cosO +M sinO R2 c o s ~ - V 2 s i n ~ M cosO - R2 sinO M






represent a rotation of angle rJ> of r about some axis it

as R ' = URUt, where R ' is the Pauli algebra representa

tion of r ' (the result of the rotation), and

U=u ol+u ' a

== exp[(-1irJ»it· a] ,

where

u ~ = c o s i Q ,

(48)

(49)

(50)

UN = (UI)N "" exp[ - i ( ~ N r J » u 0 a] .

We would like to convert this to the form

UN=uN.ol +uN ' a .

But. using Eqs. (50) and (55).

un •o= cosONrJ» = cos[N cos-1(cosO cosn)]

= cos(N cos-1Ut,o) ,

(56)

(57)

(58)

u = - i sini¢ii = - i( t - u ~ ) j / 2 u .Fo r the dipolar evolution,

(51) where we define UI,O,,"UO(T -P -T). Now, by Eq. (51),

UN == - i s i n ( ~ N r J » u uT=z , rJ>=T/; UT=exp[ - i (n /2)u z]'

For the pulse,

uP=Y. rJ>=2e; Up=exp( - iBu) .

Thus,

= - i sin(N COS-IUI,olu

=sin(Ncos-1Ul,O)Ul(1-ui,ot1l2,

where UI =U(T -P - T).

We define

(59)

U( T - P - T) = UTUpUT = [cos(n/2)1 - i sin(T//2)u.)uN,y,,"sin(Ncos-1ul,O)(-isin8)(1-uLo)"I/Z, (60)

x [cosln - i sinOuy)[cos(n/2)1 - i sin(n/2)u.]

= cos8 cosT/I - i sin8uy - i cos8 sinnu • (52) Then

(62)here we have used the relation upuq " " iE NrU. and u; "" 1.

Thus we can represent the (T -P - T) sequence by

lto =cosB cosT/ , (53)

(54)

Now, Pmj(O)=aW2; a = - w ~ s i n 8 0 ' Thus, R(t=O)=acJ y •

and therefore,

U= - i sinBy - i cosO sin17z •

with u i u( l - u ~ r t l 2 . We obtain the precession angle of the (T - P - T) se

quence by examining uo. By Eqs. (50) and (53),

R(2NT) = UNR(O)U%

1rJ>=cos-1(cosBcosn). (55)

Thus, by Eq. (49),

U j =U( T - P - T) = exp{l- i ( ~ rJ»}u • u} ,

USing the relation ,0 - ,y - " = I, we obtain

R(2NT) =a[(1 + 2u1,.)uy - 2i uN,oUN,/ Jx - 2uN,yUN,.UZ ] ' (63)

Returning to operator notation:

where irJ> and uare given above. Now the result of N

such sequences will be to multiply the angle by N,

C. System magnetization

Pml(2NT) = a[( l + 2u1,,)w2

First we calculate (My,pml)' the magnetization due to Pml, using Eqs. (61), (64), (53), and (32):

(M y ,p,)2NT» =Tr[p",t(2NT)(cOS8 Lly,z)] = cosO L)a(l + 2 u ~ . z ) Tr(W21y ,Z)

2 2 2 2 • • 2 -1 ( co s20 sin

21/ ) ]

=acosBL(1+2uN.hTr ly2=-3wysmOOcosl iL 1-2sm[Ncos (coslicosn)] 1 2/1 2 •, , -cos cos 11

(64)

(65)

Substituting in the preceding calculation Pm2(O) =dV2 fo r Pml(O) =aW2, fo r n, and {VZ, R z, M} fo r {WZ, 52' Q} in Eq.

(64), we obtain the magnetization due to Pm2:

(M y ,.,)2NT» = Tr[PmZ(2NT)(cos8L1".Z)] = (cose L)d(1 + 2 u ~ ... ) Tr(V 2I y,z)

z 1 2 1 a . r,. { -I[ (1.) ]} ( c o s 2 ( t 8 ) s i n 2 ~ ),1=dcosBL(1 +2uN •)( 3 Tr1y.2) = - 3WysmeOcoSliL e 2 sm Ncos cos 2 0 c o s ~ 1 - c o S 2 ( t 8 ) c o s 2 ~ J '

since d =a = - sine. The total magnetization is

(M ) = MY'Pmt> + (M Y ,Pm2) •

We can easily calculate the initial magnetization

(My(O» = Tr[Pm(O) cosBLl,,2] = - sinO 0 cosO L Tr(W 2 + V 2)Iy•2 = - sin8 0 cos8L ,

J. Chem. Phys., Vol. 73, No.3, 1 August 1980

(66)

(67)




R. S. Cantor and J. S. Waugh: Pure nuclear quadrupole resonance

where eO=eMoCOse L. Thus, using Eqs. (65)-(67),

(My (2NT» == (My(0»(1 - A) ,

with

. A'" Hsin2[N cos-1(cose COS17)] cos2e sin217(1 - cos2e COS217rl

+ t sin2{N c o s - l [ c o s ( ~ e ) c o s ~ ] } c o s 2 ( ~ e ) s i n 2 ~ [1 _ c o s 2 ( ~ e ) c o s ~ ]-1} ,

where e == eM cose L; eM is the integrated rf pulse intensity. Now, using Eqs. (19) and (23), we obtain

== (w DT)(+ v'f)[(3 cos2eD- 1) - si n

2eDCOS2<PD], 17 == ~ ( w DT)( - 4)[M3 cos2eD - 1) + si n

2e Dcos2<pD] .

1061

(68)

(69)

(70)

The dependence of the magnetization on the parameters characterizing JeD' {WD' eD' <PD}' is contained completely in

and 17. I t is assumed that even for nearest neighbors (i . e., for largest WD) WDT« 1. Thus, 17« 1 and ~ « 1 for all

possible choices of pairs of spins within each crystallite, and we can replace 17 and by average values Tj and twhich are of order (w DT) . Also, e depends upon crystallite orientation; it relates the lab and quadrupole PAS's.

Thus, to obtain the total magnetization due to al l the spin pairs of a given (w D, eD' <PD) in the entire sample, we do an

ensemble average over eL;

(My(2NT» == (My(O» - A(My(O» .

Therefore, using Eq. (68), the fractional loss in magnetization is

F' "<MTo» -(M y(2NT» == A(My(O» •

(M)O» (M/O»

We ca n calculate (M/O» easily, using Eq. (67),

(My(O» == - sin(e MO cose L) cose L •

Thus,-- 1 j2< j8L=< wa

(My(O» == -4 d<pL sineL de L[(My(O»] == i?-(e M coseM - sineMO) •7T 0 8L =0 IJMOO 0

(71)

(72)

(73)

We can determine e O from this expression, because in the experiment, 2 the pulse intensity is chosen to maximize

(My(O»; setting (d /de Mo)(My(O» == 0, we obtain the relation

tane Mo ==2eM/(2 -e10) . (74)

eMo :::;2. 08 :::;119° is the smallest such eMO. It is not 90°, as MK state. 2 This gives

(My(O» == + w ~ [ - eM/(4 + et/12] "" - O. 4 3 6 w ~ .Now, since Tj and «1 , we can approximate

c o s T j " " 1 _ ~ T j 2 " " 1 , sinTj""Tj, c o s ~ " " 1 _ ~ ~ 2 " " 1 , s i n ~ " " ~ , and substituting into Eq. (69), we obtain

A"" ~ s i n 2 N e ( c o t 2 e ) Tj2 + t si n2( i N e ) [ c o t 2 ( ~ e ) H2 ,

where e = eM cose L . Thus, using Eqs. (67) and (76),

A(My(O» ==A Tj2 +BP ,

with

1 wa I8 M

(Ne) (8)- "3 it -8M de si n22"" cot

2 2" (e sine) .

(75)

(76)

(77)

Now in the experiment of MK, eM =eMO ::::119°. In that case, A and B, although they depend on N, are never of

greater magnitude than of order ( w ~ ) . Thus, using Eqs. (70), (72), (75), and (77), the fractional loss of magnetiza

tion is of order (WOT)2« 1.






VI. DISCUSSION

We have seen that this model predicts almost com

plete refocusing of the magnetization, i . e., spin locking

when observed stroboscopically. Thus it reproduces the

short-time behavior observed experimentally. 2 The

long-time behavior (magnetization decay) is no t pre

dicted by this model. This is not surprising considering

that the decay depends intrinsically on the many-spinnature of the real system, which, of course, the model

cannot handle, by it s definition. However, we could ap

ply to the model a technique similar to that used by

Waugh and Wang, 5 a kind of molecular-chaos approxi

mation, which leads to the conclusion that the charac

teristic decay time increases as the time between

pulses decreases, which is observed experimentally,

although the functional dependence of on T is not ob

tained.

APPENDIX A: FICTITIOUS SPIN·Y2 OPERATORS

Vega and Pines 6 and Shattuck? have developed a nota

tionwhich

is very useful for describing those spin-l

quadrupolar systems in which the quadrupolar Hamilto

nian is dominant. We adopt this notation, with some

additions and modifications, summarized as follows:

Define, for each spin,

We will need to know various commutation and anti

commutation relations among these operators, as fol

lows:commutators:

[I/>.I,lp,J]=ilp•k , i , j , k cyclic,

[1/>.1,1/>.4]=0,

[I/>.I,I •• t l=t i I r•1 \

[Ip.l,I••2]=-tiIr.2 , p,q,rcyclic,

[1/>.2,1•• 2]= - t i Ir . l

[Ip.l,I•• 3]=tilp.2 l[ ]

1 P*q ,I p• 2, I • 3 = - z i l p• 1 '

[IP. 3' 1 • 3] = 0 ,

[Ix •1, I y•4] = i l x•2 ,

[Ix.2,I y•4]= iIx.l ,

[Ie.t>Iy•4]= iI z•2 ,

[Iz.2,I y•4]=%iI • 1 ;

anticommutators:

[Ip.I,Ip.J]t=O, i * j ,

[IP.l,I•• tlt= t lr •2 !

[Ip.l,I • 2]t=tlr.l ,

[Ip•2, I •2]t= -t lr .2

[Ip•1, ! • 3P = tlp.lt

[Ip.2,I •• 3)t = t IM), p, q cyclic,

[IY.i,Iy.4]t=Iy.i, i= I ,2

[Ip.i,IY.4)t=-tlp.i, p*y , i= I , 2 .

Also, we note the following additional properties:

Ix,3+ IY,3+ I z,3=0,

1;,1 =1;.2=1;,3= ~ ( 1 +I/>,4)' p=X, y, z

21;,1 +I x,3=2I;,1 -Iz,3= t (1-2I y ,4) .

APPENDIX B

We have defined the following operators:

L = I ~ , l I ~ , l + I ~ , 2 I ~ , 2 + I ; , I I ~ , 1 + I ; , 2 I ~ , 2 ,M= - I ~ , I I ~ , 1 - I ~ , 2 I ~ , 2 + I ; , I I ~ , 1 + I ; , 2 I ~ , 2 ,N = 2 ( I ~ , I I ~ , 1 + I ; , 2 I ~ , 2 ) ,Y i = 1 " I ~ , i + I;,i 1

8

Ui =I;.i I ~ , 4 + I ; . 4 I ~ . j Wi = t( Y i + Ui)

Vi = t( Yi -2U;)

, i = 1,2,3 ,

5 2= 2 ( I ; . I I ~ , 3 + I ~ , 3 I ~ , I ) ,

Q 2 = 2 ( I ~ , I I ~ , 1 - I ; , 3 I ~ , 3 ) ,

5 1 = 2 ( I ~ , 2 I ~ , 3 + I ; , 3 I ~ , 2 ) ,

Rl=Ix",1 I ~ , l + I : , I I ~ , I - I : . 2 I ~ , 2 - I : , 2 I ~ . 2 ' 5 3 = 2 ( I ~ . l I e , 2 + I ~ , 2 I ~ , l ) ,

where

Je D= ( 1 / T ) ( ~ L + +T/N)

First we need to show that L, M, and N mutually

commute, and then to determine the commutators of L,

M, N, and Y2 with each of the 14 operators compriSing

p. Since these operators are sums of bilinear opera

tors, the commutators will be broken down into sums of

commutators of bilinear operators. We need then to

calculate commutators of the form [A" B8, C"DB], where

A, B, C, D ar e Single fictitious spin-t operators. Now

[A" B8, C"DB] = t([A", C"][BB, DB]t + [A", C"]t[BB, D8]) ,

so we n ~ e d only the single spin commutation and anticommutation relations, which ar e listed in Appendix A,

in order to calculate the bilinear commutators. The

calculations are quite lengthy, and are performed else

where. 11 We summarize the results by listing only the

nonzero commutators of L, M, N, and Y2 with the 14

operators comprising p:

[M, vtl = - iR(, [Y 2, wtl = - iW 3 ,

[M, V2]=iR 2 , [Y2, W3]=iWl ,

[M,R 2]= - iV2 , [Y 2, V1]= - iV 3 ,

[Y 2, V 3]=iVl ,

[Y2,5 2]=i2Q2,






[N, wtl=-iS t , [Y2,R 2]=iM,

[N, W2]=iS2 , [Y2, M]= -iR2 •

[N, S2] = - iW2 , [Y2•Q2] = - i2S 2 •

[N,Stl=iW t , [Y2,Stl=iS 3 •

[Y2,S3]=-iS t

IE . D. Ostroff and J. S. Waugh, Phys. Rev. Lett. 16, 1097

(1966).

2R. A. Marino an d S. M. Klainer, J. Chern. Phys. 67, 3388

(1977).

3U. Haeberlen and J. S. Waugh, Phys. Rev. 175, 453 (1968).

4J. S. Waugh and C. H. Wang, L. M. Huber, and R. L. VoId,

J. Chern. Phys. 48, 662 (1968).

5J. S. Waugh and C. H. Wang, Phys. Rev. 162, 209 (1967).

6S. Vega and A. Pines, J. Chern. Phys. 66, 5624 (1977).

7T. W. Shattuck, Ph . D. thesis (Univ. California, Berkeley,

LBL-5458, 1976) (unpublished).

BU. Haeberlen, High Resolution NMR in Solids (Academic, New

York, 1976).

sM. Goldman, Spin Temperature and Nuclear Magnetic Reso

nance in Solids (Oxford U. P., Oxford, 1970).

IOL. Tisza, "Applied Geometric Algebra," Course Notes fo r

Physics 8.352, M.LT. , 1976) (unpublished).

I I R. S. Cantor, Ph. D. thesis (M. LT. , 1979) (unpublished).


quadrapole_spinlocking

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